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Transcript
DARTMOUTH MAGNETIC EVOLUTIONARY STELLAR TRACKS AND RELATIONS
A Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
in
Physics and Astronomy
by
Gregory Alexander Feiden
DARTMOUTH COLLEGE
Hanover, New Hampshire
July 11, 2013
Examining Committee:
Brian C. Chaboyer (Chair)
Robert A. Fesen
John R. Thorstensen
F. Jon Kull
Dean of Graduate Studies
Sarbani Basu
Copyright © 2013 by Gregory A. Feiden
This work is licensed under the Creative Commons Attribution 3.0 Unported License.
To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/.
Abstract
Strong evidence exists showing that stellar evolution models are unable to accurately predict
the fundamental properties of low-mass stars. Observations of low-mass stars in detached
eclipsing binaries (DEBs) indicate that stellar models under-predict real stellar radii by 5 –
10% and predict effective temperatures that are 3 – 5% too hot. This dissertation provides a
careful examination of this problem using the Dartmouth stellar evolution code.
Accurate models of the three stars in KOI-126 are presented. These models represent the
first successful stellar evolution models of fully convective stars. I then introduce a novel
method for estimating the ages of young, low-mass DEBs. The method takes advantage
of apsidal motion to enable the use of stellar interior structure to predict ages instead of
stellar surface properties, which are prone to significant uncertainty. Next, a reanalysis of
the magnitude of the mass-radius discrepancies is performed with models that account for
realistic metallicity and age variation. Results suggest that discrepancies are about a factor
of two smaller than previously believed, although the problem is not entirely resolved.
Lastly, I describe the development of a new one-dimensional stellar evolution code that includes effects of a globally pervasive magnetic field. This is done within the framework of
the existing Dartmouth code. I find that model radius and effective temperature discrepancies can be reconciled with a magnetic field in stars with a radiative core. The predictions
from these models can be observationally tested. Fully convective stars appear insensitive
to the influence of magnetic fields, in contradiction with previous studies. I suggest that
deficiencies in fully convective stars may instead be related to metallicity.
ii
Looking back to all that has occurred to me since that
eventful day, I am scarcely able to believe in the reality
of my adventures. They were truly so wonderful that
even now I am bewildered when I think of them.
— Jules Verne
Preface
My time at Dartmouth truly has been an adventure. The adventure would not have been
successful, however, without the support of a great many people. There is much talk of
standing on the shoulders of giants, but I would be remiss if I did not acknowledge those
who provided the boost necessary to reach those shoulders.
The Dartmouth stellar evolution code is the product of the hard work of successive generations of faculty, postdocs, and graduate students from Yale University and Dartmouth
College. This thesis is an extension of their work and is a testament to how far the field of
stellar evolution has come. Without the untiring dedication of those before me, this project
would not have been possible. To all of my predecessors, you are the giants. Even if we
have not met, thank you.
On a more personal level, there are many people who have helped support me over the past
five years. To all of the hockey folks in the Upper Valley, I will miss the competition on the
ice and the pub visits that inevitably followed. This is especially true of my teammates on
K2, the past four years have been a blast and I am truly grateful to have been a part of the
franchise. To my graduate school friends and colleagues, past and present, it has been an
absolute pleasure to have gotten to know you and work with you. I have cherished all of
the conversations and discussions, from late night homework sessions to thought-provoking
conversations about your research and mine. And who could ever forget the powerhouse
that is the Absolute Zeros hockey team? Without a doubt you have all been an integral part
iii
of the completion of this thesis. Thank you for the boost. It’s been one hell of a ride.
The faculty and staff at Dartmouth have been wonderful. The guidance and mentoring I
have received was crucial to my development as a scientist. This is especially true of the astronomers: Gary Wegner, Ryan Hickox, Rob Fesen, John Thorstensen, and Brian Chaboyer.
You have all been an inspiration. I can only hope to one day be as good of a scientist as
you all are. Special thanks is owed to Rob and John for serving on my thesis committee
and reading through the pages that are contained within. I also have to thank Sarbani Basu,
from Yale, who made the trip up to Hanover to sit as the external committee member and
who provided excellent feedback on my science. Of course, so many thanks are owed to my
adviser, Brian, for his continuing support, guidance, and excellent defense. I always knew
the front of the net would be well guarded when he was on the ice.
I must also mention those others that helped me along my way. Bruce Zellar, Aaron Dotter,
the faculty during my tenure at SUNY Oswego, and the astronomers in the Physics and
Astronomy Department at Uppsala University. Swedish hospitality is second to none. To
my family, you have always been my strongest advocates. Mom and Dad, this thesis is a
testament to your success at raising me.
Finally, to my wife, Meghan. No words in a preface can express how grateful I am to have
you in my life.
On a less personal level, I’d like to thank the William H. Neukom 1964 Institute for Computational Science for their generous financial support. The science in this thesis has also been
supported by the National Science Foundation (NSF) grant AST-0908345. This research has
made use of NASA’s Astrophysics Data System (ADS), the SIMBAD database, operated at
CDS, Strasbourg, France, and the ROSAT data archive tools hosted by the High Energy Astrophysics Science Archive Research Center (HEASARC) at NASA’s Goddard Space Flight
Center.
iv
Contents
Abstract
ii
Preface
iii
Contents
v
List of Tables
xi
List of Figures
xiii
1
Introduction
1
1.1
Low-Mass Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
Properties of Low-Mass Stars . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.2
Low-Mass Stars as Hosts for Exoplanets . . . . . . . . . . . . . . . . .
6
1.2
Detached Eclipsing Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
The Mass–Radius(–Teff ) Problem . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.1
Metallicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3.2
Opacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.3.3
Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.3.4
Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Magnetic Fields in Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . .
22
1.4.1
22
1.4
Modified Adiabatic Gradient . . . . . . . . . . . . . . . . . . . . . . . .
v
1.5
1.6
2
3
Reduced Mixing Length . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.4.3
Star Spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.4.4
A Need for New Models? . . . . . . . . . . . . . . . . . . . . . . . . . .
26
The Dartmouth Stellar Evolution Code . . . . . . . . . . . . . . . . . . . . . .
27
1.5.1
Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.5.2
Solar Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Thesis Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Accurate Low-Mass Stellar Models of KOI-126
36
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.2.1
Stellar Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.2.2
Mass-Radius Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.2.3
Relative Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.2.4
Apsidal Motion Constant . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Using the Interior Structure Constants as an Age Diagnostic
52
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.2
Interior Structure Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.3.1
Single Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.3.2
Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.4.1
Observational Considerations . . . . . . . . . . . . . . . . . . . . . . .
66
3.4.2
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.4
4
1.4.2
The Mass-Radius Relation for Low-Mass, Main-Sequence Stars
vi
70
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.2
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
4.3
Isochrone Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.3.1
Isochrone Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.3.2
Model Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.3.3
Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.3.4
Age & Metallicity Priors . . . . . . . . . . . . . . . . . . . . . . . . . .
82
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.4.1
Standard Stellar Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.4.2
Variable Mixing Length Models . . . . . . . . . . . . . . . . . . . . . .
93
4.4.3
Peculiar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4.4
4.5
4.6
5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5.1
Radius Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5.2
Radius-Rotation-Activity Correlations . . . . . . . . . . . . . . . . . . 103
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Magnetic Perturbation within the Framework of DSEP
116
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2
Magnetic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3
5.4
5.2.1
Magnetic Field Characterization . . . . . . . . . . . . . . . . . . . . . . 119
5.2.2
Stellar Structure Perturbations . . . . . . . . . . . . . . . . . . . . . . . 124
5.2.3
Thermodynamic Structure . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.4
Magnetic Mixing Length Theory . . . . . . . . . . . . . . . . . . . . . 136
5.2.5
The Parameter f and the Frozen Flux Condition . . . . . . . . . . . . 154
Implementation in the Dartmouth Code . . . . . . . . . . . . . . . . . . . . . . 156
5.3.1
Magnetic Field Strength Distribution . . . . . . . . . . . . . . . . . . . 156
5.3.2
Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 159
Testing and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
vii
5.5
5.6
6
5.4.1
Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.4.2
Solar Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4.3
Ramped Versus Single Perturbation . . . . . . . . . . . . . . . . . . . . 162
Case Study: EF Aquarii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.5.1
Standard Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.5.2
Magnetic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.6.1
Field Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.6.2
Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Magnetic Models of Low-Mass Stars with a Radiative Core
178
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.2
Analysis of Individual DEB Systems . . . . . . . . . . . . . . . . . . . . . . . . 182
6.3
6.4
6.5
6.2.1
UV Piscium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.2.2
YY Geminorum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2.3
CU Cancri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Magnetic Field Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.3.1
Surface Field Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.3.2
Interior Field Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Reducing the Magnetic Field Strengths . . . . . . . . . . . . . . . . . . . . . . 213
6.4.1
Magnetic Field Radial Profile . . . . . . . . . . . . . . . . . . . . . . . 214
6.4.2
Finite Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . 218
6.4.3
Presence of E-fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.4.4
Dynamo Energy Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.5.1
Interior Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.5.2
Comparison to Previous Work . . . . . . . . . . . . . . . . . . . . . . . 228
6.5.3
Starspots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
viii
6.6
7
Implications for Asteroseismology . . . . . . . . . . . . . . . . . . . . 232
6.5.5
Exoplanets & the Circumstellar Habitable Zone . . . . . . . . . . . . 235
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Magnetic Models of Low-Mass, Fully Convective Stars
246
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.2
Magnetic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.3
7.4
7.5
8
6.5.4
7.2.1
Dipole Radial Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.2.2
Gaussian Radial Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.2.3
Constant Λ Turbulent Dynamo . . . . . . . . . . . . . . . . . . . . . . 249
Analysis of Individual DEB Systems . . . . . . . . . . . . . . . . . . . . . . . . 250
7.3.1
Kepler-16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.3.2
CM Draconis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.4.1
Magnetic Field Radial Profiles . . . . . . . . . . . . . . . . . . . . . . . 266
7.4.2
Surface Magnetic Field Strengths . . . . . . . . . . . . . . . . . . . . . 267
7.4.3
Comparison to Previous Studies . . . . . . . . . . . . . . . . . . . . . . 269
7.4.4
Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Conclusions & Future Investigations
280
8.1
Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
8.2
Future Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
A DMESTAR User’s Guide
286
B Additional Model Physics
295
B.1
Turbulent Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
B.2
Internal Structure Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
ix
B.3
Convective Overturn Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
B.4
Nuclear Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
B.5
Gough & Taylor Magnetic Inhibition . . . . . . . . . . . . . . . . . . . . . . . . 302
B.6
Electron Conduction Opacities . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
B.7
τ=100 Boundary Condition Fit Point . . . . . . . . . . . . . . . . . . . . . . . . 304
Bibliography
305
x
List of Tables
1.1
Solar calibration parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.1
Properties of the KOI-126 system. . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.2
Influence of the adopted EOS on low-mass stellar radii. . . . . . . . . . . . . .
42
2.3
Apsidal motion constants (k 2 ) for fully convective stars. . . . . . . . . . . . .
46
4.1
DEBs with at least one low-mass component with precise masses and radii. .
75
4.2
Age and [Fe/H] priors for low-mass DEBs. . . . . . . . . . . . . . . . . . . . .
82
4.3
Best fit isochrone using a solar calibrated mixing-length. . . . . . . . . . . . .
90
4.4
Best fit isochrone using a mass dependent convective mixing length. . . . . .
95
5.1
Fundamental stellar properties of EF Aquarii. . . . . . . . . . . . . . . . . . . . 118
5.2
Comparison of estimated magnetic field strengths (in G). . . . . . . . . . . . . 173
6.1
Sample of DEBs whose stars possess a radiative core. . . . . . . . . . . . . . . 181
6.2
X-ray properties for the three DEB systems. . . . . . . . . . . . . . . . . . . . 206
6.3
Surface magnetic field properties of the stars in UV Psc, YY Gem, & CU Cnc 210
6.4
Peak interior magnetic field strengths. . . . . . . . . . . . . . . . . . . . . . . . 213
6.5
Low-mass stars with direct magnetic field measurements. . . . . . . . . . . . 242
7.1
Fundamental properties of Kepler-16. . . . . . . . . . . . . . . . . . . . . . . . 251
7.2
Fundamental properties of CM Draconis. . . . . . . . . . . . . . . . . . . . . . 261
xi
A.1 Magnetic namelist variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
B.1
Updated DSEP nuclear reaction rates. . . . . . . . . . . . . . . . . . . . . . . . 303
xii
List of Figures
1.1
The noted mass-radius and mass-Teff discrepancies with the BCAH98 models. 12
1.2
The influence of αMLT on the BCAH98 models. . . . . . . . . . . . . . . . . . .
1.3
Figure 4 from López-Morales (2007) plotting radius deviations against [Fe/H]. 16
2.1
Age determination of KOI-126 A. . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.2
Model Fit to KOI-126 B & C . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.3
Comparison of three popular equations of state. . . . . . . . . . . . . . . . . .
43
2.4
The Kepler filter transmission profile. . . . . . . . . . . . . . . . . . . . . . . .
44
3.1
Time evolution of k 2 for different stellar masses. . . . . . . . . . . . . . . . . .
59
3.2
The influence of model inputs on the predited k 2 . . . . . . . . . . . . . . . . .
61
3.3
The evolution of the theoretical k 2 for a binary system. . . . . . . . . . . . . .
63
4.1
Comparison of solar-calibrated and variable mixing length isochrones. . . .
80
4.2
Relative radius errors between known DEBs and DSEP . . . . . . . . . . . . .
87
4.3
Relative radius errors between known DEBs and DSEP with variable αMLT . .
94
4.4
Correlation between radius deviations and P orb for standard models. . . . . 104
4.5
Theoretical Rossby Number versus Relative Radius Error . . . . . . . . . . . . 105
4.6
Correlation between radius deviations and P orb for variable αMLT models. . 108
4.7
Rossby number versus relative radius error for variable αMLT models. . . . . 109
4.8
X-ray Luminosity of Low-Mass DEB Systems . . . . . . . . . . . . . . . . . . 111
xiii
13
5.1
The assumed radial profile of the magnetic field strength. . . . . . . . . . . . 157
5.2
Numerical stability tests for the magnetic Dartmouth code. . . . . . . . . . . 160
5.3
The influence of ramping the magnetic field perturbation. . . . . . . . . . . . 163
5.4
Effect of altering the perturbation age on magnetic models. . . . . . . . . . . 165
5.5
Mass tracks for EF Aqaurii in the age–radius plane. . . . . . . . . . . . . . . . 167
5.6
Mass tracks for EF Aqaurii in the Teff –radius plane. . . . . . . . . . . . . . . . 168
6.1
Standard DSEP mass tracks of UV Psc at two different metallicities. . . . . . 185
6.2
Magnetic mass track of UV Psc B with a 4.0 kG surface magnetic field. . . . 186
6.3
Magnetic mass track of UV Psc B with a reduced αMLT . . . . . . . . . . . . . . 187
6.4
Magnetic mass tracks of UV Psc A and UV Psc B with [Fe/H] = −0.3. . . . . 190
6.5
Mass tracks of YY Gem in the age-radius plane with [Fe/H] = −0.1. . . . . . 192
6.6
Mass tracks of YY Gem in the Teff -radius plane with [Fe/H] = −0.1. . . . . . 193
6.7
Mass tracks of YY Gem in the age-radius plane with [Fe/H] = −0.2. . . . . . 196
6.8
Mass tracks of YY Gem in the Teff -radius plane with [Fe/H] = −0.2. . . . . . 197
6.9
Standard DSEP mass tracks for CU Cnc. . . . . . . . . . . . . . . . . . . . . . . 201
6.10 Magnetic stellar evolution mass tracks for CU Cnc. . . . . . . . . . . . . . . . 202
6.11 Temperature profile within a model of CU Cnc A. . . . . . . . . . . . . . . . . 203
6.12 Surface magnetic flux Φ versus X-ray luminosity. . . . . . . . . . . . . . . . . 208
6.13 Theoretical surface magnetic flux Φ versus X-ray luminosity. . . . . . . . . . 211
6.14 Dipole versus Gaussian field profile. . . . . . . . . . . . . . . . . . . . . . . . . 215
6.15 Dependence of the stellar model radius evolution on the radial field profile. . 216
6.16 (∇s − ∇ad ) as a function of the logarithmic density. . . . . . . . . . . . . . . . 217
6.17 Influence of the parameter f on model radius predictions. . . . . . . . . . . . 220
6.18 Effect on the model radius predictions assuming a turbulent dynamo. . . . . 224
6.19 Interior density profile using different magnetic field prescriptions. . . . . . . 227
6.20 Interior density profile of a magnetic and reduced αMLT model. . . . . . . . . 230
6.21 Sound speed profile for a magnetic and non-magnetic model of equal mass.
xiv
233
6.22 Habitable zone boundaries for magnetic models. . . . . . . . . . . . . . . . . . 237
7.1
Standard DSEP models for Kepler-16 A. . . . . . . . . . . . . . . . . . . . . . . 252
7.2
Standard DSEP models for Kepler-16 B . . . . . . . . . . . . . . . . . . . . . . 255
7.3
Magnetic models of Kepler-16 B w/ D11 mass and a dipole radial field. . . . 258
7.4
Magnetic models of Kepler-16 B w/ D11 mass and a Gaussian radial field. . . 258
7.5
Magnetic model of Kepler-16 B w/ D11 mass and a constant Λ profile. . . . . 259
7.6
Magnetic models of Kepler-16 B w/ B12 mass and a Gaussian radial field. . . 259
7.7
Standard stellar evolution models of CM Dra A and B. . . . . . . . . . . . . . 262
7.8
Magnetic models CM Dra with a dipole radial profile. . . . . . . . . . . . . . 263
7.9
Magnetic models CM Dra with a Gaussian radial profile. . . . . . . . . . . . . 265
7.10 Constant Λ models for CM Dra A and B. . . . . . . . . . . . . . . . . . . . . . 265
7.11 (∇s − ∇ad ) as a function of the logarithmic density. . . . . . . . . . . . . . . . 267
7.12 Radius residuals for fully convective stars as a function of stellar properties. 276
7.13 Radius residuals for fully convective stars versus metallicity with fixed-Y . . 278
xv
Chapter 1
Introduction
It was a dark and stormy night, so he decided to be a theorist.
There are approximately 100 billion stars in the Milky Way Galaxy. Of those 100 billion
stars, nearly 90% are thought to have a mass equal to or less than the mass of the Sun
(Research Consortium On Nearby Stars, RECONS;1 Henry et al. 2006). Furthermore, nearly
75% of the stars in the Galaxy, or at least in the local galactic neighborhood, belong to a
particular class of low-mass stars known as M-dwarfs (also called red dwarfs; Henry et al.
2006; Lépine & Gaidos 2011). Indeed, stars less massive than the Sun dominate the stellar
population of the Milky Way. Low-mass stars appear to also dominate the total stellar
population of large, elliptical galaxies, suggesting that low-mass stars are the most common
type of star in the Universe (van Dokkum & Conroy 2010; Conroy & van Dokkum 2012).
The sheer number of low-mass stars makes them worthy of intensive study, but significant
interest in low-mass stars has grown in recent years for other reasons. A resurgence of
interest in these stars has been motivated by the search for habitable extra-solar planets.
Low-mass stars turn out to make better targets when searching for small, rocky planets orbiting in the “habitable zone,” or the region of space surrounding a star where life, as we
1
Number counts taken from their January 1, 2012 census. See http://www.recons.org/.
1
know it, has a chance to exist (Charbonneau 2009; Gillon et al. 2010). Low-mass stars also
play host to a wide range of interesting physical processes. This is due to the extreme physical conditions that exist in and around these stars, conditions which are nearly impossible
to achieve within an Earth-based laboratory. Complex physics such as the condensation
of molecules and dust grains (see, e.g., Allard et al. 1997), non-ideal gas thermodynamics
at high pressure and low temperature (see, e.g., Saumon et al. 1995), turbulent convection
(Wende et al. 2009; Trampedach & Stein 2011), and multiple magnetic dynamo mechanisms
(see, e.g., Parker 1979; Mestel 1999) are thought to be associated with low-mass stars. It is
for these reasons that make investigations of low-mass stars worthwhile and rewarding.
The information that can be gathered from observations of low-mass stars, however, is
rather limited. Low-mass stars are intrinsically faints objects, as we will see, which makes
them difficult to study. Investigations involving low-mass stars often have to rely on the
construction of theoretical models to aid in the interpretation of the observations. The
models allow us to predict the physical properties of low-mass stars, which in turn allows
for a complete characterization of the star. Given that studies of low-mass stars often require
a model dependent characterization of the stars, it is crucial that the models be accurate. The
thesis that follows assesses our present ability to construct accurate theoretical models of
low-mass stars and proceeds to introduce new physics that may improve model predictions.
Before we delve into investigations involving low-mass stars, we would be remiss if we did
not first explore the fundamental concepts and themes that will be present throughout the
text.
1.1 Low-Mass Stars
The definition of what constitutes a low-mass star typically depends on whom you ask.
However, it is only the upper mass limit that can be debated. The lower boundary is “fixed”
2
to the lowest possible mass where a self-gravitating body can fuse hydrogen in its core
(∼ 0.08M⊙ ; Kumar 1963; Baraffe et al. 1998). Below that critical mass, electron degeneracy
pressure prevents the star from collapsing prior to the onset of hydrogen fusion.2 While the
precise numerical value defining the hydrogen burning mass limit may vary slightly, it is
safe to say that a star near the hydrogen burning mass limit is low-mass.
To avoid any ambiguity or personal preference that might exist about what defines the
upper mass limit, we shall define a low-mass star as one that has a mass less than 80% that
of the Sun (0.80M⊙ ). While our motivation may not be clear now, the choice of 0.80M⊙ as
an upper mass limit will be revealed in later sections. If there needs to be any more reason
for selecting 0.80M⊙ at the moment, let it be for purely aesthetic purposes. After all, 0.80 is
an order of magnitude larger than 0.08.
Much to astronomers’ chagrin, observations of single low-mass stars cannot reveal their
mass in a model-independent fashion. Therefore, defining the class of stars we are interested
in studying by their mass is not very beneficial. How will we know when we have found
one? Instead, we take advantage of the fact that the low-mass that characterizes these stars
directly influences their observable properties.
1.1.1 Properties of Low-Mass Stars
Stars that fuse hydrogen to helium in their core belong to a class of stars known as main
sequence stars, or “dwarfs.”3 The properties of main sequence stars are largely governed
by their mass. In fact, one can derive simple scaling relations, either from theory or from
empirical data (Weiss et al. 2004; Andersen 1991). A low-mass star’s radius is approximately
2
Deuterium burning can, however, occur in sub-stellar objects.
The use of the word dwarfs is the correct lexical construction for the plural of “dwarf.” Some may be
tempted to invoke “dwarves” as the plural form, but this appears to be an alternative construction popularized
by Tolkein in his books. See http://grammarist.com/usage/dwarfs-dwarves/ for a further discussion.
3
3
proportional to its mass,
M
R
∝
,
R⊙
M⊙
(1.1)
where R ⊙ is the radius of the Sun, or 6.9598 × 1010 cm (Neckel 1995; Bahcall et al. 2005).
Using this relation, we infer that low-mass stars have typical radii between 0.10R ⊙ and
0.80R ⊙ , for a 0.10M ⊙ and 0.80M ⊙ star, respectively. Additionally, low-mass stars are found
to have a luminosity, or intrinsic brightness, that varies with mass according to the relation
µ
¶
M 3
L
∝
,
L⊙
M⊙
(1.2)
with L ⊙ being the luminosity of the Sun, here defined as 3.8418×1033 erg s−1 (Bahcall et al.
2005). We see that low-mass stars have luminosities between approximately 10−3 L ⊙ and
0.5L ⊙ , for a 0.10M ⊙ and 0.80M ⊙ star, respectively. Invoking the Stefan-Boltzmann equation
relating a star’s luminosity to its radius and effective (surface) temperature,
4
,
L = 4πσR 2 Teff
(1.3)
we find that the surface temperature of low-mass stars is approximately
µ
¶
Teff
M 1/4
∝
,
Teff,⊙
M⊙
(1.4)
where Teff,⊙ = 5 776 K is the effective temperature of the Sun from the Stefan-Boltzmann
equation and our definitions of R ⊙ and L ⊙ . Low-mass stars are then expected to have
effective temperatures in the range of ∼ 3 250 K to ∼ 5 500 K. In reality, the lower bound on
the effective temperature is below 3 200 K, but the scaling relation is approximately valid.
These effective temperatures imply that the spectral types of low-mass stars are in the range
of mid-G to mid-M.
The low temperatures and luminosities associated with low-mass stars permit a variety of
4
unique observational characteristics. Photometrically, low-mass stars appear fainter and
redder than stars like the Sun. Therefore, identifying low-mass stars can be as simple as
identifying faint, red stars. However, very distant red giant stars may have similar effective
temperatures (and thus colors) as low-mass main sequence stars. Red giants can therefore
masquerade as nearby low-mass stars. Identifying low-mass stars by color alone is not sufficient, at least without a distance estimate to provide information regarding the intrinsic
brightness of the star. Instead, photometric information must be combined with spectroscopic data.
Spectra of low-mass stars are dominated by the presence of broad absorption features created by the dense forest of available molecular transitions (Kirkpatrick et al. 1991; Allard et al.
1997). That is, for stars of sufficiently low-mass. Stars with effective temperatures between
about 4 000 K and 5 500 K (spectral types late-K to mid-G) have spectra similar to the Sun.
Titanium oxide (TiO) bands begin to appear around 4 000 K (late-K) and grow stronger with
decreasing effective temperature before saturating around 3 000 K (mid-M). While TiO begins to saturate at the coolest effective temperatures, vanadium oxide (VO) bands begin to
form and grow stronger with decreasing temperature (Kirkpatrick et al. 1991). In the nearinfrared (NIR), the dominant source of opacity is due to water (H2 O), creating the “steam
bands.” Additional contributions in the optical and NIR from metal-hydrides such as calcium hydride (CaH), magnesium hydride (MgH), and iron hydride (FeH) are also present in
low-mass stellar spectra owing to their cool effective temperatures.
Ambiguity may occur when comparing the spectra of two cool stars with different luminosities (e.g., giants versus dwarfs). To discern between the two, optical and NIR atomic features
are generally used (Kirkpatrick et al. 1991). Neutral sodium doublets (Na i; λ = 5890, 5896 Å
and 8183, 8195 Å), neutral potassium doublets (K i; λ = 7665, 7699 Å), and neutral calcium
(Ca i; λ = 4227 Å) optical absorption lines are gravity- (or luminosity-) sensitive features
that appear very strong in the spectra of dwarf stars, but not giant stars. The optical Ca
5
i line is less favorable than Na i as there is considerably less flux for an M-dwarf short of
6000 Å. Additionally, the NIR Ca ii triplet and Na i doublet may be used as gravity diagnos-
tics, where the Ca ii triplet (λ = 8498, 8542, 8662 Å) is weakest in dwarf spectra and the Na
i doublet is strongest for dwarfs (Kirkpatrick et al. 1991).
Owing to their low mass, low-mass stars fuse hydrogen at a relatively slow rate. The internal temperature profile is proportional to the internal pressure gradient. It is the latter
that supports the star against gravity. Less mass means that less of a pressure gradient is
required, leading to cooler temperatures throughout the star. The proton-proton (p-p) chain
produces energy at a rate proportional to the temperature to the fourth-power, ǫp−p ∝ T 4 .
Due to the cooler temperatures inside low-mass stars, they primarily burn hydrogen via the
first channel of the p-p chain (Chabrier & Baraffe 1997), and they do so slowly. Low-mass
stars thus have main sequence lifetimes that range from 12 Gyr for an 0.80M⊙ star to over
3 Tyr for a 0.1M⊙ star (Laughlin et al. 1997). After their main sequence life, most low-mass
stars will progress up the red giant branch. However, below a given (model-dependent)
mass between 0.1M⊙ and 0.2M⊙ , low-mass stars never ascend the red giant branch and are
doomed to slowly fade away. This makes models of low-mass stars possible tools to study
why stars become red giants (Laughlin et al. 1997).
1.1.2 Low-Mass Stars as Hosts for Exoplanets
Interest in low-mass stars has seen a revival due to interest in discovering extra-solar planets
(henceforth exoplanets). However, it is not just any planet that researchers are interested in
finding. The ultimate goal is to discover the first Earth-sized rocky planet in the habitable
zone of its host star. Why, though, should low-mass stars be of interest to those searching
for the next Earth? We know that at least one Earth exists around an early G-dwarf. Would
not other solar-type stars provide the best opportunities based on our existing knowledge?
The advantages provided by low-mass stars over solar-type stars are numerous. First, M6
dwarfs are the most common type of star and thus provide a large number of potential
exoplanet hosts. Within 10 pc of the Sun, there are 248 M-dwarfs and only 20 solar-type
G-dwarfs (RECONS4 ). Other advantages rely on the lower mass and cooler effective temperatures of low-mass stars compared to solar-type stars. Consider the two primary techniques
for detecting exoplanets—radial velocity and transit methods. Let us imagine two stars,
one an M-dwarf and one a G-dwarf, each with an identical Earth-sized planet orbiting in
the circumstellar habitable zone (HZ). It is important to mention that the exoplanet will be
orbiting closer to the M-dwarf than the G-dwarf. The low luminosity and cool effective
temperature of the M-dwarf requires an Earth-like planet to orbit closer to the star if it is
to receive enough incident flux to support liquid water and a suitable greenhouse effect
(Kasting et al. 1993).
Measuring the radial velocity induced on the two stars by their respective planets depends
primarily on the semi-major axis of the planet’s orbit and the planet’s mass relative to the
star (we assume the inclination to be i = 90◦ ). The semi-amplitude of the radial velocity
signal is (assuming a circular orbit),
2πG
K=
P orb
µ
¶1/3
Mp
¡
M p + M⋆
¢2/3 ,
(1.5)
where P orb is the orbital period, M p is the planet mass, and M⋆ is the stellar mass. The
closer the planet and higher the planet-to-star mass ratio, the larger the radial velocity
signal. In both situations, M-dwarfs provide an advantage over G-dwarfs. For a similar
mass planet around an M-dwarf compared to a G-dwarf, the ratio of the semi-amplitude is
¶ µ
¶
µ
A g 1/2 M g 2/3
Km
,
=
Kg
Am
Mm
(1.6)
where the subscripts m and g refer to the values for an M-dwarf and G-dwarf, respectively.
Here, A is the semi-major axis of the planet orbit and M is the mass. For a planet orbiting in
4
http://www.chara.gsu.edu/RECONS/census.posted.htm
7
the habitable zone, K m /K g ∼ 2 to 15 for the range of M-dwarf masses. M-dwarfs do suffer
from significant radial velocity noise (Saar & Donahue 1997), decreasing the radial velocity
sensitivity, but there is considerable effort to mitigate this problem (see, e.g., Barnes et al.
2011; Muirhead et al. 2011).
Detecting an Earth-like planet via the transit method requires precise photometry and the
fortuitous alignment of a planet orbiting across the face of the star from the observer’s
point-of-view. Precise photometry is necessary because the dip in the flux received from a
star due to the transit of a planet is proportional to the ratio of the projected planet surface
area to the projected stellar surface area—assuming the planet contributes negligible flux
and can be considered dark. Explicitly, the change in flux is
∆F ∝
µ
Rp
R⋆
¶2
.
(1.7)
In other words, the key variable is the square of the ratio of the planet-to-star radius. Therefore, an Earth-like planet transiting an M-dwarf will create a change in flux that is a factor
of 4 to 100 times larger compared to transiting a G-dwarf, depending on the size of the
M-dwarf.
The second requirement is actually observing a transit. There is a finite probability of
detecting a planet transit if we assume that the inclination of the planet orbit from the
observer’s point-of-view is completely random. This geometric probability is proportional
to the sum of the stellar and planetary radius and inversely proportional to the orbital semimajor axis (Borucki & Summers 1984). The probability can be simplified such that
p∝
R⋆
,
A
(1.8)
where p is the probability. The planetary radius can usually be ignored because it is a small
fraction of the stellar radius. The proximity of the HZ to low-mass stars compensates for
8
the small stellar radius, giving an observer a greater geometric probability, by up to a factor
of 2, of detecting the transit of an Earth-like planet around a low-mass star.
We can now see why low-mass stars are considered preferable when it comes to exoplanet
target selection. But, there is one glaring caveat to locating exoplanets around low-mass
stars: the stellar properties are required to characterize the planet. This work largely falls
on the shoulders of stellar evolution theory. There are serious issues with low-mass stellar
evolution theory that have arisen due to the study of eclipsing binary systems. Throughout
the rest of this chapter we will introduce these topics and the known issues with low-mass
stellar evolution theory.
1.2 Detached Eclipsing Binaries
Eclipsing binary systems (EBs) are binary stellar systems in which the two components
are observed to pass in front of—or eclipse—one another. Here, we are concerned only
with detached eclipsing binaries (DEBs). EBs are considered detached when the two stars
have a binary separation (or semi-major axis, A ), that is large enough to prevent the stars
from undergoing mass exchange or from having undergone mass exchange in the past.
This condition also suggests that the individual stars are not strongly distorted by tidal
interactions with the companion (see, also, Chapter 3).
What makes DEBs important for the present work is that the characteristics of these systems allow for the masses and radii of the individual stars to be determined with very high
precision (see reviews by Popper 1980; Andersen 1991; Torres et al. 2010). Typical random
uncertainties of well-studied systems are routinely below 2% in both quantities. Furthermore, the mass and radius determinations are very nearly model-independent. This makes
DEBs phenomenal tools for testing stellar structure and evolution theory. Effective temperatures may also be derived for these systems, but the results are less robust as additional
9
assumptions, such as the temperature scale, must be made (Torres et al. 2010).
Rigorous discussions of the data quality needed to obtain precise (and accurate) stellar
properties from DEBs are presented in the literature (Popper 1980, 1984; Andersen 1991;
Torres et al. 2010). For our purposes, we are more concerned with using DEBs as tools to
test the validity of stellar evolution models and to assess the current state of low-mass stellar evolution theory. We therefore choose to avoid a discussion of the actual data products
and refer the reader to the referenced works for further information. The important information to take away is that DEBs provide a powerful, direct test of the predictions of stellar
evolution models.
1.3 The Mass–Radius(–Teff) Problem
Historically, the development of sophisticated low-mass stellar evolution models (Osterbrock
1953; Limber 1958a,b)5 has been accompanied by discussions that the observed radius and
luminosity of stars in DEBs do not compare well to the predictions of stellar evolution theory. Although stellar models continued to become more physically realistic by including
updated opacity sources, detailed equations of state, and rigorous atmosphere calculations
(Hoxie 1970), it was recognized that observational errors and model uncertainties were too
large to provide a meaningful comparison (Hoxie 1973). Additionally, prior to the discovery and characterization of CM Draconis (Zwicky 1966; Lacy 1977), most comparisons were
made to only a single low-mass DEB (YY Geminorum; Kron 1952). Until more recently, the
state of the low-mass stellar evolution field was rather stagnant.
Investigations over the past two decades have made it clear that stellar evolution models are
unable to accurately reproduce the properties of low-mass stars. This fact has been the result
of significant reductions in observational uncertainties (Andersen 1991; Torres et al. 2010)
5
Here we refer to the inclusion of basic nuclear reaction networks, opacity effects, and the inclusion of
convective envelopes.
10
and the development of sophisticated stellar models (Baraffe et al. 1998; Dotter et al. 2008).
The primary line of evidence stems from studies of DEB systems, where it has been found
that stellar evolution models systematically under-predict stellar radii by 5% – 10% and
over-predict stellar effective temperatures by 3% – 5% at a given mass (e.g., Metcalfe et al.
1996; Popper 1997; Torres & Ribas 2002; López-Morales & Ribas 2005; López-Morales et al.
2006; Bayless & Orosz 2006; López-Morales & Shaw 2007; Morales et al. 2009b,a; Torres et al.
2010). Further evidence has been supplied by the direct measurement of stellar radii using
interferometry (Berger et al. 2006), although the uncertainty in the quoted stellar masses
makes comparisons with models tenuous.
Figures 1.1(a) and 1.1(b) demonstrate the discrepancies observed between models and observations in the mass-radius and mass-Teff planes. These figures represent the cutting-edge
data and models prior to the work for this dissertation (i.e., prior to 2011). The data shown
are only those DEB systems with masses and radii measured to better than 3% and that
passed the screening process for inclusion in the Torres et al. (2010) review.6 The models are
those from the seminal work of Baraffe et al. (1998). A full description of the physics used in
the models is presented elsewhere (Chabrier & Baraffe 1997; Baraffe et al. 1997; Baraffe et al.
1998), but it is worth remarking that they represent the most physically realistic low-mass
models that had undergone rigorous evaluation in the literature prior to 2011. The models
include a highly detailed equation of state that accounts for numerous non-ideal gas effects
(Saumon et al. 1995), they use the latest low-temperature opacities that account for complicated atomic and molecular species, and they prescribe the surface boundary conditions
using non-gray model atmospheres that rigorously solve the equations of radiative transfer
(Allard & Hauschildt 1995; Allard et al. 1997).
It is clear from Figures 1.1(a) and 1.1(b) that the models under-predict the stellar radii and
over-predict the Teff at a given mass. This result is independent of the adopted isochrone
6
The total number of DEBs included in the Torres et al. (2010) review is only a fraction of the total number
of DEBs identified and characterized.
11
0.8
BCAH98 Isochrones
0.7
1 Gyr
5 Gyr
4400
0.6
Teff (K)
Radius (R⊙)
BCAH98 Isochrones
4800
1 Gyr
5 Gyr
0.5
0.4
4000
3600
0.3
3200
[M/H] = 0.0
0.2
0.2
0.3
0.4
0.5
0.6
Mass (M⊙)
0.7
0.8
[M/H] = 0.0
0.2
0.3
0.4
0.5
0.6
Mass (M⊙)
0.7
0.8
Figure 1.1: (a) The mass-radius relation as defined by DEBs with precise masses and radii
(grey points) and by the Baraffe et al. (1998) low-mass stellar models (solid lines). The models have a solar metallicity and are shown at two ages: 1 Gyr (maroon) and 5 Gyr (lightblue). (b) Mass-Teff relation for the same set of DEBs and isochrones.
age. The data above 0.8M⊙ , on the other hand, appears to be better fit by the models—one
star even has a radius smaller than model predictions. However, the BCAH98 models shown
in Figure 1.1 adopted a convective mixing length of αMLT = 1.0. The reduced mixing length
disproportionately affects models with masses above ∼ 0.6M⊙ , making the models appear
significantly larger than when a solar calibrated αMLT is adopted.
Figure 1.2 shows the difference in model radius predictions when a solar calibrated mixing
length is adopted. The relative insensitivity of models to αMLT is the primary reason why
we define the upper limit of “low-mass” to be 0.8M⊙ . There are additional benefits to
this selection, including relative insensitivity to age and metallicity, but these are largely
secondary to the influence of αMLT .
To understand the implications of the discrepancies in radius and Teff , consider the example of an exoplanet discovered transiting a low-mass star. The properties of the exoplanet depend on the properties inferred for the low-mass star, which are typically model
dependent (see e.g., Charbonneau et al. 2009). Of particular importance is the stellar radius, which sets the radius and average density of the planet (Seager & Mallén-Ornelas
12
BCAH98 Isochrones
0.7
αMLT = 1.9
αMLT = 1.0
BCAH98 Isochrones
4800
αMLT = 1.9
αMLT = 1.0
4400
0.6
Teff (K)
Radius (R⊙)
0.8
0.5
0.4
4000
3600
0.3
3200
[M/H] = 0.0
0.2
0.2
0.3
0.4
0.5
0.6
Mass (M⊙)
0.7
0.8
[M/H] = 0.0
0.2
0.3
0.4
0.5
0.6
Mass (M⊙)
0.7
0.8
Figure 1.2: The influence of αMLT on the Baraffe et al. (1998) models above 0.6M⊙ . The
isochrones were calculated at 1 Gyr with [M/H] = 0.0 and have αMLT = 1.0 (solid line) and
αMLT = 1.9 (dashed line). (a) The mass-radius relation. (b) The mass-Teff relation.
2003; Charbonneau et al. 2009). The latter permits an estimate of the planet composition.
Therefore, a 10% underestimate of the stellar radius (as we see for low-mass stellar models) translates into a 10% under estimate of the planet radius (R p ) and a 40% overestimate
of the average density (ρ ∝ R p−3 ). Both effects create a situation where a planet will be
characterized as more “Earth-like.” To avoid such deceptions, models must be made more
reliable.
The relative insensitivity of low-mass stellar evolution models to various modeling parameters (e.g., age and αMLT ) compared to solar-type stars is both a curse and a blessing. On the
one hand, the predictions from low-mass stellar evolution models are robust. This suggests
the model predictions are precise. However, this also means that the radius and Teff deviations observed in Figures 1.1 and 1.2 are robust, calling into question the accuracy of the
models. The high precisions with which the measurements are quoted exacerbate the situation. There must be another reason for the appearance of the deviations. Several hypotheses
have been advanced, which we shall now describe.
13
1.3.1 Metallicity
Stellar composition is the second most important property, after mass, governing the evolution of a main sequence star. The precise breakdown of the chemical composition—the mass
fractions of hydrogen, X , helium, Y , and other “metals,”7 Z —can affect the evolution of a
star in multiple ways. The two primary channels are through altering the nuclear reaction
network and through altering the plasma’s opacity. Therefore, stellar composition is a key
physical ingredient that must be specified in stellar models.
It is sufficient to specify only two of the mass fractions listed above knowing that X + Y +
Z = 1. In practice, Y and Z are specified. Some simple assumptions are made regarding
the composition of helium: it is either fixed to a pre-determined value or it is allowed to
scale with the metal abundance. The latter approach is typically used to cut down on the
number of models required to compare theory to observation and to reduce the specified
mass fractions to just Z . The standard approach is to allow Y to scale linearly with Z ,
Y = Yprim +
dY
∆Z ,
dZ
(1.9)
where Yprim is the primordial helium abundance adopted from Big Bang Nucleosynthesis
calculations (Alpher et al. 1948) or determined semi-empirically (Peimbert et al. 2007), ∆Z =
Z − Zprim = Z ( Zprim = 0), and the slope of the linear relation is determined empirically (see
e.g., Casagrande et al. 2007).
Unfortunately, metallicities of low-mass stars are also notoriously difficult to measure (Woolf & Wallerstein
2005). The same complex molecular bands that allow for the classification of low-mass
stars hamper metallicity and temperature analyses, especially at optical wavelengths where
TiO and VO bands dominate stellar spectra (Kirkpatrick et al. 1991; Reid et al. 1995). Current photometric (Johnson & Apps 2009; Schlaufman & Laughlin 2010) and spectroscopic
7
All elements heavier than helium are known as “metals”.
14
(Woolf & Wallerstein 2005; Bonfils et al. 2005; Bean et al. 2006; Rojas-Ayala et al. 2010) techniques for estimating the metallicity of low-mass stars are beginning to converge, but those
are only valid for single stars. DEB systems add the complication that the contributions of
the two stars must be accurately decomposed, unless we are lucky enough to find that one
of the eclipses is total. All told, there does not exist any robust metallicity determinations
for low-mass DEBs.
Not only are metallicities difficult to measure, but previous state-of-the-art low-mass stellar
evolution models were only computed for a limited number of metallicities (Chabrier & Baraffe
1997; Baraffe et al. 1997; Baraffe et al. 1998). It was argued that variations in the stellar composition would affect the properties of low-mass stars at the few percent level and that
uncertainties at that level could be tolerated (Chabrier & Baraffe 1997). This view was acceptable, at the time. However, the availability of high quality data and improved methods of analyzing DEB systems dramatically increased in the decade following Baraffe et al.
(1998). This meant that the precision with which stellar properties could be determined was
considerably higher (see Section 1.2). Therefore, it was becoming increasingly essential that
stellar models used in comparisons with observations account for variations in metallicity
(Burrows et al. 2011).
The neglect of metallicity variations in stellar models became a natural candidate to explain the observed radius and Teff deviations. The metallicity hypothesis was advanced by
Berger et al. (2006). They used the CHARA8 Array on Mount Wilson to measure the radii
of six single M-dwarfs with interferometry. The key result was that for the objects they
measured, the radius deviations with the BCAH98 stellar models correlated with metallicity
(Berger et al. 2006).
Shortly after the Berger et al. (2006) study, López-Morales (2007) investigated the origin of
the radius deviations in single and DEB stars. She found that radius deviations correlate
8
Center for High Angular Resolution Astronomy
15
Figure 1.3: Figure 4 from López-Morales (2007) plotting the deviations between models and
observations against metallicity. The models had an age of 1 Gyr with [Fe/H] = 0.0. (top)
Single stars with radii determined using interferometry. (bottom) Binary stars. Used by
permission of M. López-Morales.
with metallicity for single field stars, but the two properties do not correlate for DEBs.
However, the single stars that showed radius discrepancies with stellar models were all
from Berger et al. (2006), which may explain the supporting evidence. Regardless, Figure 1.3
demonstrates that there appears to be no correlation with metallicity and radius deviations
in DEB systems (López-Morales 2007).
Although no correlation is present in Figure 1.3, it is difficult to provide a definitive interpretation for two reasons: (1) all of the stars were compared to a 1 Gyr, solar metallicity
isochrone, and (2) there are only a few DEBs with metallicity estimates, all of which are
highly uncertain. Metallicity should therefore still be considered a plausible explanation,
though it may not be able to induce radius variation in the models at the 10% level. We
address metallicity issues specifically in Chapters 2 and 4.
16
1.3.2 Opacity
Closely related to the aforementioned metallicity hypothesis is the idea that there are important sources of opacity missing from stellar interior and atmosphere models (Baraffe et al.
1998). These missing sources of opacity are presumed to belong to molecular species that
are able to condense in the near-surface and atmosphere regions of low-mass stars. The
molecules typically singled out as possible candidates for reevaluation are TiO and VO in
the optical and H2 O in the NIR.
The motivation for discussion about opacity, however, does not stem from the discrepancies observed in the mass-radius plane. Instead, they are largely motivated by discrepancies
observed in the mass-magnitude and color-magnitude diagrams. Converting model predictions for stellar temperatures and luminosities to observable properties (i.e., colors and
magnitudes) requires accurate non-gray atmosphere models, such as those described in Section 1.5 (Allard & Hauschildt 1995; Allard et al. 1997; Hauschildt et al. 1999a). The accuracy
of the resulting colors and magnitudes are dependent on the ability for the atmosphere
model to correctly determine the distribution of flux across the electromagnetic spectrum.
Several studies have shown that the theoretical atmosphere models perform rather poorly
when it comes to predicting fluxes in the optical. However, they perform adequately in
the NIR (Baraffe et al. 1997; Baraffe et al. 1998; Delfosse et al. 1998). The consensus is that
the treatment of TiO opacity likely needs improvement while VO may be adequate. In the
NIR, the predicted colors, and thus the broad distribution of flux, appears to be accurate,
but the modeling of individual H2 O line profiles needs to be improved (Baraffe et al. 1998;
Delfosse et al. 1998). Improvements in the theoretical atmosphere models are ongoing, but
a large factor in the model accuracy may have to do with the solar abundance adopted
(Allard et al. 2011).
We elect to avoid pursuing opacity as a solution for the mass-radius problem. This is because
studies discussing opacity have been largely focused on problems with theoretical colors
17
and magnitudes, and only indirectly on stellar radii. Our decision is reinforced by the
acknowledgement that opacity sources are unlikely to significantly alter the atmosphere
structure (Baraffe et al. 1998; Allard et al. 2011). It is the latter feature of atmosphere models
that would have the greatest impact on stellar interior models.
1.3.3 Convection
One of the greatest unsolved problems in (astro)physics is the theory of convection. This is
particularly true in the context of stellar evolution, where a one-dimensional prescription
of convection is required. While there are several different local (e.g., Böhm-Vitense 1958)
and non-local (e.g., Canuto & Mazzitelli 1991) theories of convection, they still largely rely
on the mixing length parameter, αMLT = ℓm /HP , relating the radial extent of a convective
eddy, ℓm , to the pressure scale height, HP .9
The notion of a convective mixing length (ℓm ) is to set a distance over which a convecting
bubble travels before mixing into its surroundings. Thus, if a convecting bubble travels a
large distance, it is efficiently transporting energy from one point in a fluid to the other.
Conversely, if the bubble travels a small distance such that multiple bubbles are required to
transfer excess heat the same distance as our first example, it can be deemed inefficient. In
that sense, αMLT is a measure of the convective efficiency.
The precise numerical value of the mixing length parameter is the subject of considerable
debate. Standard practice is to avoid making an arbitrary choice by calibrating a 1.0M⊙
stellar evolution model to the Sun, as we will do later in Section 1.5.2. However, it has long
been recognized that there is no a priori reason to leave αMLT set to the solar calibrated
value. Stars with a mass different than that of the Sun may convect with greater or less
efficiency.
A lower convective efficiency, interpreted as a lower αMLT , would act to inflate a low-mass
9
The distance over which the pressure within a star changes by a factor of e .
18
star. Reducing the flux of heat due to convection forces the star to compensate by developing a steeper radiative temperature gradient. Therefore, the star will inflate to conserve
flux. The effects of this were shown in Figure 1.2. Since the radii of low-mass stars appear
to be inflated compared to theoretical models, one might expect that convection is a naturally inefficient process in low-mass stars. This idea motivated Baraffe et al. (1998) to adopt
αMLT = 1.0 for all of their models below 0.6M ⊙ .
Testing the inefficient convection hypothesis is difficult, as we do not yet have the ability to peer inside low-mass stars. Furthermore, the current sample of low-mass stars in
well-studied DEBs totals 8 stars (Torres et al. 2010). This can be noted in Figure 1.1. A
considerably larger sample is required to develop an understanding of how radius inflation
might correlate with mass and other intrinsic properties, such as metallicity. If naturally
inefficient convection is the culprit leading to inflated radii, then two stars of similar mass
and metallicity part of two different DEB systems should have similar radii and Teff s. For
further discussion of this matter, see Chapter 4.
Inefficient convection may also arise for another reason: the presence of a magnetic field.
This leads us to the final and most prominent hypothesis.
1.3.4 Magnetic Fields
Magnetic fields and magnetic activity are currently the leading explanation for the inflation
of stellar radii and suppressed Teff s. The hypothesis was advanced by Ribas (2006) after
he presented evidence that radius and temperature disagreements were largely apparent in
systems where the stellar parameters were determined with exquisite precision. All of those
well-characterized DEB systems were also found to have orbital periods under three days.
It was theorized that tidal synchronization of the components could be driving strong magnetic fields, and thus the altering of observable stellar properties. Mutual tidal interactions
between the two components would act to both circularize the orbit and synchronize their
19
rotation periods (e.g., Zahn 1977). Considering that the orbital periods were relatively short,
the stellar rotational periods would also be short, forcing the stars to rotate more rapidly
than if they were isolated single stars.
Stellar magnetism is thought to be governed by the dynamo mechanism (Parker 1955, 1979;
Mestel 1999). At the most fundamental level, the stellar dynamo is theorized to source
the energy for a magnetic field from the Coriolis force acting on convecting fluid motions
in a rotating star and also from shear generated at the boundary between the convective
envelope and radiative core (Parker 1955, 1979; Mestel 1999). Conventional wisdom dictates
that the more rapid the rotation, the stronger the dynamo action and, as a result, the stronger
the magnetic field.10 Therefore, tidal synchronization is theorized to help maintain a strong
dynamo within the stars in DEB systems.
The existence of strong magnetic fields is thought to affect stellar structure through two primary channels: (1) the suppression of thermal convection (Thompson 1951; Chandrasekhar
1961; Gough & Tayler 1966) and (2) by the emergence of surface spots (Hale 1908; Spruit
1982a,b). Ultimately, the energy flux across a given surface within the star is reduced, forcing the star to inflate and cool to maintain the necessary energy flux required for global
hydrodynamic stability (Spruit 1982a,b; Mullan & MacDonald 2001; Chabrier et al. 2007).
Magnetic fields therefore appear to provide an attractive solution.
However, the reduction in flux caused by magnetic fields is only part of the picture. Observations of total solar irradiance (TSI)—the total flux of solar radiation received at Earth—
show that it is variable at the 0.1% level on two timescales: (1) the timescale of the solar rotation period, and (2) on the timescale of the 11 year solar cycle (see review by
Foukal et al. 2006, and references therein). Reductions in flux caused by spot explain the
variation over the rotational period of the Sun. Spots rotating into view will decrease the
TSI by a fraction related to their projected surface coverage and their temperature contrast
10
A fluid composed of mostly neutral species can be highly resistive, allowing for rapid dissipation of
magnetic energy inhibiting the generation of favorable current networks (Mohanty et al. 2002).
20
with the non-active photosphere (Spruit 1982a,b). The second variability timescale is related
to the appearance of faculae, which are brighter than the non-magnetic photosphere (Spruit
1977). Observations reveal that the TSI increases with the number of sun spots and therefore increases with magnetic activity. Whether the changes in TSI translate to changes in
the fundamental solar properties (radius, Teff , and luminosity) is still a point of contention
(Li & Sofia 2001; Foukal et al. 2006).
Observationally, there is evidence to support the hypothesis that magnetic fields may be
at the origin of the discrepancies. Studies have suggested that correlations exist between
observed radius and Teff discrepancies and the intensity of particular emission features
(López-Morales 2007; Morales et al. 2008). The emission features, collectively referred to
as “magnetic activity,” are thought to be the result of magnetic energy being dissipated in
the stellar atmosphere. This energy heats the very tenuous atmospheric layers. The chromosphere, heated to nearly 104 K, produces strong Hα (Young et al. 1989) and Ca ii H &
K emission (Skumanich et al. 1975), while the stellar corona is heated to temperatures in
excess of 106 K, generating significant emission at X-ray wavelengths. Radio emission is
observed as well, correlated with the X-ray emission (Güdel et al. 1993).
Additional, indirect evidence for existence of strong magnetic fields has also been observed.
Low-mass stars are known to undergo energetic flares across the entire spectrum (see, e.g.,
Osten et al. 2005). These events are associated with magnetic activity on the Sun. Lowmass stars have also been monitored for periodic amplitude fluctuations in their light curves,
thought to betray the presence of star spots. As the star rotates, different spot configurations
on the stellar disc rotate into and out of view.
Theoretically testing the magnetic hypothesis is difficult, owing to the inherently threedimensional nature of magnetic fields. Stellar evolution codes, being calculated in only a
single dimension, therefore require a substantially simplified approach. Nevertheless, attempts have been made to model the effects of magnetic fields on low-mass stellar structure
21
(Lydon & Sofia 1995; D’Antona et al. 2000; Mullan & MacDonald 2001; Chabrier et al. 2007).
We will review these approaches in the next section.
1.4 Magnetic Fields in Stellar Evolution
The identification of the magnetic hypothesis made the inclusion of a magnetic field in lowmass stellar evolution calculations desirable. Two groups have developed different methods
for including magnetic fields and the effects on convection (Mullan & MacDonald 2001;
Chabrier et al. 2007). An additional ingredient was also proposed by Chabrier et al. (2007)
to include the effects of photospheric spots. Here, we will briefly introduce the various
methods so that they may be familiar to the reader.
1.4.1 Modified Adiabatic Gradient
Placing a vertical magnetic field in a plasma helps to stabilize the plasma against convective
instability (Thompson 1951; Chandrasekhar 1961). Knowing this, Gough & Tayler (1966)
derived modifications to the Schwarzschild criterion that may be used in stellar evolution
models to mimic the presence of a magnetic field. The Schwarzschild criterion is a local condition that stellar evolution models use to determine whether a location within the model
transports energy primarily through radiation or convection. In a non-magnetic fluid the
condition states that a fluid is stable against convection if
∇s − ∇ad < 0,
(1.10)
where ∇s is the temperature gradient (d ln T /d ln P ) of the fluid and ∇ad is the adiabatic
temperature gradient. If a magnetic field is included, however, the right-hand-side of the
22
equation is greater than zero,
∇s − ∇ad <
B2
,
B 2 + 4πγP gas
(1.11)
where B is the magnetic induction, γ is the ratio of specific heats c P /cV , and P gas is the
pressure due to the fluid or gas particles. This is one of several conditions that can be
formulated depending on various assumptions (Gough & Tayler 1966). Hereafter, B will be
referred to as the magnetic field strength.
The new stability criterion advanced by Gough & Tayler (1966) has since been implemented
in stellar evolution models of low-mass stars on the pre-main-sequence (D’Antona et al.
2000) and main sequence (Mullan & MacDonald 2001). The authors of the latter study have
since used their models in a variety of situations, including testing if magnetic fields can prevent stars from being fully convective (Mullan & MacDonald 2001), looking at the influence
of magnetic field on the lithium abundances of stars in β-Pic (MacDonald & Mullan 2010),
and investigating how magnetic fields affect the structure of brown dwarfs (MacDonald & Mullan
2009; Mullan & MacDonald 2010). Additionally, the authors have attempted to quantify how
finite electrical conductivity might alter the approach, which is crucial for work on brown
dwarfs (MacDonald & Mullan 2009).
1.4.2 Reduced Mixing Length
The second method to include magnetic field effects was proposed by Chabrier et al. (2007).
Their approach is qualitatively different from that of Mullan & MacDonald (2001). Instead
of looking at the stabilization of convection, Chabrier et al. (2007) attempted to mimic the
development of cooling flows along magnetic flux tubes. These cooling flows would transport energy away from a convecting bubble, reducing the efficiency of convection (see Section 1.3.3). To mimic the reduction of convective efficiency the authors reduced the convec-
23
tive mixing length, αMLT .
The method has been used by Chabrier et al. (2007) and Morales et al. (2010) to explore
the effect of a magnetic field on models of low-mass stars in DEBs. Significant reductions to αMLT were required, but the authors in both papers were able to reconcile their
models with the observations. The problem with the approach, however, is that it has no
predictive power. There is no means of calibrating a reduction in αMLT to magnetic field
strengths. Consequently, it is not possible to test the validity of these models. The ambiguity of whether the reduction in αMLT is the result of magnetic fields or naturally inefficient
convection only further complicates the matter.
1.4.3 Star Spots
In addition to the reductions in αMLT , Chabrier et al. (2007) introduced a means of accounting for the emergence of photospheric spots. They did this by reducing the total bolometric
flux at the stellar surface (Spruit 1982a,b; Spruit & Weiss 1986). Spots were included by
imagining a given fraction of the total surface flux is blocked by spots with some temperature contrast relative to an immaculate photosphere. Specifically,
¶ ¸
· µ
Ss
Ts 4
β=
1−
S
Teff
(1.12)
where S s /S is the fractional surface coverage by spots, Ts is the temperature within the
active region, and Teff is the effective temperature of the immaculate (i.e., spot free) photosphere. The bolometric flux is then modified such that
¡
¢
F = 1 − β F⋆
(1.13)
with F and F⋆ representing the total flux and the immaculate photospheric flux, respectively. By combining the above star spot correction and a reduced αMLT formalism,
24
Chabrier et al. (2007) and Morales et al. (2010) were able to effectively inflate low-mass stellar radii and suppress effective temperatures.
There are observational techniques developed to measure spot filling factors and temperature contrasts (O’Neal et al. 1998, 2004; O’Neal 2006; Catalano et al. 2002). Unfortunately,
the large majority of stars in their samples are evolved stars. Two very active K-dwarfs were
observed (O’Neal et al. 1998, 2004), but the extraction of spot properties requires accurate
knowledge of the stellar reference spectra, in particular the Teff of their M-dwarf calibrators
(O’Neal et al. 1998). Photometric color indices were used to assign M-dwarf Teff s (Bessell
1991) and appear to be several hundred degrees too cool. We determined the latter by
cross-referencing their M-dwarf calibration stars with stars that have had their temperature
determined using interferometry (Berger et al. 2006). Therefore, caution must be exhibited
when benchmarking spot properties to the results of these studies.
Including star spots in this fashion also raises several issues. First, by including spots with
other mechanisms for convective suppression (either reduced αMLT or δMM ), there is an inherent risk that one might “double count” the effects of magnetic fields. Star spots are the
physical manifestation of localized vertical magnetic fields suppressing convection. Therefore, methods that include the suppression or inhibition of convection and star spots conflate the actual ability of magnetic fields to suppress convection. Ultimately, star spots have
a physical origin in the suppression of convection. This leads us to question why reductions
in flux must be accounted for in “spots,” when methods like those described above ought to
perform the same task.
This leads into the second issue, which is that it is difficult to verify to what degree the
total flux of a star is being bottled up by the presence of spots. Stellar evolution models are
computed in 1-dimension, a severe restriction. Accounting for the 2- or 3-dimensional flow
of heat around or near spots and active regions is extremely difficult. Thermal modeling
of the regions near spots does suggest that a spot creates a bottleneck and traps flux in the
25
immediate region below it (Foukal et al. 1983). It appears that heat does not necessarily flow
efficiently around spots. However, such reductions in flux observed at the surface of a star
may be counteracted by the appearance of bright faculae, such as those observed on the Sun
(Foukal et al. 2006). Small depressions near the stellar photosphere allow flux from deeper
in the star to escape causing those small regions to appear brighter (Spruit 1977). This has
been cited as the reason for the increase in TSI with sun spot number (Foukal et al. 2006).
It is not clear if the solar-stellar comparison is completely valid, but sun spot data and the
effects of solar activity on the solar properties should not be neglected.
1.4.4 A Need for New Models?
With the availability of the magnetic field models described above, do we need another
set of models using yet another approach? We feel that there is a need, predicated on the
desire for accurate low-mass stellar models with strong predictive power. The example we
have used throughout this chapter is that of exoplanet host star characterization. Currently
available magnetic stellar models do not provide a complete package necessary for accurate
exoplanet characterization.
The models of Baraffe et al. (1998) are commonly used to predict the properties of low-mass
stars under the assumption that magnetic fields are not required. Correcting for magnetic
fields with the Baraffe et al. (1998) models has been explored, but only through the use of a
reduced αMLT and the star spot parameterization (Chabrier et al. 2007; Morales et al. 2010).
However, there is currently no means of assuming what value of αMLT is required, since
no observable has been posited to correlate with the necessary αMLT . The variables needed
in the star spot approach can be verified observationally, but must be used in combination
with the reduced αMLT approach.
On the other hand, the magnetic models of Mullan & MacDonald (2001) can be verified
observationally. Their magnetic inhibition parameter, δMM , has the value of the magnetic
26
field strength, which is defined at the surface. Observations of stellar magnetic fields can
then be used to validate the proposed magnetic field strengths (Reiners & Basri 2007, 2009;
Donati & Landstreet 2009). However, the models advanced by Mullan & MacDonald (2001)
have not been benchmarked against observations of low-mass stars without their magnetic
perturbation, nor have they been compared to the BCAH98 standard models. Thus, the
changes in radius as a function of magnetic field strength should be used with caution to
correct the radius predictions of other model sets.
We see a need for a set of magnetic models that can be trusted as accurate in all situations—
magnetic and non-magnetic—across multiple mass regimes. The goal is to create a set of
robust and versatile low-mass stellar models. To achieve this goal, we have opted to pursue the development of magnetic models within the framework of the Dartmouth stellar
evolution code.
1.5 The Dartmouth Stellar Evolution Code
The Dartmouth Stellar Evolution Program (DSEP),11 is one of the latest progeny of the
Yale Rotating Evolution Code (Guenther et al. 1992). Since the code’s inception, the physics
used in the models have been updated and improved. These additions and updates have
been described extensively in the literature (Chaboyer & Kim 1995; Chaboyer et al. 2001;
Bjork & Chaboyer 2006; Dotter et al. 2007, 2008). Below, we will summarize the physics
used in the code that are pertinent for the present work and also discuss the calibration of
the models.
11
http://stellar.dartmouth.edu/models/
27
1.5.1 Physics
The conditions in the optically thin atmosphere of low-mass stars precludes the use of gray
atmosphere approximations and the use of radiative T(τ) relations (Allard & Hauschildt
1995; Chabrier & Baraffe 2000, and references therein). Instead, non-gray radiative transfer
calculations are required to accurately describe the thermal structure of the atmosphere. We
therefore define the surface boundary conditions of our models using the phoenix amescond theoretical atmospheres (Hauschildt et al. 1999a,b). While the validity of gray atmosphere approximations increases as the model mass increases, all of our models use the
phoenix model atmospheres for consistency.
The surface boundary condition is defined by the gas pressure at a pre-selected optical depth.
The surface gas pressure is found by interpolating in pre-compiled tables that list the gas
pressure as a function of log g and Teff at different metallicities. These tables are compiled
using the atmosphere structures generated by the phoenix code. The interpolation quantities (log g , Teff , and metallicity) are defined by the interior structure model (i.e., DSEP). The
choice of where to attach the model atmosphere to the interior model—the pre-selected optical depth that was alluded to—is an important one. Our experience indicates that attaching
the atmosphere at T = Teff is reasonable for M & 0.2M⊙ (Dotter 2007). The optical depth
is then set by the phoenix structures, which compute the optical depth at which T = Teff .
Below that threshold mass it is imperative to attach the model atmosphere deeper into the
interior (say τ = 100) because the temperature gradient in the outer layers is significantly
non-adiabatic (Chabrier & Baraffe 2000; Dotter 2007).
Arguably, the most crucial piece of physics in stellar evolution codes is the equation of state
(EOS). In general, the EOS is mass dependent, with the level of non-ideal contributions
gaining importance towards lower masses. There are two primary EOSs used by DSEP: an
analytical EOS and the FreeEOS, a numerical EOS created by Alan Irwin.12
12
http://freeeos.sourceforge.net
28
Model comparisons have shown that models of main seqeuence stars with masses above
0.8M ⊙ can be accurately calculated using the analytical EOS (Prather 1976; Dotter 2007).
This analytical EOS takes into account the particular ionization state of the stellar plasma
using the Saha ionization equation. Furthermore, ion-charge shielding is included via a
Debye-Hückel correction (Chaboyer & Kim 1995).
The EOS for stars below 0.8M⊙ must include a variety of non-ideal contributions. For
this job, DSEP uses the FreeEOS in the EOS4 configuration (Dotter 2007; Dotter et al. 2008).
We selected the FreeEOS for three primary reasons. First, it includes several non-ideal
contributions to the EOS, such as Coulomb interactions and pressure ionization, that become
important in the dense plasma of low-mass stars. Second, the FreeEOS calculates the EOS
for hydrogen, helium, and eighteen heavier elements as opposed to only calculating the EOS
for hydrogen and helium. One can therefore alter the specific solar abundance reference
data as well as the particular stellar metallicity. Finally, the FreeEOS may be called directly
from within the stellar evolution code. This feature avoids the need to interpolate within
EOS tables, thereby minimizing numerical errors.
Opacities are an integral component within stellar evolution models. Radiative opacity governs, almost entirely, the size of the convective envelope in low-mass stars. The efficiency of
energy transport, in turn, largely determines the star’s physical properties. Radiative opacities used by DSEP are temperature dependent. The OPAL opacities are used at temperatures
above 104 K (Iglesias & Rogers 1996) while the Ferguson et al. (2005) opacities are used at
lower temperatures. Contributions to the opacity also arise from the conduction of heat
through the mobility of free electrons in the stellar plasma (Hubbard & Lampe 1969; Canuto
1970; Sweigart 1973). Updates to the standard electron thermal conduction calculations were
presented by Cassisi et al. (2007) and have been included (see Appendix B). However, they
a negligible effect on the main-sequence evolution of low-mass stars.
As stellar mass increases there is a decrease in the size of the surface convective enve-
29
lope. Accompanying this decrease is an increase in the importance of diffusion physics,
which occurs in the radiative zone below the convection zone boundary (Michaud et al.
1984). Diffusion physics governs the separation of chemical species through processes such
as gravitational settling and thermal diffusion. DSEP follows the diffusion of helium and
heavy elements following the formulation of Thoul et al. (1994), unless the star is fully convective. Fully convective stars are assumed to be completely and homogeneously mixed
throughout since the convective timescale dominates the diffusion timescale (Michaud et al.
1984). We note that an extra diffusion term due to turbulent mixing is also included
(Proffitt & Michaud 1991; Richard et al. 2005). This extra mixing is described in Appendix B
and replaces the method of partially inhibiting diffusion (Chaboyer et al. 2001).
Many of the stars in low-mass DEBs are likely to have their rotation periods synchronized with their orbital periods. This means they are likely to be rotating rapidly (v sin i >
5 km s−1 ). It is then natural to wonder whether we should consider the effects of rotational
deformation in our models. As a first approximation, we applied Chandrasekhar’s analysis of slowly rotating polytropes to our DEB sample. Chandrasekhar (1933) derived an
expression for the stellar oblateness analytically. He defined slowly rotating to be when
χ≡
ω2
≪1
2πGρ c
(1.14)
where ω is the stellar angular velocity and ρ c is the central mass density. As an example we
will assume rather extreme values for a 0.80M⊙ star under consideration. With a rotational
velocity of 102 km s−1 and a radius of 0.80R ⊙ , this star would have ω ≈ 10−4 . If the star has
ρ c ∼ 50 g cm−3 , it will have χ ∼ 10−3 .
Based on the assumed polytropic index, n , Chandrasekhar derived that the deviation from
30
sphericity (oblateness) could be approximated as
F≡
r eq − r pole
r eq




5.79χ



=
9.82χ





 41.81χ
for n = 1.5
for n = 2.0
for n = 3.0
where F is the oblateness and r eq and r pole are the equatorial and polar radius, respectively. We approximated the polytropic index for each star using the interior density profile
predicted by DSEP. The density profile for low-mass stars only slightly deviates from the
polytrope prediction over the inner 98% of the star (by radius). Specifically, we found that
below 0.4M⊙ , the models were best represented by an n = 1.5 polytrope. Above 0.4M⊙ but
below 0.65M⊙ , an n = 2.0 polytrope was appropriate. For all masses greater than 0.65M⊙ ,
we defaulted to assuming an n = 3.0 polytropic index.
The deviation from sphericity for our imagined case above would then be around 5%. Note
that these conditions are exaggerated. We’ll see in Chapter 4 that for a main sequence star,
χ < 10−4 is more typical. We therefore feel justified in neglecting rotation. At times we will
return to this calculation to re-justify our assumption with more specific stellar properties.
The final pieces of physics worth mentioning are the updated nuclear reaction rates and the
introduction of the convective overturn time. We have updated the previous set of nuclear
reaction rates (Adelberger et al. 1998; Dotter 2007) with the latest nuclear reaction rate recommendations (Adelberger et al. 2011). Although modifications have been made to the p-p
chain reaction rates, they have a negligible impact on low-mass main sequence evolution.
Further discussion of the updated rates are provided in Appendix B. Lastly, DSEP now computes the “local” convective overturn time at each model time step (Kim & Demarque 1996).
As with other physics that have been added or updated, details are located in Appendix B.
31
Table 1.1
Solar calibration parameters.
Parameter
Adopted
Model
Age (Gyr)
4.57
···
M ⊙ (g)
1.989 × 1033
R ⊙ (cm)
6.9598 × 1010
R bcz /R ⊙
0.713 ± 0.001
L ⊙ (erg s−1 )
(Z /X )surf
Y⊙, surf
†
···
Reference
Bahcall et al. (2005)
IAU 2009†
log(R) = 8 × 10−5
Neckel (1995); Bahcall et al. (2005)
3.8418 × 1033
log(L ) = 2 × 10−4
0.714
Basu & Antia (1997); Basu (1998)
0.0231
0.0230
Grevesse & Sauval (1998)
0.2485 ± 0.0034
0.2455
Basu & Antia (2004)
Bahcall et al. (2005)
http://maia.usno.navy.mil/NSFA/NSFA_cbe.html
1.5.2 Solar Calibration
The primary input variables for stellar evolution models are defined relative to comparable
solar values. These input variables include the stellar mass, the initial mass fractions of
helium and heavy elements (Yi and Zi , respectively), and the convective mixing length
parameter, αMLT . The latter defines the length scale of a turbulent convective eddy relative
to the pressure scale height. Since they are all defined relative to the solar values, we must
first define what constitutes the Sun for the model setup. To do this, we require a 1 M⊙
model to reproduce the solar radius, luminosity, radius to the base of the convection zone,
and the solar photospheric (Z /X ) at the solar age (4.57 Gyr; Bahcall et al. 2005). These
values are presented in Table 1.1.
We use an iterative procedure to find the optimal combination of model input variables that
yield the solar properties at the solar age. First, we supply the model with an initial guess
for X i , Zi , and αMLT . Four models are then automatically generated. One of these is the
“base model” where the user-supplied values are used. The three additional models are run
with one of the input variables perturbed slightly.
Second, we calculate the difference between the actual solar radius, luminosity, and photospheric (Z /X ) and those predicted by the base model. Since the radius and luminosity
32
of the base model are quoted in solar units, this is straightforward. Note that the solar
photospheric (Z /X ) is dependent on the heavy element composition adopted. This process
yields δR , δL , and δ(Z /X ), where we have defined R = R/R⊙ and L = L/L ⊙ . Ultimately,
we seek to identify the set of X , Z , and αMLT that produces a model where δR , δL , and
δ(Z /X ) are all within some pre-defined tolerance of zero. This tolerance was set to 4×10−4
in the appropriate units.
It is unlikely that the first base model will yield a correct solar calibration. We therefore
want to improve upon our guess, which we do using the three additional models with perturbed input variables. The three perturbed models allow us to compute how the predicted
radius, luminosity, and photospheric (Z /X ) change as a function of the individual input
variables. For each perturbed model, we compute the difference between the model radius,
luminosity, and photospheric (Z /X ) at the solar age relative to the base model. We then
setup a set of three linearly independent equations:
¶
µ
¶
µ
¶
∂R
∂R
∂R
δX +
δZ +
δαMLT
δR =
∂X
∂Z
∂αMLT
µ
(1.15)
µ
¶
¶
µ
¶
µ
∂L
∂L
∂L
δL =
δαMLT
δX +
δZ +
∂X
∂Z
∂αMLT
(1.16)
¶
µ
¶
µ
¶
∂(Z /X )
∂(Z /X )
∂(Z /X )
δX +
δZ +
δαMLT ,
δ(Z /X ) =
∂X
∂Z
∂αMLT
(1.17)
µ
where the partial differentials in the above equations are determined by comparing the
perturbed models to the base model, as described above. We then solve the set of linear
33
equations in matrix form



³
´
³
´
³
´
−1 
∂R
∂R

 δX   ∂R
∂Z
∂αMLT 

  ∂X

  ³ ´
³ ´
³
´


 
∂L
∂L

 δY  =  ∂L
∂Z
∂αMLT 

  ∂X

 ³
´
´ ³
´ ³


 
∂(Z /X )
∂(Z /X )
∂(Z /X )
δαMLT
∂X
∂Z
∂α
MLT

 δR 






·  δL  .






δ(Z /X )
(1.18)
Solving this system of equations yields δX , δZ , and δαMLT . These values are used to iterate
on the values for our initial guess. This procedure is repeated until the code converges on a
solution for δR , δL , and δ(Z /X ) that are within the defined tolerance. The final model is
considered calibrated and the code prints the final values of X i , Zi , and αMLT .
Results of the calibration procedure using the model physics described in Section 1.5.1 are
give in Table 1.1. The final set of input variables was X i = 0.70625, Yi = 0.27491, Zi =
0.01884, and αMLT = 1.938. The adopted heavy element abundance used throughout this
work is the solar photospheric abundance determined by Grevesse & Sauval (1998).
1.6 Thesis Outlook
The body of work contained in this thesis represents our analysis and approach to addressing the low-mass stellar mass-radius problem. Our aim from the outset was to develop a
set of self-consistent magnetic stellar evolution models using methods independent from
previous investigations (Mullan & MacDonald 2001; Chabrier et al. 2007). However, along
the way we discovered that other factors had not been adequately addressed in previous
studies. This realization had its genesis after the results from Kepler began to trickle in with
the announcement of a new DEB KOI-126 (Carter et al. 2011). We present the theoretical
modeling effort of KOI-126 in Chapter 2. Our investigation in Chapter 2 led to an interest in
the application of stellar interior structure constants in stellar evolution, as seen in Chapter
3.
34
Shortly after the announcement of KOI-126, several more low-mass DEBs were discovered
and characterized (e.g., Kraus et al. 2011; Doyle et al. 2011). Chapter 4 concerns using the
enlarged DEB data set to look at the different hypotheses suggested for the mass-radius
problem.
The following three chapters present the development, testing, and initial results of a new
set of magnetic stellar evolution models. Our models are introduced and tested in Chapter
5. A detailed look at modeling DEB stars is presented in Chapters 6 and 7. We not only
study whether or not magnetic models can sufficiently inflate stellar radii, but also discuss
how physically realistic our models are so that we may attempt to enhance their predictive
power. Finally, in Chapter 8 we present the current state of the low-mass star field and ideas
for future projects involving our magnetic models.
We also include two appendices that do not present any significant scientific content. Appendix A provides a short user’s guide for the Dartmouth Magnetic Evolutionary Stellar
Tracks And Relations, DMESTAR. Appendix B gives a summary of additional model physics
introduced to DSEP during the course of the development of DMESTAR. The intention of
this appendix is to provide a reference for new model input variables.
35
Chapter 2
Accurate Low-Mass Stellar Models of
KOI-126
This chapter presents and expands on material that appeared in The Astrophysical Journal
(2012, 740, L25) in an article titled “Accurate Low-Mass Stellar Models of KOI-126.”
2.1 Introduction
Prior to 2011, only nine low-mass, detached eclipsing binary (DEB) systems with precisely
measured masses and radii were known (Popper 1997; Torres & Ribas 2002; Ribas 2003;
López-Morales & Ribas 2005; Bayless & Orosz 2006; López-Morales et al. 2006; López-Morales & Shaw
2007; Morales et al. 2009b,a). Of those nine, only CM Draconis (Lacy 1977) contains a star
thought to lie below the fully convective boundary (M . 0.35M⊙ ). In fact, it contains two.
CM Draconis displays radii that are discrepant with stellar evolution models (Morales et al.
2009a). However, because it is the only system for which we have precisely measured masses
and radii, it is unclear whether the models are insufficient or whether CM Draconis is peculiar. This scenario changed with the launch of the NASA’s Kepler Space Telescope (hereafter
36
Kepler).
Kepler was launched in May of 2009 with the express purpose of constraining the the frequency of Earth-sized planets orbiting solar-type stars in the habitable zone (Borucki et al.
2010). To acheive this goal, Kepler was designed to continuously stare at the same galactic location—in the direction of the constellation Cygnus—and photometrically monitor the
brightness of over 150,000 stars for transiting exoplanets. High precision photometry of each
star is acquired with a 30 minute cadence, 1 providing extremely detailed light curves that
are continuous over nearly the entire lifetime of the mission.
The Kepler mission created a revolution in identifying DEB candidates—or any stars showing variability, for that matter—resulting in a catalog of 1 879 DEB candidates, 231 of which
are potentially low-mass, from only 44 days of data (Prša et al. 2011; Coughlin et al. 2011).
Among these candidates was a system that displayed a highly irregular pattern of eclipses.
The system, KOI-126, was announced by Carter et al. (2011) as a triply eclipsing, hierarchical triple system with two low-mass, fully convective stars.
KOI-126 has a G star primary2 with two low-mass, fully convective companions (Carter et al.
2011). The two low-mass stars (KOI-126 B and C) are in a tight 1.77 day orbit that is orbiting the more massive primary star (KOI-126 A) every 33.92 days on a fairly eccentric
path (e = 0.3043 ± 0.0024). The authors were able to derive fundamental properties for all
three stars using a photometric-dynamical model that finds the most probable stellar and
orbital properties by reconstructing the detailed light curve (see supplementary material
for Carter et al. 2011). They found that M A = 1.347 ± 0.032M⊙ , R A = 2.0254 ± 0.0098R ⊙ ,
M B = 0.2413 ± 0.0030M ⊙ , R B = 0.2543 ± 0.0014R ⊙ , MC = 0.2127 ± 0.0026M ⊙ , and RC =
0.2318 ± 0.0013R ⊙ . Spectroscopy of the more massive primary reveiled it has an effective
temperature of 5 875±100 K with a slightly super-solar metallicity of [Fe/H] = +0.15±0.08.
Lastly, a non-zero eccentricity of the (B, C) system (e = 0.02234) allowed the authors to
1
2
Short cadence observations of 1 minute are also available for select targets.
In truth, the G star is actually the tertiary, but was deemed the primary based on its relative brightness.
37
Table 2.1
Properties of the KOI-126 hierarchical triple system.
Property
Mass (M⊙ )
Radius (R⊙ )
Teff (K)
Relative Flux
[Fe/H] (dex)
KOI-126 A
KOI-126 B
KOI-126 C
1.347 ± 0.032
0.2413 ± 0.0030
0.2127 ± 0.0026
2.0254 ± 0.0098
0.2543 ± 0.0014
0.2318 ± 0.0013
1.0
(3.26 ± 0.24) × 10−4
(2.24 ± 0.48) × 10−4
5875 ± 100
···
+0.15 ± 0.08
···
Properties of the (A, (B, C)) binary orbit
Period (day)
33.9214 ± 0.0013
Eccentricity
0.3043 ± 0.0024
Semi-major axis (R ⊙ )
54.63 ± 0.37
Properties of the (B, C) binary orbit
Period (day)
1.76713 ± 0.00019
Eccentricity
0.02234 ± 0.00036
Semi-major axis (R ⊙ )
4.726 ± 0.019
place a weak constraint on the apsidal motion constant for the two low-mass stars. They
determined that it is below 0.6 at the 95% confidence level. The properties of the individual
stars and the orbits are summarized in Table 2.1.
Carter et al. (2011) derived an age for the system of 4 ± 1 Gyr by fitting the properties
of KOI-126 A to theoretical Y2 isochrones (Demarque et al. 2004). They then compared
theoretical isochrones from Baraffe et al. (1998, henceforth BCAH98)3 to observations of the
low-mass stars in the mass-radius plane to investigate the low-mass, stellar mass-radius
relationship. The models were seen to underpredict the radii of the two low-mass stars by
2% – 5%, even though several ages (1, 2, 4, and 5 Gyr) were adopted. It is important to
note that no super-solar metallicity isochrones are available for the BCAH98 models. This
is in part due to limitations of their equation of state. Carter et al. (2011) suggested that
3
Two separate isochrone sets were used due to specialization of each set in the different mass regimes.
38
the observed radius discrepancy could be the result of both the super-solar metallicity and
possible magnetic activity of the system.
Here, we test the hypothesis proposed by Carter et al. (2011) that metallicity underlies the
observed discrepancies between the BCAH98 models and the observations of the two lowmass stars. Results are presented in Section 2.2 of theoretical stellar modeling of the KOI126 components using the Dartmouth Stellar Evolution Program, which was described in
Chapter 1.5. If the metallicity hypothesis is confirmed, it would represent the first accurate
modeling of low-mass, fully convective stars using standard stellar evolution models.
In Section 2.2.1 we estimate the age of the stellar system using the Dartmouth models to
provide consistency between the properties of the primary and the two low-mass stars. We
then compare models of the two low-mass stars to the observations in the mass-radius plane
in Section 2.2.2. To further improve the comparison between the models and observations
we compute the relative fluxes expected for the theoretical models in the Kepler bandpass
in Section 2.2.3 and derive the theoretical apsidal motion constants in Section 2.2.4. Finally,
in Section 2.3, we discuss the implications of this study with regards to the only other well
known low-mass, fully convective, detached eclipsing binary, CM Draconis.
2.2 Results
2.2.1 Stellar Age
The age of the system was constrained by evolving a 1.347M⊙ model with a metallicity of
[Fe/H] = +0.15 and a solar calibrated αMLT and matching the model radius with the observed
radius of KOI-126 A. This is illustrated in Figure 2.1 by the black, solid line. We derive an
age for KOI-126 of 4.1 ± 0.6 Gyr, consistent with the age derived by Carter et al. (2011). The
uncertainty in the age is dominated by the uncertainty in the observed mass and metal-
39
2.4
1.347 M⊙, [Fe/H] = +0.15
1.315 M⊙, [Fe/H] = +0.23
1.379 M⊙, [Fe/H] = +0.07
Radius (R⊙)
2.2
2.0
1.8
1.6
1.4
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Age (Gyr)
Figure 2.1: Stellar evolution mass tracks used to constrain the age of KOI-126 A. The lightblue horizontal band represents the observed radius with the associated 1σ uncertainty.
licity of the system. Figure 2.1 shows two additional mass tracks with mass-metallicity
combinations chosen to produce the largest variations in the possible age of the primary
while remaining with the observational constraints. The observational uncertainty in the
primary’s radius contributes little to the overall uncertainty budget as the star appears to
be at an age where its radius is undergoing rapid evolution. We also examined the effect of
changing the reference temperature for turbulent diffusion, which alters the chemical abudance profile within the star. It was found to play a negligible role in the age determination,
even when turbulent diffusion was completely removed.
2.2.2 Mass-Radius Relation
The primary results of this chapter are demonstrated in Figure 2.2. DSEP accurately reproduces the observed radius for each of the low-mass stars at 4.1 Gyr with [Fe/H] = +0.15.
40
0.26
Radius (R⊙)
0.25
0.24
0.23
DSEP, [Fe/H] = +0.00
DSEP, [Fe/H] = +0.15
BCAH98, [Fe/H] = +0.00
0.22
0.20
0.21
0.22
0.23
0.24
0.25
Mass (M⊙)
Figure 2.2: Mass-radius relationship as defined by CM Draconis (light gray – points) and
KOI-126 (maroon – points). Overlaid are 4.1 Gyr theoretical isochrones from DSEP with
[Fe/H] = 0.0 (black – solid) and [Fe/H] = +0.15 (light blue – dashed) and from BCAH98 with
[Fe/H] = 0.0 (gray – dash-dotted).
We find that for model masses4 M2 = 0.2410M⊙ and M3 = 0.2130M⊙ the predicted radii
from DSEP are R2 = 0.2544R ⊙ and R 3 = 0.2312R ⊙ . The relative error between the model
and observed radii is less than 0.3%. At solar metallicity, the models predict radii that are
approximately 1% smaller than those predicted by the super-solar metallicity models. Figure
2.2 shows this 1% radius deflation caused by lowering the model metallicity to a solar value
is enough to destroy the agreement between the models and the observations.
The predicted model radii appear to be robust. Artificially reducing the mixing length
(αMLT = 1.00), as is typically done for low-mass stellar models (Chabrier & Baraffe 1997;
Baraffe et al. 1997; Baraffe et al. 1998), introduces radius variations on the order of 0.5%.
Similarly, fitting the atmosphere to a deeper point in the stellar envelope (at τ = 100 instead
4 Observational
component indicies j = {A, B, C } map directly to i = {1, 2, 3}, which are the equivalent
model component indicies.
41
of where T = Teff ) changes the model radius by less than 0.5%.
The 1% reduction in the model radius caused by reducing the metallicity from [Fe/H] =
+0.5 to [Fe/H] = 0.0 is not as large as the 2% – 5% difference noted by Carter et al.. This is
because low-mass, solar metallicity DSEP models are approximately 1% larger than those
of BCAH98, at a given mass. The difference is likely a consequence of the equation of state
(EOS) adopted by each group. Whereas the EOS used by BCAH98 is calculated for a pure
hydrogen/helium plasma, FreeEOS calculates the EOS for an arbitrary metal abundance.
Therefore, our models can be more reliably calculated above solar metallicity. This effect of
metals on the EOS was estimated by Chabrier & Baraffe (1997) to create radius changes of
about 1% – 2%, consistent with our simulations.
To explicitly test the influence of metals on the EOS, and on the model radii, we generated
DSEP models with the Saumon et al. (1995, hereafter SCVH95) EOS in an attempt to reproduce the BCAH98 model radii predictions. In addition, models were computed using the
OPAL 2001 EOS (Rogers & Nayfonov 2002). The OPAL equation of state accounts for heavy
elements, but using a different numerical technique and different physics than FreeEOS. Results are compiled in Table 2.2 and a detailed look at the internal density profile produced
by each EOS is shown in Figure 2.3. Models using the SCVH95 EOS produced radii within
0.5% of the BCAH98 models at the same mass and composition. Those models using the
OPAL EOS had radii intermediate between the FreeEOS predictions and the SCVH95 EOS.
Table 2.2
Radii (R ⊙ ) of low-mass stars derived using DSEP and
BCAH98 at 4.1 Gyr with [Fe/H] = 0.0.
DSEP
BCAH98
Mass
FreeEOS
OPAL
SCVH
SCVH
0.20 M⊙
0.2180
0.2174
0.2155
0.2168
0.2591
0.2588
0.2578
0.2567
0.25 M⊙
42
7.0
OPAL ’01
6.5
SCVH ’95
FreeEOS
log10(T) [K]
6.0
0.25 M⊙
5.5
0.15 M⊙
5.0
4.5
4.0
3.5
-6
-4
-2
0
-3
log10(ρ) [g cm ]
2
Figure 2.3: Comparison of three popular equations of state.
2.2.3 Relative Fluxes
Stellar effective temperature is another property against which to compare the predictions
of stellar models. Measuring the effective temperature of an M-dwarfs, however, is a nontrivial task beset by numerous complications and uncertainties. The molecule-ridden spectrum presented by most M-dwarfs masks the presence of the background continuum and
any line features present in the spectrum are typically blends of multiple atomic and molecular species. It is not surprising, then, that while Carter et al. (2011) could determine the
effective temperature of KOI-126 A to be 5 875±100 K from spectroscopy they were unable
to measure the effective temperature of the two M-dwarfs.
However, the dynamical/photometric model used by Carter et al. (2011) did yield the flux of
each M-dwarf relative to the primary. In fact, this is a requirement to be able to reconstruct
the precise lightcurve eclipse profiles. Their results showed that f B / f A = (3.26±0.24)×10−4
and fC / f A = (2.24 ± 0.48) × 10−4 .
43
Normalized Flux
1.0
Kepler Filter Transmission Curve
0.8
0.6
0.4
0.2
0.0
4000
5000
6000
7000
8000
9000
10000
7000
8000
9000
10000
Normalized Flux
1.0
0.8
Kepler Filter Transmission Curve
0.6
0.4
0.2
0.0
4000
5000
6000
Wavelength (Å)
Figure 2.4: The Kepler filter transmission profile overplotted on synthetic spectra from the
phoenix ames-cond model set. The synthetic spectra are for a typical A star (log g = 4.1,
Teff = 9 600 K, and [M /H] = 0.0; blue, solid line) and an mid-M-dwarf that is similar to
KOI-126 B and C (log g = 5.0, Teff = 3 500 K, and [M /H] = +0.15; maroon solid line).
To compare our stellar models, we derived a synthetic color-Teff transformation for the
Kepler bandpass. This was performed by convolving the Kepler spectral response function
provided in the Kepler Instrument Handbook5 (Van Cleve & Caldwell 2009) with synthetic
spectra created using the phoenix ames-cond model atmosphere code. The Kepler filter
profile is illustrated in Figure 2.4 with two synthetic spectra computed using the phoenix
online simulator.6 The two synthetic spectra are reprsentative of a typical A star (log g = 4.1,
Teff = 9 600 K, and [M /H] = 0.0) and a mid-M-dwarf similar to KOI-126 B and C (log g = 5.0,
3 500 K, and [M /H] = +0.15).
Given the properties ([Fe/H], luminosity, Teff ) of the interior structure models for each KOI126 star, we were able to predict the flux ratio for each of the two low-mass stars relative to
KOI-126 A. We found f 2 / f 1 = 4.77 × 10−4 and f 3 / f 1 = 3.71 × 10−4 . These results are differ5
6
See also http://keplergo.arc.nasa.gov/CalibrationResponse.shtml
http://phoenix.ens-lyon.fr/simulator/index.faces
44
ent from the observations at the 6σ and 3σ level, respectively. In addition to transforming
directly from model properties to the fluxes in the Kepler bandpass, we used an alternative,
semi-empirical method. The semi-empirical method relied on transforming the model properties into fluxes in the Sloan Digital Sky Survey (SDSS) g and r bandpasses and then using
an empirical relation to convert to the flux in the Kepler bandpass. The empirical relation
is available on the Kepler Guest Observer website.7 SDSS fluxes for our models were calculated using a synthetic color-Teff transformation (Dotter et al. 2008). The semi-empirical
transformation yielded flux ratios of f 2 / f 1 = 3.07×10−4 and f 3 / f 1 = 2.36×10−4 . Curiously,
these flux ratios are consistent with those measured by Carter et al. (2011).
Diagnosing why the flux ratios are discrepant for the purely theoretical transformation is
difficult. There could be two possible reasons for the errors: (1) our models over predict the
M-dwarf fluxes, meaning the Teff s we derive are too hot, or (2) the flux of the primary is
underpredicted. Addressing the first reason would likely require us to adopt a new set of
theoretical atmospheres, such as the updated phoenix bt-settl models (Allard et al. 2001,
2011) and is beyond the scope of this chapter. Instead, we elect to examine the possibility
that we underpredict the flux of the primary.
In an effort to increase the flux of the primary, we varied several properties of the primary
star. Specifically, we considered changes in the stellar mass, metallicity, and the amount of
convective core overshoot (CCO). Note that we still required the models to be consistent
with the observational data. When considering the largest possible mass (1.379M⊙ ), lowest
metallicity (+0.07), and a relatively high amount of CCO, the flux ratios were reduced to
within 2σ of the accepted values. As with the age determination, the tight observational
constraint imposed on the radius of the primary made the range of relevant radii insignificant in deriving our results.
7
See http://keplergo.arc.nasa.gov/CalibrationZeropoint.shtml
45
Table 2.3
Apsidal motion constants (k 2 ) for fully convective stars with
[Fe/H] = -0.50, 0.0, and +0.15. Values are quoted at 7 different ages (in Gyr).
Mass (M⊙ )
0.15
..
.
..
.
0.20
..
.
..
.
0.25
..
.
..
.
0.30
..
.
..
.
[Fe/H]
0.5
1.0
2.0
3.0
4.0
5.0
10.0
-0.50
0.1561
0.1561
0.1561
0.1561
0.1560
0.1560
0.1560
0.00
0.1560
0.1560
0.1560
0.1560
0.1560
0.1560
0.1559
+0.15
0.1557
0.1557
0.1557
0.1557
0.1556
0.1556
0.1556
-0.50
0.1523
0.1523
0.1522
0.1522
0.1522
0.1522
0.1521
0.00
0.1521
0.1521
0.1521
0.1521
0.1521
0.1520
0.1519
+0.15
0.1518
0.1517
0.1517
0.1517
0.1517
0.1517
0.1516
-0.50
0.1499
0.1498
0.1498
0.1498
0.1498
0.1497
0.1496
0.00
0.1496
0.1496
0.1496
0.1496
0.1495
0.1495
0.1494
+0.15
0.1496
0.1496
0.1496
0.1495
0.1495
0.1495
0.1493
-0.50
0.1482
0.1482
0.1481
0.1481
0.1481
0.1480
0.1479
0.00
0.1480
0.1480
0.1479
0.1479
0.1478
0.1478
0.1477
+0.15
0.1479
0.1479
0.1479
0.1478
0.1478
0.1477
0.1476
2.2.4 Apsidal Motion Constant
By the end of the nominal Kepler mission, Carter et al. (2011) predict they will know the
mean apsidal motion constant of the two low-mass stars with a relative precision of nearly
1%. From our low-mass stellar models, we can predict an apsidal motion constant for each
star, quantifying the degree to which the mass within the star is centrally concentrated. To
do this we followed the prescription of Kopal (1978) and solved Radau’s equation with j = 2,
a
¡
¢
¢
d η 2 6ρ(a) ¡
η2 + 1 + η2 η2 − 1 = 6
+
〈ρ〉
da
(2.1)
where
η 2 ≡ η 2 (a) =
46
a d ǫ2
ǫ2 d a
(2.2)
is the value of a particular solution to Radau’s equation, which is related to the stellar
deviation from sphericity, ǫ2 . Additionally, ρ(a) is the density of the plasma and 〈ρ〉 is the
average density. In the above equations, a is the radial distance to an equipotential surface
(a = r when the surface is spherically symmetric). We used a fourth-order Runge-Kutta
integration method to solve Radau’s equation to obtain a particular solution at the surface
of each star. Having determined η2 (R), we were able to determine the apsidal motion
constant since
k2 =
3 − η 2 (R)
4 + 2η 2 (R)
(2.3)
As a check on our apsidal motion constant integrator, we generated polytropic models characterized by the polytropic constant n = 1.0 and 1.5 and compared the results of our code
with the results of Brooker & Olle (1955), who tabulated a large number of apsidal motion
constants for polytropes with various n values.
The apsidal motion constants derived from the interior structure of our models for KOI-126
B and C are k 2, 2 = 0.1499 and k 2, 3 = 0.1512. This suggests our models are slightly less
centrally condensed than an n = 1.5 polytrope, which is characterized by k 2 = 0.1433. In
fact, the run of density and pressure for our models indicate our models are best described
by a polytrope with n ∼ 1.45. A set of theoretical apsidal motion constants for fully convective stars, generated using DSEP, is given in Table 2.3 and may be compared to future
observations. Finally, we note the effects of rotation on the derived k 2 values are negligible.
Employing the formulae of Stothers (1974), we find that rotation affects our k 2 values at the
0.02% level for stars rotating with a period of 1.7 days.
2.3 Discussion
Low-mass stars below the fully convective boundary are a wonderful tool to test fundamental physics. Their low mass affords theorists a stable, long lived (τage > 1011 yr) laboratory
47
with which to test the physics incorporated in the models. These fully convective stars have
relatively simple structures and uncertainties in the opacities, surface boundary conditions,
and the treatment of convection have relatively small effects on the predicted properties of
the models (see Section 3.2 of Dotter 2007). Modelers are relieved of having to specify a free
parameter since the internal structure is largely insensitive to the prescribed mixing length.
The age of these stars is also of little consequence, as the radii of low-mass stars is nearly
constant over their main sequence lifetime. These effects imply a unique mass-radius relation for stars of a given composition below the fully convective boundary. The discovery of
two new data points in this regime indicate the mass-radius relation is not unique and that
there is significant dispersion amongst the data.
Although we find agreement between our models and the stars in KOI-126, discrepancies
are still seen between our models and the components of CM Draconis (hereafter CM Dra;
Morales et al. 2009a). This fact is visible in Figure 2.2. Comparing KOI-126 and CM Dra, we
notice that they are strikingly similar. Both systems have low-mass components with nearly
identical masses, estimated ages, orbital periods (KOI-126: 1.77 d; CM Dra: 1.27 d), and
both systems are thought to be tidally synchronized, though not necessarily circularized.
Their similar characteristics suggest the M-dwarfs in both systems should be very much
alike. Apart from the differences caused by chemical composition, the stars should have
nearly identical radii. Not only do they not, but we observe the opposite relationship of
what is theoretically expected. CM Dra has a sub-solar metallicity (Viti et al. 1997, 2002;
Morales et al. 2009a) and should therefore possess smaller radii than KOI-126. Figure 2.2
indicates that the stars in CM Dra actually have larger radii than those in KOI-126.
It is possible that the KOI-126 system has been caught in a period of inactivity, similar to a
solar minimum. CM Dra appears to have undergone such a period of quiescence in 2000.
Morales et al. noted that no corrections due to starspots were needed in the analysis of the
light curve data from that year. If KOI-126 B and C are in a magnetically quiescent state, it
48
is possible that their radii would not show signs of starspots or inflated radii. We note that
starspots would likely be of negligible importance since variations due to spots would be
reduced to noise amongst the signal of KOI-126 A.
We find it encouraging that DSEP is able to predict the mass-radius relation suggested by
KOI-126 B and C. Although, more data points are needed to allow for a more complete
understanding of the reliance of the mass-radius relation on physics (standard and nonstandard) incorporated in current models. It is clear from this work that no standard stellar
evolution model can simultaneously fit both CM Dra and KOI-126 and that progress on
non-standard stellar evolution models will be required to fit CM Dra. As suggested by
numerous authors, magnetic activity is likely the culprit and must be incorporated into the
next generation of models. However, the effects of a magnetic field on the interior structure
of stars has previously been considered as a reduction in the prescribed mixing length,
mimicking the reduction in convective efficiency that should accompany the presence of a
magnetic field (Chabrier et al. 2007). With this and the discussion presented at the beginning
of this section in mind, it is not clear how magnetic activity would affect stars below the
fully convective boundary. Self-consistent magnetic stellar models should help lend insight
into the discrepancies with CM Dra.
Concerning the relative flux discrepancies observed between the purely theoretical transformation and the photometric models of Carter et al. (2011), the color-Teff transformations are
naturally suspect. The theoretical transformations might not provide a fully accurate transformation to the observational plane. In the low-mass regime, opacities are complicated by
the formation of molecules, which have presented previous problems for low-mass model
atmospheres (Delfosse et al. 2000). This is particularly relevant since the Kepler bandpass is
very broad and covers a large portion of the optical spectrum, where molecular opacities are
the most uncertain for M-dwarfs. Though the SDSS g and r filters provide nearly the same
wavelength coverage, the fact that they are two narrow bands instead of single broad band
49
may explain why we find better agreement using the semi-empirical method. However, we
must be cautious with this result as the systematic errors are not well constrained for the
transformation from SDSS magnitudes to a Kepler magnitude. It must also be noted that
the flux of the primary star is sensitive to the details of CCO. We observed that increasing
the amount of CCO brought the purely theoretical fluxes closer to the observational values.
More investigation will be required to accurately diagnose the discrepancies.
Finally, we note that the determination of the apsidal motion constant will provide a crucial
test of low-mass stellar evolution models. Morales et al. found k 2 ≈ 0.11 using the BCAH98
models, whereas our models predict a larger value of approximately 0.15. The difference
between these two values is directly attributable to the EOS, which determines the run
of density within a stellar model. If Carter et al. are able to accurately derive the apsidal
motion constant to within 1%, it will provide a stringent benchmark against which to test
the interior physics of low-mass stellar evolution models.
2.4 Summary
In their discovery paper, Carter et al. reported that the triply eclipsing hierarchical triple
KOI-126 appeared to support the mounting evidence that current standard low-mass stellar
models are unable to reproduce the observed mass-radius relation. However, we have generated stellar models and theoretical isochrone tracks using the Dartmouth Stellar Evolution
Program and find our model radii agree with the observations. Combining the KOI-126
measurements with previous observations of the low-mass binary system CM Dra, we find
that the dispersion in the observed fully convective mass-radius relation is significant and
stands in contrast to theoretical predictions. The fact that CM Dra, a system with sub-solar
metallicity, lies on the super-solar side of the theoretical mass-radius relation is indicative
of physics currently not incorporated in standard stellar models. We predict the apsidal
50
motion constant for each of the KOI-126 low-mass stars and find k 2 ≃ 0.15. Carter et al.
postulate that they will be able to determine the apsidal motion constant with a relative
precision of 1% by the end of the nominal Kepler mission. This will provide a crucial test
for our models and will provide a stringent constraint against which to test all current and
future low-mass standard stellar evolution models.
51
Chapter 3
Using the Interior Structure Constants as
an Age Diagnostic
The following chapter appears in the Astrophysical Journal (2013, 765, 86) under the title
“The Interior Structure Constants as an Age Diagnostic for Low-Mass, Pre-Main-Sequence
Detached Eclipsing Binary Stars.” It appears chronologically out of sequence, but in sequence
from a conceptual standpoint.
3.1 Introduction
In the study of the structure and evolution of low-mass stars, there are a variety of different methods capable of yielding reasonably accurate age estimates; for a thorough discussion of the different methods, and their strengths and weaknesses, see the review by
Soderblom (2010). One such method uses detached eclipsing binaries (DEBs) to assign an
age. DEBs are fantastic systems for studying stellar evolution. Observations can provide
precise masses and radii for the component stars that are nearly model independent (see
reviews by Andersen 1991; Torres et al. 2010). Tight constraints on the stellar masses and
52
radii allow for stringent tests of stellar evolution models. Furthermore, the age of a DEB
system can be derived if stellar models can predict the radius of each star in the binary at a
common age and with a single chemical composition.
Deriving an age estimate from a DEB is straightforward once precise masses and radii are
extracted from the data. However, the reliability of the age estimate is contingent upon the
accuracy of the stellar models. Recently, pre-main-sequence (pre-MS) and MS models of
low-mass stars (< 0.8M⊙ ) have received substantial criticism for not accurately predicting
the radii of stars in DEBs. As the number of DEBs with precisely measured masses and
radii has increased, it has become clear that stellar models under-predict the radii of stars
in DEBs by upward of 10% (see, for example, Chapter 4; Mathieu et al. 2007; Jackson et al.
2009; Torres et al. 2010; Feiden & Chaboyer 2012a). The discrepancies between model and
observed radii have been largely attributed to the effects of magnetic fields and magnetic
activity (Ribas 2006; Chabrier et al. 2007; Jackson et al. 2009; Morales et al. 2010). When
present, the discrepancy between observations and model predictions severely limits the
use of stellar evolution models to derive the age of individual DEB systems.
We propose a novel method to use low-mass, pre-MS DEBs to estimate the ages of young
stellar systems. Instead of comparing individual stellar surface properties to stellar evolution
models, we propose to use the dynamics of the DEB system. That is, comparing the observed
rate of apsidal motion to stellar model predictions computed using the interior structure
constants. This technique is less sensitive to the surface effects of magnetic fields than are
methods that only invoke the stellar radius or effective temperature. Our method has the
potential to provide a more reliable age estimate.
In this chapter, we outline the computation of the interior structure constants in Section
3.2 and present results concerning their time evolution (Section 3.3). The analysis covers
mass-dependency, as well as the associated uncertainties due to stellar model inputs. We
conclude with a discussion on the observational considerations and the limitations of the
53
present study in Section 3.4.
3.2 Interior Structure Constants
The distribution of mass within a star in a close binary system is influenced by the star’s
rotation and by tidal interaction with its companion. Imagine two stars, A and B. The
rotation of star A and the tidal interaction of star B with star A distorts the shape of star
A. Instead of remaining spherically symmetric, the equilibrium configuration of star A will
become ellipsoidal. Subsequently, the gravitational potential of star A will also become
ellipsoidal. The same can be said from the perspective of star B.
If the binary orbit is elliptical, the distorted gravitational potential will cause the orbit to
precess. This precession may be likened to the precession of Mercury’s orbit about the Sun,
where Mercury’s precession also requires general relativity and not just classical gravitational interactions. The precession of the binary orbit is known as apsidal motion.
The rate of apsidal motion (ω̇; measured in degrees per cycle), or the rate at which the orbit
precesses, is governed by the shape of the gravitational potential, which may be deformed
as discussed above. Therefore, ω̇ depends on the properties of the stars and of the orbit.
Explicitly,
¢
¡
ω̇ = 360 c 2, 1 + c 2, 2 k 2 ,
(3.1)
where
c 2, i =
·µ
Ωi
ΩK
¸ µ ¶5
¶2 µ
¶
Ri
m 3−i
15m 3−i
g (e)
1+
f (e) +
.
mi
mi
A
(3.2)
In the above equation, ΩK is the mean orbital angular velocity, Ωi , m i , and R i are the rotational velocity, the mass, and the radius of the i -th component in the binary, respectively.
54
Additionally, A is the semi-major axis of the orbit,
f (e) = (1 − e 2 )−2 ,
(3.3)
and
g (e) =
(8 + 12e 2 + e 4 )
f (e)5/2 ,
8
(3.4)
with e being the eccentricity of the orbit. Finally, the last term in Equation (3.1), k 2 , is the
weighted interior structure constant observed for the two binary stars. In general,
k2 =
c 2, 1 k 2, 1 + c 2, 2 k 2, 2
.
c 2, 1 + c 2, 2
(3.5)
Here, k 2, 1 and k 2, 2 are the interior structure constants for each star.
Equations (3.1)–(3.5) are derived from a j -th order solid harmonic expansion of the gravitational potential (see, for example, Kopal 1978). The interior structure constant for a given
star, k 2 , is the second-order term from a more general set of expansion coefficients, k j .
These second-order coefficients quantify the central concentration—or radial distribution—
of mass within a star (Kopal 1978). Lower values of k 2 correspond to a higher level of central
mass concentration. Point sources, for example, have k 2 = 0.
Observationally, we can not solve for the individual k 2, i values. However, in the above
equations, every variable is a direct observable, except for k 2 . The latter must be inferred
from observational determinations of ω̇ and the c 2, i coefficients. Since the individual k 2, i
values depend on the stellar density distribution, they can provide deep insight into the
validity of stellar evolution models. To bypass the restriction that only k 2 can be inferred
from observations, we use k 2 values from stellar evolution models in combination with the
observed c 2, i coefficients to derive a theoretical k 2 .
DSEP is equipped to calculate the general interior structure constants, k j , for a star at every
55
evolutionary time step. This is achieved by solving Radau’s equation after each model
iteration. Following the formalism outlined by Kopal (1978),
a
dηj
da
+
¡
¢
¢
6ρ(a) ¡
η j + 1 + η j η j − 1 = j ( j + 1)
〈ρ〉
(3.6)
with j ∈ {2, 3, 4, . . .} being the order of the solid harmonic, a is the radius of an equipotential
surface (a = r when the surfaces are spherically symmetric), and
η j (a) =
a dǫj
.
ǫj d a
(3.7)
In the above equation, ǫ j is the stellar deviation from sphericity. We also introduced ρ(a),
the density of the stellar plasma at radius a , and 〈ρ〉, the volume averaged density at each
radius,
3
〈ρ〉 = 3
a
Za
0
ρ(a ′ )a ′2 d a ′ .
(3.8)
If we assume that tides and rotation do not significantly alter the shape of a star, we are permitted to use a spherically symmetric model to compute the interior structure constant. Our
code employs a 4th -order Runge-Kutta integration scheme to obtain a particular solution of
Radau’s equation at the stellar photosphere. Interior structure constants are then directly
related to the particular solutions at the surface, η j (R), through
j + 1 − η j (R)
¤.
kj = £
2 j + η j (R)
(3.9)
But, is it reasonable to assume that the stellar mass is not significantly redistributed due to
rotation and tides?
The effect of rotation on the central mass concentration of stars with a total mass greater
than 0.8M⊙ was investigated in several previous studies (Stothers 1974; Claret & Giménez
1993; Claret 1999). For stars less massive than 0.8M⊙ , the effect of rotation should be neg56
ligible because they have a higher mean density compared to solar-type stars. To test this
assumption, we used Chandrasekhar’s analysis of slowly rotating polytropes to estimate the
amount of oblateness—or deviation from spherical symmetry—rotating low-mass stars may
be expected to have.
Chandrasekhar (1933) derived an analytical expression for the stellar oblateness for slowly
rotating polytropes. The oblateness was defined to be the relative difference between the
equatorial radius and the polar radius,
F≡
r eq − r pole
r eq
,
(3.10)
with r eq and r pole being the equatorial and polar radius, respectively. Polytropes were
considered slowly rotating when
χ≡
Ω2
≪ 1,
2πGρ c
(3.11)
where Ω is the stellar angular velocity, G is the gravitational constant, and ρ c is the mass
density in the stellar core. Does this criterion apply to real low-mass stars? If we assume
a rotation period of 1.0 day and a core density of 10 g cm−3 (realistic for pre-MS low-mass
stars), then χ ≈ 10−3 .
The result of Chandrasekhar’s analysis was a relation between the stellar oblateness and χ
for different values of the polytropic index, n ,




5.79χ



F = 9.82χ





 41.8χ
for n = 1.5
for n = 2.0 .
(3.12)
for n = 3.0
Stars with masses below 0.65M⊙ are best represented by a polytrope with 1.5 < n < 2.0.
Assuming n = 2.0, a low-mass star will have F ∼ 0.01. This treatment indicates that the
57
effect of rotation on the sphericity of low-mass stars is a 1% effect; rotation will not be
addressed in this study.
Assessing the influence of tides is difficult. We expect spherically symmetric models to
provide accurate estimates of the mass distribution if R∗ ≪ R roche , where R roche is the
Roche lobe radius.
We also caution that the configuration of the binary must be considered. If the rotational
axes are not aligned, as found with DI Her (Albrecht et al. 2009), then the validity of the
assumptions required to derive of k 2 no longer hold. See Section 3.4 for a further discussion.
General relativistic distortion of the gravitational potential also plays a role in determining
the rate of apsidal motion (Giménez 1985). This contribution to the apsidal motion rate can
be added to the classical contribution (i.e., Equation 3.1)
ω̇tot = ω̇N + ω̇GR ,
(3.13)
where ω̇tot , ω̇N , and ω̇GR are the total apsidal motion rate, classical apsidal motion rate,
and the rate predicted from general relativity. The general relativistic contribution does not
depend on the mass distribution of the stars. Thus, general relativity does not affect the
theoretical derivation of the interior structure constants, but must be accounted for prior to
comparing the theoretical and observational determinations of k 2 .
3.3 Results
3.3.1 Single Stars
Individual, solar-metallicity evolutionary tracks were computed for several masses ranging
from the fully-convective regime (0.25M⊙ ) up to masses where stars have thin convective
58
0.25 M⊙
0.35 M⊙
0.55 M⊙
0.65 M⊙
0.27 M⊙
0.29 M⊙
0.33 M⊙
0.35 M⊙
0.45 M⊙
0.75 M⊙
0.31 M⊙
0.37 M⊙
0.15
0.15
[Fe/H] = 0.0
k2
(b)
0.10
0.14
0.05
(a)
[Fe/H] = 0.0
1
10
0.13
100
1000
Age (Myr)
1
10
100
1000
Age (Myr)
Figure 3.1: The time evolution of the interior structure constant, k 2 , for stars of various
masses. Stars that develop a radiative core during the pre-main-sequence exhibit rapidly
decreasing k 2 values between roughly 10 and 100 Myr. (a) Total range of masses considered
in this study shown in increments of 0.05M⊙ . (b) A detailed view of the transition to fullyconvective interiors. Only the 0.37M⊙ model does not ultimately become fully-convective.
envelopes (0.75M⊙ ).1 Near the boundary where the transition from a radiative core to a
fully-convective interior is expected (∼ 0.35M⊙ ), a finer grid of mass tracks was generated to
allow for further exploration. The evolution of k 2 with age for each mass track is presented
in Figure 3.1. The full collection of mass tracks (including analysis routines) used hereafter
have been made available on online2 .
Stellar models that develop a radiative core have a rapidly-changing interior structure constant between the age of 10 and 100 Myr. This can be observed in Figure 3.1(a). As the
convection zone recedes, the central regions of the collapsing pre-MS star create a more
centrally-concentrated mass profile, lowering the derived value of k 2 . The result is that the
value of k 2 decreases by about 5% – 10% every 10 Myr for masses above 0.45M⊙ . Masses
below approximately 0.45M⊙ undergo variations up to about 5%.
1
Pre-MS models were initialized using a simple polytrope model that was relaxed onto the Hyashi track.
Uncertainties in the initial seed model largely effect the model properties younger than about 1 Myr.
2 https://github.com/gfeiden/k2age/
59
This period of rapid contraction continues until a small convective core develops, producing
a star with three energy transfer zones: a convective core, a radiative shell, and a convective
outer envelope. The star settles onto the MS once the small convective core subsides with
the equilibration of 3 He burning. In Figure 3.1, this process is manifested by the upward
turn of the mass tracks near 100 Myr, followed by a flattening of k 2 as the star enters the
MS.
After first developing a small radiative core on the pre-MS, stars with masses between
0.29M ⊙ and 0.35M ⊙ eventually maintain a fully-convective interior (see Figure 3.1(b)).
These small radiative cores manifest themselves as small dips in the evolution of k 2 , just
as for stars that maintain a radiative core on the MS. While the rate of change in k 2 is rapid
during the core contraction, the relative change in k 2 with stellar age is small (< 5%). It
is evident from Figure 3.1(b) that near the fully-convective transition, stars that eventually
end up with fully-convective interiors exhibit a degeneracy in the age-k 2 plane. There is
no differentiating between their pre-MS core contraction and the eventual reduction of the
radiative shell using k 2 alone. Below 0.29M⊙ , a radiative core does not develop on the
pre-MS.
Additional models were generated to investigate the effect of specific stellar properties on
the predicted values of k 2 . In Figures 3.2(a)–(f) we illustrate the results of changing the
scaled-solar stellar metallicity, helium abundance (Y ), and the effects of artificially inflating
and deflating the stellar radius. For this exercise, two masses were selected to study the
effects on the two broad categories of low-mass stars: fully-convective stars and stars with
radiative cores.
It is apparent from Figure 3.2 that changes to the chemical composition have the largest
effect on the value of the interior structure constant on the pre-MS. Variations in metallicity
of 0.2 dex translate into a 5% difference in the calculated k 2 for a 0.55M⊙ star at a given
pre-MS age. Similarly, large variations in the helium abundance have the ability to produce
60
Metallicity
He Abundance
Stellar Radius
0.15
k2
M = 0.55 M⊙
(a)
M = 0.55 M⊙
(b)
M = 0.55 M⊙
(c)
0.10
+20% Psurf
-20% Psurf
0.05
k2
0.15
M = 0.25 M⊙
(d)
M = 0.25 M⊙
(e)
M = 0.25 M⊙
(f)
0.14
αMLT = 0.50
αMLT = 1.00
αMLT = 1.94
αMLT = 3.00
[Fe/H] = -0.5
[Fe/H] = -0.3
0.13
[Fe/H] = 0.0
[Fe/H] = +0.2
1
10
100
Age (Myr)
Y = 0.27
Y = 0.33
1
10
100
Age (Myr)
1
10
100
Age (Myr)
Figure 3.2: The influence of various model properties on the predicted evolution of k 2 for
two different stellar masses. From left to right, the properties investigated are: scaled-solar
metallicity in (a) and (d), helium mass fraction (Y ) in (b) and (e), and artificial radius changes
in (c) and (f). Models in the top panels ((a)–(c)) show a 0.55M⊙ star while the bottom series
((d)–(f)) show a fully-convective 0.25M⊙ star. Note, the legend for panels (c) and (f) is split
between the top and bottom panels due to space restrictions. The separate tracks indicated
by both legends are presented for each mass.
changes in k 2 at the 5% level.
Without some prior knowledge of the stellar composition, the effects of such variations
may be confused with a difference in stellar age. The effects on k 2 are lessened as mass
decreases, until variations nearly vanish in the fully-convective regime (panels (d) and (e)
of Figure 3.2). Note that a change in Y pushes k 2 in the same direction in both mass regimes,
whereas a change in metallicity produces a change in one direction for the radiative core
case but the opposite direction in the fully-convective case. While this behavior does not lift
the age-composition degeneracy entirely, it could prove a useful diagnostic in DEB systems
whose components straddle the fully-convective boundary.
61
We attempted to mimic the possible effects of magnetic fields on the structure of our models
by computing models at solar-metallicity with αMLT = 0.5, 1, and 3. The modified mixinglength represents magnetic suppression of convection in the deep interior. Additionally,
models were run where we artificially changed the surface pressure by ±20%. The altered
surface pressure represents the possible influence of star spots on the stellar photosphere.
The magnetic Dartmouth models (Chapter 5; Feiden & Chaboyer 2012b) were not used as
they have yet to be evaluated for stars on the pre-MS.
Figures 3.2(c) and 3.2(f) show how k 2 varies with changes to αMLT and the surface pressure.
During the pre-MS contraction, a 20% change in the surface pressure results in a ∼ 2%
change to the stellar radius. The accompanying change in k 2 was found to be 0.2% and
1.0%, for a positive and negative change to the surface pressure, respectively. In the 0.55M⊙
models, increasing αMLT to 3 yielded a stellar radius that was 2% smaller than our solarcalibrated model with k 2 variations under 1% throughout the star’s pre-MS contraction.
Decreasing αMLT , however, produced larger variations in k 2 . At an age of 60 Myr the
model radii appeared inflated by 5% and 15% for αMLT = 1.0 and 0.5, respectively. The
corresponding changes in k 2 were, respectively, 4% and 12%. Panel (f) of Figure 3.2 indicates
that the 0.25M⊙ , fully-convective models experience the greatest differences at the youngest
age and that these differences diminish until the different tracks converge on the MS. These
changes to αMLT and surface pressure are for illustrative purposes only. It is not at all
clear that simply reducing αMLT is a suitable approximation for the presence of an interior
magnetic field nor that altering the surface pressure is a good approximation for star spots
at the surface.
3.3.2 Binary Systems
The rapid evolution of k 2 for a single star can provide an accurate age estimate for that star,
but what about for a binary system? We stated in Section 3.2 that observations are only
62
0.55 M⊙ + 0.25 M⊙ Binary
—
k2
0.14
0.12
0.10
[Fe/H] = 0.0
1
10
100
1000
Age (Myr)
Figure 3.3: The evolution of the theoretical k 2 for a binary having a 0.55M⊙ primary and
a 0.25M⊙ secondary. The “observed” stellar radii are fixed at 0.62R⊙ and 0.41R ⊙ for the
primary and secondary, respectively (see Section 3.3.2). The orbit was chosen to have an
eccentricity e = 0.2.
able to provide the weighted mean value of k 2 for two stars in a binary. To derive an age
estimate for a binary system, we must find the theoretical k 2 .
Temporal evolution of the theoretical k 2 values can be obtained by combining two k 2 mass
tracks using Equation (3.5). Computation of the c 2, i coefficients requires precise knowledge
of the stellar masses, radii, and the orbital eccentricity (see Equation (3.2) and note that A
becomes irrelevant). The angular velocity term in Equation (3.2) can either be measured or
approximated using the orbital eccentricity, assuming pseudo-synchronization (Kopal 1978),
Ω2i =
(1 + e)
(1 − e)3
Ω2K .
(3.14)
Hence our reason for focusing on DEBs: they can yield precise estimates of the stellar and
orbital properties.
The evolution of k 2 is simplest for an equal mass binary. In this case, both stars contribute
63
equally to k 2 , meaning the k 2 track is exactly equal to the two individual k 2 tracks. The
discussion from Section 3.3.1 on single star tracks would then apply to the binary system.
A binary with unequal mass components is not so simple. Two mass tracks are required—
one computed for the mass of each star—and must be combined as a single track using
Equation (3.5). How does this effect our ability to extract an age estimate?
We have provided an example of a weighted k 2 evolutionary track in Figure 3.3. The masses
selected are those of the two stars presented in Figure 3.2, M1 = 0.55M⊙ and M2 = 0.25M⊙ .
For this example, we arbitrarily adopted an orbit with an eccentricity e = 0.2. Weighting of
the theoretical k 2 values is insensitive to the semi-major axis. The stellar radii (R1 = 0.62R ⊙
and R 2 = 0.41R⊙ ) were selected from a solar metallicity mass track at an age of 40 Myr. To
simulate the type of data available to an observer, we elected to fix the radius, and thus, fix
the c 2, i coefficients (used to weight the value of k 2 ) at each age in Figure 3.3.
The rapid evolution of k 2 that is taking place in the 0.55M⊙ star largely dominates the
relatively slow k 2 evolution of the 0.25M⊙ star. While the weighted k 2 value for the binary
does not evolve as rapidly as for a single 0.55M⊙ star, the weighted value still changes
by about 5% every 10 Myr. This is comparable to models of single stars and should not
significantly hinder any age analysis.
3.4 Discussion
DEBs provide an excellent laboratory for testing stellar structure and evolution theory in
different mass and evolutionary regimes. The mass-radius plane is the strictest test of stellar
models because these two quantities are best constrained by the observations (Torres et al.
2010). Results from studies performing such comparisons have led to the consensus that
standard stellar evolution models are unable to accurately reproduce the observed stellar radii for masses below ∼ 0.8M⊙ (e.g., Chapter 4, Torres et al. 2010; Feiden & Chaboyer
64
2012a). The model inaccuracies are particularly evident among pre-MS binaries (Mathieu et al.
2007; Jackson et al. 2009).
Radius discrepancies of approximately 10 – 15% are routinely quoted between pre-MS DEBs
and models. This makes it difficult to derive an age with less than 50% uncertainty from
stellar positions in the mass-radius plane. Magnetic effects, particularly surface spots, are
thought to belie the observed radius deviations. Canonical stellar evolution models are
non-magnetic and are therefore unable to properly account for magnetic modifications to
convective energy transport and for the presence of magnetic spots on the stellar photosphere.
We therefore advocate the inclusion of the interior structure constant, k 2 , to overcome these
age determination inaccuracies whenever possible. While this study does not lead to us to
conclude that individual k 2 values (and thus k 2 ) are entirely insensitive to magnetic activity
in low-mass stars, it is evident that k 2 has the potential to be a better diagnostic of the
pre-MS evolutionary state than the stellar radius when surface magnetic activity is present.
For instance, setting αMLT = 0.5 increases the radius and k 2 of a single star by 15% and
12%, respectively. Fixing the radius to determine an age leads to an age that is upward of
180% greater than if we assume a solar calibrated αMLT . On the other hand, fixing k 2 leads
to only a 20% greater age. The age errors are then compounded when we consider both stars
in the binary. The decreased sensitivity of the individual k 2 values makes the mean k 2 a
superior choice compared to surface quantities like the radius and effective temperature.
The age precision returned from observational determinations of k 2 is dependent on the
precision with which the observational k 2 and the system’s metallicity are known. For
example, given an equal mass binary with 0.5M⊙ stars, knowing k 2 with 5% uncertainty
and the metallicity to ±0.2 dex yields a pre-MS age with an uncertainty of approximately
33%. Constraining the metallicity uncertainty to ±0.1 dex improves this age uncertainty to
20%. Furthermore, to obtain an age with 5% precision would require k 2 to be measured with
65
near 1% precision for a metallicity known to within 0.1 dex.
3.4.1 Observational Considerations
The results presented and discussed above show that it is possible to precisely derive the
age of a binary system from measurements of apsidal motion. However, obtaining precise
observations of apsidal motion and measuring k 2 with 5% precision is a painstaking task.
The binary system must meet several criteria and the data must be of high quality.
Foremost is that the binary must be a double-lined, eclipsing system. While this has been
alluded to, we have yet to illuminate precisely why this is so. The reason for requiring a
DEB stems from the need for extremely accurate and precise stellar and orbital properties.
Equations (3.1) and (3.2) reveal that derivation of k 2 requires exquisite knowledge of the
stellar mass ratio, the stellar radii, and the orbital properties (eccentricity and semi-major
axis). Only data from DEBs can provide these quantities in a (nearly) model-independent
fashion (Andersen 1991; Torres et al. 2010).
A meticulous examination of the binary light curve and radial velocity curve is of the utmost
importance. The light curve must be assembled from multi-epoch, time-series differential
photometry and must provide nearly complete phase coverage. This latter feature is essential. Not only must the primary and secondary eclipses be captured, but also the behavior of
the light curve out of eclipse. Such data is becoming increasingly available with the latest
generation of photometric surveys (e.g., Kepler, CoRoT3 , SuperWASP4 ) and those to come
(e.g., LSST5 , BRITE6 Constellation Mission).
Similar to the photometry, a large number of high dispersion, high signal-to-noise spectra are needed to construct a detailed radial velocity curve. Attempts must be made to
3 Convection,
Rotation, & planetary Transits
Angle Search for Planets
5 Large Synoptic Survey Telescope
6 Bright Target Explorer
4 Wide
66
provide adequate phase coverage (see Andersen 1991) and uncover deviations due to the
Rossiter-McLaughlin effect (Rossiter 1924; McLaughlin 1924) for reasons we shall discuss
momentarily. Torres (2013) points out that the spectroscopy is still the limiting factor of
quality in DEB analyses.
Only with the quality of data described above and within Andersen (1991) and Torres et al.
(2010) can a truly adequate analysis of a DEB be performed. However, acquisition of data of
such quality is very rewarding and allows for a rigorous examination and characterization
of the stellar system. It would permit the measurement of the stellar masses and radii with
extreme precision (below 2%). This is imperative considering that Equation (3.2) depends
upon the fractional radii (R/A) to the 5th power. Additionally, actual measurement of ω̇
requires careful monitoring of eclipse times of minimum. This can only be performed if one
has a densely populated light curve.
The data would also permit measurement of the binary eccentricity and semi-major axis
with high precision. These properties affect both Equation (3.2) and the general relativistic
contribution discussed in Section 3.2 (Giménez 1985). Recall that the contribution from the
latter must be removed from the total apsidal motion rate to derive the classical contribution
given in Equation (3.1). Detailed radial velocity curves not only provide accurate mass and
eccentricity estimates, but may also be used to investigate the inclination of the system.
The theory presented in Section 3.2 relies on the assumption that the rotational axes of the
two stars be parallel to one another and perpendicular to the orbital plane. It is possible
to evaluate this restriction by using the Rossiter-McLaughlin effect in a manner similar
to that presented for DI Her (Albrecht et al. 2009). A detailed radial velocity curve may,
additionally, betray the presence of a third body. The presence of a tertiary may affect the
binary orbit, altering the derived apsidal motion.
One added benefit is that lengthy observations may help to identify—and thus correct for—
the impact of star spots and magnetic activity on the eclipse profiles. Star spots have the
67
ability to distort the light curve, which can diminish the accuracy of the derived stellar
properties (Windmiller et al. 2010). Removing the effects of spots is critical to obtain not
only precise but also accurate stellar properties. The easiest means of obtaining detailed
time-series photometry is through space-based satellites, such as CoRoT and Kepler. However, long-term ground-based observational efforts are beginning to produce apsidal motion
detections (e.g., Zasche 2012), demonstrating that it is feasible to carry out the necessary
observations using ground-based telescopes.
Finally, by acquiring a large number of quality spectra, it may be possible to extract the
projected rotational velocities (v sin i ) and a modest estimate of the chemical composition.
Spectroscopic determinations of cool star metallicities is notoriously complicated, but most
pre-MS DEBs will likely reside near or in a cluster from which metallicity estimates can be
extracted using the higher mass stellar population. In the event there is no known association from which to draw a metallicity, techniques based on low- and medium-resolution
spectra are encouraging (see, e.g., Rojas-Ayala et al. 2012). While the validity of such techniques along the pre-MS is unclear, they provide a viable starting point and typically produce metallicities with uncertainties below 0.2 dex, the limit we recommend.
3.4.2 Limitations
The usefulness of k 2 as an age estimator is limited to the pre-MS, in particular, during
the evolutionary phase where the radiative core is rapidly contracting. This typically corresponds to an age between 10 Myr and 100 Myr (see Figure 3.1). At the 5 – 10 % measurement
level, this technique is also restricted to binaries where one of the stars has a mass above
∼0.40 M ⊙ . This restriction ensures the radiative core contraction and relative change in k 2
for the more massive star is rapid enough to dominate the theoretical k 2 evolution. Reducing the observational uncertainty in k 2 below 5% enables a more accurate age estimate and
would allow for DEBs with lower mass components to be reliably analyzed.
68
Precise measures of apsidal motion and metallicity are challenging to obtain, but are already feasible and should only improve over time. Instead of observational limitations, the
greatest limiting factor for using k 2 as an age indicator is the circularization of the DEB
orbit. Mutual tidal interactions will circularize binary orbits over time (Zahn 1977). Apsidal
motion requires an elliptical orbit. If an equal-mass binary is to maintain an elliptical orbit
for the duration of its pre-MS contraction, the orbital period must be at least 2.4 days (Zahn
1977). The probability of discovering an elliptical binary decreases with time, but this provides an additional consistency check. The age suggested by k 2 should not be significantly
older than the orbital circularization timescale.
Finally, the age estimate is only as accurate as the stellar models. Verifying that a given
stellar model produces the proper mass distribution, hence k 2 , may at first seem rather
unreasonable. However, at least one known system, with the possibility of a second (KIC
002856960; Lee et al. 2013), is capable of providing validation of the physics incorporated
in low-mass stellar evolution models. Carter et al. (2011) have indicated that by the end
of the nominal Kepler mission, they will know the interior structure constants of KOI-126
B and C with about 1% precision. Interior structure constants known with this precision
can place stringent constraints on the equation of state of the stellar plasma (Chapter 2;
Feiden et al. 2011). The veracity of low-mass models, and therefore the validity of their
predicted interior structure constants, may be assessed according to the results from KOI126 and similar systems.
69
Chapter 4
The Mass-Radius Relation for Low-Mass,
Main-Sequence Stars
The content of this chapter is largely based on material from the paper entitled “Reevaluating
the Mass-Radius Relation for Low-Mass, Main-Sequence Stars,” published in the Astrophysical Journal (2012, 757, 42).
4.1 Introduction
We saw in Chapter 2 that low-mass stellar evolution models accurately reproduced the stellar properties of all three stars in the KOI-126 triple system. The inclusion of the super-solar
metallicity within the stellar evolutionary calculations permitted the accurate modeling.
The agreement between stellar models and observations of a low-mass system based on
metallicity forced us to reconsider whether the routinely quoted 5 – 15% disagreement is
representative of true radius discrepancies or whether there are other factors contributing
to the derivation of such large radius errors.
One such factor derives from the fact that previous studies focusing on the comparison
70
between models and observations have generally applied a limited sample of isochrones to
their data. Largely, these sets are comprised of 1 Gyr and 5 Gyr, solar metallicity isochrones.
This is predominantly a consequence of the limited age and metallicity range of currently
available low-mass stellar models. Age and metallicity effects are less important in the lowmass regime, but the stringent uncertainties quoted by observational efforts preclude the
use of such a limited set of isochrones. For example, Burrows et al. (2011) discovered nonnegligible radius variations in brown dwarfs and very-low-mass stars (< 0.1M⊙ ) when allowing for a more comprehensive set of metallicities. Isochrones with metallicities spanning
a range characteristic of the local galactic neighborhood are therefore essential to accurately
assess the validity of stellar evolution models.
Furthermore, one must also consider that the population of well-characterized DEBs has,
until recently, consisted of eight systems. While unlikely, it is not unimaginable that those
eight systems were more the exception than the rule in terms of their lack of consistency
with stellar evolution models. Since publication of the Torres et al. (2010) review, the population of well-characterized, low-mass DEBs has more than doubled. The availability of this
new data allows for a more accurate statistical characterization of the agreement (or lack
of) between the MR relationship defined by models and observations.
When discrepancies are observed, they are typically attributed to the effects of a large scale
magnetic field (e.g., Ribas 2006; López-Morales 2007; Morales et al. 2008, 2009a; Torres et al.
2010; Kraus et al. 2011) as DEBs are often found in tight, short-period orbits with periods
under three days. Tidal interactions and angular momentum conservation act to synchronize the orbital and rotational periods of the components, increasing the rotational velocity
of each star in the process. The dynamo mechanism, thought to be responsible for generating and sustaining stellar magnetic fields, is amplified as a result of the rotational spin-up
and enhances the efficiency of magnetic field generation within the star. Each component in
the binary system is then more able to produce and maintain a strong, large-scale magnetic
71
field than a comparable single field star.
The effects of a large-scale magnetic field are thought to be two-fold: convective motions
within the star are suppressed and the total surface coverage of starspots is increased. In
both cases, a reduction in the total energy flux across a given surface within the star occurs,
forcing the stellar radius to inflate in order to conserve flux (Gough & Tayler 1966). Recent
attempts at modeling these effects have indicated that an enhanced magnetic field is a plausible explanation, although the primary physical mechanism affecting the structure of the
star is still debated (Mullan & MacDonald 2001; Chabrier et al. 2007; MacDonald & Mullan
2012).
Regardless of the precise physical mechanism, magnetic fields should betray their presence through the generation of magnetic activity in the stellar atmosphere. If magnetism
is responsible for the observed inflated stellar radii, then correlations should be expected
between individual stellar radius deviations and magnetic activity indicators (i.e., chromospheric Hα and Ca ii H & K emission, coronal x-ray emission, etc.). Tantalizing evidence of
such correlations has been reported previously by López-Morales (2007) and Morales et al.
(2008).
However, recent evidence appears to stand in contrast with the current theory. Two systems,
LSPM J1112+7626 (Irwin et al. 2011) and Kepler-16 (Doyle et al. 2011), were discovered that
have wide orbits with approximately forty-one day periods. Despite this, both appear to
display discrepancies with stellar evolution models. In these systems, the component stars
should be evolving individually with mutual tidal interactions playing a negligible role in
the overall angular momentum evolution. The stars should be spinning down over time
due to magnetic breaking processes (Skumanich 1972), meaning the stars should not be as
magnetically active compared with short period binary systems. The contrast is particularly
evident for LSPM J1112+7626, where a rotation period of sixty-five days was detected via
starspot modulation in the out-of-eclipse light curve. Gyrochronology suggests that the sys-
72
tem has an age of approximately 9 Gyr (Barnes 2010) and implies further that the secondary
is likely slowly rotating and should, therefore, not shown signs of strong magnetic activity
or an inflated radius.
A third system also appears to defy the current hypothesis. KOI-126 (Carter et al. 2011)
is a hierarchical triple system with two low-mass, fully convective stars in orbit around a
1.35 M⊙ primary. The two low-mass stars are orbiting each other with a period of 1.77
days. Therefore, they should show signs of inflated radii due to enhanced magnetic activity.
However, it has been shown that the two low-mass, fully convective stars were in agreement
with model predictions when considering their super-solar metallicity and the age of the
higher mass primary (Chapter 2; Feiden et al. 2011). This agreement was further confirmed
by Spada & Demarque (2012).
Metallicity has been proposed previously as a solution to the observed MR discrepancies, but
for the case of single field stars (Berger et al. 2006). This was contradicted shortly thereafter
by López-Morales (2007), most notably for DEBs. Although, we must consider that the radius discrepancy-metallicity correlation is severely complicated by the fact that metallicities
of M dwarfs are notoriously difficult to determine observationally.
Finally, developments in light curve modeling of spotted stars has generated interesting
results. The presence of large polar spots may alter the light curve analysis of DEBs by
modifying the eclipse profile. These modifications lead to 2 – 4% uncertainties in the derived
stellar radii (Morales et al. 2010; Windmiller et al. 2010; Kraus et al. 2011). Thus far, only
two DEBs (GU Boo and CM Dra) have been thoroughly tested for their sensitivity to spots.
Systematic uncertainties may therefore dominate the error budget, casting a shadow of
doubt on the observed radius discrepancies, which are often made apparent due to the
minuscule random uncertainties.
The uncertainties and developments outlined above have motivated us to reevaluate the
current state of the low-mass MR relationship. In what is to follow, we use a large grid of
73
theoretical stellar evolution isochrones to compare the low-mass models of the Dartmouth
Stellar Evolution Program (DSEP) with DEB systems that have well constrained masses
and radii. We then explore how potentially unaccounted for systematic uncertainties have
the ability to create the appearance of discrepancies when neglected and mask real ones
when considered. Section 4.2 will present the DEB sample followed by a description of the
isochrone grid and fitting procedures in Section 4.3. Results will be presented in Section 4.4
and a discussion of the implications of our findings is in Section 4.5.
4.2 Data
Our selection criteria mimic those of Torres et al. (2010, hereafter TAG10) in so far as we require the random uncertainties in the mass and radius measurements to be less than 3%. No
data was disqualified due to perceived data quality issues or the original author’s attempts
to constrain possible systematic uncertainties. We also applied the criterion that the DEB
system must include at least one component with a mass less than 0.8 M⊙ . This cut-off in
mass is used to designate “low-mass” stars.
The mass cut-off was selected for two main reasons. First, the effects of age and metallicity
on the structure of low-mass main sequence stars are suppressed compared to stars with
masses of approximately 1.0 M⊙ or above. Overall, this allows for less flexibility in fitting
models to the observations, providing a more critical analysis of the stellar evolution models.
Second, some of the largest discrepancies between observations and models are seen in the
low-mass regime. This is likely a consequence of the former reason: true discrepancies
become more apparent as the models become less sensitive to the input parameters.
Stars used in this study are listed in Table 4.1 along with their observationally determined
properties and original references. Our final sample consisted of eighteen DEB systems for
a total of thirty-six stars. Six of these systems are taken from TAG10 who reanalyzed the
74
available data using a common set of reduction and parameter extraction routines in an
effort to standardize the process. While the original references are cited, the parameters
listed are those derived by TAG10 whose results were similar to the original values.
A majority of the systems, ten in total, were published after the release of the TAG10 review. Six are drawn from the study performed by Kraus et al. (2011), three are products of
recent results from the Kepler Space Telescope mission (Carter et al. 2011; Doyle et al. 2011;
Bass et al. 2012), and the final post-TAG10 system was discovered by the MEarth survey
(Irwin et al. 2011). The remaining two systems from our sample were announced before
TAG10 (López-Morales et al. 2006; López-Morales & Shaw 2007), however, they were not
included in the review for reasons related to either data availability or data quality.
One final note: the Kepler systems KOI-126 (Carter et al. 2011) and Kepler-16 (Doyle et al.
2011) were not analyzed in a similar manner to the rest of the double-lined DEB population. They are DEB systems whose parameters were derived from Kepler photometry using
a dynamical-photometric model (see Supporting Online Material from Carter et al. 2011).
However, they still satisfy the criteria for comparison with stellar evolution models and
have been included for this reason.
Table 4.1
DEBs with at least one low-mass component with precise masses and radii.
Star
P orb
Mass
Radius
P rot
Name
(day)
(M ⊙ )
(R ⊙ )
(day)
UV Psc A
0.86
0.9829 ± 0.0077
1.110 ± 0.023
···
0.76440 ± 0.00450
0.8350 ± 0.0180
0.80
0.981 ± 0.012
1.061 ± 0.016
···
0.6644 ± 0.0048
0.6810 ± 0.013
1.31
0.924 ± 0.008
0.8807 ± 0.0017
···
0.683 ± 0.005
0.6392 ± 0.0013
···
UV Psc B
IM Vir A
1.309
IM Vir B
KID 6131659 A
KID 6131659 B
17.528
continued on next page
75
Source†
1
2
3
Table 4.1 – continued
Star
P orb
Mass
Radius
P rot
Name
(day)
(M ⊙ )
(R ⊙ )
(day)
RX J0239.1-1028 A
2.072
0.7300 ± 0.0090
0.7410 ± 0.0040
···
0.6930 ± 0.0060
0.7030 ± 0.0020
···
0.6897 ± 0.0034
0.6489 ± 0.0013
···
0.20255 ± 0.0007
0.22623± 0.0005
···
0.61010 ± 0.00640
0.6270 ± 0.0160
0.49
0.59950 ± 0.00640
0.6240 ± 0.0160
0.54
0.59920 ± 0.00470
0.6194 ± 0.0057
0.87
0.59920 ± 0.00470
0.6194 ± 0.0057
0.82
0.584 ± 0.002
0.560 ± 0.005
···
0.544 ± 0.002
0.513 ± 0.009
···
0.567 ± 0.002
0.552 ± 0.004
···
0.532 ± 0.002
0.532 ± 0.004
···
0.557 ± 0.001
0.569 ± 0.002
···
0.535 ± 0.001
0.500 ± 0.003
···
0.5428 ± 0.0027
0.5260 ± 0.0028
···
0.4982 ± 0.0025
0.5088 ± 0.0030
···
0.527 ± 0.002
0.505 ± 0.008
···
0.491 ± 0.001
0.471 ± 0.009
···
0.499 ± 0.002
0.457 ± 0.010
···
0.443 ± 0.002
0.427 ± 0.008
···
0.469 ± 0.002
0.441 ± 0.004
···
0.382 ± 0.001
0.374 ± 0.004
···
0.43490 ± 0.00120
0.4323 ± 0.0055
···
0.39922 ± 0.00089
0.3916 ± 0.0094
···
0.3946 ± 0.0023
0.3860 ± 0.0052
···
RX J0239.1-1028 B
Kepler-16 A
41.08
Kepler-16 B
GU Boo A
0.49
GU Boo B
YY Gem A
0.81
YY Gem B
MG1-506664 A
1.55
MG1-506664 B
MG1-116309 A
0.827
MG1-116309 B
MG1-1819499 A
0.630
MG1-1819499 B
NSVS 01031772 A
0.368
NSVS 01031772 B
MG1-78457 A
1.586
MG1-78457 B
MG1-646680 A
1.64
MG1-646680 B
MG1-2056316 A
1.72
MG1-2056316 B
CU Cnc A
2.77
CU Cnc B
LSPM J1112+7626 A
41.03
continued on next page
76
Source†
4
5
6
7
8
8
8
9
8
8
8
10
11
Table 4.1 – continued
Star
P orb
Mass
Radius
P rot
Name
(day)
(M ⊙ )
(R ⊙ )
(day)
0.2745 ± 0.0012
0.2978 ± 0.0046
···
0.2413 ± 0.0030
0.2543 ± 0.0014
···
0.2127 ± 0.0026
0.2318 ± 0.0013
···
0.23102 ± 0.00089
0.2534 ± 0.0019
···
0.21409 ± 0.00083
0.2398 ± 0.0018
···
LSPM J1112+7626 B
KOI-126 B
1.77
KOI-126 C
CM Dra A
1.27
CM Dra B
†
References:
(1) Popper (1997);
(2) Morales et al. (2009b);
Source†
12
13
(3) Bass et al. (2012);
(4) López-Morales & Shaw (2007); (5) Doyle et al. (2011); (6) López-Morales & Ribas (2005);
(7) Torres & Ribas (2002); (8) Kraus et al. (2011); (9) López-Morales et al. (2006); (10) Ribas (2003);
(11) Irwin et al. (2011); (12) Carter et al. (2011); (13) Morales et al. (2009a)
4.3 Isochrone Fitting
4.3.1 Isochrone Grid
Isochrones were computed for a wide range of age and metallicity values. The parameter space was defined to encompass a vast majority of stars typical of the local galactic
neighborhood. Seven ages and seven scaled-solar metallicities were adopted for a total of
forty-nine individual isochrones. The sets of values used in this study were: [Fe/H] = {−1.0,
−0.5, −0.3, −0.1, 0.0, +0.1, +0.2} dex and age = {0.3, 1.0, 3.0, 5.0, 6.0, 7.0, 8.0} Gyr. For
completeness, we also generated a set of isochrones that employed a smoothly varying convective mixing length.
A second set of isochrones was generated in order to address the idea that low-mass stars
may possess inflated radii due to inefficient convective energy transport. It has been posited
that convection within low-mass stars may either be inherently inefficient or that other
physical processes (i.e., magnetic fields) act to reduce the ability of convection to effectively
transport energy. We do not attempt to prescribe a physical mechanism associated with this
77
conjecture. Instead, we attempt to parametrize convection in such a way so as to reduce
the efficiency of convection in the lower mass regimes while still maintaining our solar
calibration, necessary to properly model the Sun.
The second grid of isochrones was computed with a mass-dependent mixing length, henceforth referred to as “variable αMLT ” models. Convection for these models was parametrized
with a smooth quadratic function of the form
αMLT
µ
¶
µ
¶
M 2
M
=a
+ b.
M⊙
M⊙
(4.1)
Selection of a quadratic was arbitrary and carries no physical justification, except to produce
low-mass stars with relatively inefficient convection compared to those in the standard
model case.
Coefficients were determined by matching the convective mixing length to predetermined
values at two different masses. Since the overall structure of very low-mass stars is rather
insensitive to the precise value of the mixing length, we anchored αMLT = 1.00 at M =
0.1M ⊙ . The other end of the mass spectrum is constrained by our need to satisfy our solar
calibration. Thus, at M = 1.0M⊙ , the convective mixing length was fixed to αMLT = 1.94.
Subsequently,
αMLT
µ
¶
µ
¶
M 2
M
= 0.949
+ 0.991.
M⊙
M⊙
(4.2)
A relative comparison between models computed with a solar calibrated mixing length and
those generated with a parametrized mixing length is presented in Figure 4.1.
Our isochrone grid for these models covered a fraction of the parameter space compared to
the standard model set. There was a total of 12 isochrones for the variable αMLT models:
[Fe/H] = {−0.5, 0.0, +0.2} dex with ages = {1.0, 3.0, 5.0, 8.0} Gyr. Below the fully convective
boundary, the models are rather insensitive to the mixing length, as stated above. For higher
masses, the reduced mixing length increases the model radii by up to 3%. As designed, a
78
Sun-like star is unaffected by this mixing length prescription. We also see that above 1M⊙
the model radii begin to decrease due to an increased mixing length as a result of our
parametrization.
4.3.2 Model Rotation
Rotation is not included in our models, as was discussed in Section 1.5.1. Recalling that
discussion, we find that the stars in the present sample all have χ < 10−4 , except for the two
stars in NSVS 01031772. They have χ ∼ 10−3 as a consequence of their short orbital period.
Regardless, all of the stars satisfy the slowly rotating approximation.
For all of the systems in question we found F < 0.001, except for NSVS 01031772. This
system is deviating slightly from sphericity with F ≈ 0.011. We feel justified in neglecting
the physics of rotation in our stellar models, with the caveat that attempting to probe model
precisions below 1% for NSVS 01031772 will likely require a more detailed treatment of
rotational deformation.
4.3.3 Fitting Procedure
Judging agreement between observations and models was performed on a system-by-system
basis, as opposed to fitting individual stars. Critical to the process was ensuring that both
components of a given system were consistent with isochrones of a common age and metallicity. For each object, corresponding model radii were derived by linearly interpolating in
each isochrone using the observationally determined mass. A linear interpolation scheme
was sufficient since the mass resolution along the isochrones was small (∆M = 0.02M⊙ ).
Relative errors between the model radii and the observationally determined radius were
79
1.20
Standard Model
Variable αMLT
Radius (R⊙)
1.00
8 Gyr
0.80
1 Gyr
0.60
0.40
0.20
δR/R
0.05
[Fe/H] = 0.0
1 Gyr
8 Gyr
0.00
0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Mass (M⊙)
Figure 4.1: (Top) A mass-radius comparison of isochrones computed using a solar-calibrated
mixing length (solid line) and the mass dependent mixing length, described in Equation (4.2), (dashed line) at 1 Gyr (light-blue) and 8 Gyr (maroon). (Bottom) Radius variations
induced by our prescription of a mass-dependent αMLT . Shown are 1 Gyr (light-blue, solid
line) and 8 Gyr (maroon, dashed line) isochrones computed with a solar heavy element composition (Grevesse & Sauval 1998). Positive values indicate that the variable αMLT models
are inflated compared to the solar-calibrated models. The “blip” observed near the fully convective boundary is likely due to a 3 He instability described by van Saders & Pinsonneault
(2012).
80
then calculated. The relative error was defined as
R obs − R model
δR
=
R obs
R obs
(4.3)
and will be presented as such throughout the rest of the paper. Formal agreement was
determined by analyzing the number of standard deviations outside of the accepted range
our model radii were located. Explicitly,
# σR =
R obs − R model
σR
(4.4)
where σR is the random uncertainty of the observational radius.
Different levels of compatibility were assigned to each system based on the individual agreement of each component in the system. For instance, if an isochrone matched both components at a common age within the 1σ limits set by the quoted random uncertainties, the
system as a whole was designated as being “fit” by the models. Only within the 1σ level
was a system considered to be accurately represented by the models. The analysis was applied to two sets of data, which differ in the adopted radius uncertainties (formal quoted
uncertainties or fixed 3% uncertainties) that will be justified later.
The process detailed above generated a list indicating the level of agreement between each
isochrone and each DEB system. Narrowing the list to a single “best fit” isochrone involved
minimizing the root mean square deviation (RMSD)
v
u 2 µ
u 1 X δR i ¶2
RMSD =t
.
2 i =1 σR,i
where the sum is performed over the components of the DEB system.
81
(4.5)
Table 4.2
Age and [Fe/H] priors for low-mass DEBs.
DEB System
Age (Gyr)
[Fe/H]
Status
UV Psc
7.9
Rejected
IM Vir
···
···
-0.3±0.3
≤ 1.0
···
Kepler-16
GU Boo
YY Gem
CU Cnc
KOI-126
CM Dra
···
0.4 ± 0.1
0.4 ± 0.1
≤ 0.0
Accepted
+0.1±0.2
Accepted
+0.15±0.8
Accepted
+0.1±0.2
4.0 ± 1.0
4.0 ± 1.0
Rejected
≤ 0.0
Rejected
Rejected
Accepted
4.3.4 Age & Metallicity Priors
DEBs that have been well studied have additional constraints that allow us to restrict the set
of isochrones used in the fitting procedure. Specifically, our sample contains eight systems
that have the added constraint of either an estimated age or metallicity, and in a couple
cases, both. Before accepting the quoted age and metallicity priors, however, we performed
a qualitative analysis to ensure that the estimates we adopted were reliable. We judged four
of the eight systems with quoted age or metallicity priors to have been determined reliably.
A summary of the adopted priors is given in Table 4.2.
UV Psc
The age of the UV Psc system is quoted by the TAG10 review to be 7.9 Gyr. Popper’s original
paper describing the characteristics of the UV Psc system does not provide any evidence to
support an age estimate (Popper 1997). An age of approximately 8 Gyr is typically assigned
to the system due to the fact that stellar evolution models predict the physical properties
of the primary star, UV Psc A, at that age. However, there has yet to be any set of models
that can place both components on a consistent, coeval isochrone. Since the estimated age
82
of UV Psc A appears to be derived from stellar evolution models, we allowed our models to
independently determine the most acceptable age.
IM Vir
Morales et al. (2009b) attempt to determine the metallicty of IM Vir by applying various
photometric metallicity relations. The result of their efforts was that they found all of
the various empirical methods quote different values with large uncertainties. Values for
[Fe/H] vary from the metal-poor end with [Fe/H] = -0.8 up to a super-solar value of [Fe/H]
= +0.15. This range also fits nicely within the set of model metallicity values selected for
this study. Since the cited metallicity range would not provide any additional constraint on
our analysis, we rejected the metallicity prior.
Kepler-16
Kepler-16, the first confirmed binary system with a circumbinary planet, was provided with
a metallicity estimate in its discovery paper (Doyle et al. 2011). A metallicity of [Fe/H] =
−0.3±0.2 was determined spectroscopically. The authors indicated the spectroscopic anal-
ysis was performed on the K-dwarf primary and that the general method was similar to
that applied to KOI-126 using Spectroscopy Made Easy (SME; Valenti & Piskunov 1996;
Carter et al. 2011). Reliability is lent to the method, in general, due to its success at deriving the metallicity of KOI-126 A. We therefore accepted the quoted metallicity prior.
GU Boo
The age estimate provided by López-Morales & Ribas (2005) in their characterization of GU
Boo was primarily based on kinematics. Specifically, they conclude that GU Boo is an isolated system and the vertical component of its motion provides a hint that it has undergone
83
perturbations due to disk heating processes. Assuming that the system has been subjected
to disk heating, one can only infer that the system has an age greater then 108 yr, the typical
timescale for dynamical perturbations associated with an objects orbit around the galactic
center (Soderblom 2010). Unfortunately, no further constraints were able to be placed on
the age of the system. However, it is not possible to rule out the scenario that GU Boo
was dynamically ejected from its stellar nursery. It would then appear that no reliable age
estimation exists, prompting us to reject the age prior for this system.
YY Gem
YY Gem is thought to be physically associated with the Castor AB (α Gem) quadruple
system. All three systems were found to be gravitationally bound based on a statistical
analysis performed using a three-body interaction code (Anosova et al. 1989). While the
Anosova et al. (1989) analysis was performed with pre-Hipparcos proper motions as their
initial conditions, it is unlikely that the results will be effected at the level necessary to
unbind the systems.
Castor A and B are themselves both binaries. The primary in both systems is an A star and
both are thought to have an M dwarf companion. Therefore, physical properties derived
from spectroscopic and photometric observations of the two A stars will be essentially unaffected by the presence of their companions. Placing the two primaries on an MV -(log(Teff )
H-R diagram, we used DSEP to derive an age of about 400 Myr. This age is consistent with
the average age of 370 Myr derived by Torres & Ribas (2002), who modeled the A-star primaries using multiple stellar evolution codes.
The A stars also lend themselves well to a spectroscopic determination of the metallicity.
Unfortunately, there appears to be only one result reported. Smith (1974) estimates an average metallicity for the two A stars to be about [Fe/H] = +0.7 relative to Vega. As described
in Torres & Ribas (2002), this implies a rather uncertain metallicity relative to the Sun of
84
[Fe/H] = +0.1 ± 0.2. Despite this, we adopt this metallicity constraint due to its rather large
uncertainty, which should presumably encompass the true value.
CU Cnc
When CU Cnc was originally investigated by Ribas (2003), it was found to have a space motion very similar to that of the Castor sextuple system (α Gem, YY Gem) and, subsequently,
the proposed “Castor moving group.” For this reason, CU Cnc was deemed to be associated
with the Castor moving group implying an age and metallicity similar to the Castor sextuple. However, determining membership of a moving group is complicated and has often
lead to ambiguous results concerning the coeval nature of the group. It is not uncommon
for members of the same kinematic moving group to have different metallicities, implying
that members of a defined moving group may not have been born in the same galactic environment and, as such, are not necessarily coeval. The lack of an age estimation beyond its
kinematic similarity to the Castor system led us to reject the age prior of CU Cnc.
KOI-126
KOI-126 is a hierarchical triple eclipsing binary recently whose discovery was recently announced (Carter et al. 2011). The quoted metallicity prior of [Fe/H] = +0.15 ± 0.08 was
determined using SME, as was mentioned above in the discussion of Kepler 16. Assuming
this metallicity allowed for a relatively precise age constraint (∼ 4 ± 1 Gyr) to be placed
on the primary, a 1.35M⊙ subgiant. Combining both the age and metallicity information
led to two low-mass companions to also be fit by stellar models (Chapter 2; Feiden et al.
2011). Thus, there appears to be little question about the validity of the age and metallicity
estimations, leading us to adopt the given priors for our study.
85
CM Dra
Finally, we consider the very well studied CM Dra system. Spectroscopic observations of the
system have produced varying results for the metallicity of the system, but all appear to be
consistent with −1 ≤ [Fe/H] ≤ 0 (Viti et al. 1997, 2002; Morales et al. 2009a; Kuznetsov et al.
2012). Due to the difficulties in modeling, and thus fitting, the entire SED of an M dwarf,
we determined there was no particular reason to strongly favor one metallicity result over
another.
An age was determined for the system by assuming that the common proper motion white
dwarf (WD) companion, WD 1633+572, is bound to the CM Dra system. It is the possible to
derive an age from WD cooling tracks combined with stellar evolution models of the approximate WD progenitor. The WD cooling age was found to be 2.38 ± 0.37 Gyr. Morales et al.
(2009a) predicted a mass for the progenitor star of M = 2.1±0.4M⊙ and derived a total (stellar model + WD cooling) age of 4.1 ± 0.8 Gyr. For consistency, we calculated the lifetime of
the progenitor star using DSEP with the full available suite of physics. Accounting for the
metallicity constraint defined above, we derived an age of 3.7 ± 1.2 Gyr and, in doing so,
found no reason to reject, or modify, the age prior.
4.4 Results
4.4.1 Standard Stellar Models
Direct Comparison
Figure 4.2 demonstrates that, in general, our models reduce the observed radius discrepancies to below 4% for 92% of the stars in our sample. Only a few outliers are seen to be largely
discrepant. Across the entire sample, we find a mean absolute error of 2.3%. This broadly
86
δR/Robs (obs - model)
0.12
Quoted Uncertainties
A)
Fixed 3% Uncertainties
B)
0.08
0.04
0.00
-0.04
δR/Robs (obs - model)
-0.08
0.12
0.08
0.04
0.00
-0.04
-0.08
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mass (M⊙)
Figure 4.2: Relative errors between stellar evolution models and observationally determined
radii. Reliable age and metallicity priors were accounted for in the statistical analysis. (Top)
The adopted measurement uncertainties are the cited observational random uncertainties
derived from the light curve fitting procedure. (Bottom) A fixed 3% uncertainty is adopted
to represent possible systematic uncertainties (i.e., starspots).
represents a factor of two reduction in the previously cited radius discrepancies. The age
and metallicity of the best fit isochrone for each DEB system is listed in Table 4.3 along with
the level of agreement between the best fit isochrone and the individual stars comprising
each DEB system.
Contrasting with previous studies, we do not observe an overwhelming systematic trend of
the models grossly under predicting stellar radii. Instead, systems discrepant by more than
5% represent an exception to the broad agreement between the models and observations.
While the agreement is admittedly not perfect, it is apparent that most observed discrepancies must now be judged according to the precision with which their radii were measured.
87
As we will see in the next subsection, systematic uncertainties have the potential to blur our
interpretation of the agreement between models and observations, simply due to the factor
of two improvement described above.
Ignoring systematic uncertainties, for now, we are still confronted with the need to explain
both the slightly discrepant as well as the largely discrepant radii. Discussion and comments
pertaining to plausible explanations of the inflated radii will be addressed later in Section 4.5.
The presence of stars that appear undersized compared to our models is likely the result of
our statistical fitting method and not a physical phenomenon.
Fixed Uncertainties
Recent work in modeling eclipse profiles of DEB systems has led to the realization that
large, polar starspots can potentially affect observationally derived radii and subsequently
dominate over the quoted random uncertainties. Windmiller et al. (GU Boo; 2010) and
Morales et al. (CM Dra; 2010) have found that systematic uncertainties may be present in
DEB radii on the order of 2 – 4%. This is to be compared with quoted random uncertainties
that are typically on the order of 1% or less.
Similarly, Kraus et al. (2011) included an estimate of systematic uncertainties for their radius
measurements and found typical uncertainties on the order of 2 – 3%. With the radius
uncertainties potentially dominated by often unquoted systematics, we were curious to see
what effect such uncertainties would have on comparisons with theoretical models. Quite
obviously, observed discrepancies would be reduced by the level of systematic uncertainty.
However, it has not been clear whether those systematics have the ability to completely
mask the observed radius residuals, particularly when combined with isochrones that cover
a wide age-metallicity parameter space. Since the radius uncertainty cut-off for our DEB
sample was set at 3%, we elected to adopt a fixed 3% radius uncertainty in our analysis to
mimic potential systematics.
88
Demonstrated in Figure 4.2 B is the effect of including a fixed 3% radius uncertainty. The
overall distribution of points appears to be similar to the previous case, except that the larger
uncertainty enables more systems (eleven in total) to be considered fit by our analysis (see
Table 4.3). This result is clearly expected when introducing larger error bars, as mentioned
above. Introducing a fixed systematic uncertainty also relieves the problem that some stars
appear smaller than the models in Figure 4.2 A.
What is important to realize is that we are now presented with a scenario where the systematic uncertainties mask our ability to draw firm conclusions about whether the observed
radius discrepancies are inherently real or merely a consequence of neglected uncertainties.
Such ambiguity was previously not present since the observed radius residuals were substantially larger than any potential systematic uncertainties.
89
Table 4.3
Best fit isochrone using a solar calibrated mixing-length.
Quoted
Star Name
Age
UV Psc A
8.0
[Fe/H] δR/R obs
-0.10
UV Psc B
IM Vir A
7.0
-0.10
IM Vir B
KID 6131659 A
3.0
-0.50
90
KID 6131659 B
RX J0239.1-1028 A
8.0
-0.10
RX J0239.1-1028 B
Kepler-16 A
1.0
-0.10
Kepler-16 B
GU Boo A
8.0
-0.10
GU Boo B
YY Gem A
YY Gem B
0.3
-0.10
Fixed 3%
# σR
Fit
Age
-0.0285
-1.375
No
8.0
0.1031
4.785
-0.0037
-0.248
0.0408
2.136
-0.0014
-0.750
0.0004
0.200
0.0297
5.510
0.0285
10.007
0.0004
0.210
0.0299
5.510
0.0292
1.144
0.0414
1.616
0.0813
8.834
0.0813
8.834
No
Yes
No
No
No
No
continued on next page
7.0
3.0
8.0
0.3
8.0
0.3
[Fe/H] δR/R obs
-0.10
-0.10
-0.50
-0.10
-0.10
-0.10
-0.10
# σR
Fit
-0.0285
-0.950
No
0.1032
3.439
-0.0038
-0.125
0.0408
1.359
-0.0014
-0.047
0.0004
0.013
0.0297
0.991
0.0285
0.949
0.0099
0.330
0.0225
0.749
0.0292
0.973
0.0414
1.381
0.0813
2.710
0.0813
2.710
No
Yes
Yes
Yes
No
No
Table 4.3 – continued
Quoted
Star Name
Age
MG1-506664 A
1.0
[Fe/H] δR/R obs
-0.10
MG1-506664 B
MG1-116309 A
7.0
-0.50
MG1-116309 B
MG1-1819499 A
8.0
-0.50
MG1-1819499 B
91
NSVS 01031772 A
8.0
-1.00
NSVS 01031772 B
MG1-78457 A
5.0
+0.20
MG1-78457 B
MG1-646680 A
1.0
+0.20
MG1-646680 B
MG1-2056316 A
3.0
+0.20
MG1-2056316 B
CU Cnc A
8.0
+0.20
Fixed 3%
# σR
Fit
Age
0.0035
1.951
No
1.0
-0.0031
-1.612
-0.0084
-1.162
0.0150
1.993
0.0296
8.425
-0.0528
-8.792
-0.0021
-0.396
0.0412
6.991
-0.0008
-0.049
-0.0001
-0.006
-0.0226
-1.725
0.0184
1.309
-0.0074
-1.632
0.0067
1.248
0.0233
1.832
No
No
No
Yes
No
No
No
continued on next page
7.0
7.0
8.0
5.0
1.0
3.0
8.0
[Fe/H] δR/R obs
-0.10
-0.50
-0.10
-0.50
+0.20
+0.20
+0.20
+0.20
# σR
Fit
0.0035
0.116
Yes
-0.0031
-0.105
-0.0084
-0.281
0.0150
0.500
0.0404
1.348
-0.0408
-1.360
-0.0134
-0.447
0.0395
1.317
-0.0008
-0.026
-0.0001
-0.004
-0.0226
-0.755
0.0184
0.613
-0.0074
-0.247
0.0067
0.222
0.0233
0.777
Yes
No
No
Yes
Yes
Yes
Yes
Table 4.3 – continued
Quoted
Star Name
Age
[Fe/H] δR/R obs
# σR
0.0020
0.084
-0.0005
-0.035
0.0290
1.875
-0.0011
-0.205
-0.0003
-0.053
0.0316
4.221
0.0360
4.798
CU Cnc B
LSPM J1112+7626 A
8.0
+0.20
LSPM J1112+7626 B
KOI-126 B
1.0
+0.10
KOI-126 C
CM Dra A
92
CM Dra B
5.0
Fixed 3%
0.00
Fit
No
Yes
No
Age
8.0
3.0
5.0
[Fe/H] δR/R obs
# σR
0.0020
0.067
-0.0005
-0.016
0.0290
0.966
-0.0036
-0.120
-0.0004
-0.013
0.0316
1.055
0.0360
1.200
+0.20
+0.20
0.00
Fit
Yes
Yes
No
4.4.2 Variable Mixing Length Models
Non-standard, variable mixing-length models lead to only slightly better results over the
standard models when considering the quoted random uncertainties (see Table 4.4 and Figure 4.3). We find that radius deviations are largely reduced to below 4%, seen clearly in
Figure 4.3. The residuals do appear to be more tightly clustered around the zero point
than they do in Figure 4.2. Although, only slightly. This is suggestive of an overall better
agreement between models and observations when we vary the mixing length, which is
reinforced by a slightly lower MAE of 2.1% across the entire sample.
There was evidence in Figure 4.3 of a possible tendency for the best fit theoretical isochrone
to over predict the observed radius at higher masses and under predict the radius for
lower mass stars. A least squares regression performed on the data finds a linear slope
of −0.031 ± 0.013, significant at the 2.5σ level (not shown). Half of the DEB systems that
were not formally fit are characterized by an isochrone that over-predicts the higher mass
primary but under-predicts the lower mass secondary. This is only an artifact of our functional parametrization of αMLT , suggesting convection was too heavily suppressed at higher
masses and under-suppressed in the low-mass regime.
Artificially fixing the uncertainties at 3% adds ambiguity as to whether there are any real
discrepancies for a majority of the systems. This is illustrated in Figure 4.3. As was suggested by our study of the standard models, systematic uncertainties must be minimized to
below 2% before an accurate comparison between models and observations may occur.
93
δR/Robs (obs - model)
0.12
Quoted Uncertainties
A)
Fixed 3% Uncertainties
B)
0.08
0.04
0.00
-0.04
δR/Robs (obs - model)
-0.08
0.12
0.08
0.04
0.00
-0.04
-0.08
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mass (M⊙)
Figure 4.3: Relative errors between stellar evolution models with a mass-dependent convective mixing length and observationally determined radii. Reliable age and metallicity
priors were accounted for in the statistical analysis. (Top) The adopted measurement uncertainties are the cited observational random uncertainties derived from the light curve fitting
procedure. (Bottom) A fixed 3% uncertainty is adopted to represent possible systematic uncertainties (i.e., starspots).
94
Table 4.4
Best fit isochrone using a mass dependent convective mixing length.
Quoted
Star Name
Age
UV Psc A
8.0
[Fe/H] δR/R obs
0.00
UV Psc B
IM Vir A
8.0
+0.20
IM Vir B
KID 6131659 A
3.0
-0.50
95
KID 6131659 B
RX J0239.1-1028 A
8.0
-0.50
RX J0239.1-1028 B
Kepler-16 A
1.0
0.00
Kepler-16 B
GU Boo A
8.0
-0.50
GU Boo B
YY Gem A
YY Gem B
0.4
0.00
Fixed 3%
# σR
Fit
Age
-0.0072
-0.349
No
8.0
0.0866
4.019
-0.0032
-0.209
0.0238
1.245
-0.0113
-5.876
-0.0190
-9.352
-0.0075
-1.394
0.0039
1.356
-0.0085
-4.227
0.0244
11.061
0.0085
0.333
0.0190
0.740
0.0757
8.226
0.0757
8.226
No
No
No
No
Yes
No
continued on next page
8.0
3.0
8.0
1.0
8.0
0.4
[Fe/H] δR/R obs
0.00
+0.20
-0.50
-0.50
0.00
-0.50
0.00
# σR
Fit
-0.0072
-0.241
No
0.0866
2.888
-0.0032
-0.105
0.0238
0.792
0.0113
-0.378
-0.0190
-0.634
-0.0075
-0.251
0.0039
0.129
-0.0085
-0.282
0.0245
0.815
0.0085
0.283
0.0190
0.632
0.0757
2.523
0.0757
2.523
Yes
Yes
Yes
Yes
Yes
No
Table 4.4 – continued
Quoted
Star Name
Age
MG1-506664 A
1.0
[Fe/H] δR/R obs
0.00
MG1-506664 B
MG1-116309 A
8.0
+0.20
MG1-116309 B
MG1-1819499 A
5.0
-0.50
MG1-1819499 B
96
NSVS 01031772 A
8.0
+0.20
NSVS 01031772 B
MG1-78457 A
3.0
0.00
MG1-78457 B
MG1-646680 A
1.0
0.00
MG1-646680 B
MG1-2056316 A
1.0
+0.20
MG1-2056316 B
CU Cnc A
8.0
+0.20
Fixed 3%
# σR
Fit
Age
-0.0017
-0.964
No
1.0
-0.0099
-5.065
-0.0126
-1.743
0.0115
1.534
0.0240
6.839
-0.0575
-9.587
-0.0121
-2.276
0.0352
5.965
-0.0023
-0.143
0.0005
0.025
-0.0319
-2.428
0.0159
1.129
-0.0083
-1.829
0.0112
2.095
0.0127
0.998
No
No
No
Yes
No
No
Yes
continued on next page
8.0
5.0
8.0
3.0
1.0
1.0
8.0
[Fe/H] δR/R obs
0.00
+0.20
0.00
+0.20
0.00
0.00
+0.20
+0.20
# σR
Fit
-0.0017
-0.057
Yes
-0.0099
-0.329
-0.0126
-0.421
0.0115
0.384
0.0413
1.376
-0.0396
-1.320
-0.0121
-0.404
0.0352
1.172
-0.0023
-0.075
0.0005
0.016
-0.0319
-1.063
0.0159
0.529
-0.0083
-0.277
0.0112
0.373
0.0127
0.423
Yes
No
No
Yes
No
Yes
Yes
Table 4.4 – continued
Quoted
Star Name
Age
[Fe/H] δR/R obs
# σR
-0.0028
-0.116
-0.0038
-0.278
0.0255
1.652
0.0020
0.365
-0.0017
-0.307
0.0282
3.757
0.0333
4.441
CU Cnc B
LSPM J1112+7626 A
8.0
+0.20
LSPM J1112+7626 B
KOI-126 B
3.0
+0.20
KOI-126 C
CM Dra A
97
CM Dra B
5.0
Fixed 3%
0.00
Fit
No
Yes
No
Age
8.0
5.0
5.0
[Fe/H] δR/R obs
# σR
-0.0028
-0.092
-0.0038
-0.125
0.0255
0.850
0.0010
0.033
-0.0014
-0.047
0.0282
0.939
0.0333
1.111
+0.20
0.00
0.00
Fit
Yes
Yes
No
4.4.3 Peculiar Systems
Before continuing with a further interpretation of our results, we wanted to briefly comment on several individual DEB systems. These systems stood out in our mind as worth
remarking upon separate from the ensemble.
Our models never fit the UV Psc system to a coeval age. While we were able to fit UV Psc A,
the secondary component was always found to have an 8 – 10% larger radius than would
be expected from stellar evolution theory, consistent with Popper (1997). Both stars may
realistically be discrepant with the models, however, if the age is truly younger than 8 Gyr.
Constraining the actual magnitude of the discrepancies is difficult without observational
age and metallicity estimates. Fittingly, the system is known to be very active based on
spectroscopic analysis of Hα cores, multi-epoch photometric monitoring (Kjurkchieva et al.
2005), and the derived X-ray luminosity (this work, Section 4.5.2). Further observations of
UV Psc to constrain its metallicity and to provide data for a more rigorous starspot analysis
would be a worthwhile endeavor.
MG1-189499, characterized by Kraus et al. (2011), was also never found to be in total agreement. The primary and secondary were found to deviate by 3.0% and -5.3%, respectively.
This appeared to be a cause for concern, but Kraus et al. were one of the few groups
to provide an estimate of potential systematic uncertainties. Uncertainties in the primary
star’s radius were elevated to 4.6% with a comparatively large uncertainty of 3.4% in the secondary’s radius. Applying these uncertainties and rerunning the isochrone fitting procedure
allows the predictions of the models to nearly fall inline with the observations. An age of 8
Gyr was again derived but with a new metallicity of [Fe/H] = -0.3. This isochrone yielded
relative radius errors of 4.8% and -2.9% for the primary and secondary, respectively. Hence,
the secondary is now considered fit by the models and the primary is only 0.2% outside the
bounds of uncertainty.
The last two systems we would like to discuss are two that have long posed problems for
98
modelers: YY Gem and CM Dra. YY Gem (Torres & Ribas 2002) is an equal mass, equal
radius DEB system that is effectively represented by a single point at (0.6, 0.08) in Figure 4.2.
Beginning with the low-mass models of Hoxie (1970), YY Gem has never been adequately
reproduced by stellar models.
The difficulty with YY Gem seems to be in the estimated age of 400 Myr, determined by
its association with the Castor AB quadruple system. An older age is logically preferred
by the models considering the notably inflated radius. Interestingly, this would have led
previous studies to underestimate the observed radius discrepancy. Assuming either a 1 or
5 Gyr isochrone naturally results in models with larger radii than those computed with an
age of 400 Myr. Further study of this system, particularly its distance (Section 4.5.2) and
association with Castor AB (Section 4.3.4), will be beneficial toward fully understanding the
nature of its inflated radii.
Finally, CM Draconis is another constant thorn in theoreticians sides. Here, the primary
and secondary deviate by 3.2% and 3.6%, respectively. CM Dra is one of the most studied
DEB systems and has very well constrained physical properties measured from data spanning multiple decades (Morales et al. 2009a). Our models indicate that a solar composition
is preferred1 . However, the composition of CM Dra, while not known positively, is very
likely subsolar (Viti et al. 1997, 2002; Morales et al. 2009a; Kuznetsov et al. 2012), meaning
our models are probably more discrepant than indicated by this study. Morales et al. (2010)
found that large polar spots produce systematic uncertainties in the radius measurements
of around 3%, potentially bringing CM Dra more in line or more out of line with model
predictions. For the time-being, this system will continue to test our knowledge of stellar
evolution.
1 A super-solar metallicity is actually favored, as was also noted by Spada & Demarque (2012), but the
application of the metallicity prior in our analysis restricted the models to only solar or subsolar compositions.
99
4.5 Discussion
Data presented in this study hint at two competing explanations for the occurrence of the
differing model and observational MR relations. It is entirely plausible that stellar evolution models are not incorporating key physical processes that can account for the observed
discrepancies. This view is not new and has always accompanied discussions of low-mass
models (Hoxie 1970, 1973; Chabrier & Baraffe 1997; Baraffe et al. 1997; Baraffe et al. 1998).
Typically cited is our incomplete knowledge dealing with the complex array of molecules
present in M-dwarf atmospheres as well as the lack of structural changes induced by a
large-scale magnetic field. Included in the latter are both the effects on convective energy
transport and the emergence of spots on the stellar photosphere.
The other scenario is one in which the neglect of systematic uncertainties is driving the
apparent discrepancies. We have shown that by allowing for realistic variation in age and
metallicity of the models, the radius residuals are of the same magnitude as the potential
systematic uncertainties. Note, this is without any modification to the solar calibrated mixing length or the inclusion of any non-standard physics. Previous studies have indicated that
systematics may help alleviate the size of the radius residuals, but have required additional
modifications to the models in order to fully reconcile models with observations. It is now
clear that we must work to constrain and minimize systematic uncertainties in observations
of DEBs to allow for them to provide an accurate test of stellar evolution models.
4.5.1 Radius Deviations
The MAE between our models and the observations was 2.3%, a factor of two improvement
over the canonical minimum of 5%. We found deviations of no more than 4%, with the
exception of a few stars, instead of ubiquitous 5 – 15% errors, as is often quoted. Despite
improving the situation, panel A of Figure 4.2 illustrates that the models are still unable to
100
fully reproduce the observed stellar radii.
Accepting the factor of two improvement presented in Section 4.4.1, the paradigm of broad
disagreement between models and observations is shifted to one where agreement is broad,
and large discrepancies are an exception. With the radius deviations typically less than 4%,
an evaluation of the systematic errors becomes imperative. Formerly, systematic uncertainties of about 4% were incapable of relieving radius deviations greater than 5%. Stellar
evolution models still appeared to disagree with DEB observations even after the inclusion
of systematic errors.
The reason for the factor of two improvement is twofold. First, we have calculated models
with a finer grid of metallicities. DSEP utilizes an EOS that enables models to be more
reliably calculated for super-solar metallicities, allowing for a greater range of stellar compositions to be considered. Low-mass stellar models with super-solar metallicity have previously been unavailable for comparison with low-mass DEB data. Thus, the ability to extend
our model set to super-solar metallicities allows for more flexibility in attempting to match
the observed properties of DEB components.
Second, a far larger number of low-mass DEBs with precisely measured radii were available
to us as opposed to previous studies. Before the publication of TAG10, there were only eight
systems that met the criteria necessary to accurately constrain stellar evolution models.
Following the TAG10 review, the total number of systems that met the necessary criteria
more than doubled with the addition of ten newly characterized DEBs. These additional
systems appear to be more in line with the results of standard stellar evolution theory.
However, well-known discrepant systems remain noticeably discrepant and still require
further explanation.
We must now ask, “what belies the current discrepancies between models and observations?” Figure 4.2 favors the hypothesis that non-standard physics are absent from current
stellar evolution models. Our larger data set allows us to notice that stars of similar masses
101
from different DEB systems appear to be discrepant at varying levels, an effect a single set
of standard models cannot correct. However, this is contingent upon the accuracy our age
and metallicity predictions. Until we have better empirical age and metallicity estimates
for the various systems, it is too difficult to ascertain the true level of discrepancy for any
individual star.
The efficiency of convective energy transport is of greatest interest. It is possible that convection is naturally inefficient. Although, we gather from Figures 4.2 and 4.3 that suppression of convective energy transport must be tied to a stellar property that is largely independent of mass. Simple parametrization of the suppression of convection is too uniform
over a given mass regime to fully account for the observed differences in stellar radii for
stars with similar masses (see Section 4.4.2). Any effort, either theoretical or observational,
to constrain the physics of convection in low-mass stars will lend crucial insight.
The most favored option, is that convection is not intrinsically inefficient, but that a largescale magnetic field acts to suppress convective motions (Mullan & MacDonald 2001; Chabrier et al.
2007; MacDonald & Mullan 2012). Stellar evolution models self-consistently incorporating
the effects of large-scale magnetic fields will help on this front. Observations of cool-star
magnetic field strengths and topologies will then provide a means of judging the validity of
any new models.
One final hypothesis is that starspots may affect the structure of stars and generate the
inflated radii we observe. Chabrier et al. (2007) investigated such a possibility by artificially
reducing the total stellar bolometric luminosity in an effort to mimic the effects of spots.
They found that radius discrepancies were relieved with their parametrization. However,
it is still not apparent whether spots reduce the total bolometric flux or if they locally shift
flux to longer wavelengths (Jackson et al. 2009), preserving the total luminosity. Ultimately,
if starspots are required in stellar evolution models, their inclusion is necessitated in the
analysis of DEB light curves.
102
Quantifying the effect starspots may have on observed DEB light curves is extremely difficult. Obtaining accurate knowledge of the total surface coverage of spots, the total number
of spots, their individual sizes, temperature contrasts, and their overall distribution on the
stellar surface is nearly an impossible task given only a light curve. From a theoretical
perspective, finding a proper parametrization to mimic the effect of spots on a three dimensional volume within the framework of a one dimensional model provides its own complications. Currently, the only feasible method to include spots, is to include their potential
effects on the radius measurement uncertainties.
Unfortunately, while the inclusion of fixed 3% radius uncertainties in our analysis was able
to alleviate many of the noted radius discrepancies, it also created a situation where the
measurement uncertainty was on the order of the typical radius deviation. We are presented with a case where the observations are no longer effective at testing the models and
the manifestation of most radius discrepancies can be attributed to under estimated error
bars. At this point, we require observations that have been rigorously vetted for potential systematic uncertainties and are able to still provide mass and radius measurements to
better than 2%.
4.5.2 Radius-Rotation-Activity Correlations
Direct measurements of low-mass magnetic field strengths are rare, especially among fast
rotating stars (Reiners 2012). Without a direct measure of the magnetic field strength, we
are forced to rely on indirect measures to probe correlations between stellar magnetism
and the appearance of inflated stellar radii. Ideally, these indirect measures are intimately
connected with the dynamo mechanism, thought to generate and maintain stellar magnetic
fields, or are the product of magnetic processes in the stellar atmosphere. The preferred
indirect measures are typically stellar rotation or the observation of magnetically driven
emission (Hα, Ca ii H and K, X-ray).
103
A)
δR/Robs (obs - model)
0.12
B)
0.08
0.04
0.00
-0.04
-0.08
0
10
20
30
Porb (day)
40
1
2
Porb (day)
3
4
Figure 4.4: Observed correlation between radius deviations and DEB orbital periods. Maroon asterisks are stars with known X-ray flux measurements. Dashed vertical lines represent the various period cuts used in our statistical analysis to divide the sample into two
subsamples. (Left) The full range of periods present in the current sample. (Right) Highlighting the short period regime as it encompasses a majority of the DEB systems.
Rotation
Typically, low-mass DEBs are found in tight, short period orbits (< 3d; see Figure 4.4). Tidal
interactions spin-up the individual components and allow them to remain rapidly rotating
throughout their life cycle. Stellar dynamo theory dictates that the large-scale magnetic
field strength is tied to the rotational properties of a star (i.e., a rapidly rotating star should
be more magnetically active than a comparable star that is slowly rotating; Parker 1979;
Reiners et al. 2012), providing a natural starting point for our investigation.
We performed two independent statistical tests on the distribution of radius residuals as
a function of the orbital period (P orb ). Our primary objective was to determine whether
rapidly rotating systems produce, on average, larger radius deviations than systems perceived to be slow rotators. Rotational periods were assumed to be synchronous with the
orbital period unless a separate value for the rotational period was cited in the literature. The statistical tests performed were a Kirmogorov-Smirnov (K-S) test and another
whereby we tested the probability of obtaining a given distribution of residuals via a Monte
Carlo method. Both tests were performed on the observed difference in mean absolute error
104
δR/Robs (obs - model)
0.12
0.08
0.04
0.00
-0.04
-0.08
0.01
0.1
1
Rossby Number, Ro
Figure 4.5: The theoretical Rossby number, Ro = P rot /τconv , versus the relative radius error.
Ro is tied directly to the theoretical stellar dynamo mechanism and is empirically related to
the ratio of a star’s X-ray to bolometric luminosity (a magnetic activity indicator). Asterisks
in maroon are stars with known X-ray flux measurements.
(MAE)2 between two data bins. The data bins were divided at preselected values of P orb ,
identified visually as vertical dashed lines in Figure 4.4.
Comparing the radius deviations with the rotational periods, we find no evidence of any
dominant correlation. Figure 4.4 displays the residual data as a function of the orbital period,
with frame A showing the full range of observed periods and frame B highlighting the “short
period” regime. The correlation of radius deviations with orbital period has been studied
previously (e.g., Kraus et al. 2011) where a significant difference between the two bins was
observed around P orb = 1.5 days. We performed the statistical tests using three values for
the orbital period (1.0d, 1.5d, and 2.0d) that defined the two period bins.
We confirm the results of Kraus et al. (2011) and find a 3.1σ difference in the distribution of
radius deviations around 1.5d. Systems with P orb < 1.5d had a MAE of 3.4% while the longer
2 We
selected the MAE over the RMSD as a measure of the mean radius deviation of a given ensemble in
order to reduce the weight of any individual outlier in the final mean.
105
period systems had a MAE of 1.0%. While this is tantalizing, we can not necessarily attribute
any physical significance to this particular division. We should expect this difference to be
present for any two subsamples. However, we fail to find any evidence for a statistically
meaningful difference when we divide the subsamples at 1.0d and 2.0d. Therefore, the
significant difference noted at 1.5d is likely a spurious statistical result3 . There does not
appear to be a physically meaningful explanation for why the divide should be made at
1.5d, but then also not hold for a division at 1.0d.
The rotational (or orbital) period is not necessarily the most appropriate proxy for the magnetic field strength or potential magnetic activity level. For instance, two stars may have
identical rotation periods, but they may have different rotational velocities depending on
their radius. The latter variable may play a more important role in governing the stellar
magnetic field. It is therefore ideal for our chosen rotational variable to have some connection to intrinsic stellar properties. Optimally, the rotational variable in question would also
provide a direct link to either observable magnetic activity or a theoretical description of
stellar magnetism.
Accordingly, we advocate the use of the Rossby number (hereafter Ro). Ro is defined as the
ratio of the rotational period of the star to its convective overturn time, Ro = P rot /τconv , and
measures the strength of the Coriolis force acting on the vertical motion of convection cells.
The dimensionless quantity Ro appears directly in standard mean-field dynamo theory (α-ω
dynamo; Parker 1979)4 and is intimately related to the ratio of the stellar X-ray luminosity
to bolometric luminosity (Wright et al. 2011). The latter quantity has been shown to be a
strong indicator of a stellar corona heated to over 106 K by magnetic activity (Vaiana et al.
1981; Pallavicini et al. 1981; Noyes et al. 1984).
3 Kraus
et al. (2011) posit the difference may actually be a by-product of the DEB light curve analysis
methods. Providing a further examination is outside the scope of this study.
4 For fully convective stars, an α-ω dynamo cannot operate due to the lack of a tachocline. However, it
is thought that an α2 dynamo can efficiently generate a magnetic field. Since it is the α mechanism that is
related to the strength of the Coriolis force, the Rossby number should be just as applicable to fully convective
stars (Chabrier & Küker 2006; Browning 2008).
106
We find no significant difference in the distribution of data points as a function of Ro. One
cut in Ro was performed at Ro = 0.1, illustrated in Figure 4.5. The selection of Ro = 0.1 approximately corresponds to Rosat , or the value of Ro associated with an observed saturation
in the ratio of the stellar X-ray luminosity to the bolometric luminosity (Wright et al. 2011).
This suggests that all points with Ro values less than 0.1 should presumably be sufficiently
magnetically active so as to display inflated radii. We find this is not the case. Unfortunately, the selection of Ro = 0.1 creates one bin with a small sample size which could affect
the statistics. Caution must further be taken with the above results as secondary stars in the
longer period (> 15d) systems do not have independently measured rotation periods, meaning they may actually have lower Ro values. A visual inspection of Figure 4.5 leads us to
the same conclusion, if we ignore the two most discrepant points. Since we were comparing
MAE values, the outliers do not significantly affect the results of the statistical analysis.
Arguably, the MAE is not an effective measure of the degree of inflation of each sample as it
treats deflated radii the same as inflated radii. Instead, the actual direct mean may provide
a more compelling statistical measure. Thus, we ran our statistical analysis on the direct
mean error. Overall, the typical degree of inflation among the radii of low-mass stars was
found to be about 1.6%. No statistically significant correlations with either P orb or Ro were
uncovered. The most significant result was found for the period cut at 1.5d, where the K-S
test indicated a significant difference. However, the MC method produced a difference in
the mean error of the two populations at 2.2σ, below our significance threshold of 3.0σ. All
other bin divisions yielded results significant at only about the 1σ level.
Curiously, if we consider only data points with measured X-ray fluxes (see below), there is
evidence that systems with lower Ro values may have larger radius discrepancies. However,
we are then prompted to explain why the trend does not continue to higher Ro values,
a question we are currently not equipped to answer. There is still some ambiguity with
the presence of the points near Ro = 0.01, which appear to contradict the presence of any
107
δR/Robs (obs - model)
0.12
0.08
0.04
0.00
-0.04
Variable αMLT
Variable αMLT
-0.08
0.0
10.0
20.0
30.0
Porb (day)
40.0
1.0
2.0
Porb (day)
3.0
4.0
Figure 4.6: Observed correlation between radius deviations and DEB orbital periods.
Dashed vertical lines represent the various period cuts used in our statistical analysis to
divide the sample into two subsamples. (Left) The full range of periods present in the current sample. (Right) Highlighting the short period regime as it encompasses a majority of
the DEB systems.
definite correlation. Deeper X-ray observations of X-ray faint low-mass DEBs will clarify
this ambiguity.
We subjected the data produced from fits to the variable αMLT models to the same statistical analyses described above. Again, we compared the radius deviations to the rotational
period and Rossby number. The resulting distribution are shown in Figures 4.6 and 4.7. The
strongest hint of a correlation was found when applying a cut at a period of 1.5d, although
it was only significant at the 2.7σ level, below our 3σ significance threshold. Performing
the period cut at 1.0d and 2.0d yielded results that were even less significant (2.4σ and 1.9σ,
respectively), providing the same qualitative result as standard model scenario. Similarly,
no strong correlation is observed between the radius deviations and the Rossby number. As
with the standard models, we conclude that there are no strong correlations between radius
deviations and rotational properties.
X-ray Activity
An interesting extension of our discussion in the previous subsection, is to compare our
derived radius deviations with the observed X-ray to bolometric luminosity ratio (hereafter
108
Variable αMLT
δR/Robs (obs - model)
0.12
0.08
0.04
0.00
-0.04
-0.08
0.01
0.1
1
Rossby Number, Ro
Figure 4.7: The theoretical Rossby number, Ro = P rot /τconv , versus the relative radius error
for models computed with a mass-dependent mixing length. Ro is tied directly to the theoretical stellar dynamo mechanism and is empirically related to the ratio of a star’s X-ray to
bolometric luminosity (a magnetic activity indicator).
R x = L x /L bol ). Since no correlation was noted with Ro, we expect that no correlation will
be observed with R x , as it has been shown to be intimately connected with Ro (Wright et al.
2011).
López-Morales (2007) previously performed such a comparison and found a clear correlation
between R x and radius deviations, in contradiction with our predictions. Her comparison
was performed under different modeling assumptions, which may have influenced the results. Specifically, she compared all observations to a 1 Gyr, solar metallicity isochrone
from Baraffe et al. (1998). Without variation in age and metallicity, the observed radius discrepancies may have been over estimated (or under estimated in the case of YY Gem). We
therefore perform the same comparison as López-Morales (2007) to probe whether age and
metallicity variations affect the outcome.
A subsample of sixteen DEBs from this study have identified X-ray counterparts in the
ROSAT All-Sky Survey Bright and Faint Source Catalogues (Voges et al. 1999; Voges et al.
109
2000). Our analysis follows that of López-Morales (2007) whose previous analysis contained
a fraction of our current X-ray detected sample. X-ray count rates were converted to X-ray
fluxes according to the formula derived by Schmitt et al. (1995),
F x = (5.30HR + 8.31) × 10−12 X cr
(4.6)
where HR is the X-ray hardness ratio5 , X cr is the X-ray count rate, and F x is the X-ray
flux. Luminosities in the X-ray spectrum were computed with distances determined from
either Hipparcos parallaxes (preferred when available; van Leeuwen 2007) or near-infrared
photometry.
Photometric distances were estimated using the luminosity calculated from the StefanBoltzmann relation assuming the observationally determined radius and effective temperature. Absolute magnitudes were then derived using the phoenix ames-cond model atmospheres, adopting the best fit isochrone metallicities. In an effort to reduce errors introduced by the theoretical atmospheres, distances were computed using the average distance
modulus derived from Two Micron All Sky Survey (2MASS) J - and K -band photometry
(Cutri et al. 2003).
We derived R x values for all sixteen stars in our X-ray sample to ensure internal consistency. Slight discrepancies between values presented here and those of López-Morales
(2007) are due entirely to differences in the adopted distances. Attributing the X-ray flux
contribution from each star in an DEB system is a difficult task. As such, López-Morales
performed her analysis using three reported empirical scaling relations (Pallavicini et al.
1981; Fleming et al. 1989):
• Case 1 – each component contributes equal weight.
• Case 2 – proportional to the respective rotational velocity, v sin i , of each component.
5 There
are typically two hardness ratios listed in the ROSAT catalog, HR1 and HR2. The Schmitt et al.
(1995) formula requires the use of HR1.
110
ROSAT Detections
δR/Robs (obs - model)
Linear Regression
0.12
Case 1
0.08
0.04
0.00
δR/Robs (obs - model)
-0.04
0.12
Case 2
Case 3
0.08
0.04
0.00
-0.04
0.0000
0.0005
0.0010
Rx = Lx/Lbol
0.0015
0.0000
0.0005
0.0010
Rx = Lx/Lbol
0.0015
Figure 4.8: Relative error between observational and model radii for stars with detected
X-ray emission. X-ray data is drawn from the ROSAT All-Sky Survey and combined with
the radius residuals derived from this study. Data are shown as maroon filled circles. Illustrated are two least-square regressions performed on the data. The light-blue, dashed line
demonstrates a non-negligible slope of ∼ 25 ± 17 across all of the data, while the indigo,
dash-dotted line excludes the two most discrepant points, UV Psc B and YY Gem.
• Case 3 – proportional to the square of the rotational velocity, (v sin i )2 , of each component.
We present results from all three cases in Figure 4.8. Immediately we notice that the size of
the stellar radius deviations appears to correlate with R x . A linear least-square regression
performed on each data set (light-blue dashed line in Figure 4.8) suggests the slope for each
case (1 – 3) is 26, 24, and 22 (± 17), respectively, all with reduced-χ2 values of ∼8. Pearson r
coefficients were 0.72, 0.69, and 0.63, respectively. The statistics suggest that the likelihood
of uncorrelated sets of data producing these particular correlations are 0.2%, 0.3%, and 0.9%,
for case 1, 2, and 3, respectively. We can therefore rule out the null hypothesis with greater
than 99% confidence.
111
Visually, however, we notice that the correlation is largely driven by the presence of YY Gem
and UV Psc B, located in the upper-right region of each panel in Figure 4.8. If we were to
remove those three points (YY Gem appears as a single point), the correlation vanishes
and we only observe an offset from the zero point (Indigo dash-dotted line in Figure 4.8).
Furthermore, the distance to YY Gem is highly uncertain. Assuming it is associated with
Castor AB provides a distance of about 15 pc (discussed in Section 4.3.4), but our photometric
analysis, as described above, places YY Gem at a distance of approximately 11 pc, 4 pc closer
to the Sun than the Castor AB system. Instead of selecting a single distance estimate, we
averaged the two estimates and adopted d = 13±2 pc, which also happens to be the distance
assumed by Delfosse et al. (2000).
The fact that our statistical correlation critically hinges upon three points, two of which are
strongly distance dependent, is a cause for concern and implies that the statistics should be
interpreted with care. If we believe the strong statistical correlation, then we are presented
with a situation where the rotational data and the X-ray data disagree. This may be due to
the physics underlying the stellar dynamo or those underlying X-ray saturation. However,
there are two further interpretations that are contingent upon the role systematics play.
First, we have that the correlation is entirely real and the presence of non-inflated stars with
R x ∼ 0.0007 are outliers in the relation. Second, systematics play a large role, as proposed in
Section 4.4.1. Here, the truly deviant stars exhibit very strong X-ray emission (R x > 0.001),
while non-inflated stars show lower, varying levels of X-ray emission. For this view to hold,
the relation between the level of radius inflation and magnetic field strength can not be
linear. Strong magnetic fields would induce significant radius inflation while moderate and
weak fields would produce little or no inflation.
Accepting, on the other hand, that the statistical correlation is spurious, we are left with a
picture that is coherent with our rotation analysis. Namely, magnetic activity may not be
the leading cause for all of the observed inflated radii. Here, systematics may still play a
112
role in producing stars that appear inflated, but that are consistent with the models, leaving
a few discrepant stars. Naturally inefficient convection, potentially dependent on particular
stellar properties, may be operating. However, magnetic activity cannot be fully ruled out,
as we do not yet have a fully self-consistent description of the interaction of magnetic
fields with the stellar interior and atmosphere. It may be that magnetic fields acting within
YY Gem and UV Psc have a more noticeable affect on stellar structure, as higher mass stars
are affected more by changes to the properties of convection (see Figure 4.1).
We tend to favor a hybrid interpretation. Here, most of the observed “inflation” is an artifact of unaccounted-for systematics, but significantly discrepant stars (YY Gem, UV Psc) are
associated with very strong magnetic activity. We believe that CM Dra probably fits into
the latter category due to a push in the literature towards subsolar metallicity. Effects of a
large-scale magnetic field are presumably mass dependent, with higher mass stars showing
a greater propensity to become inflated owing to their sensitivity to changes in convective properties. In other words, higher mass stars are more sensitive to weaker magnetic
fields than lower mass, fully convective stars. Whether there is a characteristic magnetic
field strength that induces substantial radius deviations is unclear. Dynamo saturation and
the saturation of magnetic activity, as evidenced by the flattening of the Ro-R x relation in
Wright et al. (2011), is not yet fully understood, but may provide further insight into the
apparent disagreement between our rotation and X-ray analyses.
Finally, clarity will be obtained with a better distance measurement to YY Gem, either from
ground-based parallax programs or with eventual results from Gaia. Of all the points in
Figure 4.8, the existence of a positive correlation is most dependent upon the distance to
YY Gem. An accurate distance, as well as a reanalysis of its association with the Castor
system, has the ability to not only relieve the ambiguity present in the X-ray data, but
also to provide insight into the necessary constraints for the system (i.e., is YY Gem really
about 400 Myr old?). As further low-mass DEB systems are discovered, X-ray observations
113
are strongly encouraged in order to develop a coherent picture of how radius deviations
correlate with magnetic activity.
4.6 Summary
This chapter focused on reevaluating the current state of agreement between the theoretical
and observational low-mass, main sequence MR relationship. The DEB systems used in
the analysis were required to have quoted random uncertainties in the mass and radius
below 3% in order to provide an effective test of stellar evolution models. A large grid of
DSEP models were computed with variation in age and metallicity characteristic of the local
galactic neighborhood. Best fit isochrones were derived by allowing the age and metallicity
to be optimized while maintaining the constraint that the system be coeval with a single
composition. DEBs with reliably determined ages or metallicities were compared with a
restricted set of isochrones in the range allowed by the observational priors.
Overall, we find 92% of stars in our sample are less than 4% discrepant with the models,
largely representing a factor of two improvement over the canonical 5% – 15% deviations.
Our results suggest that low-mass stars with radii that deviate significantly from model
predictions are exceptions to general agreement. Discrepancies may also be the result of
unaccounted for systematic uncertainties (i.e., starspots) that may as large as 4%. With
uncertainties as large as the typical radius deviations, we find it difficult to draw the firm
conclusion suggesting that models are in broad disagreement with observations. Instead, we
are left with a situation where the observational uncertainties may be too large to provide
an adequate test of stellar models. The combination of random and systematic uncertainties
for the sample of low-mass DEBs must be constrained and minimized below the 2% level
before accurate model comparisons may be made.
Radius correlations with orbital (rotational) period, Ro, and R x were also considered. No dis-
114
tinct trends were identified with either orbital period or Ro. However, we find evidence for
a strong correlation between radius deviations and R x (previously noted by López-Morales
2007) in contradiction with our Ro analysis. The trend is not as tight as that derived by
López-Morales (2007), owing to the age and metallicity variations allowed by our analysis. This correlation is also largely contingent upon the veracity of the distance estimate
to YY Gem. Accurately determining the distance to YY Gem and evaluating its association
with the Castor AB quadruple would alleviate much of the uncertainty.
Finally, we must not leave the theoreticians out of the spotlight. The degree to which a
magnetic field can alter the interior structure of low-mass stars is still only partially known
and further investigations are required. Development of models with self-consistent magnetic field perturbations will begin to shed light on this unknown. Comparisons between
predicted magnetic field strengths from self-consistent models and observational data (either direct or indirect) will provide a measure of the validity of the ability of magnetic fields
to inflate stellar radii. Whatever the final solution may be to this long-standing problem, it
is now apparent that the level of inflation required by theory is not a ubiquitous 5% – 15%,
but only so in extreme cases.
115
Chapter 5
Magnetic Perturbation within the
Framework of DSEP
Material presented in the chapter to follow provides mathematical derivations of the magnetic perturbations now contained within DSEP. The mathematical formalism and the general approach to the problem were originally presented by Lydon & Sofia (1995) and within
the context of DSEP in a publication in The Astrophysical Journal under the title “SelfConsistent Magnetic Stellar Evolution Models of the Detached, Solar-Type Eclipsing Binary
EF Aquarii.”
5.1 Introduction
Despite the identification of magnetic field as a potential culprit responsible for inflating
the radii of DEB components, only ad hoc procedures for treating the effects of magnetic
fields have been introduced (Mullan & MacDonald 2001; Chabrier et al. 2007). The method
examined by Chabrier et al. (2007) included artificially decreasing the convective mixing
length parameter, so as to mimic the effect of a global magnetic field within the star, as
116
well as artificially reducing the star’s bolometric flux in an effort to reproduce the effects of
photospheric spots. A second method, proposed by Mullan & MacDonald (2001), altered the
Schwarzschild criterion by perturbing the adiabatic gradient in a manner consistent with
the work of Gough & Tayler (1966).
Investigations by both groups appear to be at odds with one another. Chabrier et al. (2007)
and Morales et al. (2010) claim that starspots appear to be the dominate mechanism inflating stellar radii, and that modifications to convection require unrealistic magnetic field
strengths (i.e., reductions in the mixing length in their formulation). On the other hand,
Mullan & MacDonald (2001) and MacDonald & Mullan (2012) conclude the opposite that
reduction in convective efficiency is ultimately the dominant mechanism. Regardless of
which is really the dominant mechanism, both approaches are inherently ad hoc, yet both
are capable of reproducing the observed inflated stellar radii.
In this chapter, we introduce a self-consistent treatment of a globally pervasive magnetic
field embedded in the framework of the Dartmouth stellar evolution code (Dotter et al. 2007,
2008). Our approach follows the outline provided by Lydon & Sofia (1995), though we deviate from their method in a number of ways that are described below. All of the stellar
structure equations, including those in the equation of state, are self-consistently modified,
as opposed to arbitrarily altering a single quantity. In this way, modifications to the efficiency of thermal convection are accounted for in a more complete fashion, owing to the
full thermodynamic treatment of the magnetic field. Overall, the approach used to model
magnetic effects can be considered analogous to the parameterized mixing length treatment
of convection.1 The viability of the models is tested against results from a recent study that
characterized the DEB EF Aquarii (Vos et al. 2012).
EF Aquarii (HD 217512; henceforth EF Aqr) is a solar-type DEB found to contain two components displaying drastically inflated radii (Vos et al. 2012). Fundamental properties of the
1
In so far as reducing an inherently nonlinear, three-dimensional process into terms suitable for a onedimensional model.
117
Table 5.1
Fundamental stellar properties of EF Aquarii.
Property
EF Aqr A
EF Aqr B
M (M ⊙ )
1.244±0.008
0.946±0.006
R (R ⊙ )
Teff (K)
1.338±0.012
[Fe/H]
6150±65
0.956±0.012
0.00±0.10
5185±110
system are quoted in Table 5.1. What is most striking is the similarity of the secondary to
the Sun and the entire system to α Centauri A and B, in terms of the stellar masses and
composition. Although the secondary appears similar to both α Cen B and the Sun, its
radius appears to be about 10% larger than one would expect based on standard stellar evolutionary calculations. The effective temperature of the primary is nearly 300 K cooler than
standard models predict, further revealing that both components suffer from substantial
radius inflation.
In an effort to reconcile the observations with predictions from theoretical models, Vos et al.
(2012) reduced the value of the convective mixing length. They found αMLT of 1.30 and
1.05 were required for the primary and secondary, respectively, compared to their solarcalibrated value of 1.68. They concluded that fine-tuning the models allows for an accurate
description of the observed properties.
Reduction of the required convective mixing length may be physically motivated in two
ways: (1) naturally inefficient convection and (2) magnetically suppressed convection. While
we must be careful to not read too much into the reality of mixing length theory, in stellar
evolution models the mixing length is an intrinsic “property” of convection. Thus, reducing
the mixing length is akin to saying convection is not very efficient at transporting excess
energy.
Bonaca et al. (2012) calibrated the convective mixing length for solar-like stars using as-
118
teroseismic results provided by the Kepler Space Telescope. They found that the value of
αMLT in stellar models is tied to stellar properties (i.e., log g , log Teff , and [M /H]). Applying
the Bonaca et al. (2012) empirical calibration to the stars in EF Aqr, we find that the primary
and secondary component require αMLT = 1.68 and 1.44, respectively. Again, compared with
their solar calibrated value of αMLT = 1.68. The asteroseismically adjusted mixing lengths
are significantly larger than the fine-tuned values determined by Vos et al. (2012). Therefore, it appears that naturally inefficient convection is insufficient to explain the inflated
radii of EF Aqr. We are left with the option that magnetic fields may possibly be to blame.
In what is to follow, we describe a self-consistent approach to modeling the effects of a globally pervasive magnetic field with application to the EF Aqr system. A complete description
of the magnetic perturbation is provided in Sections 5.2 – 5.4. Section 5.5 demonstrates the
ability of the invoked perturbation to reconcile the models with observations through a case
study of the EF Aqr system. We conclude with a further discussion of the results and their
implications in Section 5.6.
5.2 Magnetic Perturbation
5.2.1 Magnetic Field Characterization
Investigating the effects of a global magnetic field on the interior structure of a star over
long time baselines, requires formulating a purely three-dimensional (3D) phenomenon in
terms suitable for a one-dimensional (1D) numerical model. Unfortunately, full 3D magnetohydrodynamic (MHD) models are not yet capable of modeling stellar magnetic fields over
the long time baselines required for stellar evolutionary calculations. This is in part due
to the rapid2 diffusion of the magnetic field and the immense computational time required.
Therefore, in order to probe the effects of a magnetic field, we seek to avoid directly solving
2 Relative
to a typical stellar lifetime.
119
the induction equation
∂B
= ∇ × (u × B) + η∇2 B.
∂t
(5.1)
While not actively seeking a solution to the full suite of 3D MHD equations, it is possible
to use the theoretical framework of MHD to provide a reasonably accurate 1D description
of a magnetic field and its associated properties. Ultimately, we are able to describe a
magnetic field in terms of the MHD equations and then project out the radial component,
the component necessary for stellar evolutionary model computations.
The spatial and temporal evolution of a given magnetic field are governed, quite naturally,
by Maxwell’s equations, presented here in Gaussian-cgs,
∇·E = 0
∇×E = −
(5.2)
1 ∂B
c ∂t
∇·B = 0
∇×B =
4π
J
c
(5.3)
(5.4)
(5.5)
where within the stellar plasma, we assume any regions of excess charge inducing an electric
potential will rapidly neutralize owing to the mobility of other charges (Debye shielding).
Thus, we can safely assume that the plasma is electrically neutral, ρ e = 0. For simplicity, we
here made another assumption, that temporal variations of the large-scale field are small,
suggesting that the conduction current dominates the displacement current.
Now, let us consider the interactions between the electric and magnetic fields within a
dense, ionized fluid moving with arbitrary velocity, u. For slow temporal evolution, nonrelativistic dynamics may be described by a single conducting fluid that obeys the classical
equations of hydrodynamics coupled with the equations of electromagnetism; the MHD
equations (Jackson 1999). Considering a perfectly conducting, non-viscous, non-rotating,
compressible fluid in the presence of a gravitational field, the MHD equations governing
120
the system are Ohm’s law for a moving fluid,
¶
µ
u×B
,
J = σ E+
c
(5.6)
∂ρ m
+ ∇ · (ρ m u) = 0,
∂t
(5.7)
the equation of mass continuity,
where ρ m is the mass density, and the fluid equation of motion,
ρm
du J × B
←
→
=
− ∇ · P + ρm g
dt
c
(5.8)
←
→
with g being the gravitational field vector and P representing the gas pressure tensor. The
electromagnetic term in the fluid equation of motion is associated with the assumption that
a magnetic field permeates the plasma. However, we have neglected forces associated with
any electric fields, for reasons detailed above.
Since, a priori, we have no knowledge of the current density within a given fluid, we replace
the current density within Equation (5.8) using Equation (5.5). The equation of motion may
now be written as
ρm
1
du
←
→
=
(∇ × B) × B − ∇ · P + ρ m g.
d t 4π
(5.9)
The magnetic force term may be rewritten with the help of the vector identity
∇(a · b) = (a · ∇)b + (b · ∇)a + a × (∇ × b) + a × (∇ × b),
which for a = b yields
1
− a × (∇ × a) = (a · ∇)a − ∇a 2
2
(5.10)
(5.11)
Knowing that magnetic fields are divergenceless by Equation (5.4), the momentum equation
121
becomes
ρm
1
1
du
←
→
=
(B · ∇)B −
∇B 2 − ∇ · P + ρ m g
d t 4π
8π
(5.12)
as a result. The equation of motion may be further simplified by recognizing that the magnetic force terms may be rewritten as the divergence of a tensor (Gurnett & Bhattacharjee
2005). Let us define the components of this tensor as
Tr,s = −
Br B s
B2
+ δr,s
4π
8π
(5.13)
where δr,s is the Kronecker delta. Then, in vector notation,
BB ←
→B2
←
→
+ I
T =−
4π
8π
(5.14)
←
→
with I representing the identity tensor. Taking the s component of the divergence of this
newly formed tensor,
¸
·
X ∂
B2
Br B s
←
→
− (∇ · T )s = −
+ δr,s
,
−
4π
8π
r ∂x r
(5.15)
¸
µ ¶
·
∂ B2
∂B r
∂B s
1 X
←
→
−
+ Bs
Br
.
− (∇ · T )s =
4π r
∂x r
∂x r
∂x s 8π
(5.16)
which becomes
Equation (5.4) imposes the condition that
∇·B =
X ∂B r
r
∂x r
= 0.
(5.17)
reducing Equation (5.16) to
¶
µ ¶
µ
∂ B2
1 X
∂B s
←
→
−
− (∇ · T )s =
Br
.
4π r
∂x r
∂x s 8π
122
(5.18)
In vector notation, the above equation is precisely the same as the Lorentz force term in the
equation of motion,
1
1
←
→
(B · ∇)B −
∇B 2 .
− (∇ · T )s =
4π
8π
(5.19)
The final form of Equation (5.8) can then be written in terms of the divergence of two
tensors,
ρm
³←
du
→ ←
→´
= −∇ · T + P + ρ m g.
dt
(5.20)
The magnetic tensor introduced above may be considered an anisotropic pressure tensor,
where the pressures it describes are intrinsic properties of the magnetic field. Further justification of this interpretation is provided below.
Stars, however, are considered to be in hydrostatic equilibrium, suggesting that the macroscopic variables of the system change only on timescales longer than the free-fall time
(Weiss et al. 2004). Therefore, the lefthand side of the equation of motion vanishes,
³←
→ ←
→´
∇ · T + P = ρ m g,
(5.21)
which is a statement of magnetohydrostatic equilibrium.
Let us now analyze the two components of the electromagnetic contribution to the equation
of motion. There are two terms associated with the Lorentz force. The first term,
−∇
B2
8π
(5.22)
is the force generated by a gradient in the square of the magnetic field strength. Drawing
an analogy with the gas pressure of the system (∇P gas ), we recognize the above term represents the force associated with the gradient of a “magnetic pressure.” This force is directed
isotropically from regions of strong magnetic field to regions of weak magnetic field.
123
The second term in the Lorentz force,
1
(B · ∇)B
4π
(5.23)
is directed opposite along the radius of curvature of the magnetic field line. This is analagous
to tension in a string and may be interpretted as the magnetic tension force. Together, the
magnetic pressure and tension exert a force on the plasma in which the magnetic field is
threaded. Since the magnetic field exerts a force on the plasma, there is an energy associated
with the magnetic field,
Umag =
B2
.
8π
(5.24)
It is now possible to consider how the introduction of a magnetic field would affect the
canonical stellar structure equations.
5.2.2 Stellar Structure Perturbations
At the most fundamental level, one-dimensional stellar evolution codes simultaneously
solve a set of four coupled, first-order differential equations.3 They are
dr
dm
dP
dm
dT
dm
dL
dm
1
4πr 2 ρ
Gm
= −
4πr 4
T
dP
=
∇s
P dm
dU P d ρ
= ǫ−
+ 2
dt
ρ dt
=
(5.25)
(5.26)
(5.27)
(5.28)
the equation of mass conservation, hydrostatic equilibrium, energy transport, and energy
conservation, respectively. Qualitatively, we can easily predict how these equations will be
3
There are additional equations often included to account for atomic diffusion. While included in DSEP,
we do not seek perturbations to these equations at the present time. See Mathis & Zahn (2005) for a rigorous
treatment of mixing associated with magnetic fields.
124
altered by the presence of a magnetic field which may then be translated into a quantitative
description.
The equation of mass conservation should be unaltered by any magnetic perturbation. Of
course, this is assuming that mass removed by stellar winds is negligible and that transient
events that may remove mass (i.e., flares, coronal mass ejections) are neglected. The stated
conditions hold for our approach, thus,
1
dr
=
.
d m 4πr 2 ρ
(5.29)
Note that we have dropped the subscript m on the density and assume all references to
density are specifically to the mass density, unless otherwise noted.
Hydrostatic equilibrium, as we saw earlier in Equation (5.21), is modified through the inclusion of the magnetic pressure and tension. Projecting out the radial component of the
magnetic pressures, we are able to adapt the three-dimensional concept for one-dimensional
models. Therefore, we have
µ 2 ¶¸
·
Gm
1
B
(B · ∇)B
dP
=−
−∇
+
· r̂.
4
2
dm
4πr
4πr ρ
4π
8π
(5.30)
The precise handling of the vector magnetic field within the code will be discussed later.
The final form of the energy transport equation is the same as if there were no perturbation.
Namely,
T
dP
dT
= ∇s
dm P dm
(5.31)
where ∇s is the local temperature gradient of the ambient plasma. Magnetic perturbations to
the stellar structure equations will self-consistently alter the temperature gradient through
various thermodynamic considerations. Of greatest importance will be the affects on the
treatment of convection. The full treatment will be discussed in the next section.
125
Finally, there are changes to the quantities present in the canonical equation of energy conservation in stellar evolution. Modifications to these quantities arise from the treatment of
the specific thermodynamic equations (discussed in the next section). Additionally, there are
terms that are purely electromagnetic in origin. The final form of the energy conservation
equation is
dU P d ρ Q ohm F Poynt
dL
= ǫ−
+ 2
+
+
.
dm
dt
ρ dt
ρ
ρ
(5.32)
Aside from the first three standard terms on the right-hand side, there are two additional
electromagnetic terms. First, there is a Poynting flux associated with the field,
F Poynt =
c
E × B,
4π
(5.33)
although as discussed above, we assume the E -field is zero everywhere. Next, energy is also
associated with the Ohmic dissipation of electric currents brought about by the resistive
nature of the plasma.
Q ohm ∝ I 2 R,
(5.34)
where I is the electric current and R is the resistance of the medium. Here, electrical
currents are converted to heat that then is transmitted to the surrounding plasma. Since
we have assumed an infinitely conducting plasma, this energy term goes immediately to
zero.
5.2.3 Thermodynamic Structure
The effects of a global magnetic field are introduced into the thermodynamic framework
supplied by DSEP following the approach outlined by Lydon & Sofia (1995, hereafter LS95).
Our aim in this section is to provide a detailed mathematical description of the magnetic
perturbation introduced by LS95. Places where we diverge from their original prescription
126
will be noted.
At the core of the LS95 method is the specification of a new thermodynamic state variable,
χ, such that
χ = χ(r, ρ) =
Umag
ρ
=
B (r )2
.
8πρ
(5.35)
The state variable χ is the magnetic energy per unit mass and B (r ) is the magnetic field
strength at radius r . Unlike LS95, our definition of χ depends on the radial distribution of
the magnetic field strength and also on the density of the stellar plasma. Originally, LS95
favored a mass-depth-dependent function, χ(Mr ). However, we moved away from this
prescription when we realized several thermodynamic derivatives became divergent. Once
the magnetic field strength is specified throughout the star, it is straightforward to calculate
χ at each point within the model.
The energy associated with the magnetic field arises due to forces exerted by the magnetic
field on the plasma. These forces are represented by the anisotropic pressure tensor presented in Equation (5.12). To handle the stress tensor in 1D, we convert the pressure tensor
to a scalar pressure, by taking the trace of the pressure tensor to yield the mean magnetic
pressure,
1 ³←
→´
〈P mag 〉 ∼ Tr T .
3
(5.36)
Substituting the expression for the magnetic stress tensor from Equation (5.14),
µ
¶
1
BB ←
→B2
〈P mag 〉 ∼ Tr −
+ I
.
3
4π
8π
(5.37)
Since we are not solving the full set of MHD equations, we look, instead, to set approximate
upper and lower limits on the scalar pressure. Assuming a Cartesian coordinate system, if
we imagine the magnetic field is parallel to the z -axis, or for a star, the rotational axis, then
127
we may expand the pressure tensor to read

0
0
 B 2 /8π


←
→ 
T =
0
B 2 /8π
0



0
0
−B 2 /4π + B 2 /8π





.



(5.38)
Note that there is an isotropic magnetic pressure associated with each diagonal element
along with the additional magnetic tension term in the final element. Since tension is directed along the field line, the tension exists in the z -direction only, in this instance. Taking
the trace, we find
µ
¶
1 B 2 3B 2
〈P mag 〉 ∼ −
+
3 4π 8π
(5.39)
since the negative tension term only acts along a single direction at any point along a given
field line while the pressure acts isotropically. Therefore,
µ ¶
1 B2
1
〈P mag 〉 ∼
= χρ.
3 8π
3
The above equation is satisified for a field which has a strong magnetic tension component.
However, if we assume that there is no tension at all, then, following the same procedure,
µ
¶
1 3B 2
= χρ.
〈P mag 〉 ∼
3 8π
These two approaches limit the strength of the scalar magnetic pressure within the onedimensional framework. Explicitly,
1
χρ ≤ 〈P mag 〉 ≤ χρ.
3
(5.40)
Defining a “geometry parameter,” akin to LS95, allows us to emulate the effects of having
128
a strongly curved field or a field with no curvature, and varying degrees between the two
extremes. This geometry parameter is defined such that we recover the average magnetic
pressure for each case above,
¡
¢
〈P mag 〉 = γ − 1 χρ ≡ P χ
(5.41)
where the geometry parameter
γ=


 2

 4/3
tension-free
.
(5.42)
maximum tension
In both cases, the appropriate expression for the scalar magnetic pressure is returned. With
the magnetic energy density and pressure formulated as scalars, we have successfully converted the inherently three-dimensional magnetic field into a one-dimensional magnetic
perturbation. In the process, we have also reproduced the scalar parameters originally presented by LS95.
As we have seen, magnetic fields carry along an associated pressure, tension, and energy
density. The introduction of magnetic terms into the equations of stellar structure then
necessitates the inclusion of these quantities in the EOS of the system. Using the definitions
introduced above for the magnetic energy density (χ) and the magnetic pressure (Pχ ) it is
possible to develop a magnetic version of the thermodynamics involved in standard stellar
evolution. Beginning with a statement of the first law of thermodynamics,
dQ = T d S = dU + P dV,
(5.43)
we recognize that each term contains both the standard gas and radiation terms as well as
129
a new magnetic contribution,
dQ = T (d S 0 + d S χ ) = (dU0 + dUχ ) + P 0 dV,
(5.44)
where no work is performed by the magnetic perturbation. Hereafter it is assumed that
any unsubscripted variables refers to the total quantity while gas and radiation are lumped
together under the subscript 0 (zero) convention and magnetic variables carry a subscript χ.
To write the first law in terms of the total pressure, we substract off the magnetic contribution,
£
¤
T d S = dU + P − (γ − 1)χρ dV.
(5.45)
Stellar evolution calculations are performed using specific quantities, meaning that ρ = V −1 .
So,
T d S = dU + P dV − (γ − 1)
χ
dV
V
(5.46)
Equation (5.46) is the new, “non-standard” first law of thermodynamics. It suggests that
¡
¢
V or ρ = f P, T, χ
(5.47)
¡
¢
U or S = f ρ, T, χ .
(5.48)
and
Following the derivation by LS95, explicitly writing out the other state variables illustrates
the effects of adding the magnetic perturbation. The total pressure is the sum of the individual contributions,
1
P = RρT + aT 4 + χρ(γ − 1),
3
(5.49)
where R is the specific gas constant and a = 4σ/c is the radiation constant. Rearranging
130
and solving for the density gives,
ρ=
P − aT 4 /3
.
RT + (γ − 1)χ
(5.50)
Additionally, the total specific internal energy of the system is the sum of the individual
energies,
3
aT 4
U = RT +
+ χ.
2
ρ
(5.51)
These thermodynamic quantities are subject to the EOS,
d ln ρ = αd ln P − δd ln T − νd ln χ.
(5.52)
The coefficients in the EOS are thermodynamic derivatives that complete the total derivative
d ln ρ ,
∂ ln ρ
α=
∂ ln P
µ
¶
(5.53)
,
T,χ
µ
∂ ln ρ
δ=−
∂ ln T
¶
,
(5.54)
∂ ln ρ
ν=−
∂ ln χ
¶
.
(5.55)
P,χ
and
µ
P,T
The coefficients δ and ν carry a negative sign in anticipation that without it, they would be
negative quantities. Note that α carries no subscript and should not be confused with the
convective mixing length parameter, αMLT .
An immediate consequence of altering the thermodynamic variables are the effects on the
specific heats. The specific heat at constant pressure,
µ
dQ
cP =
dT
¶
µ
dU
=
dT
P,χ
¶
µ
¶
£
¤ dV
+ P − (γ − 1)χρ
,
d T P,χ
P,χ
131
(5.56)
and at constant volume,
cV =
µ
dQ
dT
¶
V,χ
=
µ
dU
dT
¶
(5.57)
,
V,χ
The specific heats may be rewritten by deriving expressions for each of the terms in the
above equations. Assuming a small infinitesimal change in the total specific energy,
µ
∂U
dU =
∂V
¶
µ
∂U
dV +
∂T
T,χ
¶
µ
¶
∂U
dT +
∂χ
V,χ
(5.58)
d χ,
V,T
we can divide take the derivative with respect to temperature at constant pressure and
magnetic energy (d P = d χ = 0),
µ
dU
dT
¶
µ
∂U
=
∂V
P,χ
¶
T,χ
µ
dV
dT
¶
µ
∂U
+
∂T
P,χ
¶
(5.59)
.
V,χ
The difference between the two specific heats is then
c P − cV =
(µ
∂U
∂V
¶
T,χ
£
¤
+ P − (γ − 1)χρ
)µ
dV
dT
¶
(5.60)
.
P,χ
Simplifying the above equation is possible using the reciprocity relation (Weiss et al. 2004).
Starting with the first law, Equation (5.46),
T d S = dU + P dV − (γ − 1)
χ
dV
V
(5.61)
we may expand the internal energy term producing the equation,
µ
∂U
T dS =
∂V
¶
µ
∂U
dV +
∂T
T,χ
¶
µ
∂U
dT +
∂χ
V,χ
¶
V,T
d χ + P dV − (γ − 1)
χ
dV.
V
(5.62)
In Equations (5.62) and (5.58), d S and dU are total differentials. Therefore, we must have
∂2 S/(∂V ∂T ) = ∂2 S/(∂T ∂V ) and ∂2U /(∂V ∂T ) = ∂2U /(∂T ∂V ). Utilizing this fact, we may
132
substitute Equation (5.62) into Equation (5.58) to find
µ
∂U
∂V
¶
µ
∂P
=T
∂T
T,χ
¶
− P + χV −1 (γ − 1).
V,χ
(5.63)
This is the reciprocity relation mentioned earlier. With the reciprocity relation, the difference in specific heats may be simplified. Substituting Equation (5.63) into Equation (5.60),
µ
∂P
c P − cV = T
∂T
¶
V,χ
µ
∂V
∂T
¶
.
(5.64)
P,χ
Writing the EOS in Equation (5.52) in terms of the volume reveals that we may further
simplify Equation (5.64),
µ
¶
µ
¶
µ
¶
dP
d T ∂ lnV
dχ
∂ lnV
∂ lnV
dV
=
+
.
V
∂ ln P T,χ P
∂ ln T P,χ T ∂ ln χ P,T χ
(5.65)
At constant volume and magnetic energy,
µ
dP
dT
¶
µ
∂V
=−
∂T
V,χ
¶
P,χ
µ
∂V
∂P
¶−1
T,χ
=
Pδ
,
Tα
(5.66)
which yields our final equation for the difference in specific heats
c P − cV =
P δ2
.
ρT α
(5.67)
The difference in specific heats is written just as it would be without any magnetic perturbation. However, the individual quantities have been self-consistently modified to include
the contribution from the magnetic perturbation.
Knowledge of the specific heats enables the computation of the change in energy through a
133
change in heat within a given stellar layer. Recall,
dQ =
"µ
∂U
∂V
¶
T,χ
+ P − (γ − 1)χV
−1
#
µ
∂U
dV +
∂T
¶
V,χ
d T + d χ.
(5.68)
The change in heat may be simplified to read
µ
∂P
dQ = cV d T − T
∂T
¶
V,χ
dV + d χ,
(5.69)
which is further reducable using Equation (5.66) and by using the relation dV = ρ −2 d ρ ,
giving,
dQ = cV d T −
Pδ dρ
+ d χ.
ρα ρ
(5.70)
Rewriting the preceding equation using the EOS from Equation (5.52),
·
¸
dT
dχ
dP
Pδ
−δ
−ν
α
+ d χ,
dQ = cV d T −
ρα
P
T
χ
(5.71)
which can be rearranged to be
µ
µ
¶
¶
P δ2
Pδ dP
P δν
dQ = cV +
+
+ 1 d χ.
dT −
ρT α
ρ P
ραχ
(5.72)
Finally, recognize that the quantity multiplying d T is c P according to Equation (5.67), meaning the change in heat becomes
µ
¶
δ
P δν
dQ = c P d T − d P +
+ 1 d χ.
ρ
ραχ
(5.73)
An alternative form of this equation is used by the stellar evolution calculation, where the
infinitesimal changes are logarithmic quantites. To be explicit,
µ
¶
P δν
Pδ
d ln P +
+ χ d ln χ.
dQ = c P T d ln T −
ρ
ρα
134
(5.74)
We now have an equation for the change in heat within a stellar layer. However, the presence of a purely magnetic term must be considered. The introduction of the pure magnetic
term suggests that energy is removed from the internal energy of the system and converted
into magnetic energy. If we assume that magnetic phenomena are generated through a rotationally driven dynamo action, then we instead infer that rotational energy is the source of
pure magnetic term. Since our models do not specifically treat rotation, we must discard the
pure magnetic energy term when computing the change in heat in the stellar layer. Thus,
the change in heat to be considered in the stellar luminosity equation is
dQ = c P T d ln T −
Pδ
P δν
d ln P +
d ln χ,
ρ
ρα
(5.75)
or for the purposes of the derivations to follow,
P δν
δ
d χ.
dQ = c P d T − d P +
ρ
ραχ
(5.76)
This concludes the changes to the general thermodynamic structure of a stellar evolution
model. The rest of this section contains the effects a magnetic perturbation would have on
the equations for convective motion. The changes made are a direct consequence of the
changes made to the thermodynamic structure and are, thus, closely related. We preface
with the adiabatic temperature gradient. Adiabaticity requires constant entropy (no heat
exchange in the system) and constant magnetic energy,
d ln T
∇ad =
d ln P
µ
¶
.
(5.77)
S,χ
Therefore,
dQ = 0 = c P T d ln T −
135
Pδ
d ln P.
ρ
(5.78)
This can be easily solved to give the definition of the adiabatic gradient,
d ln T
∇ad =
d ln P
µ
¶
S,χ
=
Pδ
,
ρT c P
(5.79)
which has precisely the same form as the non-magnetic derivation.
5.2.4 Magnetic Mixing Length Theory
Convection is determined to occur in regions where a given fluid parcel is unstable to a
small displacement in the radial direction. The primary method of determining convective
stability is to analyze the density of a generalized fluid parcel. Parcels that are less dense
than their surroundings will travel radially outward until they reach a height within the
star at which the surrounding fluid has the same density as the parcel.
Upon reaching this point, the fluid parcel is assumed to fully mix with its surroundings
becoming indistinguishable from the rest of the fluid. Conversely, if a fluid element is more
dense than surrounding fluid, it will sink down to a greater depth in the star, following the
same trend as a rising convective element. In either case, gravity is the restoring force. So
long as the velocity of the unstable convective parcel is slower than the local sound speed,
the parcel will remain distinguishable from its surroundings. Namely, information about
pressure changes outside the parcel can be transmitted throughout the parcel sufficiently
fast to keep the parcel in constant pressure equilibrium.
The distance over which a fluid parcel travels before mixing is the well-known “mixing
length.” Mixing length theory (MLT) has been well established as a local means of prescribing convection for a one-dimensional stellar evolution code (Vitense 1953; Böhm-Vitense
1958). At locations where various differences in prescriptions of MLT occur, we will specify
our assumptions.
Stability of a fluid parcel against convection is determined by comparing the density of a
136
fluid parcel against its surroundings. For the rest of the document, let us use the definition
that
(5.80)
D A = Ae − A s
where e and s represent values for the fluid element and surroundings, respectively. Then,
Dρ =
·µ
dρ
dr
¶
µ
dρ
−
dr
e
¶¸
∆r.
(5.81)
s
To ensure stability, we require Dρ > 0, meaning the element is stable against small radial
displacements when
µ
dρ
dr
¶
µ
dρ
−
dr
e
¶
s
> 0.
(5.82)
Calling upon the EOS from Equation (5.52) allows us to expand the above equation. The
change in density of the element is
µ
¶
µ
¶
µ
¶
µ
¶
1 dρ
d ln P
d ln T
d ln χ
=α
−δ
−ν
ρ dr e
dr e
dr e
dr e
(5.83)
while the change in density of the surrounding material is
µ
¶
¶
¶
¶
µ
µ
µ
1 dρ
d ln P
d ln T
d ln χ
=α
−δ
−ν
ρ dr s
dr s
dr s
dr s
(5.84)
Included in the previous two equations is the radial gradient of the magnetic energy. We
can calculate how the background magnetic energy changes with radius, but it is difficult to
specify how the magnetic energy changes for the fluid parcel. This is because the parcel may
exchange magnetic energy with its surroundings. If the parcel carries its original magnetic
energy as it moves, then
µ
d ln χ
dr
¶
e
= 0.
(5.85)
On the other hand, if the magnetic energy in the parcel is always equal to that of its sur-
137
roundings, there must be a flux of χ as the parcel moves. Thus,
µ
d ln χ
dr
d ln χ
=
dr
e
¶
µ
¶
(5.86)
.
s
It is therefore advantageous to relate the spatial gradient of the magnetic energy of the
parcel to that of the surroundings by introducing a free parameter, f , such that
µ
d ln χ
dr
d ln χ
=f
dr
e
¶
µ
¶
(5.87)
s
where f varies between 0 and 1. Later, we shall attempt to physically motivate a fixed value
for f .
Substituting Equations (5.83), (5.84), and (5.87) into Equation (5.82) gives
d ln T
−δ
dr
d ln χ
+ (1 − f )ν
dr
e
µ
¶
µ
¶
d ln T
+δ
dr
s
µ
¶
s
>0
(5.88)
The pressure derivatives cancelled due to our assumption that the parcel is in pressure equilibrium with its surroundings. If we multiply the above equation through by the local
pressure scale height,4
HP = −
dr
d ln P
(5.89)
we end up with an expression for stability against convection,
d ln T
δ
d ln P
µ
d ln χ
− (1 − f )ν
d ln P
e
¶
µ
¶
d ln T
−δ
d ln P
s
µ
¶
s
>0
(5.90)
Before we go any further, let us now define three new gradient variables. The temperature
gradient of the ambient plasma,
d ln T
∇s ≡
d ln P
µ
4
¶
,
(5.91)
s
Other variations of MLT use the temperature or density scale height, but our assumption of pressure
equilibrium makes the pressure scale height a natural choice.
138
which is the same gradient that appears in Equation (5.31), the temperature gradient of the
displaced fluid parcel,
d ln T
∇e ≡
d ln P
µ
¶
(5.92)
,
e
and the gradient of the magnetic energy per mass unit,
d ln χ
∇χ ≡
d ln P
µ
¶
(5.93)
.
s
This final gradient we can expand to also read
d ln χ
∇χ =
dr
µ
¶ µ
s
dr
d ln P
¶
(5.94)
s
This allows us to write the stability criterion as
ν
∇e − (1 − f ) ∇χ − ∇s > 0.
δ
(5.95)
The stability criterion above can be simplified further by considering the temperature gradient of the element, ∇e . Our selection of the free parameter f is of importance. By assuming
f = 0, we are at liberty to assume complete adiabaticity because no energy is exchanged
between the fluid parcel and its surroundings. Therefore, ∇e → ∇ad . When f 6= 0, then
any heat transferred away from the parcel will be in the form magnetic energy (dQ = d χ).
Equation (5.73) then tells us that
P δν
δ
d χe .
0 = c P d Te − d P e +
ρ
ραχ
(5.96)
Multiplying through by the pressure scale height gives
µ
d ln T
d ln P
¶
µ
P δν d ln χ
Pδ
−f
=
,
T ρc P
T ραc P d ln P s
e
¶
139
(5.97)
which can be reduced using the gradient definitions,
∇e = ∇ad − f
ν
∇ad ∇χ .
α
(5.98)
Substituting this back into the stability criterion given by Equation (5.95) yields
∇ad − f
ν
ν
∇ad ∇χ − (1 − f ) ∇χ − ∇s > 0.
α
δ
(5.99)
This is the final form of the stability criterion. For a fluid parcel to be stable against convection, this criterion must be satisfied. Under the assumption that convection will occur
whenever a fluid parcel is convectively unstable, convection will occur when
³
ν ´
ν
∇s − ∇ad 1 − f ∇χ + (1 − f ) ∇χ > 0.
α
δ
(5.100)
Expressed compactly with the definition of ∇e ,
ν
∇s − ∇e + (1 − f ) ∇χ > 0
δ
(5.101)
This completes our discussion of convective stability. However, it requires that we know
the value of ∇s , the temperature gradient of the surrounding material. In principle, ∇s is not
known a priori.
We may not know ∇s at any given point in a stellar model, but we do know the total
flux of energy that must be transported across any given layer. Stars are finely balanced.
The energy generated by gravitational contraction or nuclear fusion in the core must be
transferred to the surface so that it may be radiated away. We can use this to our advantage
and solve for the ambient temperature gradient.
Imagine a fictitious gradient, ∇rad , that represents what the background temperature gradient would have to be to transport all of the energy by radiation. The total energy flux is
140
then
F tot =
4acG T 4 M r
∇rad .
3 κP r 2
(5.102)
Here, a is the radiation constant, G is Newton’s gravitational constant, c is the speed of light
in a vacuum, and κ is the opacity of the plasma. Additionally, Mr is the mass contained
within the radius r . However, if convection is present
F tot =
4acG T 4 M r
∇s + F conv .
3 κP r 2
(5.103)
The convective flux may be considered a flux of enthalpy
D H = DU + P DV + V DP
(5.104)
because it is not only transporting excess internal energy, but is also doing work on the
surroundings. It has to “push” material out of the way to move. The last term in Equation
(5.104) is zero due to the assumed pressure equilibrium. Therefore,
D H = DU + P DV = DQ.
(5.105)
We may then rationalize that the convective flux is
F conv = ρu conv DQ,
(5.106)
where uconv is the velocity of the convecting element. From Equation (5.73), we may rewrite
DQ as
µ
¶
P δν
DQ = c P DT +
+ 1 Dχ,
ραχ
(5.107)
·
µ
¶
¸
P δν
F conv = ρu conv c P DT +
+ 1 Dχ
ραχ
(5.108)
which yields,
141
In the original formulation by LS95, they separated out the pure magnetic flux term,
(5.109)
F conv, χ = ρu conv Dχ,
which represents the magnetic energy transported by a convecting element when f 6= 1.
Since it is not presently clear what the value of f should be, we leave the pure magnetic
flux in the equation.
We now introduce the concept of a mixing length to aid our evaluation of the temperature
excess, DT , and magnetic energy excess Dχ of the convecting element. After the parcel
travels a distance ℓm (= αMLT HP ), we assume the fluid parcel completely mixes with the
surrounding material and deposits its excess energy. The parcels crossing any given layer
within a convection zone will not have necessarily started their ascent or descent at the same
location. Therefore, we assume that a typical parcel started at a distance ℓm /2 from the
point under consideration (Böhm-Vitense 1958). The results of MLT are relatively insensitive
to this choice (Henyey et al. 1965).
The average temperature excess taking into account all of the various parcels that may be
dissolving at a given location may be written,
D ln T =
1 ∂(DT )
1 ∂(DT ) ℓm
∆r =
,
T ∂r
T ∂r
2
(5.110)
where,
∂
∂(D ln T )
=
∂r
∂r
·½µ
d ln T
dr
¶
e
−
µ
d ln T
dr
¶¾
s
¸
∆r .
(5.111)
Multiplying the terms inside the curly braces by 1 = −(d r /d ln P )HP−1 ,
·
¸
∂(D ln T )
∆r
∂
=
,
(∇s − ∇e )
∂r
∂r
HP
142
(5.112)
which implies that
1
∂(D ln T )
= (∇s − ∇e )
.
∂r
HP
(5.113)
The final expression for the temperature excess D ln T is then
D ln T = (∇s − ∇e )
ℓm
.
2HP
(5.114)
Evaluating the magnetic energy excess in a similar manner gives
D ln χ = (1 − f )∇χ
ℓm
2HP
(5.115)
The above equation nicely illustrates that if f = 1, the magnetic energy excess between an
element and its surroundings is zero because they are continually transferring energy to one
another.
The rate at which the excess energy is deposited by a convective parcel is also governed
by the convective velocity uconv . An expression for uconv may be derived knowing that
buoyancy is the restoring force acting on the parcel. The radial buoyancy force per unit
mass is
k r = −g D ln ρ.
(5.116)
Once again, Equation (5.52) can be used to substitute for D ln ρ , again accounting for pressure equilibrium. Thus,
¡
¢
k r = g δD ln T + νD ln χ
(5.117)
It is now possible to calculate the work done on the fluid parcel. Over half a mixing length,
the work done is
µ
¶
£
¤ ℓm
1
ℓm
kr
.
= g δD ln T + νD ln χ
2
2
4
143
(5.118)
Using the definition of the energy excesses derived above,
µ
¶
h
i ℓ2
ν
ℓm
1
m
kr
.
= g δ (∇s − ∇e ) + (1 − f )∇χ
2
2
δ
8HP
(5.119)
Now make the crude assumption that half of the energy derived from the buoyancy force
is converted into kinetic energy of the fluid parcel. The other half is transferred to the
surroundings because the parcel has to move the surrounding material out of the way. The
convective velocity may then be written,
h
i ℓ2
ν
m
2
u conv
= g δ (∇s − ∇e ) + (1 − f )∇χ
δ
8HP
(5.120)
The addition of the convective velocity does not yet close the system of equations presented
thus far. We need yet another equation, which we can develop by account for the fact that
as the parcel travels, it does so non-adiabatically.
There are three ways for the parcel to change temperature: adiabatic expansion (or contraction), radiative losses, and the flux of magnetic energy. We start Equation (5.73),
µ
dQ
dr
¶
e
= cP
µ
dT
dr
¶µ
µ
¶ µ
¶
dχ
δ dP
P δν
+1
−
+
.
ρ dr e
ραχ
dr e
e
¶
(5.121)
where the change in heat on the left-hand-side is now the sum of the radiative losses and
the change of magnetic energy. In presenting the terms associated with radiative losses, we
will not provide a full derivation, but only cite the final form. The radiative losses for a
semi-opaque bubble is taken to be the average between the flux for a transparent bubble
and that of a completely opaque bubble (Henyey et al. 1965; Mihalas 1978). The resulting
expression is,
¸
·
2acT 4
ω
(∇s − ∇e ),
(radiative losses) = −
ρu conv HP 1 + ÿω2
(5.122)
where ÿ is a constant describing the shape and temperature distribution of the convective
144
parcel (Henyey et al. 1965) and ω = κρℓm , related to the mean free path of a photon. The
change of magnetic energy is given by Equation (5.87). Combining the radiative loses and
magnetic energy transfer between the parcel and the surrounding medium,
¸
µ
¶µ
¶
¶
µ
¶ µ
¶
·
µ
dχ
dχ
2acT 4
δ dP
P δν
ω
dT
(∇s − ∇e ) +
+1
−
= cP
−
+
.
ρu conv HP 1 + ÿω2
dr e
dr e ρ dr e
ραχ
dr e
(5.123)
The preceding equation is easily reducible by recognizing the appearance of the pure magnetic terms on each side cancel out,
¸
µ
¶
µ
¶ µ
¶µ
¶
·
dT
δ dP
2acT 4
P δν d χ
ω
(∇s − ∇e ) = c P
−
+
.
−
ρu conv HP 1 + ÿω2
dr e ρ dr e
ραχ d r e
(5.124)
To yield a better form for the equation of heat exchange between the parcel and the surroundings, we multiply by HP T −1 = −T −1 (d r /d ln P ) to get
¸
µ
¶
·
d ln T
P δν
ω
Pδ
2acT 3
(∇s − ∇e ) = c P
+f
∇χ
−
2
ρu conv 1 + ÿω
d ln P e T ρ
ραT
(5.125)
If we now divide by c P
¸
·
2acT 3
Pδ
P δν
ω
(∇s − ∇e ) = ∇e −
+f
∇χ
2
ρu conv c P 1 + ÿω
T ρc P
ραT c P
(5.126)
¸
·
ν
ω
2acT 3
(∇s − ∇e ) = (∇e − ∇ad ) + f ∇ad ∇χ .
2
ρu conv c P 1 + ÿω
α
(5.127)
which becomes
We now have a final set of five equations which we can use to solve for the temperature
gradient of the system. The total flux of energy transported by a fictitious radiative energy
gradient (Equation 5.102),
F tot =
4acG T 4 M r
∇ad ,
3 κP r 2
145
the total flux transported by a combination of radiation and convection (Equation 5.103),
F tot =
4acG T 4 M r
∇s + F conv ,
3 κP r 2
the convective flux (Equation 5.108),
¶
¸
·
µ
P δν
+ 1 Dχ ,
F conv = ρu conv c P DT +
ραχ
the convective velocity (Equation 5.120),
h
i ℓ2
ν
m
2
u conv
= g δ (∇s − ∇e ) + (1 − f )∇χ
,
δ
8HP
and the exchange of heat between the parcel and the surrounding as the parcel moves (Equation 5.127),
¸
·
ν
ω
2acT 3
(∇s − ∇e ) = (∇e − ∇ad ) + f ∇ad ∇χ .
2
ρu conv c P 1 + ÿω
α
Using these five equations, we are in a position to solve for the temperature gradient of the
ambient medium.
Solution of the Magnetic Mixing Length Theory Equations
Solving the system of five equations presented above is quite tedious. If we first define
several new variables, the process algebra becomes slightly more clean. First, we set the
value of the parcel shape parameter, ÿ = 1/3, as is done in the standard version of MLT
employed by our code (Böhm-Vitense 1958; Henyey et al. 1965). The new variables are
defined in anticipation of the fun to come,
∂ ln ρ
Q =δ=−
∂ ln T
µ
146
¶
,
P,χ
(5.128)
γ0 =
c P ρ 1 + ω2 /3
,
2acT 3
ω
C=
gQℓ2m
8HP
,
(5.129)
(5.130)
¸
·
³
ν ´1/2 −1
ν
1/2
V = γ0C
,
∇rad − ∇ad + f ∇ad ∇χ + (1 − f ) ∇χ
α
δ
(5.131)
9 ω2
,
A=
8 3 + ω2
(5.132)
y = u convV γ0
(5.133)
and
With these new variables defined, we will work towards a solution for the equations in a
series of seven steps. Each step will be labeled in bold lettering.
Step 1: Rewrite the convective velocity (Equation 5.120) in terms of the new variables.
µ
y
V γ0
¶2
¸
·
ν
= C ∇s − ∇e + (1 − f )∇χ
Q
(5.134)
Step 2: Equation (5.127), the heat exchange equation, may be redefined with the new variables,
ν
1
(∇s − ∇e ) = (∇e − ∇ad ) + f ∇ad ∇χ .
γ0 u conv
α
(5.135)
This enables us to derive an expression for ∇e . Replace (γ0 uconv )−1 with (V /y) using Equation (5.133) and distribute the first term,
V
ν
V
∇s − ∇e = (∇e − ∇ad ) + f ∇ad ∇χ ,
y
y
α
(5.136)
and then combine like-terms for the gradients,
¶
µ
³
V
V
ν ´
∇s + ∇ad 1 − f ∇χ = ∇e 1 +
,
y
α
y
147
(5.137)
and divide by (V /y),
∇s +
³
³
ν ´
y´
y
∇ad 1 − f ∇χ = ∇e 1 +
.
V
α
V
(5.138)
This now defines the temperature gradient of the convecting parcel as
∇e =
¡
¢
∇s + ∇ad 1 − f ν∇χ /α y/V
(5.139)
1 + y/V
Step 3: Find an expression for (∇s − ∇ad ) by substituting the Equation (5.139) into Equation
(5.134). We start by expanding Equation (5.134),
µ
y
γ0V
¶2
= C ∇s −C ∇e +C
ν
(1 − f )∇χ ,
Q
(5.140)
which becomes, upon substitution,
µ
y
γ0V
¶2
= C ∇s −C
Ã
!
¡
¢
∇s + ∇ad 1 − f ν∇χ /α y/V
1 + y/V
+C
ν
(1 − f )∇χ .
Q
(5.141)
Add the first two terms on the right hand side,
µ
y
γ0V
¶2
=
"
#
¡
¢
C ∇s +C ∇s (y/V ) −C ∇s −C ∇ad 1 − f ν∇χ /α y/V
1 + y/V
+C
ν
(1 − f )∇χ . (5.142)
Q
Cancel equal terms and divide by C ,
#
"
¡
¢
µ
¶
∇s (y/V ) − ∇ad 1 − f ν∇χ /α y/V
ν
y 2
1
+ (1 − f )∇χ .
=
C γ0V
1 + y/V
Q
(5.143)
Rearrange the terms such that we have (∇s − ∇ad ) in the numerator and bring the magnetic
contribution to the left hand side,
·
µ
¶
¸
(∇s − ∇ad )(y/V ) + ( f ν∇ad ∇χ /α)(y/V )
1
y 2 ν
.
− (1 − f )∇χ =
C γ0V
Q
1 + y/V
148
(5.144)
Divide both sides of the equation by (y/V ) to get,
µ ¶ ·
¸
(∇s − ∇ad ) + ( f ν∇ad ∇χ /α)
ν
V
− (1 − f )∇χ
=
.
y
1 + y/V
C γ20V Q
y
(5.145)
Finally, solve for (∇s − ∇ad ),
"
µ ¶# ³
ν
V
ν
y´
(∇s − ∇ad ) =
−
(1
−
f
)∇
− f ∇ad ∇χ .
1
+
χ
2
y
V
α
C γ0V Q
y
(5.146)
Step 4: Rewrite Equation (5.108) in terms of the convective velocity. Begin with Equation
(5.120) and solve for (∇s − ∇e ),
2
u conv
¸
·
ν
= C (∇s − ∇e ) + (1 − f )∇χ .
Q
(5.147)
Therefore,
(∇s − ∇e ) =
2
u conv
C
−
ν
(1 − f )∇χ .
Q
(5.148)
Now, using Equations (5.114) and (5.115) rewrite the convective flux (Equation 5.108),
·
µ
¶
¸
P δν
ℓm
F conv = ρu conv c P T (∇s − ∇e ) +
+ 1 (1 − f )χ∇χ
.
ραχ
2HP
(5.149)
Expanding the above equation and recalling the definition of ∇ad from Equation(5.79),
i ℓ
h
ν
m
.
F conv = ρu conv c P T (∇s − ∇e ) + (1 − f )∇ad ∇χ + (1 − f )χ∇χ
α
2HP
(5.150)
Replacing (∇s − ∇e ) with Equation (5.148),
F conv = ρu conv c P T
·
2
u conv
C
¸
ν
ν
ℓm
− (1 − f )∇χ + (1 − f )∇ad ∇χ + (1 − f )χ∇χ
,
Q
α
2HP
149
(5.151)
which rearranges to read
F conv = ρu conv c P T
·
2
u conv
C
+ ν(1 − f )∇χ
µ
∇ad 1 χ
− +
α
Q ν
¶¸
ℓm
,
2HP
(5.152)
Step 5: Substitute Equation (5.152) into Equation (5.103) to derive an expression for the total
flux,
· 2
µ
¶¸
u conv
4acG T 4 M r
∇ad 1 χ
ℓm
F tot =
∇s + ρu conv c P T
+ ν(1 − f )∇χ
− +
.
2
3 κP r
C
α
Q ν 2HP
(5.153)
Distribute the terms associated with the convective flux to produce,
¶
µ
3
u conv
ℓm
∇ad 1 χ ℓm
4acG T 4 M r
∇s + ρc P T
− +
+ ρu conv c P T ν(1 − f )∇χ
.
F tot =
3 κP r 2
C 2HP
α
Q ν 2HP
(5.154)
Since it will be advantageous going forward, let us define
¶
∇ad 1 χ
− +
∇χ .
Ξ ≡ ν(1 − f )
α
Q ν
(5.155)
3
u conv
4acG T 4 M r
ℓm
ℓm
F tot =
∇s + ρc P T
+ ρu conv c P T Ξ
.
2
3 κP r
C 2HP
2HP
(5.156)
µ
Therefore,
The step involves equation Equations (5.102) and (5.156) and rearranging the first term on
the right,
3
u conv
ℓm
4acG T 4 M r
ℓm
(∇
−
∇
)
=
ρc
T
+
ρu
c
T
Ξ
.
s
P
conv
P
rad
3 κP r 2
C 2HP
2HP
(5.157)
Next, we know that g = G Mr /r 2 , HP = P /(ρg ), and ω = κρℓm . This allows us to write,
u3
4acT 4 ℓm
ℓm
ℓm
(∇rad − ∇s ) = ρc P T conv
+ ρu conv c P T Ξ
.
3HP
C 2HP
2HP
150
(5.158)
Dividing by T ℓm /HP gives,
ρc P 3
ρc P
4acT 3
(∇rad − ∇s ) =
u conv +
Ξu conv .
3
2C
2
(5.159)
Isolating the gradient difference on the left hand side,
(∇rad − ∇s ) =
3 ρc P
3 ρc P
3
u conv
+
Ξu conv ,
3
4C 2acT
4 2acT 3
(5.160)
which allows us to rewrite the equation in terms of γ0 ,
(∇rad − ∇s ) =
³
´
´
3
ω
ω
3 ³
3
γ0
γ
u
+
Ξu conv .
0
4C
1 + ω2 /3 conv 4
1 + ω2 /3
(5.161)
Pulling the factor of 1/3 out of the term in parenthesis,
(∇rad − ∇s ) =
³ ω ´
9 ³ ω ´
9
3
γ0
γ0
u
Ξu conv ,
+
4C
3 + ω2 conv 4
3 + ω2
(5.162)
which is ripe in the making for using one of our predefined variables, A ,
(∇rad − ∇s ) =
2A
3
γ0 u conv
+ 2Aγ0 Ξu conv .
C
(5.163)
Finally, substitute for uconv using Equation (5.133) and simplify to find
(∇rad − ∇s ) =
2Ay 3
C γ20V 3
+
2Ay
Ξ.
V
(5.164)
Before we proceed to the next step in the solution, it is worth noting that our result for
Equation (5.164) is not the same as that of LS95. Whereas there are three terms incorporated
in our variable Ξ, LS95 have only a single term. One of the extra terms we picked up was
due to LS95 neglecting the pure magnetic term in the convective flux, as we mentioned
earlier. The second term, however, was spuriously dropped by LS95.
151
Step 6: This step is rather unintuitive, but we want to rewrite Equation (5.164) in terms of
quantities that we have previously derived or defined. Thus, we essentially add zero to the
left hand side by adding
³
ν
ν ´ ³
ν ´
ν
0 = ∇ad − f ∇ad ∇χ − (1 − f ) ∇χ − ∇ad − f ∇ad ∇χ − (1 − f ) ∇χ .
α
δ
α
δ
(5.165)
We then have that
¢ ¡
¢¤
£¡
C γ20V 3 ∇rad − ∇ad + αν f ∇ad ∇χ + (1 − f ) νδ ∇χ − ∇s − ∇ad + αν f ∇ad ∇χ + (1 − f ) νδ ∇χ
= 2Ay 3 + 2AC γ20V 2 yΞ .
(5.166)
Utilizing the definition for our variable V ,
³
ν
ν ´
V −C γ20V 3 ∇s − ∇ad + f ∇ad ∇χ + (1 − f ) ∇χ = 2Ay 3 + 2AC γ20V 2 yΞ
α
δ
(5.167)
Step 7: The final step in our solution involves rewriting Equation (5.146) and substituting it
into Equation (5.167). First we write
µ ¶
ν
ν
ν
V
y2
−
(1
−
f
)∇
−
(1
−
f
)∇
−
f
∇ad ∇χ
+
(∇s − ∇ad ) =
χ
χ
y
α
C γ20V δ
C γ20V 2 δ
y
(5.168)
and then substitute into Equation (5.167) to get
V −C γ20V 3
"
#
µ ¶
ν
y2
V
− (1 − f )∇χ
+
= 2Ay 3 + 2AC γ20V 2 yΞ.
y
C γ20V δ
C γ20V 2
y
(5.169)
Distribute the terms on the left hand side so we have that
C γ2V 4
ν
− V y 2 = 2Ay 3 + 2AC γ20V 2 yΞ.
V − V 2 y + (1 − f )∇χ 0
δ
y
152
(5.170)
Then multiply by y ,
ν
V y − V 2 y 2 + (1 − f )∇χC γ20V 4 − V y 3 = 2Ay 4 + 2AC γ20V 2 y 2 Ξ.
δ
(5.171)
If we perform a final rearrangement of the terms to set the equation equal to zero,
ν
2Ay 4 + V y 3 + (2AC γ20 Ξ + 1)V 2 y 2 − V y − (1 − f )∇χC γ20V 4 = 0.
δ
(5.172)
This is the final quadratic equation for which we need a solution. Standard MLT, in contrast to the equation above, requires finding the root of a cubic equation and lacks the
first quadratic and the zeroth order coefficients (Böhm-Vitense 1958; Henyey et al. 1965;
Weiss et al. 2004). If we are able to obtain a solution for y , then we know the rest of the
variables in the system, including the convective velocity, uconv , and the temperature gradient, ∇s .
To find the root of Equation (5.172), we make an educated guess as to the solution for y .
A series of Newton-Raphson corrections are used to converge to the proper solution with a
convergence tolerance of 10−10 selected to reduce the propagation of large numerical errors.
There are several intelligent first guesses for y , as outlined by LS95. We have found that
¤−1
£
y = 2AC γ20 Ξ + 1
(5.173)
provides the quickest convergence in our code, typically converging to a solution in approximately 10 iterations.
One might notice that our definition of f can greatly reduce the complexity of Equation
(5.172). The selection of f , as you might recall, defines the transfer of magnetic energy
from the parcel to the surroundings or vice versa. We may interpret this parameter as
determining whether magnetic flux tubes are frozen into the fluid or whether they will,
instead, efficiently diffuse through the medium.
153
5.2.5 The Parameter f and the Frozen Flux Condition
In Section 5.2.1 we specified that the plasma under consideration was perfectly conducting
and, thus, had zero resistivity. One consequence of assuming an ideal MHD plasma is
that magnetic field lines become physical objects that are transported by the plasma, the
so-called frozen flux condition (FFC; Alfvén 1942). The magnetic flux is
Φ(t ) =
Z
S(t )
B(r, t ) · d Â.
(5.174)
For an ideal plasma, the evolution of Φ in time is dependent upon not only the time rate of
change of the magnetic field, B(r, t), but also on any distortion occurring to the bounding
surface, S(t ), as the plasma moves. The net effect is that the time rate of change of the
magnetic flux is equal to zero. Therefore, we must have
∂B
= 0,
∂t
(5.175)
which may be rewritten using Equation (5.1), the induction equation, with η = 0. This
results in the well-known FFC condition,
∇ × (u × B) = 0.
(5.176)
The FFC enforces the restriction that, for a spherically symmetric bubble of plasma undergoing isotropic expansion or contraction (Kulsrud 2004),
B
ρ 2/3
= constant.
(5.177)
Applying this constraint to the magnetic energy gradient definition of a convecting fluid
element (Equation 5.87), we are able to physically motivate the definition of the parameter
f which governs the flux of energy between the fluid element and the surrounding material.
154
Imagine a region in a star where a small bubble begins to grow convectively unstable.
Initially, the bubble has the same properties as the surrounding fluid. It is only because of
the change in density that other properties begin changing as well. The FFC allows us to
write the magnetic energy contained within a fluid parcel as a function of the magnetic
energy of the surrounding material. Since a convecting fluid parcel has a slight under- or
overdensity compared to its surroundings, we perturb the element’s density
ρe = ρ s + ξ
(5.178)
where it is understood that ξ ≪ ρ s . We also drop the subscript s hereafter. Assuming,
for simplicity, that the bubble expands isotropically, the magnetic field strength within a
convectively unstable bubble is
µ
¶
¢2/3
ξ 2/3
B ¡
= B 1+
B e = 2/3 ρ + ξ
ρ
ρ
(5.179)
meaning the magnetic energy per mass may be written as
µ
¶
B2
ξ 1/3
χe =
=
1+
.
8π(ρ + ξ) 8πρ
ρ
B e2
(5.180)
We now have a direct relation between the magnetic energy density of the convecting fluid
element and the surrounding material,
¶
µ
ξ 1/3
.
χe = χs 1 +
ρ
(5.181)
Taking the radial, logarithmic derivative,
µ
d ln χ
dr
d ln χ
=
dr
e
¶
µ
¶
µ
¶·
¸
1
ξ
d ln ξ d ln ρ
−
+
.
3 ρ +ξ
dr
dr
s
(5.182)
The first of the bracketed terms goes to zero, as the density perturbation is independent of
155
radial location. Using the definition of χs (Equation 5.35) to expand the derivative, and after
rearranging the resulting terms, the equation becomes
µ
d ln χ
dr
µ 2¶ µ
¶·
µ
¶¸
B
d
d ln ρ
1
ξ
ln
=
−
1+
.
dr
8π
dr
3 ρ +ξ
e
¶
(5.183)
By definition, we know that ξ/ρ ≪ 1, meaning the perturbation term in the square brackets
is negligible. As such,
µ
d ln χ
dr
d ln χ
≈
dr
e
¶
µ
¶
(5.184)
s
allowing us to conclude that the FFC implies that f ≈ 1.
5.3 Implementation in the Dartmouth Code
In the previous sections, we developed the mathematical framework necessary to include
the effects of a globally pervasive magnetic field within a 1D stellar evolution code. Here,
we briefly outline some of the technical details of the actual implementation.
5.3.1 Magnetic Field Strength Distribution
The strength of the perturbations described in the preceding sections are determined by the
magnitude and spatial gradient of χ. Mentioned in Section 5.2.3, was that we deviate from
the prescription of χ proposed by LS95. Instead of defining
·
µ
¶ ¸
1 M D − M Dc 2
χ = χmax exp −
,
2
σ
(5.185)
¶¸
· µ
Mr
,
M D = log10 1 −
M∗
(5.186)
where
156
1
14
Magnetic Field Strength
Tachocline Location
12
10
0.6
8
0.4
|B| (kG)
Radius Fraction
0.8
6
4
0.2
2
0
0
0.2
0.4
0.6
Radius Fraction
0.8
1
0
Figure 5.1: Magnetic field strength profile for a 1.0 M⊙ star with a 5.0 kG photospheric
magnetic field strength (maroon, solid). The green dash-dotted line indicates the location of
the stellar tachocline, the interface between the radiative and convective regions. The plot
is meant only to illustrate the field strength profile. A small gap in the field strength profile
is barely perceptible near the surface of the star. This artifact is due to the separation of the
stellar interior and envelope integration regimes in the code.
we opt to directly prescribe the radial magnetic field profile. Approaching the problem in
this manner introduces the difficulty of selecting a particular radial profile. Without any
real confidence of the radial profile inside stars, we are left to our own devices.
One of the simplest profiles to select is that of a dipole configuration, where the field strength
drops off as r −3 from the magnetic field source location. This is illustrated in Figure 5.1. The
radial profile may then be prescribed as
B (R) = B (R tach ) ·


 (R tach /R)3 R > R tach

 (R/R tach )3 R < R tach
(5.187)
with the peak magnetic field strength defined to occur at the radius Rtach . The radial location described by R tach is the location of the stellar tachocline, an interface between the
157
convective envelope and radiative core. This interface region is thought to be characterized
as a strong shear layer where the differentially rotating convection zone meets the radiative
core rotating as a solid body. Theory suggests that the tachocline is the source location for
the standard mean-field stellar dynamo (i.e., the α–ω dynamo; Parker 1975).
Since DSEP monitors the shell location of the boundary to the convection zone, the tachocline
appeared to be a reasonable location, both theoretically and numerically, to base the scaling of the magnetic field strength. However, defining the magnetic field strength at the
tachocline (B (Rtach )) is required. In an effort to allow for direct comparisons between field
strengths input into the code and observed magnetic field strengths, the field strength at the
tachocline is anchored to the photospheric (surface) magnetic field strength,
B (R tach ) = B surf
µ
R∗
R tach
¶3
.
(5.188)
where B surf = B (R ∗ ) is introduced as a free parameter. The advantage of B surf as a free
parameter is that it has potential to be constrained observationally.
Fully convective stars do not possess a tachocline. However, a dynamo mechanism still has
the potential to drive strong magnetic fields through an α2 mechanism (Chabrier & Küker
2006). Full 3D MHD modeling suggests that the magnetic field strength peaks at about
0.15 R ∗ (Browning 2008). Unfortunately, the adopted micro-physics were solar-like and may
not be entirely suitable for fully convective M-dwarfs. Regardless of these shortcomings,
Browning’s investigation provides the only estimate, to date, for the location of the peak
magnetic field strength in fully convective stars. As such, we adopt Rtach = 0.15R ∗ as the
dynamo source location in our models of fully convective stars.
158
5.3.2 Numerical Implementation
Although we have laid out the mathematical construction of the magnetic perturbation,
we have yet to illuminate precisely how the perturbation is treated numerically. When a
magnetic model is first executed, the user provides a surface magnetic field strength, the
geometry parameter γ, and the age at which the magnetic perturbation will occur. The
model proceeds to evolve the same as a standard model until the initial perturbation age is
reached.
Once the perturbation age is reached, the magnetic field strength profile is prescribed based
on the assumed photospheric field strength and the location of the tachocline, as in Figure 5.1. The magnetic energy and magnetic pressure are then computed for each of the
model’s mass shells. Here, the total pressure associated with each mass shell is also perturbed.
Following the introduction of the perturbation, the code must recompute the structure of
the stellar envelope, which is separate from the stellar interior integration. The envelope
comprises the outer 2%–3%, by radius, of the stellar model. Surface boundary conditions are
determined prior to the envelope integration by interpolating within the phoenix model
atmosphere tables using log g and Teff . This interpolation returns P gas at the surface of
the star, and defines the start of the inward integration. The magnetic perturbation is then
explicitly included in the calculation of the analytic EOS.
This leads into the convection routines, where the non-standard stability criterion in Equation (5.100) is evaluated and judged. Either the equations of magnetic MLT are solved, or
the radiative gradient is selected. The envelope integration scheme proceeds until it reaches
a pressure commensurate with the pressure for the interior regime.
From the newly calculated envelope, the interior integration begins using a Henyey integration scheme (Henyey et al. 1964) with the magnetic perturbation implemented. The EOS
159
log(R/R⊙)
0.08
Non-mag
2.0 kG
0.5 kG
3.0 kG
3.0 kG
Non-mag
1.0 kG
0.04
0.00
-0.04
-0.08
(a)
ZAMS
Pre-ZAMS
0.08
log(R/R⊙)
(b)
Post-ZAMS
Non-mag
500 Models
150 Models
Non-mag
0.04
0.00
-0.04
-0.08
(c)
0.01
(d)
0.1
1
0.01
Age (Gyr)
0.1
1
Age (Gyr)
Figure 5.2: Tests of numerical stability for a 1 M⊙ magnetic model with γ = 2. (a) Influence of various magnetic field strengths. (b) Evidence of a smooth perturbation at a large
magnetic field strength with a relatively large radius change. (c) Consistency among models
with the perturbation turned on at various evolutionary stages. (d) Demonstrating the insensitivity of the perturbation to the size of the time steps after the perturbation is enabled.
Here, more models means smaller evolutionary time steps.
and convection routines are evaluated as in the envelope. Once a final solution is converged
upon, the code iterates in time and the process is repeated. For each temporal iteration, the
magnetic field profile is adjusted to adapt to changes in the location of the tachocline and
changes in the number of mass shells.
5.4 Testing and Validation
5.4.1 Numerical Stability
In Sections 5.2 – 5.3, we outlined the formulation and implementation of a magnetic perturbation within the framework of the Dartmouth stellar evolution code. With the perturbation
160
implemented, it was crucial to perform a series of numerical tests and common-sense checks
to validate that the code was operating properly.
The four key numerical tests were to ensure that:
1. The relative change in radius between magnetic models of monotonically increasing photospheric magnetic field strength must also be monotonically increasing with
respect to a non-magnetic model.
2. All model adjustments after the initial perturbation must be continuous and smooth.
3. The final perturbed model properties must be independent of the evolutionary stage
at which the perturbation is made.
4. The resulting perturbed model must be consistent, regardless of the number of time
steps taken after the perturbation.
All of these tests were performed to confirm that the code was producing consistent results
and that it was doing so in a numerically stable manner (i.e., no wild fluctuations).
Figure 5.2 demonstrates that all model adjustments to a magnetic perturbation satisfy each
of the four criteria we required for numerical validation. Panel (a) demonstrates that the radius monotonically increases as the surface field strength applied monotonically increases.
Changes are observed to be smooth, as seen in panel (b) and are independent of the number
(or size) of the evolutionary time steps taken (panel (d)). Finally, the plot in panel (c) indicates that the relative change to the model asymptotes to the same value, regardless of the
evolutionary phase at which the perturbation is applied.
5.4.2 Solar Model
Beyond testing for numerical stability, we must be assured that the code produces results
consistent with reality. Typically, a comparison with previous studies would be utilized.
161
However, the only such examples computed for evolutionary timescales are for CM Dra
(Chabrier et al. 2007; MacDonald & Mullan 2012). The stellar mass regime occupied by
CM Dra would require the implementation of FreeEOS, a task reserved for a future investigation. With the analytical EOS, it would appear there are no models generated with
which to compare. Even LS95 operate over timescales on the order of a solar-cycle and not
evolutionary times.
In the absence of previous results with which to compare, we opted to impose a weak
magnetic field (5 G) on our solar-calibrated model. Seeing as the properties of the Sun do not
require a magnetic field in order to produce an adequate solar model (Bahcall et al. 1997),
we would expect the solar properties to remain almost entirely unaffected. As expected,
the magnetically perturbed model still meets our requirements for it to be considered solarcalibrated (see Section 1.5.2). The model radius increases by 3 parts in 105 while effective
temperature decreases by 4 parts in 106 given the presence of the weak field.
5.4.3 Ramped Versus Single Perturbation
The nature of the procedure developed in this chapter requires that the magnetic perturbation be introduced in the model during runtime. Therefore, we must explore how the
method of introducing the magnetic perturbation affects the structural evolution of a model.
We consider whether the magnetic perturbation is slowly ramped from an initially very
weak field up to full strength or whether the magnetic perturbation is applied at a single
moment, regardless of the pertrubation strength.
Numerical stability of the magnetic perturbations was discussed previously in Section 5.4.1
and it was shown that the code produces smooth, consistent results independent of the
age of perturbation and the size of model time steps adopted. However, the perturbation
was enacted in a single step, regardless of the surface magnetic field strength. Stars do not
operate in this manner. We were curious to probe the effects of ramping the magnetic field
162
3.60
Non-Magnetic
Single Step
Ramped Field
0.4
3.58
0.2
3.56
log(Teff/K)
log(R/R⊙)
Non-Magnetic
Single Step
Ramped Field
0.0
3.54
3.52
-0.2
3.50
-0.4
(a)
(b)
3.48
0.1
1
10
Age (Myr)
100
1000
0.1
1
10
Age (Myr)
100
1000
Figure 5.3: Standard DSEP (black, solid) and magnetic mass tracks for a 0.4 M⊙ star are
plotted to illustrate the consequences of using a ramped magnetic perturbation (light blue,
dash-dotted) as opposed to a single-step perturbation (maroon, dashed). The effects are
considerably different depending on if we view (a) the age–radius plane or (b) the Teff radius plane.
strength from an initial weak field up to full strength.
Ramping was performed by assuming that the intial magnetic perturbation occurs along
the pre-main sequence (pre-MS) at an age of approximately 50 kyr. The initial perturbation
could be no larger than 100 G at the model photosphere. If the prescribed field strength
was larger than 100 G, the initial 100 G field was increased by 0.005 G per year. This rate
corresponds to about 50 G per model time step, thereby allowing the field to reach maximum
strength within 0.6 Myr.
A comparison between the morphology of a stellar mass track along the pre-MS with no
magnetic field, a single-step peturbation, and field ramping is illustrated in Figure 5.3. The
particular procedure for introducing the magnetic perturbation is irrelavant for the final
stellar parameters along the main sequence, once the star has settled into its final configuration. In Figure 5.3(a) the magnetic tracks converge smoothly and rapidly, regardless of
the treatment of the perturbation. This is true despite the appearance of a discontinuity
that develops in the effective temperature track for strong, single perturbation models (Fig-
163
ure 5.3(b)). We hesitate to label this spike an instability, because while the temperature does
fluctuate immediately after the perturbation, it settles quite rapidly into a stable configuration. Interestingly, there is no such discontinuity in the model radius. Ramped models do
not display this discontinuity but still produce results consistent with the single perturbation
models along the main sequence. Therefore, all stellar parameters from single perturbation
magnetic models are trustworthy beyond the zero-age main sequence (ZAMS).
Along the pre-MS and on the approach to the ZAMS, ramping tends to change to the overall
characteristics of the mass track. Changes to the track morphology are gradual and lack the
abrupt spike displayed by single perturbation models, as one would expect. Even after the
single perturbation model has relaxed, there are significant differences in the track morphology leading up to the equilibration of 3 He burning (the bump that appears around 200 Myr).
Such differences are accentuated depending on the location of the single perturbation (not
shown, but the earlier the perturbation, the worse the disagreement between the ramped
and single step models). Ramping should clearly be the preferred method to analyze how
the structure of pre-MS stars are affected. This is not surprising since pre-MS stars are still
contracting and are presumably still building up strong magnetic fields. It is not clear why
the single perturbation models do such a poor job along the pre-MS, though, as mentioned
above, the situation improves when the perturbation age is increased and we find better
agreement between the ramped and single step models.
The dependence of the pre-MS stellar properties on the initial age of the perturbation and the
rate of ramping is unclear and is outside the scope of this chapter. What we are primarily
concerned with is that the MS properties are unaffected, in line with our discussion of
numerical stability in Section 5.4.1. However, Figure 5.4 indicates that the precise details
have an ability to play a significant role in what properties are derived for pre-MS systems.
Only the effect on the effective temperature is shown as it demonstrates the differences
moreso than the effect on the radius, which appear to be negligible.
164
3.60
Non-Magnetic
τpert = 0.10 Myr
τpert = 0.04 Myr
3.58
log(Teff/K)
3.56
3.54
3.52
3.50
3.48
0.1
1
10
Age (Myr)
100
1000
Figure 5.4: The effect of altering the magnetic perturbation age for the same 0.4 M⊙ model
in Figure 5.3. Significant differences along the pre-main sequence are observed when comparing the two magnetic mass tracks that were perturbed at 0.10 Myr (light blue, dashed)
and 0.04 Myr (maroon, dash-dotted). The standard model mass track (black, solid) is provided for reference.
We note that it may be possible to use ramping techniques for modeling single stars over
their entire lifetime. Essentially, increase the field strength along the pre-MS, steadily decrease it along the MS, thereby mimicing the magnetic breaking process, and the assuming
constant magnetic flux along the red giant branch (Reiners 2012). Tying the field strength to
rotation would be ideal, however, rotation is currently not implemented in the Dartmouth
code.
5.5 Case Study: EF Aquarii
5.5.1 Standard Models
Standard, non-magnetic mass tracks with solar metallicity were computed for both EF Aqr
A and B with masses of 1.24 M⊙ and 0.95 M⊙ , respectively. Additional mass tracks were
165
also generated for a scaled-solar metallicity of +0.1 dex. The two components were unable
to be fit with a coeval age, regardless of the adopted metallicity. This is consistent with the
conclusions of Vos et al. (2012). Two fitting methods were performed to this end.
We first attempted to fit both components on an age–radius plane, which is equivalent to
fitting on the mass–radius (M –R ) plane. This is illustrated by the solid lines in Figure 5.5.
The primary star evolves much more rapidly than secondary, owing to the rather large mass
difference. Thus, the model radius of the primary inflates to the observed radius at an age
of 2.0 Gyr. The radius of the primary then quickly exceeds the observational bounds within
about 0.1 Gyr. The observational bounds on the primary radius places tight constraints on
the allowed age our models predict for the system. However, our model of the secondary
does not reach the observed radius until an age of 6.3 Gyr, an age difference of 4.3 Gyr
between the two components. This 4.3 Gyr difference is consistently present in our models,
including when mass and composition are allowed to vary within the observed limits (not
shown).
Assuming the age of the system was predicted accurately using models of EF Aqr A in
the M –R diagram, we find that the model radius of EF Aqr B underpredicts the observed
radius by 11%. Again, consistent with the findings of Vos et al. (2012), who found a radius
discrepancy of 9%. Such a disagreement is also broadly consistent with results from other
studies of active EBs (Ribas 2006; Torres et al. 2010).
A second approach was to fit the system on an HR diagram using the individual mass
tracks. Figure 5.6 demonstrates that the standard model tracks do not match the observed
Teff –luminosity of either star, despite fitting the stars individually in the M –R plane. Our
models predict temperatures that are 250 K (4%) and 430 K (8%) hotter than the observations
for the primary and secondary, respectively.
The noticeable temperature disagreements in both stars may be the result of two possible
effects. On the one hand, we might assume that the age predicted from the M –R plane
166
1.6
(a)
γ=2
γ = 4/3
Non-Mag.
Radius (R⊙)
1.4
1.2
EF Aqr A
1.0
EF Aqr B
0.8
10
100
1000
Age (Myr)
Figure 5.5: Individual stellar mass tracks representing EF Aqr A and B (labeled) in the
age–radius plane. The observational constraints on the stellar radii are represented by the
gray swaths. Non-magnetic mass tracks are shown as a red, solid line while γ = 2 and
γ = 4/3 magnetic tracks are indicated by the blue, long-dash and light-blue, short-dash
lines, respectively. The corresponding photospheric magnetic field strengths for EF Aqr A
are 1.6 kG (γ = 2) and 2.6 kG (γ = 4/3). For EF Aqr B they are 3.2 kG (γ = 2) and 5.5 kG
(γ = 4/3). Vertical dotted lines define the age constraint imposed by EF Aqr A, suggesting
the system has an age of 1.3 ± 0.2 Gyr.
is correct and that there only exists a discrepancy with the effective temperatures. This
implies that either the effective temperature from the models or the observations is incorrect.
However, since the stars are quite similar to the Sun, it is more likely the case that our
standard models underpredict the radius of the primary as well, driving up the modelderived effective temperature. This particular scenario is supported by Vos et al. (2012) who
found both components display obvious Ca ii emission that is likely the result of each star
having a magnetically heated chromosphere.
The age inferred from the M –R diagram would then be older than the true age of the
system. Unfortunately, this scenario further complicates the situation regarding EF Aqr B.
If the age of the system is younger than inferred from standard models of the primary, then
the radius discrepancy for the secondary becomes larger than originally quoted.
167
1.2
(b)
log(L/L⊙)
0.8
0.4
EF Aqr A
0.0
-0.4
γ=2
γ = 4/3
Non-mag.
6500
EF Aqr B
6000
5500 5000
Teff (K)
4500
4000
Figure 5.6: Individual stellar mass tracks representing EF Aqr A and B (labeled) in the Teff –
radius plane (HR diagram). The observational constraints are indicated by the black points.
Non-magnetic mass tracks are shown as a red, solid line while γ = 2 and γ = 4/3 magnetic
tracks are indicated by the blue, long-dash and light-blue, short-dash lines, respectively. The
corresponding photospheric magnetic field strengths for EF Aqr A are 1.6 kG (γ = 2) and
2.6 kG (γ = 4/3). For EF Aqr B they are 3.2 kG (γ = 2) and 5.5 kG (γ = 4/3).
5.5.2 Magnetic Models
Following the results discussed above, magnetic models were computed for both EF Aqr
components using a scaled-solar heavy element composition. Several models with a mass
of 1.24 M⊙ were generated with various surface magnetic field strengths. We then found
the model with the weakest field strength required to produce the observed radius and Teff
of EF Aqr A. With γ = 2, we had to prescribe a photospheric magnetic field strength of
1.6 kG, while with γ = 4/3, a more intense 2.6 kG field was necessary. The magnetic model
tracks are displayed in Figures 5.5 and 5.6 as blue dashed lines. The magnetic models of
the primary suggest a younger age of 1.3 ± 0.2 Gyr for the EF Aqr system, as opposed to
the 2.0 Gyr age determined from standard models. This younger age is consistent with the
1.5 ± 0.2 Gyr age derived for the primary by Vos et al. (2012) after fine-tuning the mixing
length.
168
We next had to select a magnetic field strength that would allow a 0.95 M⊙ model to have
a radius and Teff compatible with EF Aqr B at 1.3 Gyr, if finding that unique combination
was possible. Surface magnetic field strengths of 3.2 kG and 5.5 kG were able to produce the
required stellar parameters, with γ = 2 and 4/3, respectively, at an age of 1.35 Gyr. In both
cases, the models were able to reproduce the stellar radius and Teff within the quoted 1σ
uncertainties. Figures 5.5 and 5.6 demonstrate that the magnetic models do indeed match
both component radii and Teff s at a common age.
Structurally, the addition of a magnetic perturbation within the models reduces the radial
extent of the surface convection zone. For both stars in EF Aqr, we find the magnetic
models that are sufficient to correct the observed discrepancies have surface convection
zones that are 4% smaller than those in the standard models, at the same age. The reduction
of the surface convection zone can be attributed to the modified stability criterion as well
as modified convective velocities. While only speculation, we attribute the equality of the
percent reduction of convection zone sizes to coincidence.
5.6 Discussion
5.6.1 Field Strengths
The implementation of a magnetic perturbation within stellar evolution models is quite
capable of reconciling predicted model fundamental stellar properties with those determined
observationally, at least for EF Aqr. While it seems plausible that magnetic fields may
suppress thermal convection inside solar-type stars, how are we to be sure that magnetic
fields may be reasonably invoked for this particular system? Even if invoking magnetic
fields is rational, are the field strengths required by the models realistic?
Addressing the first question, we showed in Section 5.1 that naturally inefficient convection,
169
as described by the Bonaca et al. (2012) calibration, was unable to account for the small values of αMLT required to mitigate the observed model-observation disagreements. But, the
inability of naturally inefficient convection to provide a solution does not positively identify magnetic fields as the root cause. However, there is additional evidence that invoking
magnetic fields is reasonable.
High-resolution spectroscopy of the Ca ii H and K lines for both stars in EF Aqr reveals
incredibly strong emission cores superposed on the absorption troughs (Vos et al. 2012). A
search of the ROSAT Bright Source Catalogue (Voges et al. 1999) also shows that EF Aqr is a
strong X-ray emitter. Coupling these observations with high projected rotational velocities
extracted from line broadening measurements, suggests that there is the potential for a
strong dynamo mechanism to be operating. This evidence is only circumstantial, but does
provide tantalizing clues.
For the sake of argument, let us assume that the stars are significantly magnetically active.
It would then be worthwhile to compare the strength of the magnetic field for each EF Aqr
component to those values required by the models. Unfortunately, no direct magnetic field
strength estimates of EF Aqr are available, forcing us to base our analysis on indirect magnetic field strength estimates.
A natural first step would be to compare the EF Aqr components to other known solar-type
stars, such as the Sun and α Cen A and B (see Table 5.2). The Sun’s mean photospheric
magnetic field strength is between 0.1 G and 1 G (Babcock & Babcock 1955; Babcock 1959;
Demidov et al. 2002) with local patches of very intense fields (i.e., sunspots) with strengths
on the order of 2–3 kG (Hale 1908). Similarly, the average longitudinal field strength of α
Cen A was determined to be less than 0.2 G, after a null detection of a Stokes V polarization
signature (Kochukhov et al. 2011).
The field strengths required by the models of the EF Aqr components therefore suggest that
the stars are pervaded by magnetic fields that typically characterize the intense regions of
170
sunspots. This at first appears detrimental to the validity of the models. However, studies of
the active and quiet Sun, particularly sunspot regions, has led to multiple scaling relations
allowing for an indirect determination of stellar photospheric magnetic field strengths.
One of these scaling relations was observed to exist between the X-ray luminosity, L x of an
active region and its total unsigned magnetic flux, Φ, (Fisher et al. 1998; Pevtsov et al. 2003).
The two observables were found to exhibit a power-law relation,
L x ∝ Φp ,
(5.189)
where the power-law index, p , was determined to be 1.19±0.04 by Fisher et al. (1998). The
index was later revised by Pevtsov et al. (2003) using a more diverse data set, including both
solar and extrasolar sources.5 Their revised analysis decreased the index to p = 1.15. The
magnetic flux is defined in the usual manner (Equation 5.174),
Φ=
Z
S
B · d  =
Z
S
|B z | d A.
(5.190)
with |B z | represents the vertical magnetic field strength. Therefore, if we are able to determine the X-ray luminosity of the EF Aqr components, it is possible to place a lower limit on
the magnetic field strength at the surface of the two components.
The system has a confirmed X-ray counter-part in the ROSAT All-Sky Survey Bright Source
Catalogue (Voges et al. 1999). The X-ray count rate was converted to an X-ray flux according to the formula derived by Schmitt et al. (1995),
F x = (5.30HR + 8.31) × 10−12 X cr ,
(5.191)
where HR is the X-ray hardness ratio,6 X cr is the X-ray count rate, and F x is the X-ray flux.
5 The total unsigned magnetic flux of the stellar sources was obtained using direct observational techniques
(Saar 1996).
6 There are typically two hardness ratios listed in the ROSAT catalog, HR1 and HR2. The Schmitt et al.
171
Finally, the X-ray flux was converted to a luminosity using the 172 pc distance quoted by
Vos et al. (2012).
The count rate measured by ROSAT was 0.0655±0.0154 counts s−1 with a hardness ratio of
0.32±0.22. This yields an X-ray flux of 6.55×10−13 erg cm−2 . Weighting the contribution of
each star to the total X-ray flux is reliably performed in one of two ways: by assuming both
stars contribute equally (valid if for binaries if stars are similar in radius; Fleming et al. 1989)
or weighting proportional to v r ot sin i (Pallavicini et al. 1981; Fleming et al. 1989). Given
the similarity of the two stars in EF Aqr, the precise weighting does not affect the results.
If both stars contribute equally to the total X-ray flux, then at a distance of 172 pc, the Xray luminosity of each component is L x = 1.16 × 1030 erg s−1 . Alternatively, weighting the
two stars based on their projected rotational velocity gives L x, A = 1.25 × 1030 erg s−1 and
L x, B = 1.07 × 1030 erg s−1 .
To provide a comparison, the X-ray luminosity of a typical solar active region is on the
order of 1027 erg s−1 (Fisher et al. 1998; Pevtsov et al. 2003). The Sun, on average, has a
total X-ray luminosity of 1027 erg s−1 up to nearly 1028 erg s−1 , depending on where in
its activity cycle it is located (Ayres 2009). Similarly, Ayres (2009) monitored the X-ray
luminosity of α Cen and found the primary had an X-ray luminosity around half that of the
Sun (approximately 1027 erg s−1 ) and the secondary had about twice the X-ray luminosity
of the Sun (about 1028 erg s−1 ). Further estimates for the X-ray luminosity of α Cen A and
B comes from ROSAT, which yields luminosities between 1027 erg s−1 and 1028 erg s−1 for
each component, consistent with Ayres’ analysis. Table 5.2 provides a comparison of how
these quantities translate to magnetic field strengths.
Comparisons with the Sun and α Cen show that each component in EF Aqr has an Xray luminosity 2–3 orders of magnitude greater than “typical” G and early-K stars. Again,
while not indicative of causation, the correlation between high levels of X-ray emission and
(1995) formula requires the use of HR1.
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Table 5.2
Comparison of estimated magnetic field strengths (in G).
Star
Direct
X-Ray
Ca ii
DSEP
Sun
0.1 - 1.0
5 - 20
< 0.2
∼3
···
···
α Cen A
α Cen B
EF Aqr A
EF Aqr B
···
···
···
∼49
···
···
···
···
1300
830
1600 - 2600
2500
3300
3200 - 5500
magnetic activity strongly suggests that EF Aqr is incredibly active. Given this information,
our initial assumption that the stars are active seems valid. Therefore, the implementation
of a magnetic perturbation in stellar models of this system appears warranted.
The amount of vertical magnetic flux near the surface of each star may then be found using
the Pevtsov et al. (2003) scaling relation. This suggests Φ = 1.39 × 1026 Mx associated with
each component (given equal flux contribution). Converting to a magnetic field strength
involves dividing the unsigned magnetic flux by the total area through which the field is
penetrating. For our purposes, the area is the entire surface area of the star. Therefore,
we find the vertical magnetic field strength for the primary and secondary of EF Aqr to be
1.3 kG and 2.5 kG, regardless of the adopted flux weighting, respectively. Note, this is the
vertical magnetic field strength and sets a lower limit to the total magnetic field strength.
It should also be mentioned that the X-ray luminosities calculated for EF Aqr A and B are
near the edge of the data sample utilized by Pevtsov et al., although no extrapolation of the
relation was required.
Further estimates of the magnetic field strengths may be found by applying a scaling relation using the Ca ii K line core emission (Schrijver et al. 1989). The scaling relation was
developed by correlating Ca ii K line core emission and the magnetic flux density of solar
173
active regions and their surroundings. The relation between the two was found to be
I c − 0.13
= 0.008〈B f 〉0.6
Iw
(5.192)
where I c is the intensity of the emission line core, I w is the intensity of the line wing. While
the relation was derived for small, local active regions, Schrijver et al. (1989) suggest that
there is no reason to believe that the relation would not hold for hemispherical averages of
solar-type stars.
Using the spectra provided by Vos et al. (2012) for the Ca ii K core emission lines from each
EF Aqr component, we were able to estimate the magnetic field strength of each component.
Spectra for the primary indicate that the average magnetic field strength, 〈B f 〉, is equal to
830 G. Similarly, for EF Aqr B, 〈B f 〉 = 3.3 kG. The values quoted above are derived from
rough approximations of the core and wing intensities. However, we do not foresee the
values of 〈B f 〉 changing radically with more precise line intensity measurements. We do
caution that the results for EF Aqr B require an extrapolation of the Ca ii relation and the
data for EF Aqr A place it near the edge of the derived relation where only a few data points
exist.
Based on the scaling relations for X-ray emission and Ca ii K line core emission, the magnetic field strength for the primary and secondary is seen to be approximately 1 kG and
3 kG, respectively. The magnetic field strengths required by the models are therefore within
a factor of two of the predicted field strengths, regardless of the adopted γ value. Since we
expect the X-ray emission prediction to be a lower limit to the full magnetic field strength,
this is extremely encouraging. The models do not require abnormally large field strengths
to reconcile the model properties with those from observations, particularly when a γ value
of 2 is adopted.
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5.6.2 Implications
The introduction of self-consistent magnetic stellar evolution models has multiple applications, ranging from studies of exoplanet host stars (Torres 2007; Charbonneau et al. 2009;
Muirhead et al. 2012) to investigations of cataclysmic variable (CV) donor stars (Knigge et al.
2011), as well as to studies attempting to probe the stellar initial mass function of young
clusters, where stars are typically very magnetically active (Johns-Krull 2007; Jackson et al.
2009; Yang & Johns-Krull 2011). Although, most obvious, are the implications for studies of
low-mass eclipsing binary systems (see, e.g., Torres et al. 2010; Parsons et al. 2012, and references therein). Low-mass stellar evolution models have been highly criticized for being
unable to predict the radii and effective temperatures of DEB stars. Models incorporating
magnetic effects open the door to probing the underlying cause of the model-observation
disagreements and providing semi-empirical corrections to models.
Magnetic fields have long been theorized as the culprit, but previous generations of models
have only treated magnetic fields in an ad hoc manner. Comparing the results of these
methods with the one presented in this work, both in terms of surface parameters and the
underlying interior structure, will provide an interesting test of their validity. Ultimately,
the ad hoc models disagree on the dominant physical mechanism underlying the observed
discrepancies. The availability of self-consistent magnetic models should help to settle the
debate as to which mechanism (suppressed convection or starspots) is most at work.
For EF Aqr, the models suggest that magnetic suppression of thermal convection is sufficient
to reconcile the models with the observations. Since stars with small convective envelopes,
such as those discussed in this work, are more sensitive to adjustments of the convective
properties, it is not wholly surprising that suppressing convection is sufficient to explain
the observations. Whether this mechanism will be adequate for stars near the fully convective boundary has yet to be seen. Future work modeling the lowest-mass DEB systems
will clarify this ambiguity. Regardless, this may suggest why the largest radius deviations
175
are predominantly observed at higher masses (Feiden & Chaboyer 2012a), with the notable
exception of CM Dra (Terrien et al. 2012).
The nature of our models allows for independent verification of the magnetic field strengths
required as input. While the indirect estimates provided by X-ray emission and Ca ii
K emission are encouraging, confirmation of these results using high resolution Zeeman
spectroscopy, spectropolarimetry, or Zeeman–Doppler imaging (ZDI) is preferred. Unfortunately, these observations are difficult for fast rotators, such as those that comprise most
DEB systems. They are also difficult for distant systems, where the short integration time
required by ZDI inhibits the ability of acquiring measurements with sufficient signal-tonoise (see the reviews by Donati & Landstreet 2009; Reiners 2012). Once a magnetic field is
detected, there exists the question of whether the observed strength is indicative of the total
magnetic field strength. Field strengths derived for stars with spectral-type K and M using
Stokes V observations appear to yield only around 10% of the total magnetic field strength
compared to observations in Stokes I (Reiners & Basri 2009). This is a consequence of the
fact that regions of opposite polarity tend to cancel out in Stokes V , making it most sensitive to the large-scale component, not the small-scale fields thought to pervade low-mass
stars. How to accurately account for this when testing the models is not fully clear and will
require investigation. As more stars across all spectral types are observed in both Stokes V
and I , a more coherent picture is sure to develop.
Magnetic models may also be useful for transiting exoplanet surveys, particularly those focused on M-dwarfs (e.g., MEarth transit survey; Nutzman & Charbonneau 2008; Irwin et al.
2009). One of the largest uncertainties in deriving the properties of a transiting planet is the
radius of the host star. The lack of reliability involved in predicting low-mass stellar radii
from evolution models has deterred the use of models as predictors of exoplanet host-star
radii (Torres 2007; Charbonneau et al. 2009). Shoring up these deficiencies may lead to more
accurate predictions of host star radii from stellar models, circumventing, for a time, the
176
need for lengthy and costly observations. This would be most useful in identifying interesting follow-up targets by providing a better estimate of the habitable zone (Muirhead et al.
2012).
There are certainly caveats with models, as other large uncertainties exist in predicting the
properties of a single star from stellar evolution mass tracks (Basu et al. 2012). However,
work is being performed to alleviate some of these uncertainties by calibrating models to
asteroseismic data (Bonaca et al. 2012). Low-mass stars are also less sensitive to the input
parameters of stellar models than their solar-type counterparts, reducing the associated
uncertainties. Stellar evolution models may therefore provide a fast and reliable estimate of
the host star properties, depending on the required level of precision.
Since most M-dwarfs being surveyed are nearby, there is a good chance that they may have
an X-ray counterpart in either the Bright Source or Faint Source Catalogue from the ROSAT
All-Sky Survey (Voges et al. 1999; Voges et al. 2000). As was demonstrated in Section 5.6.1,
magnetic field strengths required by the models are within about a factor of two (or better,
if γ = 2 is adopted) of those predicted by the X-ray scaling relation of Pevtsov et al. (2003).
This will allow an intelligent choice of the magnetic field strength used as an input for the
models, thus producing more reliable results from stellar models.
All told, the introduction of a self-consistent set of magnetic stellar evolution models provides the potential for models to be used with greater reliability in a wide range of applications. There still exist several challenges that require attention (Boyajian et al. 2012), but
this is a first step in addressing key issues that have been raised in the past two decades.
177
Chapter 6
Magnetic Models of Low-Mass Stars with
a Radiative Core
6.1 Introduction
The previous chapter introduced the mathematical formulation and numerical implementation of a self-consistent magnetic field perturbation within the Dartmouth stellar evolution
code. Magnetic perturbations were added using a method formally based on the one outlined by Lydon & Sofia (1995). After the internals of the code were verified, a first test of
the magnetic models was carried out by modeling a solar-type DEB, EF Aquarii (EF Aqr;
Vos et al. 2012).
Results of the test case were positive. They indicated that the presence of a magnetic field
in a solar-type star could lead to inflated radii at 10% level observed in EF Aqr (see Chapter
5; Feiden & Chaboyer 2012b). The magnetic field strengths required to produce agreement
between the model properties and the observational properties of EF Aqr appeared consistent with estimates from activity data (Feiden & Chaboyer 2012b). Comparisons between
the Sun, α Centauri B, and the stars in EF Aqr displayed no obvious inconsistencies regard178
ing levels of magnetic activity and the predicted (or measured) magnetic field strengths.
However, the lack of direct magnetic field strength measurements among solar-type stars
(Donati & Landstreet 2009; Reiners 2012), especially among rapid rotators, means that we
are only able to speculate about the permitted magnetic field strengths of the surfaces of
solar-type stars.
The situation is different for low-mass stars. Considerable observational effort has been
directed towards directly measuring the magnetic field characteristics of low-mass stars
(Saar 1996; Reiners & Basri 2007; Johns-Krull 2007; Reiners & Basri 2009; Morin et al. 2010;
Shulyak et al. 2011; Reiners 2012). K- and M-dwarfs have been the focus of magnetic field
studies because around mid-M spectral type, about 0.35M⊙ , M-dwarfs are thought to become fully convective (Limber 1958a). The standard description of the stellar dynamo mechanism posits that magnetic fields are generated near the base of the outer convection zone
(Parker 1979). A strong shear layer, called the tachocline, forms between the differentially
rotating convection zone and the radiation zone, which rotates as a solid body. Fully convective stars, quite obviously, would not possess a tachocline. According to the standard
dynamo theory put forth by Parker, this would leave fully convective stars unable to generate or sustain a strong magnetic field through the dynamo mechanism.
Despite the lack of a tachocline, low-mass stars are observed to possess strong magnetic
fields with surface strengths on the order of a few kilogauss (Saar 1996; Reiners & Basri
2007; Reiners et al. 2009; Reiners & Basri 2010). Instead of a dynamo primarily powered
by rotation, it was suggested that turbulent convection may be driving the stellar dynamo
(Chabrier & Küker 2006). Observational evidence of the dynamo transitioning from a rotationally driven dynamo in solar-type stars to a purely turbulent dynamo in fully convective stars has been tentatively identified by studying the topology of low-mass stars.
There appears to be a shift in the large-scale field topology from non-axisymmetric in
solar-type stars to an axisymmetric field topology among low-mass stars (Morin et al. 2008;
179
Donati & Landstreet 2009; Phan-Bao et al. 2009; Morin et al. 2010). Whether this apparent
topology change is the hallmark of a transitioning dynamo or whether the shift in field
topology actually exists is still a matter of debate (Donati & Landstreet 2009; Reiners & Mohanty
2012)
An additional curiosity was identified by Reiners & Basri (2009), who found that Stokes V
polarization observations tend to only yield about 10% of the total magnetic field strength
for low-mass stars. Instead of using Stokes V , Reiners & Basri performed their observations
using a Stokes I polarization filter. Since Stokes V only measures the net, signed magnetic
flux, it is thought that Stokes I , which measures the total unsigned magnetic flux, is more
sensitive to small-scale magnetic fields of opposite polarity on the stellar surface.
The interest in low-mass stellar magnetic fields also benefits investigations of the low-mass
stellar mass-radius problem. Attempts to resolve the observed discrepancies between lowmass stellar radii and stellar evolution model predictions using magnetic fields can be confronted with empirical data. In particular, non-standard stellar evolution models that incorporate magnetic perturbations (Mullan & MacDonald 2001; Feiden & Chaboyer 2012b)
should not require unrealistically large surface magnetic field strengths to reconcile models
with observations.
With the dynamo dichotomy in mind, we have elected to divide our analysis of the lowmass stellar mass-radius problem into two parts. The first part presented in this chapter
concerns low-mass stars in DEBs that should possess a radiative core and convective outer
envelope. The second part, pertaining to fully convective low-mass stars will be presented
in the next chapter. Our reason for splitting the analysis is that the models described in
Chapter 5 assume that the energy for the magnetic field—and thus the dynamo mechanism—
is sourced from rotation. This was explicitly stated in the discussion of Equation (5.75).
With the possible transition from a rotationally driven dynamo to a turbulent dynamo at
the fully convective boundary (Reiners & Basri 2010), we risk invoking a theory that may
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Table 6.1
Sample of DEBs whose stars possess a radiative core.
DEB
Star
System
UV Psc
A
UV Psc
B
YY Gem
A
YY Gem
B
CU Cnc
A
CU Cnc
B
P orb
Mass
Radius
Teff
[Fe/H]
(day)
(M ⊙ )
(R ⊙ )
(K)
(dex)
0.86
0.9829 ± 0.0077
1.110 ± 0.023
5780 ± 100
···
0.6194 ± 0.0057
3820 ± 100
+0.1 ± 0.2
0.81
2.77
0.76440 ± 0.00450
0.59920 ± 0.00470
0.59920 ± 0.00470
0.43490 ± 0.00120
0.39922 ± 0.00089
0.8350 ± 0.0180
0.6194 ± 0.0057
0.4323 ± 0.0055
0.3916 ± 0.0094
4750 ± 80
···
3820 ± 100
+0.1 ± 0.2
3125 ± 150
···
3160 ± 150
···
not be entirely suitable for models fully convective stars.
Here, we will describe and present results of detailed modeling of three DEB systems with
radiative cores and convective envelopes. The sample of DEB systems selected for analysis
in this chapter represent only a subset of the total systems available to us. Three representative systems were chosen to avoid muddling the results. The three systems we chose were
UV Piscium (Carr 1967; Popper 1997), YY Geminorum (Adams & Joy 1920; Torres & Ribas
2002), and CU Cancri (Delfosse et al. 1999; Ribas 2003). We recall the properties of these
three systems in Table 6.1. Note that the systems cover three different mass regimes of
low-mass stars. This will provide us with information regarding the applicability of our
magnetic models as we begin to model masses closer to the fully convective boundary.
The chapter will proceed as follows. In Section 6.2 we present a detailed study for each of the
three DEB systems. We discuss how realistic model magnetic field strengths are in Section
6.3. Based on this discussion, we discuss means for reducing the model surface magnetic
field strengths, including introducing a way to include the effects of a turbulent dynamo
in Section 6.4. Section 6.5 then discusses the broad conclusions of our study and also explores possible applications of magnetic models beyond attacking the mass-radius problem.
Finally, we provide a summary of the study and of the main conclusions in Section 6.6.
181
6.2 Analysis of Individual DEB Systems
6.2.1 UV Piscium
UV Piscium (HD 7700; hereafter UV Psc) is a DEB that contains a solar-type primary with
a K-dwarf companion. It was likely discovered to be a variable star by Huth in 1959
(Kjurkchieva et al. 2005), but we were unable to locate the reference. However, this is reinforced by the fact that the first citation in NASA’s Astrophysics Data System is by Carr
(1967) who reported the observation of the eclipse ephemerides without surprise. Numerous
attempts at determining the fundamental stellar properties have been performed since its
discovery, with the most precise measurements produced by Popper (1997). These measurements were later confirmed and slightly revised by Torres et al. (2010), who standardized
the reduction and parameter extraction routines for a host of DEB systems. The mass and
radius for each component of UV Psc recommended by Torres et al. (2010) is summarized
in Table 6.1. Interestingly, no metallicity estimate exists despite the system being relatively
bright (V = 9.01).
One of the most notable features of UV Psc is that the secondary component is unable to be
properly fit by standard stellar evolution models at an age commensurate with the primary
(see e.g., Popper 1997; Lastennet et al. 2003; Torres et al. 2010; Feiden & Chaboyer 2012a).
The radius of the secondary appears to be nearly 10% larger than models predict and the
effective temperature is around 6% cooler than predicted. Metallicity and age are known to
affect the stellar parameters predicted by models, typically allowing for better agreement
with observations. However, the best fit age and metallicity of UV Psc still display the large
disagreements mentioned above (Chapter 4; Feiden & Chaboyer 2012a).
An investigation by Lastennet et al. (2003) found that it was possible to fit the components
on the same theoretical isochrone. Their method involved independently adjusting the helium mass fraction Y , the metal abundance Z , and the convective mixing-length αMLT . The
182
authors were able to constrain a range of Y , Z , and αMLT values that produced stellar models
compatible with the fundamental parameters of each component while enforcing that the
stars be coeval. Lastennet et al. (2003) found that a sub-solar metal abundance ( Z = 0.012)1
combined with an enhanced helium abundance (Y = 0.31) and drastically reduced mixinglengths for each star (αMLT = 0.58αMLT, ⊙ and 0.40αMLT, ⊙ , respectively) produced the highest
quality fit to the data at an age of 1.9 Gyr. The age inferred from their models is a factor of
4 lower than the 8 Gyr age commonly cited for the system.
Despite properly fitting the two components, the investigation by Lastennet et al. (2003) did
not provide any physical justification for the reduction in mixing-length. Furthermore, they
required an abnormally high helium abundance given the required sub-solar heavy element
abundance. Assuming that Y varies linearly with Z according to the formula
µ
¶
¢
∆Y ¡
Y = Yp +
Z − Zp ,
∆Z
(6.1)
where Y p is the primordial helium mass fraction and Z p = 0, implies that ∆Y /∆Z > 5 for
the Lastennet et al. (2003) study. Empirically determined values all converge around 2 ± 1
(Casagrande et al. 2007). The empirical relations are by no means certain and there is no
guarantee that all stars conform to this prescription. However, a single data point suggesting
∆Y /∆Z > 5 is a significant outlier, at 3σ above the empirical relation, introducing some
doubt as to whether that particular Y and Z combination is achievable. We must then look
elsewhere to reconcile stellar models with observations of the secondary.
The stars in UV Psc are known exhibit strong magnetic activity, showcasing a wide variety of phenomena associated with stars having magnetically heated atmospheres. Soft
X-ray emission (Agrawal et al. 1980), Ca ii H & K emission (Popper 1976; Montes et al.
1995a), and Hα emission (Barden 1985; Montes et al. 1995b) have all been observed and
1
We calculate this implies [Fe/H] = −0.14 considering the required Y and the fact that they were using
the GN93 heavy element abundances
183
associated with UV Psc. Flares have been recorded in Hα (Liu et al. 1996) and at X-ray
wavelengths (Caillault 1982), further suggesting the components are strongly active. Star
spots betray their presence in the modulation and asymmetries of a several light curves
(Kjurkchieva et al. 2005), although some of these modulations have also been attributed to
intrinsic variability in one of the components (Antonopoulou 1987). There does not appear
to be any further evidence supporting this claim (Ibanoglu 1987; Popper 1997), leading us to
believe any observed light curve variations are the result of spot modulation or flare activity.
The aforementioned evidence provides clues that magnetic fields may be the source of the
observed radius discrepancies. Lastennet et al. (2003)’s finding that a reduced convective
mixing-length was required could then be explained by the need for a mechanism to inhibit
the transport of thermal energy by convection (Chabrier et al. 2007).
All studies of UV Psc have found that standard stellar evolution models are entirely capable of reproducing the fundamental stellar properties of the primary star (Popper 1997;
Lastennet et al. 2003; Torres et al. 2010; Feiden & Chaboyer 2012a). Therefore, we begin by
assuming that UV Psc A conforms to the predictions of stellar evolution theory, but that
magnetic effects must be invoked to reconcile models with UV Psc B. Given this assumption, UV Psc A may be used to constrain the age and metallicity of the system. Using a large
grid of stellar evolution isochrones, Feiden & Chaboyer (2012a) found UV Psc A was best fit
by a 7 Gyr isochrone with a slightly metal-poor composition of −0.1 dex. The metallicity
estimate is commensurate with Lastennet et al. (2003), though two independent methods
were utilized to achieve the result. We adopt this sub-solar value as the initial target age
and metallicity for the system.
Standard model mass tracks are illustrated in Figures 6.1(a) and 6.1(b) for two metallicities.
The age of the system is anchored to the narrow region in Figure 6.1(a) where the models
agree with the observed radius of the primary. Figure 6.1(b) indicates that the [Fe/H] = −0.1
model yields satisfactory agreement with the observed effective temperature, as well as the
184
1.20
[Fe/H] = -0.1
1.20
[Fe/H] = -0.3
1.10
1.10
UV Psc A
Radius (R⊙)
Radius (R⊙)
UV Psc A
1.00
0.90
0.90
0.80
0.80
UV Psc B
0.70
0.01
1.00
0.1
1
UV Psc B
0.70
6000
10
Age (Gyr)
5600
5200
Teff (K)
4800
4400
Figure 6.1: Standard DSEP mass tracks of UV Psc A (maroon) and UV Psc B (light blue)
computed with [Fe/H] = −0.1 (solid line) and [Fe/H] = −0.3 (dashed line). (a) The ageradius plane. Horizontal swaths delineate the observed radii with associated 1σ uncertainty.
The vertical region indicates the age predicted by the primary. (b) The Teff -radius plane.
Shaded regions denote the observational constraints.
radius. We infer an age of 7.2 Gyr, more precise than Feiden & Chaboyer (2012a) as we are
not constrained to a discretized set of ages. The standard model for the secondary is shown
to reach the observed radius at an age of 18 Gyr, according to Figure 6.1(a), an 11 Gyr
difference between the two components. We also see that the model effective temperature
of the secondary is too hot compared to observations. Whereas the empirical effective
temperature (see Table 6.1) was measured to be 4 750 ± 80 K, standard models are hotter by
approximately 250 K.
Magnetic models of the secondary component were computed using a dipole profile, singlestep perturbation at 10 Myr (see Section 5.3 for details) for several values of the surface
magnetic field strength. A surface magnetic field strength of 4.0 kG (corresponding to a
tachocline field strength of 11 kG) produced a model radius that was commensurate with
the observed radius at an age consistent with the primary. This fact is depicted in Figure 6.2(a) by the dash line representing the magnetic model of the secondary. The vertical
lines bisecting the entire figure provide the age range at which the model must match the
185
1.20
|B|surf = 4.0 kG
Non-magnetic
1.20
[Fe/H] = -0.1
1.10
1.10
UV Psc A
Radius (R⊙)
Radius (R⊙)
UV Psc A
1.00
0.90
0.90
UV Psc B
0.80
0.80
|B|surf = 4.0 kG
UV Psc B
0.70
0.01
1.00
0.1
1
0.70
Non-magnetic
6000
10
Age (Gyr)
[Fe/H] = -0.1
5000
Teff (K)
4000
Figure 6.2: Similar to Figure 6.1 but with a single metallicity of [Fe/H] = −0.1 dex. Magnetic
mass track for UV Psc B with a 4.0 kG surface magnetic field strength (light blue, dashed
line). Standard DSEP mass tracks are plotted for comparison. (a) Age-radius plane. (b)
Teff -radius plane.
observed radius.
Next, we checked that the effective temperature predicted by the magnetic model agreed
with the temperature inferred from observations (Figure 6.2(b)). The same 4.0 kG magnetic
mass track required to fit the secondary in the age-radius plane over-suppresses the effective
temperature, forcing the model to be too cool compared to the empirical value. Intuitively,
one might suggest lowering the surface magnetic field strength so as to maintain agreement
in the age-radius plane while allowing for a hotter effective temperature. However, all field
strengths that provide agreement in the age-radius plane are unable to predict an effective
temperature sufficiently hot enough, they are all too cool.
How might we interpret the remaining temperature disagreement? One possible solution
is that the effective temperature measurement is incorrect. We feel this scenario is unlikely
considering the temperatures are hot enough where large uncertainties associated with complex molecular bands are not present. The uncertainties quoted in Table 6.1 seem to us large
enough to encompass the actual value. Another possibility is that we have not treated convection properly. Convection within the component stars may not have the same inherent
186
1.20
|B|surf = 3.0 kG
Non-magnetic
Bonaca et al. (2012) αMLT
1.20
[Fe/H] = -0.1
1.10
1.10
UV Psc A
1.00
Radius (R⊙)
Radius (R⊙)
UV Psc A
0.90
0.80
1.00
UV Psc B
0.90
0.80
UV Psc B
|B|surf = 3.0 kG
0.70
0.01
0.70
Bonaca et al. (2012) αMLT
0.1
1
Non-magnetic
6000
10
Age (Gyr)
[Fe/H] = -0.1
5000
Teff (K)
4000
Figure 6.3: Similar to Figure 6.2 except that all of the mass tracks have a αMLT reduced
according to the Bonaca et al. (2012) empirical relation. The surface magnetic field strength
used in modeling the secondary is 3.0 kG. (a) Age-radius plane. (b) Teff -radius plane.
properties as convection within the Sun. This idea was what led Lastennet et al. (2003) to
allow for an arbitrary mixing-length in each component. However, while mixing-length
theory is not entirely realistic and allows for such an arbitrary choice of the mixing-length,
arbitrary reduction without a necessary physical motivation (other than providing better
empirical agreement) is not wholly satisfying and does not fully illuminate the reasons for
the appearance of the discrepancies.
Instead of applying an arbitrary adjustment to the convective mixing-length, we may attempt to modify the convective mixing-length parameter according to the relation developed by Bonaca et al. (2012) who calibrated the mixing-length parameter in stellar models to
asteroseismic data. The physical motivation is that convection is inherently different among
various stars due to their physical properties (i.e., log g , Teff , [M/H]). Modifications to the
convective mixing-length are, therefore, no longer arbitrary and may not take on any random value that happens to allow the models to fit a particular case. The Bonaca et al. (2012)
relation ascribes a mixing-length of αMLT = 1.71 for the primary and αMLT = 1.49 for the
secondary of UV Psc, given a metallicity of −0.1 dex and our solar calibrated αMLT, ⊙ = 1.94.
The resulting mass tracks in the Teff -radius plane are plotted in Figure 6.3.
187
Directly altering convection in this manner does not provide an adequate solution. The
Bonaca et al. (2012) formulation indicates that the convective mixing-length is proportional
to metallicity and Teff while inversely proportional to log g . Essentially, convection is less
efficient in low-mass, metal-poor stars as compared to the solar case. Reducing the mixinglength in this fashion inflates both of the stellar radii and forces the temperature at the
photosphere to decrease. The mixing-length primarily affects the outer layers of the star,
where energy is transported by convection. A lower mixing-length implies that there is less
energy flux across a given surface within the convection zone. Since the star must remain
in equilibrium, the outer layers puff up to increase the energy flux, thereby reducing the
effective temperature.
Inflating the primary component means the models of the secondary must now agree with
the observational properties at an age younger than 7.2 Gyr. This is well illustrated in
Figure 6.3(a), where the vertical dashed lines anchoring the age of the system to UV Psc
A are shifted to the left by 0.5 Gyr of where they were in Figure 6.1(a). It is true that a
weaker magnetic field is required to alleviate the radius disagreements with the secondary
due to inflation caused by the reduced mixing-length. Shown in Figure 6.3(a) is a magnetic
model with a surface field strength of 3.0 kG. We do not find agreement between the model
and empirical radius, but more importantly, Figure 6.3(b) demonstrates that the effective
temperature of the secondary is already too cool. Increasing the magnetic field strength
to yield agreement in the age-radius plane would only worsen the lack of agreement in
the Teff -radius plane. Reducing the mixing-length is unable to provide relief to the oversuppression of the effective temperature induced by the magnetic field in Figure 6.2(b). We
must seek another method to rectify the effective temperature of the magnetic model.
The metallicity is an unconstrained input parameter for models of UV Psc. Recall, our
selection of [Fe/H] = −0.1 was based solely on the agreement of standard stellar evolution
models with the primary. Updating the adopted metallicity (and consequently, the helium
188
abundance) significantly affects the structure and evolution of the UV Psc components. Stars
with masses above approximately 0.45M⊙ are similarly affected by altering the chemical
composition. Both the stellar radiative opacity and the efficiency of the proton-proton (pp) chain are influenced. Helium plays an important role in governing the p-p chain by
increasing the mean molecular weight. The abundance of metals is relatively unimportant
in the nuclear energy generation as the CNO cycle does not contribute significantly to the
overall energy budget at the given masses and the impact of metals on the mean molecular
weight is small in comparison to the impact of helium. Radiative opacities are influenced
by both helium and heavy element abundances.
Decreasing the metallicity reduces the helium abundance in our models, as we assume a
linear relation between Y and Z according to equation (6.1) with d Y /d Z = 1.6. The result
is that the efficiency of energy generation via the p-p chain is reduced. However, a critical
energy generation rate must be achieved for the star to maintain hydrostatic equilibrium.
The star contracts, heating up the stellar material, causing the star to become hotter. The
decrease in metals also reduces the stellar radiative opacity, causing the layers below the
convection zone to contract in order to maintain a constant energy flux across a given
surface. Ultimately, the star is left with a smaller radius and higher effective temperature.
Adopting a lower metallicity of [Fe/H] = −0.3 for UV Psc increases the effective temperature
of both components in the standard models while shrinking their radii at younger ages2 .
Doing so also destroys the agreement between models effective temperatures and UV Psc A.
These effects are demonstrated for standard models in Figures 6.1(a) and 6.1(b). Accurately
reproducing the observed stellar properties now requires the use of magnetic models for
both components. One should have expected this from the outset considering both stars are
synchronously rotating with a period of 0.86 days.
Magnetic models with a dipole profile and single-step perturbation were constructed for
2
At older ages, evolutionary effects begin to play a role as the stellar lifetimes are decreased at lower
metallicity owing to the higher temperatures within the stellar interior.
189
1.20
Magnetic
1.20
Non-magnetic
UV Psc A
1.10
1.10
[Fe/H] = -0.3
1.00
Radius (R⊙)
Radius (R⊙)
UV Psc A
0.90
0.80
0.90
UV Psc B
0.80
UV Psc B
0.70
0.01
1.00
0.1
1
0.70
[Fe/H] = -0.3
6000
10
Age (Gyr)
Magnetic
Non-magnetic
5000
Teff (K)
4000
Figure 6.4: The UV Psc system assuming a lower heavy element abundance of [Fe/H]
= −0.3. Standard DSEP mass tracks are drawn as maroon and light blue solid lines for
UV Psc A and B, respectively. Magnetic tracks are represented by dashed lines with the
same color coding as the standard tracks. Surface magnetic field strengths are 2.0 kG and
4.6 kG for UV Psc A and B, respectively. (a) Age-radius plane. (b) Teff -radius plane.
both components. We find that it is possible to wholly reconcile the models with the observations if the primary has a 2.0 kG surface magnetic field and the secondary has a 4.6 kG
surface magnetic field. Model radii and temperatures match the empirical values within the
age range anchored to the primary. These facts are shown in Figures 6.4(a) and 6.4(b).
The revised age of the UV Psc system found from Figure 6.4(a) (bracketed by the two vertical
lines) is between 4.4 Gyr and 5.0 Gyr. Averaging the two implies the age is 4.7±0.3 Gyr.
This age is considerably younger than the 7 Gyr or 8 Gyr age commonly prescribed. While
feedback from the models was necessary to adjust and improve the initial metallicity and to
determine the required magnetic field strengths, we believe that the final result is entirely
consistent with the available observational data. In fact, due to the lack of constraint on the
stellar composition, our reliance on such a feedback cycle was inevitable. The metallicity
range allowing for complete agreement is not limited to −0.3 dex. Further reducing the
metallicity would also likely produce acceptable results, deduced from the fact that the
models of UV Psc B just barely skirt the boundaries of the empirical values. Our final
190
recommendation is that the metallicity of UV Psc is approximately [Fe/H] = −0.3 ± 0.1
dex with surface magnetic fields of 2.0 kG and 4.6 kG for the primary and secondary,
respectively. Verification of the metallicity, and possibly the magnetic field strengths, should
be obtainable using spectroscopic methods.
6.2.2 YY Geminorum
YY Geminorum (also Castor C and GJ 278 CD; hereafter YY Gem) has been the subject
of extensive investigation after hints of its binary nature were uncovered spectroscopically
(Adams & Joy 1920). The first definitive reports of the orbit were published simultaneously
using spectroscopic (Joy & Sanford 1926) and photographic methods (van Gent 1926), which
revealed the system to have an incredibly short period of 0.814 days. The photographic study
of van Gent (1926) also showed that the two components eclipsed one another with the
primary and secondary eclipse depths appearing nearly equal, suggesting the components
were very similar. Rough estimations of the component masses and radii were performed
using the available data, but the data was not of sufficient quality to extract reliable values
(Joy & Sanford 1926). The system has since been confirmed to consist of two equal mass,
early M-dwarfs with masses and radii established with a precision under 1% (Torres & Ribas
2002), presented in Table 6.1.
An age and metallicity have also been attributed to YY Gem, which has a common proper
motion with the Castor AB quadruple. Considered gravitationally bound, these three systems have been used to define the Castor moving group (Anosova et al. 1989). Spectroscopy
of Castor Aa and Ba, both spectral-type A stars, yielded a metallicity of [Fe/H] = +0.1 ±
0.2, where the uncertainty is dominated by the fact that the original metallicity estimate
was relative to Vega and not the Sun (Smith 1974; Torres & Ribas 2002). Stellar evolution
models of Castor Aa and Castor Ba provide an age estimate of 359 ± 34 Myr, obtained by
combining estimates from DSEP (400 Myr; Feiden & Chaboyer 2012a) and the values quoted
191
Dipole Profile
0.90
|B|surf = 2.5 kG
|B|surf = 3.0 kG
|B|surf = 4.0 kG
|B|surf = 5.0 kG
Radius (R⊙)
0.80
Non-magnetic
0.70
YY Gem A/B
0.60
[Fe/H] = -0.1
10
100
1000
Age (Myr)
Figure 6.5: Standard DSEP (solid, black line) and magnetic (colored, broken lines) stellar
evolution models of YY Gem in the age-radius plane. The horizontal swath represents the
observational radius with associated 1σ uncertainty. The vertical shaded region marks the
age limits set by Castor A and B.
in the Appendix of Torres & Ribas (2002).
Over half a century after its binarity was discovered, advanced (at the time) low-mass stellar
evolution models suggested that the observed radii of YY Gem may not agree with predictions (Hoxie 1970, 1973). Although a subsequent generation of models appeared to find
full agreement with the observations (Chabrier & Baraffe 1995), confirmation of true discrepancies remained veiled as the models and observations were fraught with uncertainties.
Modern observational determinations of the stellar properties for the components of YY
Gem (Ségransan et al. 2000; Torres & Ribas 2002), compared against the latest set of sophisticated stellar evolution models (Baraffe et al. 1998; Dotter et al. 2008), have now solidified
the fact that the components of YY Gem appear inflated compared to model predictions by
approximately 8% (Torres & Ribas 2002; Feiden & Chaboyer 2012a).
Figure 6.5 shows a single standard stellar evolution mass track for the components of YY
192
0.90
Radius (R⊙)
0.80
0.70
0.60
4400 4200 4000 3800 3600 3400 3200 3000
Teff (K)
Figure 6.6: Standard DSEP (solid, black line) and magnetic (colored, broken lines) stellar
evolution models of YY Gem in the Teff -radius plane. The shaded region denotes the observational constraints.
Gem as a black, solid line. The vertical dash lines border the adopted age thought to represent YY Gem. It is clear from Figure 6.5 that the standard stellar model significantly under
predicts the radius measured by Torres & Ribas (2002) (shown as the light blue horizontal
swath) and in Figure 6.6 the over predicted effective temperature is evident. The adopted
metallicity for this particular set of standard models is [Fe/H] = -0.1, found to provide the
best agreement between models and observations (Feiden & Chaboyer 2012a).
As an aside, it may be noted from Figure 6.5 that our models are consistent with the properties of YY Gem at a young age. Previous studies have considered the possibility that
YY Gem is still undergoing its pre-main sequence contraction (Chabrier & Baraffe 1995;
Torres & Ribas 2002), as would be inferred from our models if no age prior were assigned.
The standard model in Figure 6.5 predicts an age of about 60 Myr, around 50 Myr before our
models indicate a 0.6 M⊙ star settles onto the main sequence. Torres & Ribas (2002) provided a detailed analysis of this consideration and conclude that it is erroneous to assume
193
YY Gem is a pre-main sequence system, primarily due to YY Gem’s association with the
Castor quadruple. Without evidence to the contrary, this fact places YY Gem firmly on the
main sequence and insures that the system remains discrepant with stellar models.
YY Gem exhibits numerous features indicative of intense magnetic fields and activity. Light
curve modulation has been continually observed (Kron 1952; Leung & Schneider 1978; Torres & Ribas
2002), suggestive of the presence of star spots. Debates linger about the precise latitudinal location and distribution (e.g., Güdel et al. 2001) of star spots, but spots contained below midlatitude (between 45◦ and 50◦ ) appear to be favored (Güdel et al. 2001; Hussain et al. 2012).
The components display strong Balmer emission (Young et al. 1989; Montes et al. 1995b)
and X-ray emission (Güdel et al. 2001; Stelzer et al. 2002; López-Morales 2007; Hussain et al.
2012) during quiescence and have been observed to undergo frequent flaring events (Doyle et al.
1990; Doyle & Mathioudakis 1990; Hussain et al. 2012). Furthermore, YY Gem has been
identified as a source of radio emission, attributed to partially relativistic electron gyrosynchrotron radiation (Güdel et al. 1993; McLean et al. 2012). Given this evidence, it is widely
appreciated that the components possess strong magnetic fields. Therefore, it would not be
entirely surprising if the interplay between convection and magnetic fields contributed, at
least in part, to the disagreements mentioned previously.
Magnetic stellar evolution mass tracks were computed for various values of the surface magnetic field strength using a single-step perturbation with a dipole field configuration. These
tracks are plotted in Figure 6.5 and 6.6 as dashed, colored lines. The adopted metallicity is,
once again, -0.1 dex. The degree of radius inflation and temperature suppression increases
as subsequently stronger values of the surface magnetic field strength are applied with the
strongest field strength of 5.0 kG over predicting the observed stellar radius. Figure 6.5 also
demonstrates that a 4.0 kG field strength matches the lower-bound of the observed radius.
We, therefore, may confine the allowable magnetic field strength to be within the range of
4.0 kG to 5.0 kG.
194
Testing the agreement with the effective temperature, Figure 6.6 reveals that the models are
just able to match the observed effective temperature with a 4.0 kG magnetic field. Any
stronger of a field over-suppresses the effective temperature, forcing the model to be cooler
than the actual observations. Recall, we encountered this same issue when attempting to
model UV Psc in the previous section. Lowering the metallicity provided a solution for UV
Psc, but doing so for YY Gem would amount to jeopardize the metallicity prior established
by the association with the other Castor components (Smith 1974; Torres & Ribas 2002).
Before ruling out the option of a lower metallicity, we decided to recompute the approximate
metallicity of YY Gem using the Smith (1974) data. First, we needed to determine the
metallicity of Vega, the reference for the (Smith 1974) study. Vega has 21 listed metallicity
measurements in SIMBAD, of which, the 8 most recent appear to be converging towards a
common value. Using the entire list of 21 measurements, Vega has a metallicity [Fe/H] =
-0.4 ± 0.4 dex. If, instead, we adopt only those measurements performed since 1980, we find
[Fe/H] = -0.6 ± 0.1 dex. Due to the convergence of values in more recent years, we elect to
adopt the latter value for the metallicity of Vega.
The metallicities measured by Smith (1974) for Castor A and Castor B were +0.98 dex and
+0.45 dex, respectively. Taking the average of these two quantities as the metallicity for the
Castor AB system, we have that [Fe/H] = +0.7 ± 0.3 dex. The difference in metallicity of
Castor A and B might be explained by diffusion processes (e.g., Richer et al. 2000) and is not
necessarily a concern. However, the fact that we are not observing the initial abundances
for Castor A and B is a concern when it comes to prescribing a metallicity for YY Gem.
Therefore, a conservative estimate for the metallicity of YY Gem relative to the Sun would
be [Fe/H] = +0.1 ± 0.4 dex, providing more freedom in our model assessment of YY Gem.
We note that this reassessment neglects internal errors associated with the abundance determination performed by Smith (1974). However, given the radical difference in metallicity
measured for Castor A and B, the real metallicity is presumed to lie within the statistical
195
Dipole Profile
0.90
|B|surf = 4.3 kG
Non-magnetic
Radius (R⊙)
0.80
0.70
YY Gem A/B
0.60
[Fe/H] = -0.2
10
100
1000
Age (Myr)
Figure 6.7: Same as Figure 6.5 except with a metallicity of −0.2 dex.
error. The Castor sextuple system would certainly benefit from revised abundances using
modern methods. New abundance determinations would not only enhance our understanding of YY Gem, but also provide confirmation that the three binaries comprising the Castor
system have a common origin.
Presented with greater freedom in modeling YY Gem, we computed additional standard and
magnetic mass tracks with [Fe/H] = -0.2 dex. The magnetic tracks were computed in the
same fashion as the previous set to provide a direct comparison on the effect of metallicity.
Figures 6.7 and 6.8 illustrates the results of these models. While difficult to discern without
the previous [Fe/H] = -0.1 standard mass track for comparison, the standard model mass
track with [Fe/H] = -0.2 is smaller in radius by approximately 1% along the main sequence.
The standard model with the revised metallicity also shows a hotter effective temperature,
as anticipated.
The magnetic mass track with a surface magnetic field of 4.3 kG was found to best fit
the observations. At 360 Myr (indicated by the dashed vertical line in Figure 6.7), it is
196
0.90
Radius (R⊙)
0.80
0.70
0.60
4400 4200 4000 3800 3600 3400 3200 3000
Teff (K)
Figure 6.8: Same as Figure 6.6 except with a metallicity of −0.2 dex.
apparent that the magnetic models of YY Gem satisfy the radius restrictions enforced by the
observations. The precise model radius inferred from the mass track is 0.620 R ⊙ , compared
to the observed radius of 0.6194 R⊙ , a difference of 0.1%. Figure 6.8 further demonstrates
that while the model is consistent with the observed radius, the effective temperature of the
mass track is also in agreement with the observations. The model effective temperature at
360 Myr is 3773 K, well within the 1σ observational uncertainty (see Table 6.1).
There is one additional constraint that we have yet to mention. Lithium has been detected
in the stars of YY Gem (Barrado y Navascués et al. 1997). They find logN(7 Li) = 0.11, where
logN(7 Li) = 12 + log(X Li /A Li X H ). Standard stellar models, however, predict that lithium
should be completely depleted from the surface after about 15 Myr—well before the stars
reach the main sequence. Since magnetic fields can shrink the surface convection zone, it is
possible for the fields to extend the lithium depletion timescale (MacDonald & Mullan 2010).
This is precisely what our magnetic models predict. With a metallicity of [Fe/H] = −0.2 at
375 Myr and with a 4.3 kG magnetic field our models predict logN(7 Li) ∼ 0.9 while with
197
[Fe/H] = −0.1 at 360 Myr and with a 4.0 kG we find logN(7 Li) = 0.1. The latter value is
consistent with the lithium abundance determination of Barrado y Navascués et al. (1997).
In summary, we find that the best fit magnetic models have a magnetic field strength between 4.0 and 4.5 kG. A sub-solar metallicity of [Fe/H] = −0.2 provides the most robust
fit in the age-radius and Teff -radius planes, but a metallicity as high as [Fe/H] = −0.1 is
allowed. The latter result also provides a lithium abundance estimate consistent with observations, whereas the lower metallicity model predicts too much lithium leftover at 360 Myr.
All of these results should be verifiable using spectroscopic methods.
6.2.3 CU Cancri
The variable M-dwarf CU Cancri (GJ 2069A, hereafter CU Cnc; Haro et al. 1975) was recently discovered to be a double-lined spectroscopic binary system by Delfosse et al. (1998).
Follow up observations provided evidence that CU Cnc underwent periodic eclipses, making it the third known M-dwarf DEB at the time (Delfosse et al. 1999). Shortly thereafter,
Ribas (2003) published a detailed re-analysis of the CU Cnc system using more detailed
light curves in combination with the Delfosse et al. (1998). Ribas was able to derive precise
masses for the two components, similar to Delfosse et al.. However, the detailed light curves
obtained by Ribas (2003) allowed him to estimate the stellar radii with high precision. These
values were presented in Table 6.1.
Initial comparisons with the Baraffe et al. (1998) models indicated that the components of
CU Cnc were peculiar. They were found to be 1 magnitude under luminous in the V band
compared to solar metallicity models. Additionally, the prescribed spectral type was two
subclasses later than expected for two 0.4M⊙ stars (M4 instead of M2; Delfosse et al. 1999).
These oddities were used as evidence that CU Cnc has a super-solar metallicity. The increased metallicity would increase the TiO opacity at optical wavelengths thereby producing stronger TiO absorption features used for spectral classification (see Section 1.1). The
198
absolute V band magnitude would also be lowered since TiO bands primarily affect the
opacity at optical wavelengths, shifting flux from the optical to the near-infrared. Using
the BCAH98 models with metallicity 0.0 and −0.5, Delfosse et al. (1999) performed a linear
extrapolation to estimate the metallicity of CU Cnc. They found [Fe/H] ∼ +0.5.
The super-solar metallicity quoted by Delfosse et al. (1999) was supported by the space velocity of CU Cnc. It has galactic velocities U ≈ −9.99 km s−1 , V ≈ −4.66 km s−1 , and W
≈ −10.1 km s−1 . CU Cnc is therefore posited to be kinematically young and a member of the
thin-disk, which is characterized by young, more metal-rich stars. However, the space velocities were used by Ribas (2003) to refute the metallicity estimate provided by Delfosse et al.
(1999). Instead of supporting the notion that CU Cnc had a super-solar metallicity, Ribas
conjectured that CU Cnc was a member of the Castor moving group, which is defined by
U = −10.6 ± 3.7 km s−1 , V = −6.8 ± 2.3 km s−1 , and W = −9.4 ± 2.1 km s−1 ). Therefore,
Ribas prescribed the metallicity of the Castor system to CU Cnc (see Section 6.2.2). He then
assumed that CU Cnc had a near-solar metallicity
With a metallicity (and age) estimate in hand, Ribas (2003) performed a detailed comparison
between stellar models and the observed properties of CU Cnc. Models of the CU Cnc components were found to predict radii that were 10 – 14% smaller than observed. Furthermore,
the effective temperature of the models were 10 – 15% hotter than the effective temperature
estimated by Ribas (2003). CU Cnc was found to under luminous in the V band and the K
band by 1.4 magnitudes and 0.4 magnitudes, respectively. Ribas proceeds to then lay out
detailed arguments that neither stellar activity or metallicity provide a satisfactory explanation for the observed radius, Teff , and luminosity discrepancies. Instead, he proposes that
CU Cnc may possess a circumstellar disk. The disk would then disproportionately affect the
observed V band flux and not the K band. This would also force the effective temperatures
to be reconsidered, leading to a change in the observed luminosities.
The Ribas (2003) results rely heavily on the estimated effective temperature of the individual
199
components. Determining the effective temperature of M-dwarfs is fraught with difficulty.
There is a strong degeneracy between metallicity and effective temperature for M-dwarfs
when considering photometric color indicies. We will therefore return to a detailed discussion of the luminosity discrepancies later and focus on the radius deviations, for the
moment. Radius estimates will be less affected by the presence of a circumstellar disk since
radius determinations rely on differential photometry.
In Chapter 4, our models preferred a super-solar metallicity when attempting to fit CU
Cnc. The maximum metallicity permitted in that analysis was only [Fe/H] = +0.2 dex.
We begin with a standard model analysis of CU Cnc assuming a super-solar metallicity
with [Fe/H] ≥ +0.2 dex. This is valid even if CU Cnc is a member of the Castor moving
group, as suggested by Ribas (2003). Recall that in Section 6.2.2 we placed constraints on the
metallicity of the stars in the Castor sextuple of [Fe/H] = +0.1 ± 0.4.
The radius evolution of both components is presented in Figure 6.9 for multiple metallicities.
The results are nearly independent of the adopted metallicity. All of the mass tracks show
that the models do not match the observed stellar radii at the same age along the main
sequence. Models of the primary appear to deviate from the observations more than models
of the secondary. This may just be a consequence of the larger radius uncertainty quoted
for the secondary creating the illusion of better agreement.
Agreement between the models and observations for both components is found at 120 Myr.
At this age, the stars are both undergoing gravitational contraction along the pre-mainsequence (pre-MS). Can we rule out the possibility that the stars of CU Cnc are still in the
pre-MS phase? At this point we can not. Ribas (2003) tentatively detect lithium in the
spectrum of CU Cnc, which strongly suggests that the stars would be in the pre-MS phase.
However, our models predict that lithium disappears entirely around 20 Myr for the stars
of CU Cnc. We do note that the agreement between the models and observations occurs
right near the boundary of the gray vertical box (centered on 120 Myr) in Figure 6.9. The
200
0.44
0.42
Radius (R⊙)
CU Cnc A
0.40
0.38
CU Cnc B
[Fe/H] = +0.2
[Fe/H] = +0.3
[Fe/H] = +0.4
0.36
10
100
1000
10000
Age (Myr)
Figure 6.9: Standard DSEP mass tracks for CU Cnc A (maroon) and CU Cnc B (lightblue) at three different metallicities: +0.2 (solid), +0.3 (dashed), +0.4 (dotted). Horizontal
bands identify the observed radius with 1σ uncertainty while the vertical band identifies
the region in age-radius space where the models match the primary stars observed radius.
evidence for agreement is certainly not definitive.
Let us assume that the stars have reached the main sequence. There is evidence that the
stars are magnetically active. The stars show strong X-ray emission based on ROSAT observations3 (López-Morales 2007; Feiden & Chaboyer 2012a), indicative of a magnetically
heated corona. CU Cnc is also considered to be an optical flare star. It was first identified as such by (Haro et al. 1975) and optical flaring has been continually noted since then
(e.g., Qian et al. 2012). Furthermore, the stars show strong chromospheric Balmer and Ca ii
K emission during quiescence (Reid et al. 1995; Walkowicz & Hawley 2009). These tracers
point toward the presence of at least moderate magnetic activity on the stellar surface.
Magnetic models were computed using the dipole field profile. Two values for the surface
magnetic field strength were chosen, 2.6 kG and 3.5 kG. Mass tracks that include a magnetic
3
ROSAT observations actually contain both CU Cnc and its companion CV Cnc and so are difficult to
interpret.
201
CU Cnc A
0.44
Radius (R⊙)
0.42
0.40
0.38
CU Cnc B
0.36
Non-magnetic
|B|surf = 2.6 kG
|B|surf = 3.5 kG
[Fe/H] = +0.3
10
100
1000
10000
Age (Myr)
Figure 6.10: Magnetic stellar evolution mass tracks of CU Cnc A (maroon) and CU Cnc
B (light-blue) with surface field strengths of 2.6 kG (dotted) and 3.5 kG (dashed). A nonmagnetic mass track for each star is shown as a solid line. All models have [Fe/H] = +0.3
following the discussion in the text. The horizontal swaths signify the observed radius with
associated 1σ uncertainty.
field are shown in Figure 6.10. We fixed the metallicity to [Fe/H] = +0.3 since it makes
a relatively small difference to the standard model mass tracks in the age-radius plane.
Note that the perturbation time is different between the 2.6 kG and 3.5 kG tracks. The
perturbation age was pushed to 100 Myr in the 3.5 kG model to ensure convergence of
the model immediately following the perturbation. Results of the model along the main
sequence are insensitive to this change of perturbation age, as was shown in Sections 5.4.1
and 5.4.3.
Figure 6.10 shows that our model of the secondary star with a surface magnetic field strength
of 3.5 kG matches the observed radius of the secondary from 300 Myr (ignoring the pre-MS)
until about 6 Gyr. The lower field strength model is nearly identical, except that it shows
slight disagreement near the zero-age main sequence (ZAMS) around 300 Myr. Unlike the
models of the secondary, the magnetic models of the primary reproduce the observations
202
7.0
M = 0.44 M⊙
7
6.5
Li Fuses
7
log10(T) (K)
Li Stable
6.0
5.5
Radiation
Convection
5.0
0
0.2
0.4
0.6
Radius Fraction
0.8
1
Figure 6.11: Temperature profile within a model of CU Cnc A showing that the base of the
convection zone exists at a higher temperature than the 7 Li fusion temperature. Note that
the temperature profile from the stellar envelope calculation is not included.
between about 900 Myr and 6 Gyr. Finding agreement between the model and observed
radius near the ZAMS would require a stronger surface magnetic field.
The need for a stronger magnetic field in the primary depends on the real age of the system.
Ribas (2003) invoked the possible Castor moving group membership to estimate the age. The
Castor moving group is thought to be approximately 350 – 400 Myr. According to Figure
6.10, this would place CU Cnc near the ZAMS if we account for magnetic fields. It also
means that a stronger magnetic field would be needed in modeling the primary star. A
surface magnetic field strength of 4.0 kG is required to produce agreement for the primary
if it is coeval with the Castor system. However, as we discussed in Section 4.3.4, there is no
compelling argument that leads us to believe that CU Cnc is coeval with the Castor sextuple.
Kinematic membership is not sufficient for assigning an age or metallicity.
The destruction of lithium is unaffected by the presence of the magnetic field. Unlike YY
Gem, where lithium was preserved to a significantly older age, the stars in CU Cnc destroy
203
lithium in about 20 Myr independent of the magnetic field. This can be explained by the
depth of the convective envelope in the stars of CU Cnc. At ∼ 0.4M⊙ , the stars are expected
to have deep convection zones that extend from the stellar surface down to about 50% of
the stellar radius. The depth at which lithium is destroyed (T = 2.5 × 106 K) is located at
about 70% of the stellar radius. This is illustrated in Figure 6.11. Lithium will be mixed
down to the base of the convection zone and be completely destroyed. Introducing a strong
magnetic field (3.5 kG) reduces the size of the convection zone in CU Cnc A by 7% and by
9% in CU Cnc B. However, to preserve lithium the size of the convection zone would have
to be reduced by nearly 30%, corresponding to the base of the convection zone moving from
50% to 65% of the total stellar radius. If lithium does exist on the surface of CU Cnc, then
there is another process keeping lithium from being destroyed.
We conclude this section on CU Cnc by returning to the photometric issues raised by
both Delfosse et al. (1999) and Ribas (2003). Since the publication of Ribas (2003), the
Hipparcos parallax data has been revised and updated to provide more accurate solutions
(van Leeuwen 2007). The parallax for CU Cnc underwent a revision from π = 78.05 ± 5.69
mas to π = 90.37±8.22 mas, changing the distance estimate from 12.81±0.92 pc to 11.07±
1.01 pc. The absolute magnitudes must be adjusted for this revised distance. Using the
V - and K -band magnitudes listed in Weis (1991) and the 2MASS survey (Cutri et al. 2003),
we find MV, A = 12.27 mag, MV, B = 12.63 mag, and an integrated MK = 6.382 mag. From
SIMBAD we can also obtain integrated color: (J − K ) = 0.906 and (H − K ) = 0.291,4 drawn
from the 2MASS survey.
Using a theoretical color-Teff transformation (Chapter 2; Dotter et al. 2008) we converted
model surface properties to photometric magnitudes and colors. We were not able to reproduce the set of integrated colors and magnitudes or the individual V -band magnitudes
using super-solar metallicity models alone. However, combining a super-solar metallicity
4
The integrated V -band magnitude listed on SIMBAD is 0.2 mag fainter than is quoted by Weis (1991).
This is likely because the photometry listed in SIMBAD was taken during eclipse where the V -band flux drops
by ∼ 0.2 mag (Ribas 2003).
204
with a magnetic field, we were able to produce models showing the appropriate trends: total V - and K - band magnitudes were reduced due to the decrease in luminosity associated
with the magnetic field with a steeper decrease in the V -band due to increased metallicity
(greater TiO opacity) and decreased Teff due to the metallicity and the magnetic field. The
final photometric properties of our models did not exactly match the properties of CU Cnc.
However, we note there is considerable uncertainty in the color-Teff transformation using
the phoenix ames-cond theoretical models (Baraffe et al. 1998; Delfosse et al. 1998), particularly in the V -band. A larger exploration of the model parameter space and upgrading to
the latest phoenix bt-settl models would help to determine if metallicity and magnetic
fields are able to resolve the CU Cnc photometric anomalies.
6.3 Magnetic Field Strengths
6.3.1 Surface Field Strengths
Section 6.2 demonstrates that the magnetic models can reconcile low-mass stellar model
predictions with observations of stars in low-mass DEBs. The real predictive power of the
models relies on the ability of the models to do so with realistic magnetic field strengths tied
to physical observables. As with the case study of EF Aqr presented in Chapter 5, we may
estimate magnetic field strengths using X-ray emission as an auxiliary measure. For the
stars studied in this chapter, estimates based on Ca ii emission cores (Schrijver et al. 1989)
will not be possible because Ca ii line profiles are not available.
Estimates of magnetic field strength can be obtained using the scaling relation derived between magnetic flux and the total X-ray luminosity (Fisher et al. 1998; Pevtsov et al. 2003).
The relation appears to extend over several orders of magnitude, from individual solar active
regions up to hemispherical averages of stars. The basic method was presented in Section
5.6.1 using Equations (5.189), (5.190), and (5.191).
205
Table 6.2
X-ray properties for the three DEB systems.
DEB
X cr
HR
System
(cts s−1 )
UV Psc
0.92 ± 0.07
−0.10 ± 0.07
0.73 ± 0.05
−0.14 ± 0.06
YY Gem
CU Cnc
3.70 ± 0.09
π
Nstars
(erg s−1 )
(mas)
14.64 ± 1.03
d = 13 ± 2 pc
−0.15 ± 0.02
90.37 ± 8.22
Lx
2
6
4
(2.0 ± 0.3) × 1030
(9.4 ± 0.2) × 1028
(2.0 ± 0.2) × 1028
Note: Nstars is the total number of stars thought to be contributing to the total X-ray counts
detected by ROSAT. The value L x is quoted as the X-ray luminosity per star in the system.
X-ray properties of the three DEB systems studied in this chapter are determined using
X-ray data from the ROSAT All-Sky Survey Bright Source Catalogue (Voges et al. 1999),
consistent with our previous X-ray analyses in Chapters 4 and 5. ROSAT count rates ( X cr )
and hardness ratios (HR) are given in Table 6.2. We convert to X-ray fluxes using the
calibration of (Schmitt et al. 1995). The conversion to X-ray fluxes is complicated by the
fact that ROSAT has relatively poor spatial resolution, meaning any nearby companions to
these DEBs may also be contributing to X cr .
The X-ray flux quoted in Table 6.2 is therefore calculated as the estimated flux per star,
determined by dividing the total flux by the number of stars thought to be contributing to
X cr . This is not a problem for UV Psc, which appears isolated. YY Gem and CU Cnc, on
the other hand, have known, nearby companions. A search of the ROSAT Bright Source
Catalogue for Castor A and B yield the same data as is found when searching for YY Gem,
indicating that ROSAT cannot spatially resolve any of three systems. Castor A and B are
both binaries, thought to have M-dwarfs companions, meaning that ROSAT is detecting
X-ray emission from up to six sources. Similar to YY Gem, CU Cnc has a 12" companion,
CV Cnc, another M-dwarf binary system. Both systems are likely contributing to the X cr
listed in the ROSAT Bright Source Catalogue.
Distances to the systems are calculated using updated Hipparcos parallaxes (van Leeuwen
2007), except for YY Gem, for which no parallax is available. Instead, we adopt a distance of
206
13 pc, which was used earlier in Section 4.5.2. The distances then allowed for the calculation
of the total X-ray luminosity per star, quoted in Table 6.2.
Converting X-ray luminosity measures to an estimate of the magnetic field strength requires
first transforming the X-ray luminosity into an estimate of the surface magnetic flux, Φ. We
discussed this relation in Section 5.6.1, where we adopted the Pevtsov et al. (2003) relation
based on the work of (Fisher et al. 1998). Pevtsov et al. (2003) found that
L x ∝ Φ1.15 ,
(6.2)
as was mentioned in Section 5.6.1. The adopted power-law index depends on the particular
data set used in the analysis. For instance, Pevtsov et al. (2003) found a power-law index
p = 0.98 when they only considered the dwarf stars supplied by Saar (1996). When the took
the entire solar (quiet regions, active regions, sun spots) and stellar data sets together, they
found p = 1.15. The latter extends over twelve orders of magnitude in Φ and L x .
The question of which power law index to adopt is important and can strongly influence
the magnetic flux derived from X-ray luminosities. Seeing as we are focused on deriving
the approximate magnetic flux for dwarf stars, is seems a natural choice to use the relation
specifically derived from dwarf star data. However, the relation derived by Pevtsov et al.
(2003) was based on data from Saar (1996). Since then, many more stars have had their
magnetic field strengths measured, allowing us to expand the data set and derive a more
reliable scaling relation.
We computed the approximate surface magnetic flux for the dwarf stars presented in the
Reiners (2012) review. Reiners (2012) attempted to collect all of the reliably known magnetic
field measurements for cool dwarfs stars. The stars all had their magnetic field strengths
measured directly using either Stokes I or Stokes V observations. We selected only those
stars that had their fields measured using Stokes I , which is thought to yield a more accurate
estimate of the average surface magnetic field strength 〈B f 〉 (Reiners & Basri 2009), which
207
26.5
26.0
log10(Φ) (Mx)
25.5
25.0
24.5
24.0
23.5
Pevtsov et al. (2003)
23.0
22.5
26.5
This work
27.0
27.5
28.0
28.5
29.0
-1
log10(Lx) (erg s )
29.5
30.0
30.5
Figure 6.12: Stellar surface magnetic flux (Φ) as a function of total stellar X-ray luminosity for a collection of G-, K-, and M-dwarfs. Gray points are taken from Pevtsov et al.
(2003) while the maroon points were calcuated for this work. Three separate linear regressions are presented: P = Pevtsov et al. (2003) for dwarf stars only (light-gray, dotted), T =
Pevtsov et al. (2003) for (black, solid) the total comibnation of solar and stellar data, and R =
data compiled for this work drawn from the Reiners (2012) sample (maroon, dashed).
is the quantity predicted by our models.
Results of extending the sample with are presented in Figure 6.12 with the full data set
given at the end of the chapter in Table 6.5. The list of objects from the Reiners (2012) review were cross-correlated with the ROSAT Bright and Faint Source Catalogues (Voges et al.
1999; Voges et al. 2000) to extract X-ray count rates and hardness ratios. If objects did not
have an X-ray counterpart they were excluded from the data set. We then identified all of
the stars in the remaining subset that had parallax estimates from Hipparcos (van Leeuwen
2007). The final subset consisted of 25 objects with both X-ray counterparts and parallax
measurements. Two additional data points for G-dwarfs not found in Reiners (2012) review
were also included (Anderson et al. 2010) bringing the total number of objects to 27. We
208
also see in Figure 6.12 that the data points derived for this study follow the same general
trend as those in Pevtsov et al. (2003).
We caution that the data presented in Figure 6.12 suffer from uncertainty in the adopted
stellar radius, which is largely based on spectral type, and the limited spatial resolution of
ROSAT. As an example, CV Cnc is an mid-M-dwarf binary included in our final data set.
Estimates of it’s X-ray luminosity include contributions from CU Cnc, as discussed above.
Only for CV Cnc did we correct for a known binary companion. The uncertainty in the
stellar radius only affects the magnetic flux and the lack of spatial resolution only affects
the X-ray luminosity. Therefore, we believe that the estimates are robust, particularly given
that Figure 6.12 is plotted using logarithmic units, where a factor of two does not contribute
to a significant shift in the data points.
The regression line and associated uncertainty limits shown in Figure 6.12 were obtained
by performing an ordinary least squares (OLS) regression on the data. A standard OLS was
selected because we are not attempting to establish a causal relationship between Φ and
L x (Isobe et al. 1990). Our primary concern is developing a relationship that permits the
prediction of Φ given a value for L x . Following the recommendations of Isobe et al. (1990)
¡
¢
and Feigelson & Babu (1992), we perform an OLS log10 Φ| log10 L x , where log10 L x is the
predictor variable and log10 Φ is the variable to be predicted. Notice that we transformed the
variables to a logarithmic scale because the data extends over several orders of magnitude
in Φ and L x .
The result of the OLS analysis yielded the regression line
¢
¡
log10 Φ = (24.873 ± 0.004) + (0.459 ± 0.018) log10 L x − µx ,
(6.3)
where µx is the mean value of log10 L x taken over all of the data points in the sample. The
shift of the dependent variable was performed before the regression analysis. By doing this,
we were able to minimize the error associated with the y -intercept. However, the standard
209
Table 6.3
Surface magnetic field properties of the stars in UV Psc, YY Gem, & CU Cnc
DEB
log Φ
Star
(Mx)
UV Psc A
UV Psc B
YY Gem A
〈B f 〉
〈B f 〉model
25.77 ± 0.45
0.79+1.43
−0.51
2.0
25.16 ± 0.45
0.62+1.13
−0.40
4.3
(kG)
..
.
YY Gem B
..
.
CU Cnc A
24.85 ± 0.45
CU Cnc B
..
.
(kG)
1.39+2.53
−0.90
4.6
..
.
..
.
0.62+1.13
−0.40
4.0
0.76+1.38
−0.49
3.6
deviation of the mean, σx , becomes the largest source of error in the regression analysis.
We found µx = 28.34 ± 0.97 where we took
v
u
u
σx = t
N
1 X
(x i − µx )2 .
N − 1 i =1
(6.4)
We opted for computing the standard deviation with N − 1 because of our small sample
(27 data points) and to provide conservative estimates of the associated errors. The error
associated with the standard deviation of the mean and the errors associated with the OLS
coefficients in Equation (6.3) produce an uncertainty region about the regression line. We
show this with the outlined shaded region in Figure 6.12. Finally, we also note that Equation
(6.3) is strictly valid for 26.5 < log10 L x < 30.0, corresponding to the range of the data.
Our estimates for the DEB surface fluxes, surface magnetic field strengths, and their errors
are given in Table 6.3. With a non-causal relationship established between Φ and L x , we
convert the X-ray luminosities for our DEB stars presented in Table 6.2 to estimates of
surface magnetic flux. Care was taken to properly propagate the errors so that we have an
estimate of the uncertainty of our predictions. Again, the largest source of error was the
standard deviation of the mean in the shift of the dependent variable. This translates into
210
26.5
26.0
log10(Φ) (Mx)
25.5
25.0
24.5
24.0
23.5
Pevtsov et al. (2003)
This work
Model Predictions
23.0
22.5
26.5
27.0
27.5
28.0
28.5
29.0
-1
log10(Lx) (erg s )
29.5
30.0
30.5
Figure 6.13: Same as Figure 6.12 except that the results of our theoretical models are included (light-blue stars). Magnetic fluxes are derived from the results in Sections 6.2.1, 6.2.2,
and 6.2.3 while X-ray luminosities are computed from ROSAT data.
large uncertainties when we calculate the surface magnetic field strengths from the surface
fluxes.
The X-ray data reveal that the magnetic field strengths required by our models are probably
too large. We demonstrate the this visually in Figure 6.13, where we use the model surface
magnetic field strengths to compute a surface magnetic flux. The data suggest that the only
realistic magnetic fields strengths are those of UV Psc. However, the data points for UV
Psc lie outside the strict range of the data set, meaning we must assume the extrapolation
is valid. Even if we assume the extrapolation is valid, UV Psc B lies just above the error
bounds of the linear correlation. Conversely, UV Psc A lies just inside the bounds. We must
therefore be cautious with our interpretation of the accuracy of our models for UV Psc. This
is further reinforced by nothing that only two G-dwarfs (the spectral of UV Psc A) were
included in the data set and are therefore under represented. We do mention that estimates
211
of G-dwarf magnetic field strengths by Saar (1990) show that magnetic field strengths may
be on the order 500 G, consistent with the values derived for UV Psc A from the X-ray
luminosity relation.
What is clear from Figure 6.13 and Table 6.3 is that the magnetic field strengths required by
the models are too large for the stars in YY Gem and CU Cnc. The models of YY Gem need a
magnetic field strength that is around 6 times too strong. Similarly large magnetic fields are
needed by the models for the stars of CU Cnc. A more qualitative approach to evaluating
YY Gem and CU Cnc would lead us to believe that our magnetic field strengths are entirely
reasonable. Numerous studies of M-dwarfs find that surface magnetic field strengths of a
few kilogauss are quite common (Saar 1996, 2001; Reiners & Basri 2007, 2009). However,
these DEBs would then have noticeably stronger magnetic fields for their observed X-ray
luminosity than the rest of M-dwarf population. It appears that our approach to modeling
magnetic fields in low-mass stars may be incomplete.
6.3.2 Interior Field Strengths
Assessing the validity of our predicted interior magnetic field strengths is inherently more
difficult. Without any real measure currently available for the magnetic field strengths
present inside stars, we elect to compare the theoretical magnetic field strengths required by
our models with those predicted by 3D MHD models (see, e.g., Brandenburg & Subramanian
2005).
Table 6.4 presents the peak interior magnetic field strengths for the models presented above
for both the dipole and the Gaussian field profiles (see below). The peak magnetic field
strengths in Table 6.4 are pre-defined to be at the base of the stellar convection zone.
They are all on the order of a few times 103 to a few times 104 gauss. By comparison,
3D MHD models routinely find peak magnetic field strengths of a few times 103 gauss
(consistent with equipartition estimates) to 105 gauss for solar-like stars (see review by
212
Table 6.4
Peak interior magnetic field strengths.
DEB
|B |dipole
|B |Gaussian
UV Psc A
4
40
UV Psc B
12
400
YY Gem A
YY Gem B
13
..
.
500
..
.
CU Cnc A
21
1500
CU Cnc B
21
2000
Star
(kG)
(kG)
Brandenburg & Subramanian 2005). In the immediate context, “solar-like” is loosely taken
to mean stars with a radiative core and a solar-like rotation profile. The peak magnetic field
strengths of our models then appear consistent with those predicted from 3D MHD models.
Furthermore, while helioseismological investigations have yet to reveal the interior magnetic field profile for our Sun, initial indications suggest the peak magnetic field strength is
below 300 kG (Antia et al. 2000). Estimates place the strengths in the vicinity of several tens
of kG (Antia et al. 2003).
6.4 Reducing the Magnetic Field Strengths
Given the results that the model surface field strengths are likely too large, we seek to
reformulate our magnetic perturbation. Let us first consider some of the assumptions used
to formulate our magnetic models in Chapter 5. We have identified four key assumptions:
(1) our prescribed magnetic field radial profile, (2) the plasma obeys the equations of ideal
magnetohydrodynamics (MHD), (3) our neglect of electric fields, and (4) the dynamo is
driven purely by rotation.
213
6.4.1 Magnetic Field Radial Profile
All of the models presented up to this point, we have prescribed what we called a “dipole
radial profile.” The interior magnetic field is set to a maximum at the boundary of the
convection zone and falls off as r 3 towards the core and surface of the star. What if we
instead used a much steeper profile so that the peak magnetic field strength is stronger at
the base of the convection zone for a given magnetic field strength?
To this end, we define a Gaussian radial profile. Again, the peak field strength is prescribed
in the manner described above. However, instead of the field falling off as r 3 , the source
location represents the peak of a Gaussian distribution. Functionally, this may be expressed
as
µ
¶ ¸
·
1 R src − R 2
B (R) = B (R src ) exp −
2
σg
(6.5)
where the width of the Gaussian, σg , is an additional free parameter which we will discuss
in a moment, Rsrc is the radius of the peak magnetic field strength, as before. The advantage
of introducing a Gaussian profile is that the field strength increases much more rapidly
towards the source location. This rapid increase allows the field strength at the model
photosphere to be weaker than in the dipole profile for a given value of the peak magnetic
field strength.
Determining the width of the Gaussian is arbitrary. In most applications, it would seem
reasonable to set the width of the Gaussian to a constant value. A value of σg = 0.2 was
favored by Lydon & Sofia (1995), who parameterized their Gaussian as a function of mass
depth, instead of radius. However, since we are to be considering stars with a wide variety of
convection zone depths and not just the Sun, a single parameter would not be appropriate.
Our approach is to correlate the width of the Gaussian with the depth of the convection
zone. Parameterizing the width in this way allows the magnetic field to be localized in the
convection zone. Or, as for fully-convective stars, spread throughout the entire convective
214
1.2
100
15
(a)
80
4
40
2
|B|surf = 2.0 kG
0.8
10
0.4
5
Gaussian Scaled |B| (MG)
60
(b)
0.4 M⊙
Gaussian Scaled |B| (kG)
Dipole Scaled |B| (kG)
1.0 M⊙
|B|surf = 2.0 kG
Dipole Scaled |B| (kG)
6
20
0
0.0
0
0.0
0
0.2
0.4
0.6
0.8
1.0
Radius Fraction
0.2
0.4
0.6
Radius Fraction
0.8
0.0
1.0
Figure 6.14: Comparison of the dipole (light blue, dashed) and Gaussian (maroon, solid)
radial profiles for the interior magnetic field strength. (a) a 1.0M⊙ model. (b) a 0.4M⊙
model. Note that the two field profiles are plotted on different scales with the dipole on the
left y -axis and the Gaussian on the right y -axis.
region. Fixing the width of the Gaussian to σg = 0.2 in fully-convective objects and σg = 0.1
in the Sun, we find
σg = 0.2264 − 0.1776ζ
(6.6)
where ζ = R src /R ∗ is the dynamo source location in fractional units. A direct comparison
of the shape of the magnetic field profiles used in this study is given in Figure 6.14. Two
masses are shown to make clear the variable width of the Gaussian.
Figure 6.15 illustrates the influence of using the Gaussian radial profile. We plot the relative
difference in the radius evolution between the Gaussian and dipole profiles for a series of
stellar masses. All of the models have an equivalent surface magnetic field strength of 2.0
kG and solar metallicity. Despite the increased magnetic field strength at the tachocline (see
Table 6.4), using a Gaussian magnetic field profile instead of a dipole profile has a sub-1.5%
effect on model radius predictions. Therefore, we find no compelling reason to alter our
default field strength profile.
Two additional comments on the results presented in Figure 6.15. The first comment concerns the fact that the different radial profiles produce similar results. We see this despite
215
(RGaussian - Rdipole)/Rdipole
0.030
1.1 M⊙
0.025
1.0 M⊙
0.9 M⊙
0.020
0.8 M⊙
0.7 M⊙
0.015
0.6 M⊙
0.5 M⊙
0.4 M⊙
0.010
0.005
0.000
-0.005
7.0
7.5
8.0
8.5
9.0
log10(Age/yr)
9.5
Figure 6.15: The effect on stellar radius predictions caused by using a Gaussian radial profile
instead of a dipole profile. Shown is the relative radius difference as a function of age for a
series of stellar models. All of the models have equivalent surface magnetic field strenghts
(2.0 kG) but different magnetic field radial profiles.
the peak magnetic field strength in the Gaussian formalism being an order of magnitude—
at least—larger than in the dipole formalism. It would then seem that the deep interior
field strength is relatively unimportant in stars with radiative cores. Instead, the surface
magnetic field strength appears to be responsible for driving the radius inflation.
To understand why, we plot the difference between the temperature gradient, ∇s , and the
adiabatic gradient, ∇ad , as a function of density in Figure 6.16. Consider the non-magnetic
model. When the line dips below the zero point, radiation carries all of the excess energy.
Near the stellar surface, where ρ is small, ∇s is noticeably super-adiabatic, indicative of
inefficient convective energy transport. Deeper in the star, where −5 < log10 ρ < 0, the
temperature gradient is super-adiabatic, but only slightly (∇s − ∇ad < 10−8 ). This suggests
convective energy transport is highly efficient. Our approach modifies ∇ad by a factor
proportional to (ν∇χ ). Deep in the stellar interior, ∇χ ∼ 0.1, and ν ∼ 10−8 − 10−9 . This is
not sufficient to inhibit convection deep within the star. Near the surface, however, any
216
Non-magnetic
Magnetic
0.2
super-adiabatic
convection
(∇s - ∇ad)
0.1
near-adiabatic convection
0.0
0.04
radiative
zone
0.00
-0.1
-0.04
-0.08
-6.5
-6.0
-5.5
-0.2
-7
-6
-5
-4
-3
-2
log10(ρ) (g cm-3)
-1
0
1
2
Figure 6.16: The difference between the plasma temperature graident, ∇s , and the adiabatic
temperature gradient, ∇ad , as a function of the logarithmic plasma density for a M = 0.6M⊙
star. We show this for two models: a non-magnetic model (maroon, solid line) and a magnetic model (light-blue, solid). The zero point is marked by a gray dashed line, dividing
locations where convection (positive) or radiation (negative) is the dominant flux transport
mechanism. The inset zooms in on the near-surface region where radiation becomes the
dominant flux transport mechanism.
inhibition of convection causes a steepening of ∇s , forcing radiation to carry more energy.
We would then expect to see the region near the stellar surface where radiation carries all
of the flux grow. This is precisely what we see in Figure 6.16. The structural changes caused
by the steep temperature gradient near the surface are enough to reconcile the models with
the observations before the deep interior magnetic field strength becomes appreciable in
magnitude. Therefore, the outward movement of the convection zone boundary occurs
largely as a response to changes near the stellar surface.
Second, we see in Figure 6.15 that the 1.1M⊙ and 1.0M⊙ lines display an upturn. Occurring
near 1 Gyr, the models are too young to be undergoing rapid evolutionary changes. Instead,
we believe this is related to the physical properties of the stars near the boundary of the
217
convection zone. As we just discussed, the magnetic field strengths for both profiles are
typically too weak near the tachocline to affect stellar structure. However, the models indicate that the opposite is true for the 1.1M⊙ and 1.0M⊙ stars. The convection zones are thin
and convection is generally super-adiabatic throughout much of the convection zone. This
results in the Gaussian radial profile having a stronger effect on the convection properties.
It also explains why the Gaussian radial profile has a weaker influence on the stellar radius
as we decrease the stellar mass. Convection in the deep interior becomes more adiabatic (re:
efficient) at lower masses.
6.4.2 Finite Electrical Conductivity
Assuming that the stellar plasma has an infinite conductivity, and thus obeys the equations
of ideal MHD, has several implications. Therefore, it is advantageous to probe what happens
when we assume a finite electrical conductivity. Generally speaking, the introduction of a
finite electrical conductivity inhibits the generation of favorable current networks, making
it more difficult for a dynamo mechanism to generate a magnetic field compared to the case
of ideal MHD (Mohanty et al. 2002).
While magnetic field strengths would be decreased due to the plasma’s inability to set up
strong current networks, finite electrical conductivity affects the magnetic inhibition of convection. This has been treated previously by MacDonald and Mullan in a series of papers
(MacDonald & Mullan 2009; Mullan & MacDonald 2010; MacDonald & Mullan 2010, 2013)
where they applied an analytical correction factor to their magnetic suppression term discussed in Section 1.4.1. The correction factor acted to reduce the effect of their magnetic
inhibition parameter when the plasma resistivity was sufficiently high. Although important, this would ultimately force us to increase our magnetic field strengths to offset the
reduction in magnetic inhibition.
The most significant implication for our purposes is that assuming a perfectly conducting
218
plasma lead to the definition that f ≈ 1 (see Section 5.2.5). Recall that f parametrizes the
flux of magnetic energy between a convecting bubble and the surrounding plasma. If we
assume a finite electrical conductivity, then f = 1 is no longer strictly enforced. This has the
effect of introducing the Lorentz force into the magnetic buoyancy equation.
Results of changing the parameter f from f = 1 to f = 0 are shown in Figure 6.17. As with
Figure 6.15, we plot the relative difference in the radius evolution for a series of models. All
of the models were calculated with a 2.0 kG magnetic field and a dipole field profile (see
the previous section). The largest mass model shown is 0.7M⊙ because models with larger
masses failed to converge for a 2.0 kG magnetic field.
It is evident from Figure 6.17 that lowering the value of f can have a significant impact
on the model radius evolution. Differences between 2% and 9% were observed, depending
on the model mass. Therefore it is worthwhile to study the effect of setting f = 0 on the
individual DEB stars discussed above.
We tested what happened when we set f = 0 for each of the stars discussed in Section 6.2. In
each case, the strength of the surface magnetic field is reduced by nearly a factor of two. YY
Gem, for instance, now only requires a magnetic field strength of ∼ 2 500 G at the surface
instead of the 4 300 G required when f = 1. However, when we compare with the X-ray
predicted surface magnetic field strengths, the f = 0 model values are still too large by a
factor of 5, save UV Psc, but we have already stated reason to be skeptical of the X-ray
predicted values for UV Psc.
Whether or not we have reason to believe that f should be zero is a more difficult question. The stellar plasma should be highly conducting throughout most of the star because
hydrogen is completely ionized at temperatures above ∼ 104 K. In the near surface regions,
hydrogen is no longer ionized. Since finite electrical conductivity is driven by the presence
of neutral species, this would seem to favor a lower f value. But, there are other species
which contribute to the total free electron population (e.g., Ca and Na), meaning the plasma
219
0.10
0.7 M⊙
0.6 M⊙
(Rf = 0 - Rf = 1)/Rf = 1
0.08
0.5 M⊙
0.4 M⊙
0.06
0.04
0.02
0.00
7.0
7.5
8.0
8.5
9.0
log10(Age/yr)
9.5
Figure 6.17: The influence of the parameter f on model radius predictions. Plotted is the
relative difference in radius evolution between magnetic models computed with f = 1 and
f = 0. All models have a prescribed surface magnetic field strength of 2.0 kG.
is still at least partially conducting. Therefore, it is likely that near the surface f 6= 1, but
the fact that the plasma is conducting suggests that 0 < f < 1. Deriving a more realistic
value for f would be a worthwhile endeavor. We plan to investigate this free parameter in
a future study.
Ultimately, reducing f aides our modeling work, but it does not provide enough leverage
to bring our models into agreement with the observations. We then seek another method
through which we can reduce our predicted surface magnetic field strengths.
6.4.3 Presence of E-fields
It was discussed in Feiden & Chaboyer (2012b) that the presence of electric fields were neglected due to rapid neutralization of excess charge in stars. We still believe that this is a
valid assumption. Small-scale electric fields may be permitted, but it is still true that the
220
plasma is electrically neutral on scales larger than the Debye length. The Debye length is
λD =
s
kTe
4πn e q e2
≈
s
kTe m H
4πρq e2
,
(6.7)
where kTe is the temperature of the electron population, ne is the number density of electrons, and q e is the electron charge. For a plasma with a temperature of 1 eV at a density
of about 1 g cm−3 , λD ∼ 10−8 cm. We therefore feel justified in neglecting the presence of
large-scale electric fields. If we also then assume that the stellar plasma is highly conducting,
there will be little heat loss to ohmic dissipation.
6.4.4 Dynamo Energy Source
The final assumption that we identified earlier in this section was that the magnetic field is
driven purely by rotation. By doing so, we have allowed for the unintended consequence
that the magnetic field strength—both the surface and the interior—can grow without limit.5
There is no “natural” limit imposed upon the field strengths as the mechanism from which
the field is drawing energy (i.e., rotation) is neglected entirely.
Limiting the magnetic field strengths, however, does not modify in any way the results of
the models. It can only validate or invalidate the field strengths the models require (see
Section 6.3.2). Instead, we reconsider the physical source of the stellar dynamo. There are
questions as to whether the solar magnetic field is generated at the tachocline by the strong
shear induced by rotation (i.e., the standard Parker model; Parker 1955) or if it is primarily
generated within the solar convection zone by turbulent convection without explicit need
for a tachocline (Brandenburg & Subramanian 2005; Brown et al. 2010).
Consequently, the generation of a magnetic field from turbulent convection would suppress
convective velocity fluctuations thereby reducing the total heat flux transported by convec5
We note that there is a computational limit whereby too strong of a magnetic field will prevent the model
from converging to a solution.
221
tion. Early magneto-convection simulations in three-dimensions support this assessment
(Stein et al. 1992). Given that thick convective envelopes are a ubiquitous feature of lowmass stars, it is not unreasonable to posit that suppressing the total heat flux transported by
convection would strongly impact stellar structure.
Assuming that the magnetic field sources its energy directly from the kinetic energy of
turbulent convection has important consequences for a magnetic theory of convection.6
Consider a single convecting bubble. Beginning with the conservation of energy, we must
have that
1 2
B2
1 2
ρu conv, 0 = ρu conv
+
,
2
2
8π
(6.8)
where ρ is the mass density, uconv, 0 is the convective velocity prior to the introduction of
a magnetic field, uconv is the convective velocity after generation of a magnetic field, and
the final term is the magnetic energy. We have implicitly assumed that a characteristic
convecting bubble is responsible for generating the magnetic field in its vicinity. This local
treatment is an extreme oversimplification, but is a zeroth order approximation given the
framework of mixing length theory.
The result is that the characteristic convective bubble will have a lower velocity in the
presence of magnetic field,
2
2
u conv
= u conv,
0−
B2
,
4πρ
(6.9)
by a value equal to the square of the local Alfvén velocity,
u 2A =
B2
= 2χ.
4πρ
(6.10)
3
The total convective flux is proportional to uconv
, meaning that the magnetic field can sig-
nificantly alter the total flux transported by convection. To make up for the decreased
convective flux, the radiative temperature gradient will grow steeper by an amount propor6
The theory advanced here grew out of an evening of intense discussions and derivations with Axel
Brandenburg during a visit to NORDITA amidst the author’s stay in Uppsala, Sweden.
222
tional to u 2A .
There are a two different approaches that we can take to include the energy conversion
from turbulent kinetic energy to magnetic energy. The first involves using the new definition for the convective velocity give by Equation (6.9) and re-deriving the equations for
mixing length theory given in Section 5.2.4. After modifying the convective velocity and
the convective flux (to remove a term introduced by the new definition of the convective
velocity) we find that the final quartic becomes
h
i
ν
2Ay 4 +V y 3 +(1+2AC γ20 Ξ)V 2 y 2 +(γ20V 2 u 2A −1)V y + u 2A − (1 − f )∇χC γ20V 4 = 0. (6.11)
δ
The additional terms introduced are those proportional to u 2A . Unfortunately, when we
then attempt to find the root of Equation (6.11) we are unable to obtain convergence. Closer
inspection of the coefficients in the equation reveals that there is no real root in the outer
envelope. The low density present in the envelope drives up u 2A , which causes the range of
Equation (6.11) to lie above zero for all real values of y . We also attempted to modify the
“non-adiabaticity” equation, Equation (5.127), so as to model the transformation of turbulent
kinetic energy into magnetic energy as a heat loss. However, we were again unable to
converge upon a real root for the quartic equation. Further investigation into the precise
cause of non-convergence is required.
The second method we have identified is less consistent, but provides a first-order approximation. If we first solve the MLT equations as normal, we can modify the results to mimic
the conversion of turbulent kinetic energy to magnetic energy. The convective velocity is
reduced by the Alfvén velocity, as in Equation (6.9). The reduction in the convective velocity therefore reduces the amount of flux carried by convection. While energy is transferred
to the magnetic field, the energy is still confined to the local region under consideration, reducing the total flux of energy across the region. Assuming that flux is conserved, flux must
then be carried by radiation, which leads to an increase in the local temperature gradient of
223
(Rconv - Rrot)/Rrot
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
7.0
1.0 M⊙
0.9 M⊙
0.8 M⊙
0.7 M⊙
0.6 M⊙
0.5 M⊙
0.4 M⊙
7.5
8.0 8.5 9.0
log10(Age/yr)
9.5
10.0
Figure 6.18: The effect on model radius predictions when the dynamo source is turbulent
convection instead of purely rotation. Shown is the relative radius difference as a function
of age for a series of stellar models. All of the models have an equivalent surface magnetic
field strenghts of 0.5 kG.
the ambient plasma, further increasing the temperature excess over the convecting element.
The increase in temperature excess is proportional to the total energy removed from the
convecting element in the form of magnetic energy,
∆(∇s − ∇e ) ∝
u 2A
C
,
(6.12)
where C was defined in Equation (5.130). Increasing the temperature excess will create feedback onto the convective element, causing it to increase its velocity. However, since we are
treating the reduction in convective flux after solving the MLT equations, we do not concern ourselves with the feedback on the convective velocity. This would have presumably
been accounted for in the first approach we outlined.
We again computed a series of models using this new modified MLT to compare the relative
differences in radius evolution. These are shown in Figure 6.18. We assumed a 500 G
224
surface magnetic field strength with a dipole radial profile and f = 1. The 0.8M⊙ , 0.7M⊙ ,
and 0.9M⊙ tracks in Figure 6.18 show slightly different characteristics. Models assuming
a turbulent dynamo readjusted to the presence of the magnetic field at an older age than
did the models with a rotational dynamo; all perturbations were introduced at the same
age, τage = 0.1 Myr. This new formulation of the magnetic models, whereby we source the
magnetic field energy from the convective flux, has the potential to significantly reduce the
magnetic field strengths required to inflate the model radii.
We recomputed models for UV Psc, YY Gem, and CU Cnc using the best fit metallicities
found in Section 5.2. We were only able to achieve model convergence for the stars in YY
Gem and CU Cnc. Models of UV Psc with surface magnetic fields up to 650 G did converge,
but field strengths greater than that were needed to reconcile the models. The resulting
magnetic fields strengths required for YY Gem and CU Cnc A and B were 0.7 kG, 0.7 kG,
and 0.8 kG, respectively. We also attempted to recompute models for EF Aquarii (EF Aqr)
presented in (Feiden & Chaboyer 2012b), but ran into the same convergence issues as with
UV Psc.
The surface magnetic field strengths derived from sourcing the magnetic field energy from
convection are nearly identical to the X-ray estimated values. This agreement lends credence to our latest approach, at least for lower mass models. We found it difficult to attain
model convergence using this approach for models with masses greater than about 0.75M⊙ .
However, below that approximate mass limit the turbulent dynamo approach can inflate
stellar radii with relative ease compared to our original formulation. Above 0.75M⊙ , more
investigation examining the structure of the convection zones in these stars is required to
diagnose the convergence issues. The solution could be that the radial profile produces
magnetic field strengths that are simply too large compared to the predicted convective
velocities.
225
6.5 Further Discussion
6.5.1 Interior Structure
We saw in the previous section that there are two approaches to modeling the effects of magnetic fields: stabilizing the plasma against convection and reducing convective efficiency.
These approaches hint at different physical mechanisms by which stars produce magnetic
fields. They also hint at the dominant mechanism responsible for inflating stellar radii (see
Section 6.5.2). It is therefore worth exploring whether or not the different approaches affect
stellar interior structure—and as a consequence observable properties—in any discernible
manner.
Figure 6.19 shows the run of density and the development of the parameter η2 (R) through
the model interior. Recall from Equations (3.7) and (3.9) that η2 is related to the stellar
deviation from sphericity and the interior structure constant, k 2 . All three models were
computed with a mass of 0.599M⊙ , a metallicity of −0.20 dex, and were halted at an age of
370 Myr. The non-magnetic model we know has a radius smaller than the actual radius of
YY Gem (here considered a single star). The two magnetic models, on the other hand, were
calculated with the surface magnetic field strength required to reconcile the model radius
with the observational radius (4.3 kG and 0.7 kG for the LS95 prescription and turbulent
dynamo, respectively).
The presence of a magnetic field causes the density profile to be steeper deep within the
star, although the central density remains nearly constant. However, there is no noticeable
difference between the two magnetic models, despite having two vastly different magnetic
field strengths throughout the star. Given that the density profile is steeper, the density is
considered more centrally condensed, meaning the interior structure constants should be
smaller in the magnetic case than in the non-magnetic case. This is confirmed by the curves
of η(R) in Figure 6.19, which are larger throughout the magnetic models. This leads to a
226
80
Non-magnetic
LS95 prescription
2
60
η2(R)
Density, ρ(R) (g cm-3)
Turbulent dynamo
40
1
M = 0.599 M⊙
20
[Fe/H] = -0.20
Age = 370 Myr
0
0.0
0.2
0.4
0.6
0.8
0
1.0
Fractional Radius (R/R*)
Figure 6.19: Interior density profile for a M = 0.599M⊙ model with and without the presence of a magnetic field. Also shown is the η2 parameter, related to the stellar deviation
from sphericity and the interior structure constant, k 2 . The magnetic models were computed with a surface magnetic field strength strong enough to reconcile the model radius
with the observed radius of YY Gem (4.3 kG and 0.7 kG for the LS95 prescription and turbulent dynamo approach, respectively). Note that the two lines for the magnetic models
directly on top of one another.
lower value of k 2 because k j is inversely proportional to η j at constant j ,
¯
∂k j ¯
j + 2η j
¯ = −¡
¢2 ,
¯
∂η j j
j +ηj
(6.13)
where we have differentiated Equation (3.9). We confirmed that the value of k 2 in the nonmagnetic case was k 2 = 0.060 while the magnetic models had k 2 = 0.045 and k 2 = 0.046 for
the LS95 and turbulent dynamo models, respectively.
The radius to the boundary of the convection zone was also equivalent in both magnetic
models. In the non-magnetic model, the convection zone boundary was located at R/R ∗ =
0.671, but it receded to R/R ∗ = 0.691 and R/R ∗ = 0.690 for the magnetic models. Thus,
227
the only way of differentiating between the two proposed methods using stellar surface
properties is to evaluate the strength of the surface magnetic fields.
6.5.2 Comparison to Previous Work
The magnetic perturbation presented in Section 6.4.4 is qualitatively different from earlier
approaches (Chapter 5; Mullan & MacDonald 2001; MacDonald & Mullan 2012; Feiden & Chaboyer
2012b). These earlier approaches assess the ability of a magnetic field to stabilize a fluid
against convection. Qualitatively our approach in Chapter 5 is very similar to the method
favored by Mullan & MacDonald (2001). In fact, the variable ν appearing in our modified
MLT equations is essentially the Mullan & MacDonald (2001) convective inhibition parameter, δMM . We have
∂ ln ρ
ν≡−
∂ ln χ
µ
¶
P,T
=
P mag
P mag + P gas
(6.14)
whereas δMM was defined in Section 1.4.1 as
δMM =
B2
.
B 2 + 4πγP gas
(6.15)
One significant difference is that our approach also introduces the magnetic energy gradient, ∇χ , defined in Equation (5.94). This gradient reduces the effect of ν on the adiabatic gradient by an order of magnitude. We therefore expect that magnetic fields used by
Mullan & MacDonald (2001) to be about an order of magnitude smaller than ours.
The tactic used in Section 6.4.4 has attempted to quantify a reduction in convective efficiency. Chabrier et al. (2007) had previously explored this concept, although they did so
by reducing the convective mixing length via a reduction in αMLT . Note that this is conceptually different than the methods discussed above where we were concerned with the
stabilization of convection.
Convective efficiency can be defined through the framework of MLT by considering heat
228
losses that are “horizontal” to the radial motion of a bubble (Böhm-Vitense 1958; Weiss et al.
2004). Quantitatively, the efficiency can be written as
Γ=
κg δ1/2 ρ 5/2 ℓ2m
cP
(∇s − ∇e )1/2
p
1/2 T 3
P
12 2ac
(6.16)
where the individual variables were defined in Chapter 5. We see that the temperature
excess, (∇s − ∇e ), factors into the efficiency, as well as the convective mixing length, ℓm .
Rewriting the above equation in terms of the convective velocity and mixing length parameter αMLT , we have
Γ=
c P κρ 2 u conv αMLT HP
.
6ac
T3
(6.17)
Using this expression, we can attempt to compare reductions in uconv with the reductions
in αMLT used by Chabrier et al. (2007).
Consider a reduction in uconv caused by a magnetic field that has a strength that is some
fraction, Λ, of the equipartition field strength. Then the ratio of the convective efficiency
in the presence of a magnetic field to the non-magnetic efficiency is Γmag /Γ0 = (1 − Λ2 )1/2 .
Therefore, to achieve the same results by reducing αMLT , we need only multiply the solar
value by (1 − Λ2 )1/2 . We can now relate αMLT to Λ,
Λ=
s
µ
αMLT
1−
αMLT, ⊙
¶2
.
(6.18)
The drastically reduced αMLT values (αMLT = 0.1, 0.5) used by Chabrier et al. (2007) correspond to Λ ∼ 0.999 and Λ ∼ 0.966, respectively.
We find that our models require such strong magnetic fields, at least in the outer layers.
The model of YY Gem in Section 6.4.4 has Λ ∼ 1 throughout a large portion of the envelope
calculation. Λ decreases from 1 at the photosphere (where T = Teff ) to 0.3 at the point where
we match the envelope to the interior. The comparison is by no means clear given that we
have prescribed a radial profile that is independent of Λ.
229
80
Non-magnetic
2
60
η2(R)
Density, ρ(R) (g cm-3)
Turbulent dynamo
Reduced αMLT
40
1
M = 0.599 M⊙
20
[Fe/H] = -0.20
Age = 370 Myr
0
0.0
0.2
0.4
0.6
0.8
0
1.0
Fractional Radius (R/R*)
Figure 6.20: Interior density profile for a M = 0.599M⊙ model with and without the presence of a magnetic field. Also shown is the η2 parameter, related to the stellar deviation from
sphericity and the interior structure constant, k 2 . The magnetic field was prescribed using
the turbulent dynamo formuation and the reduced αMLT method (Chabrier et al. 2007). In
the turbulent dynamo model, a 0.7 kG field was used and αMLT = 0.6 was used in the reduced αMLT model. Note that the magnetic model and reduced αMLT model profiles lie on
top of one another.
To make the comparison more direct, we compare the interior structure of the magnetic
model with that from a reduced αMLT model. Our models required αMLT = 0.60 to reproduce the observed radius of YY Gem, corresponding to Λ = 0.951 for reference. Based on
Figure 1(a) of Chabrier et al. (2007) it appears that they require αMLT ∼ 0.4 to reproduce
the properties of YY Gem. When we look at the density distribution associated with our
reduced αMLT model and our magnetic model (using the turbulent dynamo formulation) we
find there is no significant difference. This is illustrated in Figure 6.20. From this it is apparent that our turbulent dynamo magnetic models produce results consistent with reduced
αMLT models.
The advantage of the magnetic models is that they can be tested observationally. Our models
230
predict a specific surface magnetic field strength whereas reduced αMLT models are difficult
to constrain. We have introduced the possibility of constraining reduced αMLT models using
the equipartition magnetic field strength, Λ. Still, the use of a magnetic field prescription
provides a more realistic solution.
6.5.3 Starspots
Up to this point we have avoided any mention of specifically incorporating effects due to
starspots. Previously magnetic investigations attempted to account for spots by reducing the
flux at the model photosphere (Chabrier et al. 2007; Morales et al. 2010; MacDonald & Mullan
2012, 2013). The details of this approach were outlined in Chapter 1. Reductions of photospheric flux were combined with the aforementioned techniques to magnetically suppress
convection to reconcile model radii with observations of DEBs.
The explicit introduction of spots into stellar models can significantly affect stellar structure
(Chabrier et al. 2007). By reducing the flux at the model photosphere the model is forced to
inflate. The prescription for incorporating spots, however, has several uncertainties including the spot filling factor, the spot temperature contrast, and the spot distribution. These are
not very well constrained for low-mass stars, particularly for M-dwarfs (Jackson & Jeffries
2013). It would therefore be advantageous to eliminate having to specify spot properties in
stellar models.
In this Chapter, we have seen that our models do not explicitly require starspots. This is
because our models effectively use the average surface magnetic field strength, 〈B f 〉, as
input. Therefore, strong localization of magnetic flux is averaged together with the typical
“quiet” regions on the stellar surface. Since measurements of stellar surface magnetic fields
generally cannot distinguish between the sources of magnetic flux (with the exception of
Zeeman Doppler Imaging), we only receive information regarding the average surface field
strength when attempting to validate our magnetic models (Reiners 2012).
231
Conceptually, starspots reduce the photospheric flux by suppressing convection. Therefore,
there is little reason to manually reduce the photospheric flux in the stellar models. A
correct treatment of magnetic suppression of convection should implicitly account for this
effect. We see that this is the case with our models. Investigations looking at the effects
on stellar colors using our models in comparison with those that explicitly include spots
(Jackson et al. 2009; Jackson & Jeffries 2013; MacDonald & Mullan 2013) are ongoing.
6.5.4 Implications for Asteroseismology
Figure 6.19 showed that magnetic models have a lower density throughout their interior
than do non-magnetic models. One result is that the magnetic models have a lower k 2 ,
indicative of the mass being more centrally condensed. This is not unexpected if we consider
that the magnetic field acts to inflate the radius of the model.
Related to the reduction in mass density throughout the stellar interior is the effect on
the sound speed profile. We plot the sound speed profile for two M = 0.6M⊙ models in
Figure 6.21. The models have a scaled-solar composition of [Fe/H] = −0.2 dex. One model
has a magnetic field with a surface field strength of 0.7 kG using the turbulent dynamo
formulation. Note that the density profile for these two models are shown in Figure 6.19.
Throughout the model interior the magnetic model has a lower sound speed. Comparing
the sound speed between the two models, we see that the magnetic model has a sound speed
that is slower by about 5%. The bottom panel in Figure 6.21 shows the relative sound speed
difference between the two models with respect to the non-magnetic models. We define
δc s c s, mag − c s, 0
=
,
cs
c s, 0
(6.19)
where c s, 0 is the sound speed in the non-magnetic model and c s, mag is that for the magnetic
232
0.5
cs (Mm s-1)
0.4
0.3
0.2
0.1
0.0
Non-magnetic
Bsurf = 0.7 kG
δcs/cs
0.00
-0.04
-0.08
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fractional Radius, (R/R*)
0.8
0.9
1.0
Figure 6.21: (Top) Sound speed profile for a M = 0.599M⊙ model with and without the
presence of a magnetic field. (Bottom) Relative difference in the sound speed profile between
the two models with respect to the non-magnetic model. The magnetic model was computed
with a surface magnetic field strength strong enough to reconcile the model radius with the
observed radius of YY Gem (0.7 kG).
model. We defined the sound speed such that
c s2
µ
¶
P P ∂ ln P
= Γ1 =
ρ ρ ∂ ln ρ ad
(6.20)
where P is the total pressure and ρ is the mass density. Using the total pressure is valid
everywhere except in the outermost layers (∼ 1% by radius) because the magnetic pressure
is significantly lower (over 4 order of magnitude lower) than the gas pressure. The change in
sound speed between the two models affects the p-mode frequencies and frequency spacing,
which consequently alters the interpretation of seismic data. We now explore these effects.
Imagine that observational data is obtained for a M = 0.6M⊙ star that has a 0.7 kG magnetic
field. These two facts about the star are unknown to the observer. Instead, the observer has
233
taken great care to obtain high resolution spectra with a high signal-to-noise of the star.
From this data they are able to ascertain the effective temperature, Teff , and the approximate composition. They also have photometric data from Kepler that allows for a seismic
analysis, where to first order, they are able to obtain the stellar radius from the p-mode frequencies. The properties they derive for the star are R = 0.619 ± 0.006R ⊙ , Teff = 3 820 ± 100
K, and [Fe/H] = −0.2 ± 0.1.7
Taking these properties in combination with the p-mode frequencies and frequency spacings, what inferences would the observer make about the star? Using the grid of nonmagnetic models presented in Chapter 4 the observer would estimate the mass of the star
to be between 0.65M⊙ and 0.69M⊙ , or M = 0.67 ± 0.02M⊙ . The age of the star would be
difficult in this particular case due to the lack of significant radius evolution of low-mass
stars on the main sequence. One concession, is that the models would not match the observed Teff within the 1σ limit quoted above. However, one can easily find a match within
2σ. Moreover, the sound speed profiles between the higher mass non-magnetic models and
the lower mass magnetic models are quite similar
With magnetic models, a slight degeneracy in mass and magnetic field strength would be
introduced, as varying the magnetic field at a given mass can alter the radius and Teff .
However, the true solution would be encompassed by the formal uncertainties. Compare
this with the case where non-magnetic models are used, where the mass estimate is too
large by ∼10% and the true solution lies 3.5σ below the derived value.
Future investigations exploring the use of magnetic models for asteroseismic analyses are
warranted. While the case above was exploratory and does not represent the typical stars
selected for asteroseismic analysis, the conclusions should be valid for all stars with an
outer convective envelope. Magnetically active stars are also not favored for asteroseismology, due to issue with removing possible spots variations in the light curve. However, the
7
Conveniently, these are the properties of YY Gem.
234
lack of photometric variability due to spots is not necessarily indicative of an inactive star
(Jackson & Jeffries 2013).
6.5.5 Exoplanets & the Circumstellar Habitable Zone
The primary effect of introducing a magnetic perturbation into a model is that the model
radius is increased and the Teff is decreased. What has not been discussed is that the luminosity of the model shows a marginal decrease of about 10%. This means the radius
increase does not precisely balance the Teff decrease, in contrast with what has been assumed previously (see, e.g., Morales et al. 2008). This reduction in luminosity occurs as the
model readjusts its internal structure to compensate for the reduction of convective heat
flux. These changes in the stellar surface properties can have a significant impact on studies
of extra-solar planets (exoplanets) around low-mass stars.
Transiting exoplanets have radii that are directly proportional to the radii of their host stars
(e.g., Seager & Mallén-Ornelas 2003). Determining the radius of a low-mass host star is difficult. Interferometry allows for the direct measurement of the stellar radius, but is only available for the brightest targets (von Braun et al. 2012; Boyajian et al. 2012). One could attempt
to perform an asteroseismic analysis, but it would suffer due to the faintness of low-mass
stars and also from the considerable variability that is characteristic of these stars. In lieu
of other methods, stellar evolution models are used to aid in the interpretation of transiting exoplanetary systems (e.g., Muirhead et al. 2012; Gaidos 2013; Dressing & Charbonneau
2013).
Studies of transiting exoplanets have relied on standard stellar evolution models (Muirhead et al.
2012; Gaidos 2013; Dressing & Charbonneau 2013). Using magnetic models can have a significant impact on these studies. For instance, Muirhead et al. (2012) measured the metallicity and Teff for the low-mass stars tagged as planet candidates in the Kepler mission
(Batalha et al. 2013). They then interpolated within DSEP isochrones to determine stellar
235
radii. Had they used magnetic stellar models, for a given Teff , the stellar radius would
be larger, with the precise factor dependent on the magnetic field strength assumed. This
would result in larger, less dense (and therefore less Earth-like) planets.
Estimating the surface magnetic field is the largest hindrance to using magnetic models.
Stars in the Kepler field are typically distant, meaning X-ray measurements will not necessarily be available. Deep observations of the Kepler field with Chandra could provide X-ray
magnetic activity information. Alternatively, it might be feasible to obtain spectra and use
the Hα equivalent width to estimate magnetic activity. Hα is less revealing, as we are not
aware of a relation linking Hα equivalent widths to magnetic field strengths. However, the
presence of Hα in emission would be indicative of an active star. At that point, several
magnetic model could be used to constrain the stellar radius.
Aside from affecting the determination of host star radii, magnetic models may affect the
edges of the circumstellar habitable zone (HZ). This can happen over two timescales: (1)
evolutionary timescales, and (2) timescales on the order of a stellar cycle. The first timescale
refers to how the luminosity of a star changes over its lifetime. We said that magnetic
fields can reduce the total luminosity of a given stellar model. Therefore, the luminosity
evolution of a particular star will be altered by the evolution of the magnetic field. The
second timescale refers to stellar activity cycle, much like the 11 yr solar cycle. Changes
in the magnetic field strength over the cycle may affect the host star’s internal structure
in such a way so as to alter the boundaries of the HZ. Regular fluctuations in the stellar
radius, effective temperature, and luminosity could cause corresponding fluctuations in the
boundaries of the HZ.
Here, we explore the second possibility. Investigations into the first affect requires more sophisticated magnetic models that have evolving magnetic fields. We compute three models
for a M = 0.6M⊙ star. One model has no magnetic field, one has a 0.5 kG magnetic field, and
the final model has a 0.7 kG magnetic field. Both of the magnetic models were calculated
236
Figure 6.22: Habitable zone boundaries for M = 0.6M⊙ models with no magnetic field, a
0.5 kG magnetic field, and a 0.7 kG magnetic field. The Teff and luminosity used to compute
the HZ were calculated at 1 Gyr. The order of the inner HZ edges correspond to the 0.7 kG,
0.5 kG, and non-magnetic model in order of increasing distance from the star. The outer
HZ edges are in the same order. Transparency was used in the coloring, meaning the green
shell is a combination of the colors used for each HZ.
assuming a turbulent dynamo. We used the Habitable Zone Calculator8 based on the most
recent HZ models by Kopparapu et al. (2013) to estimate the HZ boundaries for each of our
models. The Kopparapu et al. (2013) HZ models improve on the Kasting one-dimensional
radiative-convective climate models (Kasting 1988, 1991; Kasting et al. 1993) by including
updated H2 O and CO2 opacities.
We find that the HZ boundaries are robust. This is demonstrated in Figure 6.22 where we
show the estimated habitable zone boundaries for each of our models. We adopted the
conservative limits defined by Kopparapu et al. (2013), which place the inner edge of the
HZ at the location where their climate models predict a fully saturated troposphere and
8
http://depts.washington.edu/naivpl/sites/default/files/HZ_Calc.html
237
the outer edge of the HZ is where the maximum greenhouse warming can occur for a CO2
dominated atmosphere. In Figure 6.22, the large green region is considered habitable for all
three models.
Fluctuations of the HZ associated with possible stellar activity cycle are under 0.1 AU. Even
for the extreme case of a fluctuation between no magnetic field and 0.7 kG the HZ outer
edge changes by 0.14 AU. The reason for the stability is that, while the stellar effective
temperature is lower by nearly 300 K for the 0.7 kG model compared to the non-magnetic
model, the luminosity decreases by ∼ 0.06L ⊙ (or about 10%). Therefore, only planets very
near the to the HZ boundaries will be affected by the modest variation in luminosity.
Note that our approach considered very dramatic changes in the stellar properties throughout the magnetic activity cycle. The sizable changes in the model properties between the
different magnetic field strength occur over thermal timescales instead of short timescales
associated with stellar cycles. Fluctuations in the global stellar properties due to magnetic
fields on the timescales of a stellar cycle are expected to be less than those observed between two models with different magnetic field strengths. In fact, helioseismic studies have
indicated that the Sun does not appear to undergo significant structural changes throughout its activity cycle (Libbrecht & Woodard 1990; Foukal et al. 2006). Acoustic frequencies
do change with the solar cycle, but have their origin in the outermost layers of the Sun
(Libbrecht & Woodard 1990).
There is evidence that the total solar irradiance (TSI) does change over the course of the
solar cycle. TSI is the amount of solar radiation received by the Earth. We discussed this
previously in Section 1.3.4, that the TSI increases up to 0.1% with increasing sun spot number
(Foukal et al. 2006). Whether planets around other stars would observe similar fluctuations
in total irradiance is unclear. Addressing this question requires a better understanding of
the development of spots and faculae on active stars.
238
6.6 Conclusions
In this Chapter we addressed the question of how the presence of a magnetic field affects the
structure of a low-mass star with a radiative core. We approached this problem by taking
a careful look at three DEB systems that show significant radius and Teff deviations from
standard stellar evolution models. Using the magnetic stellar evolution models introduced
in Chapter 5 (Feiden & Chaboyer 2012b), we attempted to reproduce the observational properties of UV Psc, YY Gem, and CU Cnc. The models probe the ability of a magnetic field to
stabilize plasma against convection.
After finding that the magnetic models were able to reconcile the stellar models with the
observed radii and Teff s (consistent with the models of Mullan & MacDonald 2001), it was
show in Section 6.3.1 that the surface magnetic field strength were likely too strong. This
was done by taking the coronal X-ray luminosity as an indirect diagnostic for the surface
magnetic field strength (Fisher et al. 1998; Pevtsov et al. 2003). In contrast to the surface
magnetic fields, the interior field strengths are of a plausible magnitude, consistent with the
range of field strengths predicted within the Sun by 3D MHD models (Brandenburg & Subramanian
2005).
We then attempted to reduce the surface magnetic field strengths while still maintaining
the newfound agreement between stellar models and DEB observations. The most plausible
explanation we uncovered is that kinetic energy in turbulent convective flows is actively
converted into magnetic energy. Introducing changes to mixing length theory that mimic
this physical process yielded accurate models with surface magnetic field strengths nearly
equal to those predicted by X-ray emission. We were unable, however, to implement this
prescription for the stars of UV Psc. This is likely a consequence of the chosen magnetic
field profile, which caused the models to fail to converge. We note that by invoking the
“turbulent dynamo” approach, we are able to generate substantial radius inflation with relatively modest (sub-kG) magnetic fields. This could be a clue to solving the cases of extreme
239
radius inflation, as observed with V405 Andromeda, which appears between 70% and 80%
larger than standard stellar evolution models (Vida et al. 2009; Ribeiro et al. 2011).
Beyond direct comparisons to observational data, we explored additional implications of the
present study. We found that different theoretical descriptions leading to similarly inflated
stars produce nearly identical stellar interiors. The radial density profile within models
invoking the methods of Chapter 5, Section 6.4.4, and Chabrier et al. (2007) are all nearly
identical. The latter uses a reduced mixing length theory approach to simulate the introduction of a turbulent dynamo. We provide a variable Λ to translate a reduction in mixing
length into a magnetic field strength (normalized to the equipartition field strength, which
depends on the convective velocity).
We also argue that asteroseismic studies and planetary habitability studies should be relatively unaffected by structural changes in the stellar models. First, asteroseismic studies
typically known the stellar composition and effective temperature. Anomalous values of
the Teff due to magnetic fields should provide a clue that magnetic fields are required. For
a given density profile, it is difficult to reproduce the Teff for a star of a given mass without
a magnetic field. If Teff is ignored, then masses would typically be overestimated by about
10%.
Second, studies concerning the size of planetary habitable zones are rather insensitive to the
effects of magnetic fields on stellar structure. Stellar effective temperature are decreased, but
the stellar radius is increased, leading to only a modest decrease in the stellar luminosity in
the presence of a magnetic field. We used the habitable zone models of (Kopparapu et al.
2013) to illustrate this fact. Even if a star were to have a magnetic cycle that fluctuates
between no magnetic field and a sizable field of several hundred Gauss, only the innermost
and outermost edges of the habitable zone are affected. Increased magnetic field strength
would, however, increase the level of surface activity, which presents separate problems for
planetary habitability around low-mass stars. Since the habitable zones are relatively near
240
to low-mass stars, sizable flares and coronal mass ejections could devastate a nearby planet.
While planetary habitable zones may be relatively stable to changes in stellar structure due
to a magnetic field, the radii of transiting exoplanets are not. The radii of transiting exoplanetary radii scale with the stellar radius. This is but one reason that further exploration
of the role of magnetic field in low-mass stars is required. Interpretation of low-mass star
observations depend critically on the properties of model predictions.
241
Table 6.5
Low-mass stars from Reiners (2012) with direct magnetic field measurements used to derive our Φ − L x relation.
Star
Other
Name
Name
HD 115383
59 Vir
HD 115617
SpT
242
〈B f 〉
R⋆
log(Φ)
π
X cr
(kG)
(R ⊙ )
(Mx)
(mas)
(cnts s−1 )
G0
0.5
0.74
25.22
56.95
1.12
61 Vir
G6
0.1
0.95
24.74
116.89
σ Dra
-
K0
0.1
0.80
24.59
40 Eri
-
K1
0.1
0.75
ǫ Eri
-
K2
0.1
LQ Hya
-
K2
GJ 566 B
ξ Boo B
Gl 171.2 A
HR
Fx
log(L x )
(erg s−1 cm−2 )
(erg s−1 )
-0.14
8.48E-12
29.50
0.01
-0.96
4.58E-14
26.60
173.77
0.26
-0.80
1.06E-12
27.62
24.53
200.62
0.80
-0.28
5.46E-12
28.21
0.70
24.59
310.94
2.82
-0.44
1.69E-11
28.32
2.5
0.70
25.86
53.70
2.73
-0.04
2.21E-11
29.96
K4
0.5
0.70
25.14
149.26
2.44
-0.31
1.63E-11
28.94
-
K5
1.4
0.65
25.56
55.66
2.69
-0.04
2.18E-11
29.93
Gl 182
-
M0.0
2.5
0.60
25.74
38.64
0.65
-0.19
4.75E-12
29.58
Gl 803
AU Mic
M1.0
2.3
0.50
25.54
100.91
5.95
-0.07
4.72E-11
29.74
Gl 569 A
-
M2.0
1.8
0.40
25.24
-
0.49
-0.40
3.03E-12
-
Gl 494
DT Vir
M2.0
1.5
0.40
25.16
85.54
1.57
-0.01
1.30E-11
29.33
Gl 70
-
M2.0
0.2
0.33
24.12
87.62
0.04
-0.67
2.08E-13
27.51
Gl 873
EV Lac
M3.5
3.8
0.31
25.35
195.22
5.83
-0.16
4.35E-11
29.14
continued on next page
Table 6.5 – continued
Star
Other
Name
Name
Gl 729
V1216 Sgr
Gl 87
SpT
243
〈B f 〉
R⋆
log(Φ)
π
X cr
(kG)
(R ⊙ )
(Mx)
(mas)
(cnts s−1 )
M3.5
2.1
0.20
24.71
336.72
0.94
-
M3.5
3.9
0.30
25.33
96.02
Gl 388
AD Leo
M3.5
3.0
0.39
25.44
GJ 3379
-
M3.5
2.3
0.25
GJ 2069 B
CV Cnc
M4.0
2.7
Gl 876
IL Aqr
M4.0
GJ 1005 A
-
Gl 490 B
HR
Fx
log(L x )
(erg s−1 cm−2 )
(erg s−1 )
-0.43
5.67E-12
27.78
-
-
-
-
213.00
3.70
-0.27
2.55E-11
28.83
24.94
190.93
0.40
-0.20
2.90E-12
27.98
0.25
25.01
78.10
0.24
-0.14
1.82E-12
27.95
0.2
0.31
24.07
213.28
-
-
-
-
M4.0
0.2
0.23
23.81
191.86
-
-
-
-
G 164-31
M4.0
3.2
0.20
24.89
50.00
0.84
-0.22
6.00E-12
29.46
Gl 493.1
FN Vir
M4.5
2.1
0.20
24.71
123.10
0.14
-0.16
1.04E-12
27.92
GJ 4053
LHS 3376
M4.5
2.0
0.17
24.55
137.30
0.06
-0.49
3.43E-13
27.34
GJ 299
-
M4.5
0.5
0.18
23.99
148.00
-
-
-
-
GJ 1227
-
M4.5
0.2
0.19
23.64
120.00
-
-
-
-
GJ 1224
-
M4.5
2.7
0.18
24.73
132.60
0.21
-0.45
1.24E-12
27.93
Gl 285
YZ Cmi
M4.5
4.5
0.30
25.39
167.88
1.47
-0.21
1.06E-11
28.65
GJ 1154 A
-
M5.0
2.1
0.20
24.71
-
0.10
-0.23
7.09E-13
-
continued on next page
Table 6.5 – continued
Star
Other
Name
Name
GJ 1156
GL Vir
Gl 905
SpT
244
〈B f 〉
R⋆
log(Φ)
π
X cr
(kG)
(R ⊙ )
(Mx)
(mas)
(cnts s−1 )
M5.0
2.1
0.16
24.51
152.90
0.13
HH And
M5.5
0.1
0.17
23.24
316.70
GJ 1057
CD Cet
M5.5
0.1
0.18
23.29
GJ 1245 B
-
M5.5
1.7
0.14
GJ 1286
-
M5.5
0.4
GJ 1002
-
M5.5
Gl 406
CN Leo
Gl 412 B
HR
Fx
log(L x )
(erg s−1 cm−2 )
(erg s−1 )
-0.25
9.08E-13
27.67
0.18
0.15
1.64E-12
27.29
117.10
-
-
-
-
24.31
220.00
0.20
-0.37
1.27E-12
27.50
0.14
23.68
138.30
-
-
-
-
0.2
0.13
23.31
213.00
-
-
-
-
M5.5
2.4
0.13
24.39
418.30
0.23
-0.22
1.64E-12
27.05
WX Uma
M6.0
3.9
0.13
24.60
206.94
0.18
-0.64
8.85E-13
27.39
GJ 1111
DX Cnc
M6.0
1.7
0.12
24.17
275.80
-
-
-
-
Gl 644C
VB 8
M7.0
2.3
0.10
24.15
153.96
-
-
-
-
GJ 3877
LHS 3003
M7.0
1.5
0.10
23.96
157.80
-
-
-
-
GJ 3622
-
M7.0
0.6
0.10
23.56
221.00
-
-
-
-
LHS 2645
-
M7.5
2.1
0.08
23.91
-
-
-
-
-
LP 412-31
-
M8.0
3.9
0.08
24.18
-
-
-
-
-
VB 10
V1298 Aql
M8.0
1.3
0.08
23.70
164.30
-
-
-
-
continued on next page
Table 6.5 – continued
Star
Other
Name
Name
LHS 2924
-
LHS 2065
-
SpT
〈B f 〉
R⋆
log(Φ)
π
X cr
(kG)
(R ⊙ )
(Mx)
(mas)
(cnts s−1 )
M9.0
1.6
0.08
23.79
90.00
-
M9.0
3.9
0.08
24.18
116.80
-
HR
Fx
log(L x )
(erg s−1 cm−2 )
(erg s−1 )
-
-
-
-
-
-
245
Chapter 7
Magnetic Models of Low-Mass, Fully
Convective Stars
7.1 Introduction
The radial extent of the outer convection zone in low-mass stars increases as stellar mass
decreases. Around 0.35M⊙ , the interior of low-mass stars transitions from being partially
convective to fully convective (Limber 1958a; Baraffe et al. 1998). Fully convective stars
should be the simplest stars to describe from a theoretical perspective. The interior structure
predicted by stellar models is largely independent of many of the input variables (e.g., αMLT )
and input physics (e.g., diffusion physics) (Baraffe et al. 1998; Dotter et al. 2007). We saw
in Chapters 2, 3, and 4 that metallicity, αMLT , and age can alter the predictions of stellar
models, but only at the few percent level. Observations of significant radius inflation in
fully convective stars is therefore puzzling.
Multiple DEBs are known to host fully convective stars. Those with precise mass and radius
determinations are Kepler-38 (Orosz et al. 2012), Kepler-16 (Doyle et al. 2011; Winn et al.
2011; Bender et al. 2012), LSPM J1112+7626 (Irwin et al. 2011), KOI-126 (Carter et al. 2011),
246
and CM Draconis (CM Dra; Lacy 1977; Metcalfe et al. 1996; Morales et al. 2009a). Of these
systems, the only fully convective stars to be accurately reproduced by stellar evolution
models are KOI-126 B and C (Chapter 2; Feiden et al. 2011; Spada & Demarque 2012). The
rest display varying degrees of radius inflation compared to model predictions.
Most significant are the inflated radii of the stars in CM Dra. Historically, the stars of
CM Dra are the fully convective stars against which to benchmark stellar models. The
discovery of KOI-126 in 2011 introduced the first pair of well-characterized fully convective stars in a DEB since Lacy (1977). Over the years, the discrepancy between the model
radii of CM Dra and those determined from observations has grown. Initial modeling efforts were optimistic (Chabrier & Baraffe 1995), but this quickly changed with the introduction of more sophisticated models (Baraffe et al. 1998) and more precise observations
(Metcalfe et al. 1996; Morales et al. 2009a). Recent metallicity estimates have widened the
gap, yet again (Rojas-Ayala et al. 2012; Terrien et al. 2012).
This chapter is concerned with the second half of our magnetic analysis of the low-mass
stellar mass-radius problem: fully convective stars. Chapter 6 dealt with using the magnetic models introduced in Chapter 5 to reconcile stellar evolution model predictions with
observations of DEB stars that have a radiative core. We showed that the magnetic models
presented in Chapter 5 were likely incomplete and required the addition of terms in the mixing length equations to account for the conversion of turbulent kinetic energy into magnetic
energy.
A brief overview of the magnetic field formulations is presented in Section 7.2. Two wellcharacterized DEBs, Kepler-16 and CM Draconis, are then used as case studies for investigating the effects of the magnetic models in fully convective stars (Section 7.3). We have
elected to focus on Kepler-16 in addition to CM Dra because it has a more massive K-dwarf
companion. If there is any significant difference between a system with two fully convective stars and a system with only one, we hope that it will become apparent. The validity of
247
our model results are discussed and we provide comparisons to previous investigations in
Section 7.4. After reasoning that magnetic models are not sufficient to describe the radius
inflation observed in fully convective stars, we re-advance an old hypothesis: that metallicity is the origin of the radius discrepancies.
7.2 Magnetic Models
The magnetic models to be used in this chapter have been described in the two previous
chapters. However, we want to consolodate the terminology and remind the reader about
the various formulations after the development of several branches in Chapter 6. Below we
describe the three model formulations that will be referenced throughout this chapter.
7.2.1 Dipole Radial Profile
Models of the “dipole radial field” variety are the standard sort first introduced in Chapter 5. Given a surface magnetic field strength, the radial profile of the magnetic field is
determine by calculating the peak magnetic field strength. Stars with radiative cores have
this point defined by the tachocline (Feiden & Chaboyer 2012b), but fully convective stars
have the point placed at 15% of the stellar radius. This is loosely based on results from 3D
MHD models of fully convective stars, which find the strongest magnetic field at 0.15R ⋆
(Browning 2008). The rest of the field is then automatically defined because we assume the
magnetic field strength falls off as r 3 towards the core and surface of the star.
7.2.2 Gaussian Radial Profile
Looking to increase the peak magnetic field strength for a given surface magnetic field
strength, we used a Gaussian profile in Section 6.4.1. The peak magnetic field strength
248
was still defined at the tachocline in partially convective stars and at R = 0.15R ⋆ in fully
convective stars. However, instead of a power-law decline of the interior magnetic field
strength from the peak, the peak was set as the peak of a Gaussian distribution. The width
of the Gaussian was made variable, based on the width of the convection zone. Larger
convection zones were given broarder Gaussians compared to those with thin convective
envelopes (see Section 6.4.1). These models will be referred to as “Gaussian radial profile”
models throughout.
7.2.3 Constant Λ Turbulent Dynamo
Section 6.4.4 introduced a third method of including the effects of a magnetic field, the
so-called “turbulent dynamo” formulation. Conceptually, the turbulent dynamo formalism
seeks to source the energy for the magnetic field from turbulent convection. This is different
from our approach used in the dipole and Gaussian magnetic field models, which investigate the stabilization of convection more so than suppressing convective efficiency. For
completeness, we wanted to compute models of fully convective stars with this turbulent
dynamo formulation. However, combining this formulation with a dipole or Gaussian field
strength profile causes the models to not converge. The interior magnetic field strengths
are too large. The Alfvén velocity, u 2A , quickly dominates the convective velocity, uconv, 0 ,
term in Equation (6.9), meaning there is more magnetic energy than available kinetic energy
from convection.
To mitigate this issue, we created a new radial profile based on the equipartition field
strength parameter, Λ, introduced in Section 6.5.2. We defined
B = ΛB eq ,
(7.1)
where B is the magnetic field strength at a given point in the model and B eq is the equipar-
249
tition magnetic field strength. The latter is defined such that
B eq =
q
2
4πρu conv
,
(7.2)
where ρ is the plasma mass density and uconv is the convective velocity. With these definitions, we reformulated the magnetic mixing length theory such that
u conv = u conv, 0
p
1 − Λ2 .
(7.3)
Additionally, the term associated with the increase of the plasma temperature gradient, ∇s ,
became
∆∇s =
2
(Λu conv 0 )2 u A
=
,
C
C
(7.4)
as before, where C is defined in Chapter 5. The radial profile was then prescribed by assigning a constant value to Λ, hereafter the “constant Λ” formulation. This has the added
benefit of mimicking an α2 dynamo, where the magnetic field is generated throughout the
convection zone (Chabrier & Küker 2006; Browning 2008).
7.3 Analysis of Individual DEB Systems
7.3.1 Kepler-16
The report of the first circumbinary exoplanet was announced by Doyle et al. (2011) with
their discovery of Kepler-16b. While the planet is interesting in its own right, what made
the discovery even more remarkable was the fact that the two host stars comprise a lowmass detached eclipsing binary with a period of just over 41 days. This fact allowed for a
very precise determination of the component masses and radii using the same photometricdynamical analysis that led to the characterization of the hierarchical triple KOI-126 (Chap-
250
Table 7.1
Fundamental properties of Kepler-16.
Property
Kepler-16 A
Kepler-16 B
D11 Mass (M⊙ )
0.6897 ± 0.0034
0.20255 ± 0.00066
Radius (R⊙ )
0.6489 ± 0.0013
0.2262 ± 0.0005
B12 Mass (M⊙ )
Teff (K)
[Fe/H]
0.654 ± 0.017
4337 ± 80
Age (Gyr)
0.1959 ± 0.0031
−0.04 ± 0.08
···
3±1
ter 2; Carter et al. 2011). Kepler-16 was found to contain a K-dwarf primary with a fullyconvective M-dwarf secondary. Individual stellar parameters are presented in Table 7.1.
Following the discovery paper, Winn et al. (2011) provided a spectroscopic determination
of the primary star’s composition and a preliminary estimate of the star’s age. The age
estimation was obtained using gyrochronology and a Ca ii age-activity relation. Winn et al.
(2011) estimate that the system is 3±1 Gyr old with a near-solar metallicity of [Fe/H] =
−0.04 ± 0.08. Their study was also the first to directly compare the Kepler-16 components
to stellar evolution models in the context of the mass-radius problem. The authors found
the Baraffe et al. (1998) model predictions to be in agreement with the properties of primary.
However, the fully-convective secondary was seen to be inflated by a few percent, relative
to the models. This assessment was echoed in an independent study by Feiden & Chaboyer
(2012a, Chapter 4).
Less than a year after the initial discovery, radial velocity confirmation of the component
masses—within 2σ—was obtained (Bender et al. 2012). The results from the Bender et al.
(2012) study display slight disagreements with the original masses (Doyle et al. 2011). Of
particular note, is that the precise mass ratio is different between the two studies. Bender et al.
(2012) attempted to pin-point the origin of the discrepancy, but were unable to do so with
complete confidence.
251
0.72
[Fe/H] = -0.12
0.72
Kepler-16 A
[Fe/H] = -0.12
[Fe/H] = -0.04
[Fe/H] = -0.04
[Fe/H] = 0.04
0.70
Radius (R⊙)
Radius (R⊙)
0.70
0.68
0.66
0.64
Doyle et al. (2011)
0.62
Bender et al. (2012)
1
[Fe/H] = 0.04
0.68
Bender et al. (2012)
0.66
0.64
Doyle et al. (2011)
0.62
Kepler-16 A
4800
10
Age (Gyr)
4600
4400 4200
Teff (K)
4000
3800
Figure 7.1: Standard DSEP models computed at the exact masses measured by Doyle et al.
(2011) (maroon) and Bender et al. (2012) (light blue) for Kepler-16 A. Mass tracks for the
adopted metallicity of Winn et al. (2011) and the two boundaries of the associated 1σ uncertainty are given by the solid, dash, and dash-dotted lines, respectively. (a) Age-radius
diagram with the observed radius indicated by the gray horizontal swath. (b) Teff -radius
plane where the gray box indicates observational constraints for Kepler-16 A.
The agreement displayed between the original photometric-dynamical masses (Doyle et al.
2011) and the radial velocity masses (Bender et al. 2012) at first seems comforting, they
largely agree. However, the difference in mass for each component drastically alters the
comparison with stellar evolution models, assuming that the derived masses do not heavily
influence the derivation of the component radii and effective temperatures. The two mass
estimates will come to play a key role in our analysis. Hereafter, the masses originally
quoted by Doyle et al. (2011) will be referred to as the D11 masses, whereas the revised
values of Bender et al. (2012) will be referred to as the B12 masses. Table 7.1 provides a
comparison of the different quantities.
We focus our attention on Kepler-16 A first, as the analysis will directly influence our interpretation of the secondary. Plotted in Figure 7.1 are standard DSEP models computed
at the exact masses of the primary provided by D11 and B12. Three different tracks are
illustrated for each mass, corresponding to [Fe/H] = −0.12, −0.04, and +0.04 (Winn et al.
2011). Similar to the figures in Chapter 6, the horizontal swath designates the measured
252
radius constraints.
Given the primary mass from D11, we find that standard stellar evolution models match
the stellar radius and temperature with high accuracy at an age of 2.8 Gyr. When the
model radius equals the precise observed radius (0.6489R⊙ ), the associated model effective
temperature is 4337 K, the precise measured temperature (Winn et al. 2011). Additionally,
the age predicted by the mass track is consistent with the estimated ages (Winn et al. 2011).
The agreement between the models and observations does not guarantee the validity of
the D11 masses over the B12 masses. Agreement may be expected. The effective temperature and metallicity for the primary were measured using Spectroscopy Made Easy
(Valenti & Piskunov 1996, hereafter SME), which relies on theoretical stellar atmospheres.
DSEP also relies on theoretical atmospheres, although the phoenix model atmospheres
used by DSEP use a different line list database (see Hauschildt et al. 1999a, and references therein) than the theoretical atmospheres used by SME (VALD: Vienna Atomic Line
Database Piskunov et al. 1995; Stempels et al. 2001). In a sense, the fact that we find the precise effective temperature for a given log g and metallicity (i.e., those of Kepler-16 A) may
be a better test of the agreement between different stellar atmosphere models rather than a
test of the interior evolution models. What is encouraging, is that we derive the appropriate
stellar properties at an age consistent with the Winn et al. (2011) study. The age consistency
is not ensured by the agreement between the stellar atmosphere models.
The previous discussion may be erroneous if we are adopting a mass that is too large compared to the actual mass of the primary. B12 suggest this is the case. The effect of adopting
the lower B12 masses is displayed in Figure 7.1. Since the radius measurement has remained
constant, we derive an age of 12 Gyr for the primary star. The effective temperature associated with the lower mass model also becomes too cool compared to the observations at the
measured metallicity (−0.04 dex). Relief is found by lowering the metallicity by 0.1 dex,
which increases the temperature by 30 K. This is enough to bring the model temperature to
253
within 1σ of the measured value Winn et al. (2011).
There is on caveat with the above discussion: the spectroscopic analysis by Winn et al.
(2011) relied on fixing the stellar log g as input into SME. Thus, the temperature and metallicity are intimately tied to the adopted log g . Reducing the mass of the primary by 5%
(B12) but leaving the radius fixed to the D12 value leads to a decrease in log g of 0.02 dex.
Such a change in the fixed value of log g has the potential to decrease the derived effective
temperature by as much as 100 K (Stempels, priv. comm.). This implies the temperature
could be as cool as 4 230 K. This would bring the observed temperature into agreement with
the model derived temperature. Considering the discussion above, we believe it is safe to
assume that changing the mass of the DSEP models will not produce any large effective
temperature offset at a given radius, predicated on the fact that the temperature is always
safely above ∼ 4 000 K. Below this value, theoretical atmosphere predictions begin to degrade. More simply stated, we have no reason to doubt the DSEP predicted temperature for
Kepler-16 A, regardless of the adopted mass.
The model age of 12 Gyr for the B12 primary mass appears old given the multiple age estimates provided by Winn et al. (2011). Is it possible that the system is actually 12 Gyr old, but
appears from rotation and activity to be considerably younger? After all, 12 Gyr is not unreasonably old given the current estimate for the age of the Universe (Planck Collaboration et al.
2013). Consider that the 35.1±1.0 day rotation period of the primary was found by Winn et al.
(2011) to be nearly equal to the pseudo-synchronization rotation period, predicted to be 35.6
days (Hut 1981). Winn et al. (2011) rightly point out that intuition suggests tidal effects are
unimportant in a binary with a 41 day orbital period. However, subsequent tidal interactions briefly endured when the components are near periastron have the ability to drive the
components towards pseudo-synchronous rotation. The potential pseudo-synchronization
of the components acts to keep the stars rotating at a faster rate than they would if they were
completely isolated from one another. The time scale for this to occur is approximately 3
254
[Fe/H] = -0.12
0.24
Kepler-16 B
[Fe/H] = -0.04
Radius (R⊙)
[Fe/H] = 0.04
0.23
Doyle et al. (2011)
0.22
0.21
Bender et al. (2012)
1
10
Age (Gyr)
Figure 7.2: Identical to Figure 7.1(a), except the mass tracks are computed at the masses
measured for Kepler-16 B.
Gyr (Zahn 1977; Winn et al. 2011), meaning that the rotation period is not necessarily indicative of the system’s age. The primary will have approximately the same rotation period
at 12 Gyr as it will at 3 Gyr. Furthermore, the timescale for orbital circularization is safely
estimated to be ∼ 104 Gyr (Zahn 1977; Winn et al. 2011). Tidal evolution calculations are
subject to large uncertainties and should approached as an order of magnitude estimate. We
are therefore unable to immediately rule out the possibility that Kepler-16 is 12 Gyr.
We have so far neglected any remark on the agreement between standard models and
Kepler-16 B. This comparison is carried out in Figure 7.2. Unfortunately, no effective temperature estimates have been reported, explaining our neglect of the Teff -radius plane in the
figure. As with Figure 7.1, mass tracks are shown for multiple metallicities. The standard
models mass tracks for Kepler-16 B are unable to correctly predict the observed radius at a
realistic age. The disagreement is relatively independent of the adopted metallicity, which
introduces slight (i.e., ∼0.5%) variations in the stellar radius at a given age. Only the D11
mass track with a super-solar metallicity is able to match the observed radius at an age less
255
than 20 Gyr.
No evidence is available to support the idea that Kepler-16 B is magnetically active. Still,
we look to magnetic fields to reconcile the model predictions with the observations. All
possible scenarios relating to the various stellar mass estimates are considered. Explicitly,
we compute models for both the D11 and B12 masses and then attempt to fit the observations
using the magnetic Dartmouth stellar evolution models.
D11 Masses
The D11 primary mass implies that the age of the Kepler-16 system is approximately 3
Gyr, as shown in Figure 7.1(a). Recall, this age is reinforced by Winn et al. (2011) who
used age-activity and age-rotation relations to estimate an age of 3±1 Gyr. Due to the
consistency between the model derived age and the empirically inferred age, we see no
reason to introduce a magnetic perturbation into the model of Kepler-16 A. Winn et al.
(2011) observe moderately weak chromospheric activity coming from the primary, further
supporting our decision. Thus, we seek to reconcile models of Kepler-16 B with D11 masses
at 3 Gyr.
Magnetic models of Kepler-16 B were computed for a range of surface magnetic field strengths.
The perturbation was of the form presented in Chapter 5. A dipole radial profile was used
and the perturbation was applied in a single time step at an age of 10 Myr. Mass tracks with
a 4.0 kG and 5.0 kG surface magnetic field strength are shown in Figure 7.3. We are unable
to produce radius inflation larger than 1%, even for the strongest magnetic field strength of
5.0 kG. The peak field strength for the 5.0 kG model (located at R = 0.15R ∗ ) is approxi-
mately 1.5 MG. Discussion about how real such a magnetic field might be is deferred until
Section 7.4. For the moment, we are interested in knowing what magnetic field strength is
required to reconcile the models with the observations.
We next constructed magnetic models using a Gaussian radial profile. The magnetic field
256
was again introduced as a single perturbation. However, the model was perturbed at an
age of 1 Gyr to avoid convergence issues that arise with the Gaussian perturbation along
the pre-main sequence (pre-MS). It would also have been viable to ramp the magnetic field
strength from zero to full strength over a series of time steps. Both options lead to equivalent
results along the main sequence (Chapter 5).
Figure 7.4 presents mass tracks that include a magnetic perturbation with a Gaussian radial
profile. Two values of the surface magnetic field strength were used, 4.0 kG and 5.0 kG,
and are shown in the figure. The model with a 5.0 kG surface magnetic field strength
reconciles the model radius with the observed radius at an age of 3 Gyr. In contrast to the
results from the previous chapter, the dipole and Gaussian radial profiles produce radically
different results for a given surface magnetic field strength. This is due to the difference in
the peak magnetic field strengths. We will return to this issue in Section 7.4.
Lastly, figure 7.5 shows how the influence of the constant Λ profile on a model of Kepler-16
B. We started with a rather larger value of Λ = 0.9999 to gauge the reaction of the stellar
models to this formulation. The impact on the radius evolution of a M = 0.202M⊙ star is
negligible. Further increasing Λ had no effect on the resulting radius evolution.
257
Radius (R⊙)
0.24
[Fe/H] = -0.04
Kepler-16 B
Dipole Profile
Doyle et al. (2011)
0.23
0.22
|B|surf = 4.0 kG
|B|surf = 5.0 kG
0.21
Non-magnetic
1
10
Age (Gyr)
Figure 7.3: Standard DSEP (solid line) and magnetic models (broken lines) of Kepler-16 B
with D11 masses. The models were computed with [Fe/H] = −0.04 and a solar calibrated
αMLT . The magnetic model was calculated using a dipole radial profile with a 4.0 kG (dotted
line) and a 5.0 kG (dash-dotted line) surface magnetic field. The observed radius constraints
are shown as a shaded horizontal region and the age constraint is a vertical shaded region.
[Fe/H] = -0.04
0.24
Kepler-16 B
Doyle et al. (2011)
Radius (R⊙)
Gaussian Profile
0.23
0.22
|B|surf = 4.0 kG
|B|surf = 5.0 kG
0.21
Non-magnetic
1
10
Age (Gyr)
Figure 7.4: Standard DSEP (solid line) and magnetic models (broken lines) of Kepler-16 B
with D11 masses. The models were computed with [Fe/H] = −0.04 and a solar calibrated
αMLT . The magnetic model was calculated using a Gaussian radial profile with a 4.0 kG
(dotted line) and a 5.0 kG (dash-dotted line) surface magnetic field. The observed radius
constraints are shown as a shaded horizontal region and the age constraint is a vertical
shaded region.
258
[Fe/H] = -0.04
Radius (R⊙)
0.24
Kepler-16 B
Constant Λ
Doyle et al. (2011)
0.23
0.22
Λ = 0.9999
0.21
Non-Magnetic
1
10
Age (Gyr)
Figure 7.5: Standard DSEP (solid line) and magnetic model (dotted line) of Kepler-16 B with
D11 masses. The models were computed with [Fe/H] = −0.04 and a solar calibrated αMLT .
The magnetic model was calculated using a constant Λ = 0.9999 profile. The observed radius constraints are shown as a shaded horizontal region and the age constraint is a vertical
shaded region.
0.24
|B|surf = 5.0 kG
Kepler-16 B
|B|surf = 6.0 kG
[Fe/H] = -0.04
Gaussian Profile
Non-magnetic
Radius (R⊙)
Bender et al. (2012)
0.23
0.22
0.21
1
10
Age (Gyr)
Figure 7.6: Standard DSEP (solid line) and magnetic models (broken lines) of Kepler-16 B
with B12 masses. The models were computed with [Fe/H] = −0.04 and a solar calibrated
αMLT . The magnetic model was calculated using a Gaussian radial profile with a 5.0 kG
(dotted line) and a 6.0 kG (dash-dotted line) surface magnetic field. The observed radius
constraints are shown as a shaded horizontal region and the age constraint is a vertical
shaded region.
259
B12 Masses
Adopting the B12 masses primarily alters the age derived from stellar models. Instead of
3 Gyr, we infer an age of 12 Gyr from models of Kepler-16 A, as was shown in Figure
7.1(a). The relative radius discrepancy noted between models and Kepler-16 B is increased
by approximately 1% over the D11 case when using the B12 masses. This can be attributed
to the larger influence of age for a M = 0.65M⊙ star than for a M = 0.20M⊙ star.
Magnetic models were computed for Kepler-16 B with the B12 mass estimate using a Gaussian radial profile introduced at an age of 1 Gyr. These models are shown in Figure 7.6. We
did not generate models with a dipole radial profile models or with a constant Λ given the
lack of radius inflation observed for these magnetic field profiles. We find that a 6.0 kG surface magnetic field model was required, which translates to a nearly 50 MG peak magnetic
field strength. The stronger field strength required when using the B12 mass instead of the
D11 mass is a consequence of the 1% increase in the radius discrepancy mentioned above.
7.3.2 CM Draconis
CM Draconis (GJ 630.1 AC; henceforth CM Dra) contains two low-mass stars below the
fully-convective boundary and is arguably one of the most important systems for benchmarking stellar evolution models. Shortly after CM Dra was discovered by Luyten, Eggen & Sandage
(1967) uncovered that the star was actually a DEB. The nature of the secondary was subject
to debate. It was not clear from observations whether the secondary was a dark, very lowmass companion such that no secondary eclipse occurred or whether the two components
were of nearly equal mass. Evidence was tentatively provided in favor CM Dra being a
single dMe star with a dark companion (Martins 1975), although more observations were
encouraged as the author found a possible hint of a secondary eclipse. Any speculation that
the secondary companion to the dMe star of CM Dra was a dark, lower-mass object was
260
laid to rest by Lacy (1977) who obtained sufficient data to provide the first determination of
stellar parameters for both stars.
Following Lacy (1977)’s determination of the stellar properties, subsequent studies refined
and improved the masses and radii of the CM Dra stars (Metcalfe et al. 1996; Morales et al.
2009a). The currently accepted values (Morales et al. 2009a; Torres et al. 2010) are presented
in Table 7.2. Additional information about the CM Dra stars has been revealed in recent
years. Morales et al. (2009a) provided a detailed analysis of a nearby white dwarf (WD)
common proper motion companion and estimated the age of the WD to be 4.1 ± 0.8 Gyr.
Assuming the WD and CM Dra share a common origin, we can infer that the age of CM
Dra is comparable.
Deriving the metallicity of CM Dra has proven more difficult than estimating its age. Nearinfrared (NIR) spectroscopic studies that fit theoretical model atmospheres to atomic and
molecular features have consistently favored the interpretation that the system is metalpoor ([M/H] ≈ −0.6; Viti et al. 1997, 2002; Kuznetsov et al. 2012). Optical spectroscopy of
molecular features (CaH & TiO; Gizis 1997) and NIR photometric colors (Leggett et al. 1998),
on the other hand, have suggested that the system might have a near-solar metallicity. More
recent techniques relying on empirically-calibrated narrow-band NIR (H & K band) spectral features have started to converge on a value of [M/H] = −0.3 ± 0.1 (Rojas-Ayala et al.
2012; Terrien et al. 2012). Further support is provided by the photometric color-magnitudeTable 7.2
Fundamental properties of CM Draconis.
Property
Mass (M⊙ )
Radius (R⊙ )
Teff (K)
[Fe/H]
CM Dra A
CM Dra B
0.23102 ± 0.00089
0.21409 ± 0.00083
3130 ± 70
3120 ± 70
0.2534 ± 0.0019
Age (Gyr)
0.2398 ± 0.0018
−0.30 ± 0.12
4.1 ± 0.8
261
0.30
[Fe/H] = -0.18
[Fe/H] = -0.30
Radius (R⊙)
0.28
[Fe/H] = -0.42
0.26
CM Dra A
0.24
0.22
CM Dra B
WD Age
0.20
0.1
1
10
Age (Gyr)
Figure 7.7: Standard DSEP models of CM Dra A (maroon) and B (light-blue). The models
were computed for Fe/H] = −0.18 (dashed line), −0.30 (solid line), and −0.42 (dotted line)
with a solar calibrated αMLT . Observed radius constraints are shown as shaded horizontal
regions and the age constraint is a vertical shaded region.
metallicity relation of Johnson & Apps (2009). Their relation predicts [Fe/H] ≈ −0.4.
The components of CM Dra are known to be inflated compared to standard stellar models
(Ribas 2006; Morales et al. 2009a; Torres et al. 2010; Feiden & Chaboyer 2012a; Spada & Demarque
2012; Terrien et al. 2012). This fact has become steadily more apparent since the initial
comparative study of Chabrier & Baraffe (1995), which found little disagreement. A precise estimate of the level of disagreement depends on the adopted metallicity (see, e.g.,
Feiden & Chaboyer 2012a; Terrien et al. 2012), but the problem is robust. Figure 7.7 demonstrates the level of disagreement compared to standard DSEP mass tracks. It also illustrates
the effect of altering the metallicity in the stellar models. The level of disagreement observed in Figure 7.7 is between 5% – 6% for the primary and 7% – 8% for the secondary
(Terrien et al. 2012). The latest study by Terrien et al. (2012) has essentially doubled the
level of radius disagreement from 3% – 4% to 6% – 8%. This is purely a consequence of their
metallicity estimate.
262
0.30
|B|surf = 5.0 kG
Non-Magnetic
Radius (R⊙)
0.28
0.26
CM Dra A
0.24
CM Dra B
0.22
Dipole Profile
WD Age
[Fe/H] = -0.18
0.20
0.1
1
10
Age (Gyr)
Figure 7.8: Standard DSEP (solid line) and magnetic model (dashed line) of CM Dra A
(maroon) and B (light-blue). The models were computed with [Fe/H] = −0.18 and a solar
calibrated αMLT . The magnetic model was calculated using a dipole radial profile with a
5.0 kG surface magnetic field. Observed radius constraints are shown as shaded horizontal
regions and the age constraint is a vertical shaded region.
It is well appreciated that CM Dra is magnetically active. Balmer emission and light curve
modulation due to spots have been recognized since very early investigations (Zwicky 1966;
Martins 1975; Lacy 1977). Frequent optical flaring has also been continually noted (e.g.,
Eggen & Sandage 1967; Lacy 1977). Further details on the flare characteristics of CM Dra
may be found in the work by MacDonald & Mullan (2012). The system is also a strong
source of X-ray emission according to the ROSAT All-Sky Survey Bright Source Catalogue
(Voges et al. 1999).
The high level of magnetic activity and the short orbital period of CM Dra (1.27 days)
have been used to justify the need for magnetic perturbations in low-mass stellar evolution
models. Such studies were carried out by Chabrier et al. (2007) and MacDonald & Mullan
(2012) using the methods described in Sections 1.4 and 6.5.2. CM Dra therefore provides a
pivotal test of our models and of the magnetic field hypothesis advanced in Section 1.3.4.
263
Results from our magnetic models are displayed in Figures 7.8, 7.9, and 7.10. We computed
magnetic models using all three formulations addressed thus far: dipole radial profile, Gaussian radial profile, and the constant Λ turbulent dynamo formulation. Perturbations were
introduced over a single time step, but the precise age of the perturbations varied (dipole:
10 Myr, Gaussian: 1 Gyr, constant Λ: 100 Myr). The models all had [Fe/H] = −0.18, the
maximum allowed metallicity (Terrien et al. 2012), and a solar-calibrated αMLT .
The models performed similar to those for Kepler-16 B. A strong 5.0 kG surface magnetic
field strength (1.5 MG peak field strength) with a dipole radial profile was insufficient. It
relieved the radius disagreements but was unable to provide the necessary inflation. This is
evidenced in Figure 7.8. The Gaussian radial profile model was able to largely reconcile the
models with the observations. Surface field strengths of ∼ 6.0 kG were required (slightly
stronger for CM Dra B), which translates to a 50 MG peak field strength. These models
are displayed in Figure 7.9. Lastly, the constant Λ models were also unable to induce any
significant radius inflation. We used Λ = 0.5, 0.9, 0.99, and 0.9999. Only the Λ = 0.9999
model is shown in Figure 7.10 to more clearly illustrate the ineffectiveness of this method.
264
0.30
|B|surf = 6.0 kG
Non-Magnetic
Radius (R⊙)
0.28
0.26
CM Dra A
0.24
CM Dra B
0.22
Gaussian Profile
WD Age
[Fe/H] = -0.18
0.20
0.1
1
10
Age (Gyr)
Figure 7.9: Standard DSEP (solid line) and magnetic model (dashed line) of CM Dra A
(maroon) and B (light-blue). The models were computed with [Fe/H] = −0.18 and a solar
calibrated αMLT . The magnetic model was calculated using a Gaussian radial profile with a
6.0 kG surface magnetic field. Observed radius constraints are shown as shaded horizontal
regions and the age constraint is a vertical shaded region.
0.30
Λ = 0.9999
Non-Magnetic
Radius (R⊙)
0.28
0.26
CM Dra A
0.24
CM Dra B
0.22
Constant Λ
WD Age
[Fe/H] = -0.18
0.20
0.1
1
10
Age (Gyr)
Figure 7.10: Standard DSEP (solid line) and magnetic model (dashed line) of CM Dra A
(maroon) and B (light-blue). The models were computed with [Fe/H] = −0.18 and a solar
calibrated αMLT . The magnetic model was calculated with a constant Λ = 0.9999 turbulent
dynamo formulation. Observed radius constraints are shown as shaded horizontal regions
and the age constraint is a vertical shaded region.
265
7.4 Discussion
7.4.1 Magnetic Field Radial Profiles
The three magnetic field profiles we introduced in Section 7.2 produce different results in
the stellar models. We can understand this phenomenon by following the same line of
reasoning used in Section 6.4.1. There, we were able to diagnose why magnetic stellar
models produced nearly identical results for the dipole and Gaussian radial profiles in stars
with a radiative core. Recall that the super-adiabatic layer near the surface of those stars
was more heavily influenced by the presence of the magnetic perturbation than the deeper
near-adiabatic convective layers. Thus, the peak magnetic field strength had little effect
compared to the surface magnetic field strength, meaning the dipole and Gaussian profiles
produced similar results.
For fully convective stars, the situation is reversed. We display the run of (∇s − ∇ad ) in two
models of CM Dra A in Figure 7.11. One standard DSEP model and one magnetic model
are shown. The magnetic model is the Gaussian radial profile model with a 6.0 kG surface
magnetic field that was plotted in Figure 7.9. The surface radiative zone and the superadiabatic layer are significantly altered by the presence of the magnetic field. However,
these layers have little influence over the structure of the star. We verified this by looking
at the profile of a dipole radial profile model with a similar surface magnetic field strength.
The (∇s − ∇ad ) profile exhibited in the surface layers by the dipole radial profile model is
nearly identical to the Gaussian profile model. The difference between the dipole profile and
the Gaussian profile models was the appearance of a radiative core in the Gaussian profile
model. This is highlighted in the inset of Figure 7.11.
The final radial profile is the constant Λ profile, which also relies on the turbulent dynamo
formulation. This profile slightly perturbs the shape of the (∇s − ∇ad ) curve in the superadiabatic layer. It also produces a marginally larger surface radiative zone than the standard
266
Non-magnetic
Magnetic
0.2
(∇s - ∇ad)
0.1
super-adiabatic
convection
near-adiabatic convection
0.0
0.001
0.000
-0.1
-0.001
-0.002
1.0
1.5
2.0
-3
-2
-1
-3
log10(ρ) (g cm )
0
1
-0.2
-6
-5
-4
2
Figure 7.11: The difference between the plasma temperature graident, ∇s , and the adiabatic temperature gradient, ∇ad , as a function of the logarithmic plasma density for a
M = 0.231M ⊙ star. We show this for two models: a non-magnetic model (maroon, solid
line) and a magnetic model (light-blue, solid). The zero point is marked by a gray dashed
line, dividing locations where convection (positive) or radiation (negative) is the dominant
flux transport mechanism. The inset zooms in on the deep interior where the magnetic field
creates a small radiative core.
non-magnetic model. Convection in the deep interior is relatively unaffected, so a radiative
core does not develop. We opted to not display these features in a figure because the overall
profile is almost identical to the non-magnetic profile in Figure 7.11. Given the insensitivity
of the overall stellar structure of fully convective stars to the size of the super-adiabatic
layer, the constant Λ models have a negligible influence on the stellar radius.
7.4.2 Surface Magnetic Field Strengths
We use the relationship between the surface magnetic flux, Φ, and X-ray luminosity, L x ,
developed in the previous chapter to estimate the surface magnetic field strengths for the
267
stars of CM Dra. Surface field strength estimates can not be calculated for Kepler-16 because
it does not appear in any of the ROSAT catalogs. The ROSAT Bright Source Catalogue
(Voges et al. 1999) indicates that CM Dra has X cr = 0.18 ± 0.02 cts s−1 with HR = −0.30 ±
0.07. This translates into an X-ray luminosity per star of L x = (1.57 ± 0.40) × 1028 erg s−1 ,
where we have used the parallax, π = 68 ± 4 mas, quoted by Harrington & Dahn (1980).
Note that the X-ray luminosities are upper limits due to possible X-ray contamination in
the ROSAT data. There are several stars nearby to CM Dra in the plane of the sky, but it is
difficult to judge whether they contribute to the ROSAT count rate.
From the X-ray luminosity derived above, we find log10 (Φ) = 24.81 ± 0.45 Mx. The errors
associated with the surface magnetic field strength estimates are substantial due to the large
error on the surface magnetic flux. Converting the magnetic fluxes to a surface magnetic
+3.36
field strengths, we estimate that 〈B f 〉 A = 1.65+3.00
−1.07 kG and 〈B f 〉B = 1.85−1.19 kG for CM
Dra A and B, respectively. These estimates imply that the 6.0 kG surface magnetic field
strengths required by our Gaussian radial profile models are likely too strong. However,
this does not invalidate the magnetic field models, only our choice of radial profile. The
magnetic field strength in the deep interior is of greater consequence, so it may be possible
to construct a radial profile to greater reflect this fact (see below).
Models that use the constant Λ formalism predict surface magnetic fields strengths up to
∼ 3.0 kG. This upper limit is set by the magnetic field coming into equipartition with the
kinetic energy of convective flows (Chabrier & Küker 2006; Browning 2008). Values of 3.0
kG are consistent with the estimated field strengths. Additionally, the equipartition magnetic field strengths are consistent with typical field strengths measured on the surface of
M-dwarfs (Saar 1996; Reiners & Basri 2009; Shulyak et al. 2011; Reiners 2012). Although
the magnetic field strengths are consistent with observations and 3D MHD modeling, our
models were unable to produce radii consistent with the observations.
Although there are no X-ray measurements for Kepler-16 B, we can infer from CM Dra
268
how realistic we believe our models to be. Kepler-16 B has a similar mass to the stars in CM
Dra and may have a similar age, as well. Based on this information, it is our opinion that
Kepler-16 B should have a magnetic field strength of a similar magnitude to the stars of CM
Dra. We do not believe it would have a significantly larger field strength owing to its age
and orbital period. Therefore, we believe that the 6.0 kG surface magnetic field required to
bring the models into agreement with the observations is probably too strong.
7.4.3 Comparison to Previous Studies
Previous attempts to reconcile model radii with the observed radii of CM Dra were seemingly successful (Chabrier et al. 2007; Morales et al. 2010; MacDonald & Mullan 2012). It is
therefore prudent to compare the results of our modeling effort with these previous studies.
The methods used by the two groups were in some respects similar. Morales et al. (2010)
used the methods outlined by Chabrier et al. (2007) to include possible effects of magnetic
fields. This approach relies on reducing the convective mixing length, αMLT , to mimic a reduced convective efficiency. On the other hand, MacDonald & Mullan (2012) implemented
their convective inhibition parameter, δMM , to help stabilize regions in the model against
convection (Mullan & MacDonald 2001). While these two groups used different methods
to alter convection, they both treated star spots according to the formalism introduced by
Chabrier et al. (2007).
We will briefly summarize the results of these studies. Morales et al. (2010) found that they
were able to reproduce the observed properties of CM Dra with a reduced αMLT and a star
spot factor β = 0.17. This is after they applied a downward correction to the radii of CM
Dra to compensate for possible 3% systematic errors in the light curve analysis due to star
spots. Their value for β = 0.20 ± 0.04 combined a spot coverage fraction of 42%±8% with a
spot temperature contrast of approximately 85% (spots are 15% cooler than the unblemished
photosphere). The study by MacDonald & Mullan (2012) identified a region of δMM − β ( f
269
in their paper) parameter space that was able to reconcile their models with CM Dra (see
their Figures 13 – 15). Their investigation reinforced the work by Chabrier et al. (2007) and
Morales et al. (2010) that the spot parameter β has a greater influence on the stellar structure. The full range of β was 0.15 . β . 0.28 with 0.0 < δMM < 0.025. Using the revised
radii from Morales et al. (2010) and the recommended β = 0.17, MacDonald & Mullan (2012)
required a magnetic inhibition parameter between 0.020 < δMM < 0.025. These values of
δMM correspond to vertical magnetic field strengths of approximately 500 G.
Before we compare our models to the previous studies, we first evaluate their accuracy in
light of recent findings. The metallicity of the models adopted in both studies was [Fe/H]
= 0.0. 1 Recent measurements have measured the metallicity to be [Fe/H] = −0.30 ± 0.12
(Rojas-Ayala et al. 2012; Terrien et al. 2012). Consequently, the stellar models used in previous studies will be over-estimating the actual model-predicted radii by about 2.5%, based
on metallicity variations observed in the DSEP models. This may seem like an insignificant amount, but consider that the Morales et al. (2010) study only corrected a 2% radius
offset between the Baraffe et al. (1998) models and CM Dra (after correcting for systematic
radius errors of about 3%). Now, Morales et al. (2010) used a 1 Gyr isochrone, which is 3
Gyr younger than the estimated age of CM Dra. Therefore, the radii of their models were
about 1% smaller compared to what a 4 Gyr isochrone predicts. Including the 1% age effect,
decreasing the metallicity of their models would decrease their radius predictions by about
1.5%. This nearly doubles the original radius offset.
The resulting β needed to correct for an additional 1.5% radius offset would be ≥ 0.30
based on Figure 8 of Morales et al. (2010). This translates into a spot coverage of over
63% for a fixed temperature contrast of 85% or a temperature contrast of 73% for a fixed
spot coverage of 42%. These latter values may be consistent with TiO modeling of spot
properties (O’Neal et al. 2004) and modeling of spot characteristics of late-type stars in clus1
MacDonald & Mullan (2012) did explore varying combinations of Y , Z , αMLT , and δMM , but were unable
to find a suitable combination to reproduce the stars in CM Dra.
270
ters (Jackson et al. 2009; Jackson & Jeffries 2013). However, Jackson & Jeffries (2013) assume
randomly distributed spots whereas Morales et al. (2010) used large polar spots—spots preferentially located near the poles defined by the rotation axis. Their preference for large
polar spots was adopted to justify the 3% systematic errors used to reduce the radii of CM
Dra. Further investigations are needed to evaluate the consistency of these two studies,
as the precise spot configuration could place the 3% systematic radius correction applied
by Morales et al. (2010) in jeopardy. If the 3% radius error has to be discarded, then a
β > 0.37 will be needed to bring the models into agreement with CM Dra. It was implied
by Morales et al. (2010) that a β of this magnitude may be physically unrealistic and should
be approached with caution. Keeping this in mind, we compare our models to those of
Morales et al. (2010) and MacDonald & Mullan (2012).
Our results are largely consistent with the results of Morales et al. (2010) and MacDonald & Mullan
(2012). Comparing with MacDonald & Mullan (2012), we look at the dipole and Gaussian radial profile models (Figures 7.8 and 7.9). The magnetic perturbation applied in these models
is qualitatively similar to the one applied by MacDonald & Mullan (2012), except we neglect
star spots. In Section 6.5.2 we estimated that our magnetic field strengths would need to be
an order of magnitude larger than those of Mullan & MacDonald (2001) as a consequence
of our formulation. Our “inhibition parameter” largely boils down to the quantity (ν∇χ ).
It was previously shown that ν ∼ δMM . Since ∇χ ∼ 0.1, our magnetic field strength should
need to be about an order of magnitude larger. This is seen in our requirement of a 6.0 kG
surface magnetic field with a 50 MG peak field strength to reproduce the properties of CM
Dra. MacDonald & Mullan (2012) required a 500 G field with a 1 MG peak magnetic field
strength. It is a little surprising that this played out considering they also added a factor to
account for star spots, but is in line with our initial approximations.
The pitfall of the dipole and Gaussian radial profiles is the sizable peak magnetic field.
There is no evidence that a magnetic field on the order of a MG or greater could exist
271
within a fully convective star. There is certainly no physical mechanism to generate such
a strong magnetic field from a turbulent dynamo (Chabrier & Küker 2006; Chabrier et al.
2007; Browning 2008). To avoid invoking a turbulent dynamo to justify a strong 1 MG
magnetic field in the stars of CM Dra, MacDonald & Mullan (2012) suggest that a primordial
µG seed magnetic field was present in the proto-stellar molecular cloud. This seed field was
amplified due to flux conservation throughout the collapse of the individual proto-stars.
We see two issues with this idea. First, the magnetic field must then survive for several
billion years without diffusing through the plasma. Since the magnetic field is likely frozen
into the highly conducting stellar plasma, the timescale for diffusion is proportional to the
convective timescale. For a fully convective M-dwarf this is around 107 s, or ∼ 1 yr. If
convection were largely suppressed by the magnetic field, this convective timescale would
increase, but not by nine orders of magnitude. Even with slow convection, the molecular
diffusion timescale for such a field is, in all likelihood, quite rapid compared to the lifetime
of the star (Chabrier et al. 2007). Full 3D MHD models of fully convective stars with a large
primordial magnetic field could lend insight into a more realistic diffusion timescale.
The second problem we see with the interior MG field strength is KOI-126. Nearly identical in mass and rotational period to CM Dra, the fully convective stars of KOI-126 are
well reproduced by stellar models (Chapter 2; Feiden et al. 2011; Spada & Demarque 2012).
MacDonald & Mullan (2012) argue that model agreement based on KOI-126 should be expected given its orbital period, which is half a day longer than CM Dra’s. They calculate
that the surface magnetic fields would be significantly lower than those CM Dra. However,
if we assume the 1 MG interior field is of primordial origin, then we see no reason why
KOI-126 would not be expected to have a similar interior field strength (assuming the field
was stable for billions of years). Seeing as the interior magnetic field strength largely determines the induced radius inflation for fully convective stars, we would draw the opposite
conclusion of MacDonald & Mullan (2012). KOI-126 should appear inflated compared to
272
standard stellar evolution models. We therefore doubt the existence of super-MG magnetic
fields deep within fully convective stars. This leads us to believe the agreement between
our Gaussian radial profile model and the observations is unphysical.
Comparing our constant Λ models to the Morales et al. (2010) results, we find they are
consistent. Chabrier et al. (2007) and Morales et al. (2010) find that the influence of reducing
αMLT is negligible compared to the radius inflation created by their star spot formulation.
Section 6.5.2 demonstrated that our turbulent dynamo approach was quite similar to the
reduced αMLT approach. In fact, we formulated reductions in αMLT as a function of Λ. It is
therefore not surprising that we find agreement between the two methods.
So what are we to make of the previous discussions? On the one hand, the suppression of
convection appears to be unable to reconcile the model predictions with the observations.
But, on the other hand, reducing the total flux due to the presence of star spots is able to
do so. We find these two pieces of evidence contradictory. Star spots have a physical origin
and that origin is the suppression of convection by a vertical magnetic field (Chandrasekhar
1961; Gough & Tayler 1966). We have difficulty understanding why models that suppress
convection in a manner consistent with the physical picture of star spots (Gough & Tayler
1966; Mullan & MacDonald 2001) would be insufficient at inflating radii, yet an independent
method based on reducing the total flux—essentially the same physical effect as inhibiting
convection to model spots—would suffice (i.e., Chabrier et al. 2007). It is true that any onedimensional treatment of magnetic fields and spots will be overly simplistic, and further
work on this matter is warranted. However, even when we almost entirely suppress convection using a constant Λ profile with Λ = 0.9999 and a reduced αMLT = 0.5, there is a
negligible impact on the model radius predictions.
It may be argued that the two effects are easy to reconcile when considering that magnetic
fields can have global and local effects. A sub-kG large-scale magnetic field may exist in
combination with strong, localized magnetic field structures that form spots. The result is an
273
average magnetic field strength that appears to be only a few kG. We test this by introducing
a strong suppression factor (Λ = 0.9999) and an independent stabilization factor using a
dipole radial field with a 3.0 kG surface magnetic field in our models. This formulation
barely inflates the radius of a M = 0.23M⊙ model. We mentioned before that the deep
interior magnetic field strength is the most important factor when it comes to inflating
fully-convective stars. Thus, having strong localizations of magnetic fields on the surface is
insufficient to cause the necessary structural changes. Further testing of this hypothesis is
necessary.
The case for magnetic fields inflating the radii of fully convective stars appears to be tenuous. The more we learn about these systems, the harder it becomes to reconcile model
predictions with the observations. When we weigh the evidence, we find it harder to believe
that magnetic fields are actively inflating the radii of fully convective stars. Furthermore,
although the reduced-flux star spot formulation appears to be able to provide the necessary radius inflation, this does not include variations due to metallicity. Our models use
the largest possible value of the metallicity for CM Dra. If we consider the difficulty in obtaining small, 1% changes in the radii of these stars, moving to lower metallicities presents
new challenges. Radius changes due to metallicity have the ability to push the star spot
parameters into unphysical regimes.
We are of the opinion that magnetic fields may not be the appropriate solution. While
more work is needed to investigation the influence of star spots and improved methods for
treating a magneto-convection in 1D models, we feel that it is not too early to seek other
solutions.
7.4.4 Alternatives
If magnetic inhibition and stabilization against convection are unable to reconcile fully convective model predictions with observations, what might the solution be? As an exercise,
274
we plotted the radius discrepancies of fully convective stars against several intrinsic properties: mass, metallicity, orbital period, and Rossby number. Data for all fully convective stars
with a measured metallicity were included. This sample contains six stars: CM Dra (A, B),
KOI-126 (B, C), Kepler-16 (B), and Kepler-38 (B) (Orosz et al. 2012). LSPM J1112+7626 (B)
(Irwin et al. 2011) was not included because it lacks a proper metallicity estimate.
Figure 7.12 shows the results of this exercise. We were surprised to find that metallicity
provided the best correlation with radius deviations, as seen in panel (a). Radius deviations
clearly do not correlate with stellar mass, evidenced by Figure 7.12(b). Orbital period and
Rossby number show a hint of correlation, but the relationships are broken by the presence
of KOI-126 (B, C), which are located along the δR/R obs = 0 line. We note that the rotation
periods of the secondary stars in Kepler-38 and Kepler-16 are not known. Spin-orbit synchronization was assumed to create Figure 7.12, but is not necessarily a valid assumption
with these stars. This would also affect the calculation of the Rossby number in Figure
7.12(d).
The relationship between radius deviations and metallicity is surprising for two primary
reasons. First, it contradicts the results presented by López-Morales (2007), which we described in Section 1.3.1. The second reason is that the two variables are anti-correlated. If
a trend with metallicity had been expected, we would have predicted that the more metal
rich objects would show the largest discrepancies. With only six data points, the sample
size is admittedly small. However, that the data points follow such a tight trend is either a
remarkable coincidence or is an important clue to help unravel the radius discrepancies.
Evidence of this anti-correlation may also exist in the study by Boyajian et al. (2012). They
measured the radii of over 20 K- and M-dwarfs using the CHARA interferometer. One key
result from their study was that the empirical radius-Teff relation for low-mass stars was
independent of metallicity (see their Figure 13). Plotting the relative difference between their
measured radii and DSEP models, they found that smaller objects were typically under-
275
0.10
0.10
(a)
0.06
0.04
0.02
0.00
0.06
0.04
0.02
0.00
-0.02
-0.5 -0.4 -0.3 -0.2 -0.1 0
[Fe/H]
-0.02
0.1 0.2 0.3
0.18
0.10
0.20 0.22 0.24
Mass (M⊙)
0.26
0.10
0.08
0.06
0.04
0.02
(d)
0.08
(c)
δR/Robs
δR/Robs
(b)
0.08
δR/Robs
δR/Robs
0.08
0.06
0.04
0.02
0.00
0.00
-0.02
0.01
-0.02
0
10
20
30
40
Porb (day)
0.1
Ro
1
Figure 7.12: Radius residuals between models and observations of fully convective stars as
a function of [Fe/H], mass, orbital period (proxy for rotation period), and Rossby number.
Only fully convective stars with precise masses, radii, and metallcities were included. The
stars in the sample are CM Dra (Morales et al. 2009a), KOI-126 (B, C) (Carter et al. 2011),
Kepler-16 B (Doyle et al. 2011), and Kepler-38 B (Orosz et al. 2012).
predicted by the models. If the anti-correlation shown in Figure 7.12 is real, we would then
expect the stars with the largest observed discrepancies in Boyajian et al. (2012) to be more
metal poor. This appears to be the case (see Figure 14 in Boyajian et al. 2012). Now, they
did not find any direct correlation with metallicity and radius deviations (see their Figure
16), but precise radius deviations are difficult to determine because their mass estimates
are highly uncertain. Still, it would appear as though the data in Boyajian et al. (2012) lend
some support to the existence of a metallicity-radius anti-correlation among fully convective
stars.
We would like to offer a compelling argument to justify the existence of this relationship.
However, we do not have a definitive explanation at the present time, although we may
276
be able to rule out some scenarios. It is likely not a connection between metallicity and
the stellar convective properties. We have shown that modifications to convection can not
account for any significant radius inflation. Unless there are multi-dimensional affects of
convection that can alter the structure of the star, convection seems to be an insufficient
explanation. Opacity would also seem to be ruled out. One would expect that the significant
sources of error from opacity calculations are related to the presence of metals. An anticorrelation between model errors and metallicity would suggest that hydrogen opacities are
in error.
One possibility is that the model [α/Fe] abundances are incorrect. It has been suggested that
below solar metallicity, M-dwarfs show a total metallicity, [M/H], that is greater than the
iron abundance, [Fe/H] (Rojas-Ayala et al. 2012; Muirhead et al. 2012; Mann et al. 2013a,b).
For our models, we have been using a scaled-solar metallicity, which effectively equates
[Fe/H] to [M/H]. The differences between [M/H] and [Fe/H] noted by Rojas-Ayala et al.
(2012) are on the order of 0.1 dex. This translates, roughly, to an alpha-element enhancement of [α/Fe] ≈ +0.2 dex. Given a star with a mass similar to CM Dra A and a [Fe/H]
= −0.30, increasing [α/Fe] by 0.2 dex increases the radius by about 2%. A modest increase
compared to the 6% to 8% radius deviations we observe with CM Dra. Providing a complete solution for this anti-correlation would require a generous helping of alpha-element
enhancement.2
There is also the question of the adopted helium abundance. We noted in Equation (6.1)
that the initial helium abundance is scaled linearly with the heavy element mass fraction.
Figure 7.13 shows how the metallicity-radius deviation correlation changes when we assume a fixed helium abundance, Y = 0.33. This value represents a significant enhancement
compared to the linear-scaled value for all of the stars in our sample. It is also intermediate
2
There is also the larger issue of the overall solar composition. We use the Grevesse & Sauval (1998) solar
abundance composition, but it seems likely that the solar heavy element composition is lower (Asplund et al.
2009). The global lowering of the heavy element composition may offset the benefits of an enhanced alpha
abundance.
277
0.10
Z-scaled Yi
Yi = 0.33
0.08
δR/Robs
0.06
0.04
0.02
0.00
-0.02
-0.5
-0.4
-0.3
-0.2
-0.1
[Fe/H]
0
0.1
0.2
0.3
Figure 7.13: Radius residuals between fixed-Y models (Y = 0.33) and observations of fully
convective stars as a function of [Fe/H]. The fixed-Y points are plotted in maroon and those
with Z -scaled Y values (from Figure 7.12) are plotted in gray, for reference.
between estimated helium abundances in old star clusters (Piotto et al. 2005; Brogaard et al.
2012) and should therefore not be entirely unphysical. We see that arbitrarily increasing
the helium abundance relieves some of the radius discrepancies, but does not fully explain
the observed trend with metallicity. We must also consider that all of the stars will not
necessarily have such an enhancement in helium abundance and that altering the helium
abundance of the systems will effect other properties (most notably the age). All of the
stars have ages estimated from companions, which must also be modeled with the helium
enhancement to extract meaningful results.
7.5 Summary
This chapter addressed the question of whether magnetic fields can realistically inflate the
radii of fully convective stars. We approached the problem by modeling two individual
278
DEBs that contain fully convective stars, Kepler-16 and CM Draconis. Ultimately, we find
that magnetic fields do not provide a physically plausible solution of the observed radius
discrepancies. Multiple formulations of the magnetic perturbation in our models were used
to no avail. There was a hint that strong MG magnetic field deep in the stellar interior could
reconcile the models, but it is our opinion that this is a physically unrealistic scenario. Our
results, while largely consistent with previous studies, are in disagreement that magnetic
fields provide an adequate solution.
Without invoking magnetic fields, we are at a loss to explain the large radius discrepancies
observed with fully convective stars. However, an anti-correlation between these discrepancies and [Fe/H] was uncovered. No physical explanation for this occurrence was put
forth. Alpha-element enhancement may provide part of the solution, but can only account
for about 2% of the total radius deviations. Further investigation into this anti-correlation
may reveal the true cause of the observed radius inflation. Studies to collect more data and
generate a larger model grid, covering the available magnetic and non-magnetic parameter
space, are on-going.
279
Chapter 8
Conclusions & Future Investigations
8.1
Thesis Summary
The body of work contained in this dissertation was aimed at deepening our theoretical
understanding of low-mass stars. From the outset, an end goal of the project was to have
developed a set of new stellar evolution models with magnetic fields. Along the way, however, we found the need to evaluate the present capability of standard low-mass stellar
evolution models.
We began our journey with KOI-126, the hierarchical triply eclipsing system with two lowmass stars below the fully convective boundary (Carter et al. 2011). Chapter 2 presented
stellar evolution models that accurately reproduced the properties of all three stars. This
was the first successful modeling of fully convective stars in a DEB. Given their similarity
to CM Dra A and B, the low-mass stars in KOI-126 provide a unique constraint on theoretical descriptions of the radius inflation observed with CM Dra. Any theory mitigating the
problems with CM Dra must also not undermine the agreement with KOI-126. We also predicted theoretical interior structure constants for KOI-126 B and C. A weighted average of
these quantities helps to define the rate of apsidal motion of the binary orbit. Observational
280
constraints on the apsidal motion rate at the 1% level predicted by Carter et al. (2011) would
provide a stringent test of our stellar evolution models. Not only would the constraints
test our models, but we discussed how the observed apsidal motion could validate various
equations of state. KOI-126 B and C can therefore be used to probe the interior structure of
low-mass stars.
Chapter 3 delved deeper into applying our model predictions for the internal structure constants. We developed a novel method for estimating the ages of young DEB systems. Most
age dating techniques involve comparing the predicted surface properties of stellar evolution models to observations of young stars. The problem with this approach is that the
surface properties of stellar models and real stars may be heavily influenced by magnetic
activity. By using the interior structure constants, we can achieve an age dating method
that is more robust by avoiding the interpretation of surface features. We tested the sensitivity of this technique to specific model input properties (i.e., mass, metallicity, helium
abundance) and possible magnetic activity effects (e.g., suppressed convection). Generally
speaking, we were able to achieve an age estimate with a precision to better than 20% with
favorable observational data.
Returning to the mass-radius discrepancies in Chapter 4, we reanalyzed the severity of the
problem. Numerous literature references to the mass-radius problem quoted a systematic
5% – 10% radius inflation.1 By coupling a large data set (nearly double the number of DEBs
in previous studies) and modeling variations due to metallicity and age, we showed that the
radius problem was not systematic. Average deviations of about 2.5% were the norm and
only a couple systems exhibited significant inflation above 5%. Previous misidentification
of large systematic errors was likely the result of studies using only one or two isochrones
instead of properly fitting systems to a grid of isochrones. Furthermore, we provided a
statistical analysis comparing the observed radius deviations to potential magnetic field
strength proxies (i.e., rotation period, Rossby number, and X-ray luminosity). We found no
1
43 papers between 2007 and 2012... not that we were ever compelled to count.
281
significant difference in the distribution of radius deviations between slowly rotating and
rapidly rotating populations. However, there was evidence for a correlation between radius
deviations and X-ray luminosity.
The original idea when this project began was to develop a new set of low-mass stellar evolution models that included magnetic field perturbations. This goal was achieved in Chapter
5. We lay out, in full detail, the mathematical formalism of our approach to including magnetic field perturbations. Following the seminal work of Lydon & Sofia (1995), we tweaked
the formulation and developed our own set of magnetic stellar evolution models. A series
of tests were performed to ensure the numerical stability and overall believability of the
models. After verifying that the models were producing reliable results, we applied our
models to a recently characterized solar-type DEB, EF Aquarii (Vos et al. 2012). Specifically,
we found that models of EF Aqr A required a surface magnetic field strength between 1.6
– 2.6 kG. EF Aqr B, on the other hand, needed a stronger surface magnetic field strength
of 3.2 – 5.5 kG. The magnetic field strengths were evaluated using scaling relations based
on X-ray luminosity and Ca ii emission strengths. The derived surface magnetic fields were
consistent with the model predicted field strengths.
The final two chapters analyzed several important low-mass DEBs to test the hypothesis
that magnetic fields are inflating low-mass stars. Here, we split the analysis into two parts:
Chapter 6 dealt with stars that have a radiative core and convective envelope and Chapter
7 looked at fully convective stars. Stars with radiative cores proved to be susceptible to
significant radius inflation from magnetic fields. Three systems, UV Psc, YY Gem, and CU
Cnc, were fit by magnetic models, but only after we reformulated the mixing length theory
of magneto-convection to account for the conversion of convective energy into magnetic
energy. The resulting magnetic models were able to reproduce the radii of the stars in
YY Gem and CU Cnc with realistic surface magnetic field strengths. Estimates of the actual
surface magnetic field strengths were derived from an X-ray–magnetic-flux scaling relation.
282
Additional analysis was carried out to explore the impact of magnetic models on studies of
exoplanet habitability and asteroseismology.
Fully convective stars were not so easily influenced by the magnetic models. All but models
with super-MG magnetic fields in the deep interior were ineffective at inflating stellar radii.
This is in sharp contrast to other studies that find magnetic fields up to the task. Extensive
discussion comparing and contrasting the individual studies was provided so that the reader
could form their own opinion. We, however, are skeptical that magnetic fields are inflating fully convective stars. Instead, there appears to be an anti-correlation between radius
deviations and metallicity of fully convective stars. It is not clear to us what the physical
justification is for this anti-correlation.
8.2
Future Investigations
Throughout the course of developing this dissertation, numerous avenues for continuing
research were exposed. The most natural of these is the development of a large database
of magnetic model tracks and relations. With the magnetic models now rigorously tested
against individual DEBs, we are confident that the magnetic field perturbations are correct.
An extensive grid of models will be required to allow for the type of individual or statistical
tests performed throughout this thesis. Additionally, a functional web interface will be
necessary. Progress on both pieces to the database will begin in the near future.
Developing a database of magnetic mass tracks will enable additional studies using the
magnetic models. From Chapter 2, we learned that the apsidal motion of KOI-126 (B,C)
could be used to evaluate the validity of different equations of state. Following the publication of Feiden et al. (2011), I began work on testing the sensitivity of the theoretical interior
structure constants to the available micro-physics. This includes several equations of state
(FreeEOS; Saumon et al. 1995) and subtle variations of the FreeEOS. With the introduction
283
of reliable magnetic field models, it would be advantageous to include how variations in a
stellar magnetic field might influence the interior structure constants. Eventually we hope
to place constraints on individual components of the equation of state using observations
KOI-126 thereby allowing for the computation of more accurate equations of state suitable
for low-mass stars and giant planets.
A comprehensive database will also allow for large statistical tests of the stellar evolution
models. I am involved in a campaign to evaluate stellar evolution and stellar atmosphere
models using the Boyajian et al. (2012) sample of K- and M-dwarfs with interferometric
radii and effective temperatures. Observers are obtaining spectra for each star so that they
may determine [Fe/H] and [α/Fe] (Mann et al. 2013a,b). A fine grid of over 3700 standard
stellar evolution mass tracks is nearly complete and adding a grid of magnetic models would
provide a means of testing magnetic stellar evolution models against a sample of wellcharacterized single field stars. Single field stars are thought to not show signs of magnetic
inflation (Boyajian et al. 2012), but with uncertainties in mass and metallicity, it is difficult
to constrain at the present time.
I am also beginning an effort to directly test the validity of the magnetic stellar evolution
models in the partially convective regime. Results in Chapter 6 suggest that the influence
of a magnetic field is very particular. When we computed magnetic models for UV Psc
and YY Gem we had to shift our models to a lower metallicity than initially anticipated
to obtain the correct radii and Teff s. The predicted metallicity-Teff -magnetic field strength
combinations are particular and fairly robust. Therefore, if we can measure the metallicities and Teff s, we can test the model predictions. To get accurate metallicities and Teff s, we
will obtain high resolution optical spectra at multiple phases of the DEB orbits using the
Fibre-fed Echelle Spectrograph (FIES) on the Nordic Optical Telescope (NOT). Using SME
(Valenti & Piskunov 1996), it will be possible to decompose the spectra of the stars in each
binary system (Stempels et al. 2008). Observations for YY Gem were completed for a sep-
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arate observing program and a program to obtain spectra for UV Psc and CU Cnc will be
carried out in late 2013 or early 2014.
When it comes to the physics of the magnetic perturbation, there are other paths to explore.
For instance, the turbulent dynamo models produce the most realistic results for DEB star
with a radiative core. However, the balance between turbulent kinetic energy and magnetic energy is not necessarily dependent on previous time steps. The models calculate the
convective properties using mixing length theory and then perturb uconv and ∇s afterward.
Thus, only changes to the large scale structure affect the non-magnetic mixing length calculation. Instead of performing the perturbation in this way, it would be ideal to have feedback
between convection and the magnetic field (Chabrier & Küker 2006; Browning 2008). This
is akin to implementing an actual dynamo mechanism in the models. An extension of this
would be to then allow the magnetic field ramp up with age along the pre-MS and ramp
down along the main sequence, as single stars are thought to do (Skumanich 1972; Barnes
2010).
Finally, the issues with fully convective stars still need to be addressed. This is true from
both a magnetic and standard stellar model perspective. The contradictions between the
results of our magnetic models (Chapter 7) and the star spot formulations (Chabrier et al.
2007; Morales et al. 2010; MacDonald & Mullan 2012) need to be addressed further. Additionally, the anti-correlation that we revealed between radius deviations and metallicity deserve more attention. Exploring parameter spaces in Y , [Fe/H], and [α/Fe] would be the first
step. However, the results from a similar exploration of CM Dra in MacDonald & Mullan
(2012) suggest that this may not provide any suitable explanation. In the meantime, we
continue seek a plausible explanation.
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Appendix A
DMESTAR User’s Guide
Dartmouth Magnetic Evolutionary Stellar Tracks And Relations
Throughout the latter portion of this dissertation, the development of a magnetic version of
the Dartmouth stellar evolution code was presented. Here, we will briefly summarize the
different variations introduced above and then present a short guide for those wishing to
use the code.
Chapter 5 outlined the basic mathematical formulation of a magnetic perturbation assuming
a rotationally driven dynamo and that equipartition only occurs between the magnetic field
pressure and the surrounding gas pressure. The radial profile of the magnetic field strength
within the model was prescribed using a “dipole radial profile.” In Chapter 6, a second
radial profile was introduced, the “Gaussian radial profile.” Additionally, the mathematical
formulation was modified to include effects associated with a turbulent convection driven
dynamo. Finally, a third radial profile was added in Chapter 7. This radial profile assumed
the magnetic field strength was equal to some fraction, Λ, of the turbulent equipartition
magnetic field strength. It was coined the “constant Λ radial profile.”
The code was written in Fortran 90, which is backwards compatible with older FORTRAN
77 that forms the foundation of the Dartmouth stellar evolution code. The magnetic field
286
routines are contained in a single module, mag_field.1 There are eight (8) subroutines
within mag_field: b_initialize, b_field, find_tach_rad, b_dipole, b_gauss,
b_equip, b_envelope, and b_rescale. These subroutines perform the following functions,
• b_initialize – initializes the magnetic field properties using a combination of default values and user-supplied input.
• b_field – the primary magnetic field routine that calls all of the necessary routines
for determining the interior magnetic field strength profile.
• find_tach_rad – finds the radius of the stellar tachocline, if it exists.
• b_dipole – sets the run of the magnetic field strength using the dipole radial profile.
• b_gauss – sets the run of the magnetic field strength using the Gaussian radial profile.
• b_equip – sets the run of the magnetic field strength using the constant Λ radial
profile.
• b_envelope – calculates the run of the magnetic field strength in the model envelope
for the various radial profiles.
• b_rescale – rescales the magnetic field profile as the number of mass shells is readjusted.
Running DMESTAR may be performed by setting a single logic variable, lzmag to .true.
in the physics namelist file. By doing so, the user opts to run the magnetic code with the
magnetic field variables set to the default values. However, the default values are designed
to minimize the impact of the magnetic perturbations, such that it is equivalent to running
the code without a magnetic field. Instead, the user may create a magnetic namelist file that
specifies the desired magnetic field options. The entire list of variable options, their data
types, descriptions, and their default values are listed in Table A.1.
1
Hereafter, Fortran modules, subroutines, and variables will be written using a monospace font.
287
Table A.1
Magnetic namelist variables.
Variable
Type
Description
b_pert_age
real
Age of first perturbation (in Gyr)
0.1
b_surf
real
Surface magnetic field strength (G)
0.1
gammag
real
Magnetic field “geometry”
2.0
chi_f
real
Magnetic flux parameter, f
1.0
fc_tach
real
Fully convective “tachocline” (R⋆ )
0.15
eq_lambda
real
Equipartition fraction, Λ
0.0
b_rad_prof
char
Radial profile
‘dipole’
dynamo
char
Dynamo type
‘rot’
lbramp
logical
Magnetic field ramping toggle
Default
.false.
Again, the user should set lzmag = .true. to turn the magnetic perturbation option on
in DMESTAR. The rest of the magnetic model properties should be set in the the magnetic
namelist file, which should then be copied to the Fortran input unit 75 by a run script. The
exact purpose of lzmag is to inform the program that a magnetic model is being requested.
Instead of immediately introducing the magnetic perturbation, there is a delay between the
start of the program and the actual initiation of the magnetic perturbation. Segmentation
faults may occur if the field is introduced too early on the Hayashi track.
Currently, the default initialization age is 0.1 Gyr, but this may be altered through the use
of the variable b_pert_age. This variable defines the age (in Gyr) of the first perturbation.
Segmentation faults may also be mitigated by ramping the magnetic field from zero up to
the requested surface magnetic field strength using the logic variable lbramp. The step size
of the magnetic field ramping technique is hardwired to be 1.0×10−5 multiplied by the size
of the time step (in Gyr). So note that the rate of ramping is governed by number of time
steps taken during the model run. Given how crude this method is, it will be updated in the
future.
The strength of the perturbation is determined by the surface magnetic field strength and
288
the type of radial profile selected. These properties are controlled by the variables b_surf
and b_rad_profile. The surface magnetic field strength must be declared in gauss (G) and
is set to 0.1 G by default. Three options exist for the radial profile: “dipole,” “gauss,” and
“equip,” which instruct the program to use either a dipole, Gaussian, or constant Λ radial
profile. There is also the option to use either a rotationally driven dynamo or a turbulent
dynamo, defined by setting dynamo to either “rot” or “turb.”
If a constant Λ profile is selected, there will be no option for a rotational dynamo. The
code will automatically confirm that dynamo = ‘turb’. The fraction of the equipartition
magnetic field strength requested by the user is controlled by the variable eq_lambda. If
the users selects a dynamo = ‘turb’, then any of the radial profile options are viable. Once
a radial profile is selected, the peak magnetic field strength is prescribed to the tachocline
(except for the case of constant Λ). While the tachocline is defined by the code for models
with a radiative core, fully convective models must have one specified arbitrarily. Based on
3D MHD models by Browning (2008), the default value is set to 0.15R ⋆ . DMESTAR allows
for this point to be changed by setting fc_tach to a value between 0.001R ⋆ and 0.999R⋆ .
Lastly, there are options to control the variables f and γ, which were defined in Chapter
5. Recall, f controls the amount of flux exchanged between a convecting bubble and the
surrounding plasma and γ mimics the geometry of the magnetic field. By default, these
values are set to chi_f = 1.0 and gammag = 2.0, respectively. Cases for these values were
made within the text. If the user is unsure, it is recommended to use the default values.
A sample magnetic namelist file is provided at the end of this Appendix.
Monitoring the magnetic perturbation is possible using data contained in multiple output
files. The program tracks the magnetic field variables throughout the interior (unit 76)
and the envelope (unit 80). Both of these output files are new to the code. Explicitly, the
code tracks the magnetic pressure (p_mag) and the magnetic energy per unit mass (u_mag)
as a function of radius and density. With these variables, the magnetic field strength radial
289
profile may be extracted. Additionally, the magnetic field strength at the tachocline is output
to the model track file (unit 37) to provide a quick diagnostic.
A complete list of the input and output unit numbers allocated in DMESTAR is provided
below. The list is actually the new module that DMESTAR keeps track of allocated unit
numbers. This is different from DSEP, which used to declare all unit numbers if the subroutine parmin. By using a module, former common blocks associated with file i/o are no
longer required. The module, or individual variables, are imported where needed.
Listing A.1: Sample Magnetic Namelist File
! Example of a magnetic namelist file for DMESTAR
!
File: magnetic.nml
! Author: Gregory A. Feiden
!
Date: 07 May 2013
!---------------------------------------------------------$magnetic
lmag
= .false.
! toggle magnetic field: .true. (on) or .false. (off)
b_pert_age = 0.1
! perturbation age (in Gyr)
b_surf
= 0.6d2
! surface magnetic field (G)
gammag
= 2.0
! magnetic geometry parameter
chi_f
= 1.0
! magnetic flux parameter
fc_tach
= 0.15
! fully convective tachocline in units of R_star
eq_lambda
= 0.9
! equipartition parameter
b_rad_prof = ’equip’
! Gaussian radial profile; options: ’dipole’, ’gauss’,
’equip’
dynamo
= ’rot’
! turbulent dynamo; options: ’rot’, ’turb’
ramp_b_field = .false. ! ramp the magnetic field; options: .true. or .false.
$end
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Listing A.2: Allocated Unit Numbers for I/O
module file_lib
implicit none
integer, protected :: iowr ! output: status file
integer, protected :: igrad ! output: temperature gradients
integer, protected :: ilast ! output: last model (text)
integer, protected :: ifirst ! input: first model (text)
integer, protected :: irun ! input: physics namelist
integer, protected :: istand ! input: contral namelist
integer, protected :: ifermi ! input: Fermi tables
integer, protected :: idebug ! output: reserved for debugging
integer, protected :: itrack ! output: track
integer, protected :: ishort ! output: all diagnostic info
integer, protected :: imilne ! output: Milne invariant variables
integer, protected :: imodpt ! output: shell by shell info
integer, protected :: istor ! output: saved models
integer, protected :: iopmod ! output: pulsation, interior
integer, protected :: iopenv ! output: pulsation, envelope
integer, protected :: iopatm ! output: pulsation, atmosphere
integer, protected :: ilstbn ! output: binary last model
integer, protected :: istobn ! output: binary stored model
integer, protected :: ifstbn ! input: binary starting model
integer, protected :: idyn ! output: dynamo information
integer, protected :: illdat ! input: OPAL92 opacity table
integer, protected :: iliv95 ! input: OPAL95 opacity table
integer, protected :: isnu ! output: SNU fluxes
integer, protected :: icond ! input: conduction opacities
integer, protected :: ikur ! input: Kurucz low T opacities
integer, protected :: iiso ! output: isochrone information
integer, protected :: ioatm ! input: Kurucz atmospheres
integer, protected :: ialxo ! input: Alex Low T opacities
integer, protected :: iopale ! input: OPAL EOS
integer, protected :: io95co ! input: OPAL binary dump
integer, protected :: iolaol ! input: LAOL opacities
integer, protected :: iopurez ! input: LAOL opacities, pure CN
integer, protected :: iolaol2 ! input: opacities for diffusion
integer, protected :: ioopal2 ! input: opacities for diffusion
291
integer, protected :: ikur2 ! input: opacities for diffusion
integer, protected :: ioatma ! input: Allard atmospheres
integer, protected :: imonte1 ! input: monte carlo for SNUs
integer, protected :: imonte2 ! input: monte carlo for SNUs
integer, protected :: iscvh ! input: SCVH95 EOS
integer, protected :: iscvhe ! input: SCVH95 EOS
integer, protected :: iscvz ! input: SCVH95 EOS
integer, protected :: imag ! input: magnetic field namelist
integer, protected :: imagpro ! output: magnetic field profile
integer, protected :: iapsid ! output: apsidal motion info
integer, protected :: imenv ! output: magnetic envelope
integer, protected :: iatmphx ! input: phoenix atmosphere
logical, protected :: files_initialized
contains
subroutine file_init()
! set unit numbers for i/o files
if (files_initialized) then
write(*, *) ’Warning: unauthorized call to file setup.’
return
end if
iowr
= 6
! output: status file
igrad
= 8
! output: temperature gradients
ilast
= 11
! output: last model (text)
ifirst = 12
! input: first model (text)
irun
= 13
! input: physics namelist
istand = 14
! input: contral namelist
ifermi = 15
! input: Fermi tables
idebug = 18
! output: reserved for debugging
itrack = 19
! output: track
ishort = 20
! output: all diagnostic info
imilne = 21
! output: Milne invariant variables
imodpt = 22
! output: shell by shell info
istor
! output: saved models
= 23
iopmod = 24
! output: pulsation, interior
292
iopenv = 25
! output: pulsation, envelope
iopatm = 26
! output: pulsation, atmosphere
ilstbn = 27
! output: binary last model
istobn = 28
! output: binary stored model
ifstbn = 29
! input: binary starting model
idyn
! output: dynamo information
= 30
illdat = 32
! input: OPAL92 opacity table
isnu
= 33
! output: SNU fluxes
icond
= 35
! input: conduction opacities
ikur
= 36
! input: Kurucz low T opacities
iiso
= 37
! output: isochrone information
ioatm
= 38
! input: Kurucz atmospheres
ialxo
= 39
! input: Alex Low T opacities
iliv95 = 48
! input: OPAL95 opacity table
iopale = 49
! input: OPAL EOS
io95co = 50
! input: OPAL binary dump
iolaol = 61
! input: LAOL opacities
iopurez = 62 ! input: LAOL opacities, pure CN
iolaol2 = 63 ! input: opacities for diffusion
ioopal2 = 64 ! input: opacities for diffusion
ikur2
= 65
ioatma = 66
! input: opacities for diffusion
! input: Allard atmospheres
imonte1 = 70 ! input: monte carlo for SNUs
imonte2 = 71 ! input: monte carlo for SNUs
iscvh
= 72
! input: SCVH95 EOS
iscvhe = 73
! input: SCVH95 EOS
iscvz
= 74
! input: SCVH95 EOS
imag
= 75
! input: magnetic field namelist
imagpro = 76 ! output: magnetic field radial profile
iapsid = 77
! output: apsidal motion info
imenv
! output: magnetic envelope
= 80
iatmphx = 95 ! input: phoenix atmosphere(s), 95 --> 99
files_initialized = .true.
end subroutine file_init
subroutine file_close(file_index)
293
integer, intent(in) :: file_index
close(file_index)
end subroutine
end module file_lib
294
Appendix B
Additional Model Physics
While the majority of additions to the Dartmouth stellar evolution code surrounded the
introduction of the magnetic perturbation, other physics were either added or updated along
the way. This Appendix briefly summarizes these changes. Each section describes the
physics and any control variables added to the code, which should be specified in the physics
namelist file. Default values for these new control variables are shown in parentheses.
B.1 Turbulent Mixing
We have introduced turbulent diffusion as described by Richard et al. (2005). Turbulent
diffusion modifies the atomic diffusion coefficient and acts to extend the mixing region
below the convection zone. The magnitude of the turbulent diffusion coefficient is tied to
an adjustable reference temperature, T0 , and varies with density via:
·
ρ
D T = ωD He (T0 )
ρ(T0 )
295
¸−3
(B.1)
where ω characterizes the relative strength of turbulent diffusion and D He (T0 ) and ρ(T0 )
are the helium diffusion coefficient and density at the prescribed reference temperature,
respectively. Proffitt & Michaud (1991) motivate the ρ −3 dependence in order to reproduce
the solar beryllium abundance, which appears to be unchanged over time. Thus, any nonstandard mixing in the Sun must be localized to a narrow region below the solar convection
zone. We select ω = 400 and leave it fixed, in accordance with Richard et al. (2005). The
reference temperature used primarily in our models is T0 = 106 , which was found to best
reproduce the observed abundance trends of the globular cluster NGC 6397 (Korn et al.
2007).
The turbulent diffusion coefficient is then introduced into the gravitational setting routines
already present in DSEP. Bahcall & Loeb (1990) indicate the time rate of change of the hydrogen mass fraction satisfies the following partial differential equation,
·
¸
1 ∂ r 2 X T 5/2 ξH (r )
∂X
= 2
,
∂t
ρr ∂r
(ln Λ/2.2)
(B.2)
where the dimensionless function ξH (r ) is
¸
·
5(1 − X ) ∂ ln P
∂ ln T
∂
X (1 + X )
ξH (r ) =
+ ΦH (X )
+
ln
.
2
4
∂r
∂r
(3 + 5X )
∂r
(B.3)
The first term on the right hand side of the above equation can be identified as the gravitational settling term, the second term is term associated with atomic diffusion and the third
term accounts for thermal diffusion. We may compare the equation presented by BL with
the diffusion equation presented by Richard et al. (2005) which adds a turbulent diffusion
coefficient to the diffusion equation presented by Aller & Chapman (1960)
·
¸
¸
·
ZH mp g
A H mp
∂ lnC X
∂ ln T
v D = (D H + D T ) −
(g rad − g ) +
+ kT
+ DH
.
∂r
kT
2kT
∂r
(B.4)
Note, there is a difference in sign between the Richard et al. (2005) formulation and the
296
Bahcall & Loeb (1990) formulation which is only indicative of which direction refers to
inward or outward diffusion. Now, we may directly compare the equations presented by
both BL and RMR, but it must be noted that the diffusion equation from RMR defines atomic
diffusion in terms of the natural logarithm of the hydrogen concentration whereas BL define
atomic diffusion in terms of the hydrogren mass fraction gradient. If we simplify the second
term of Bahcall & Loeb (1990)’s dimensionless function ξH (r ), we have
¸
·
X (1 + X )
∂
=
ln
∂r
(3 + 5X )2
=
=
=
=
¸
·
(3 + 5X )2
X
10X (1 + X ) ∂X
1+ X
+
−
X (1 + X ) (3 + 5X )2 (3 + 5X )2
(3 + 5X )3 ∂r
·
¸
1
1
10
∂X
+
−
X 1 + X 3 + 5X ∂r
·
¸
(1 + X )(3 + 5X ) + X (3 + 5X ) − 10X (1 + X ) ∂X
X (1 + X )(3 + 5X )
∂r
¸
·
3 + 8X + 5X 2 + 3X + 5X 2 − 10X − 10X 2 ∂X
X (1 + X )(3 + 5X )
∂r
¸
·
∂X
3+ X
X (1 + X )(3 + 5X ) ∂r
(B.5)
To compare with Richard et al. (2005), we must determine the partial derivative of the concentration of hydrogen in terms of the hydrogen mass fraction gradient. Thoul et al. (1994)
define the concentration gradient for a particular ion species as
Ã
!−1 µ
¶
X Zi X i
X Z j X j ∂ ln X j
∂ lnC s ∂ ln X s
=
−
.
∂r
∂r
Ai
Aj
∂r
i
j
(B.6)
If we assume a composition of only hydrogen and helium ions, then we may simplify the
above equation and write it in terms of the hydrogen mass fraction gradient,
∂ lnC H
∂r
=
=
=
Ã
!−1 µ
¶
X Zi X i
X Z j X j ∂ ln X j
∂ ln X H
−
∂r
Ai
Aj
∂r
i
j
µ
¶ µ
¶
2(1 − X ) −1
∂ ln X 2(1 − X ) ∂ ln(1 − X )
∂ ln X
− X+
X
+
∂r
4
∂r
4
∂r
µ
¶
2
1 ∂X
∂X 1 ∂X
−
−
.
X ∂r 1 + X ∂r 2 ∂r
297
Thus, we find the negative gradient of the logarithm of the concentration of hydrogen
−
1
∂X
∂l nC H
=−
.
∂r
X (1 + X ) ∂r
(B.7)
Now, consider the concentration gradient of the metallicity, Z . Assume the gas is one of
fully ionized iron atoms. This was assumed for the derivation of the gravitational settling of
heavy elements currently employed in DSEP. By this assumption, ZF e = 26 and A F e = 56.
Therefore, we can write
!−1 µ
Ã
¶
X Z j X j ∂ ln X j
X Zi X i
∂ lnC Fe ∂ ln X Fe
=
−
∂r
∂r
Ai
Aj
∂r
j
i
¶
µ
∂ ln X Fe
Z H X H ZHe X He ZFe X Fe −1
=
·
−
+
+
∂r
AH
A He
A Fe
µ
¶
Z H X H ∂ ln X H ZHe X He ∂ ln X He ZFe X Fe ∂ ln X Fe
+
+
.
AH
∂r
A He
∂r
A Fe
∂r
Now, recall that X H = X , X He = Y and X Fe ∼ Z , where we may eliminate one of the mass
fraction variables knowing that the total mass fraction of elements in the solar interior is
equal to 1,
Y = 1− X − Z.
(B.8)
Then, we have
µ
¶ µ
¶
∂ lnC Fe 1 ∂Z
2Y 26Z −1 ∂X 2 ∂Y 26 ∂Z
=
− X+
+
+
+
∂r
Z ∂r
4
56
∂r 4 ∂r 56 ∂r
µ
¶ µ
¶
2X + 1 − X − Z 13Z −1 ∂X 1 ∂X 1 ∂Z 13 ∂Z
1 ∂Z
−
+
−
−
+
=
Z ∂r
2
28
∂r 2 ∂r 2 ∂r 28 ∂r
¶
µ
µ
¶
∂Z
14
14 + 14X
∂X
−
.
=
2
14Z + 14X Z − Z ∂r
14 + 14X − Z ∂r
(B.9)
We now make the approximation that the addition of Z in the denominator is negligible
298
compared to the value 14, in other words, Z ≪ 14. Therefore,
1 ∂Z
1 ∂X
∂ lnC F e
=
−
∂r
Z ∂r 1 + X ∂r
(B.10)
We can now define the atomic diffusion coefficient, D H , in the diffusion equation from
Richard et al. (2005) based on the equation presented by Bahcall & Loeb (1990). Since the
gradient of the logarithm of the concentration is defined as above, comparing with Bahcall & Loeb
(1990), we find
·
¸
r 2 T 5/2 X X + 3
DH = −
ln Λ/2.2 5X + 3
(B.11)
As defined by Richard et al. (2005), the turbulent diffusion coefficient is
·
ρ
D T = 400D He (T0 )
ρ(T0 )
¸−3
(B.12)
where T0 is selected in order to minimize Li depletion due to the diffusion of Li into the
nuclear burning region, near T = 2.5 · 106 K . Since we are following the convention of
Bahcall & Loeb (1990) in defining the diffusion coefficient for helium, we may replace the
diffusion coefficient for helium with that of hydrogen evaluated at T0 , save a minus sign to
indicate the two species are diffusing in opposite directions,
D T = 400 ·
R 02 · T05/2 · X 0
ln Λ/2.2
·
X0 + 3
5X 0 + 3
¸·
ρ
ρ(T0 )
¸−3
,
(B.13)
where R 0 and X 0 are the radius and the hydrogen mass fraction of the shell at the reference
temperature T0 , respectively. We may now rewrite the atomic diffusion term (for hydrogen)
of the diffusion equation with the new expressions for the diffusion coefficients,
µ
¶
µ
¶µ
¶ #·
¸
R 02 · T05/2 · X 0
1
X0 + 3
ρ −3
∂X
r 2 T 5/2 X + 3
−
,
+ 400 ·
vH = −
ln Λ/2.2 5X + 3
ln Λ/2.2
X (5X 0 + 3) ρ(T0 )
1 + X ∂r
"
(B.14)
which we may further simplify following the sign convention of Richard et al. (2005) and
299
Thoul et al. (1994),
·
¸"
¶ #
µ ¶2 µ ¶5/2 µ ¶ µ
X +3
T0
X 0 ρ(T0 ) 3 ∂X
r 2 T 5/2
R0
.
1 − 400 ·
vH =
ln Λ/2.2 (X + 1)(5X + 3)
r
T
X
ρ
∂r
(B.15)
To remain consistent with our initial assumption that hydrogen and helium diffuse at the
same rate in opposite directions, we can say that for helium
#
·
¸"
¶
µ ¶5/2 µ
r 2 T 5/2
∂X
X +3
ρ(T0 ) 3
T0
v He = −v H =
.
−1
400
ln Λ/2.2 (X + 1)(5X + 3)
T
ρ
∂r
(B.16)
Considering the modifications that must be made to DSEP, we already have the hydrogen,
and thus helium, atomic diffusion coefficient defined. The variable cod2 is defined in the
subroutine setup_grsett just as we have derived above, neglecting the turbulent diffusion
term. Namely,
·
¸
X +3
r 2 T 5/2
.
cod20 =
ln Λ/2.2 (X + 1)(5X + 3)
Thus, to implement the turbulent diffusion coefficient, we modify the existing cod2 term,
·
¸"
¶ #
µ ¶5/2 µ
r 2 T 5/2
X +3
ρ(T0 ) 3
T0
cod2 =
1 − 400
.
ln Λ/2.2 (X + 1)(5X + 3)
T
ρ
(B.17)
We also need to know the derivative of the newly updated diffusion coefficient defined in
DSEP,
#
·
¶
¸"
µ ¶5/2 µ
∂cod2
ρ(T0 ) 3
r 2 T 5/2 5X 2 + 30X + 21
T0
−1 .
=
400
∂X
ln Λ/2.2 (5X 2 + 8X + 3)2
T
ρ
(B.18)
These last two equations were included in the gravitational settling routine.
New Control Variables
ltdiff (.false.) – Logic variable controlling if turbulent mixing is included or not.
turbt (6.0) – Real number variable that sets the logarithmic value for the reference temperature needed for turbulent mixing.
300
B.2 Internal Structure Constant
We presented the theory behind the calculation of the internal structure constant, k 2 , in
Chapter 3. The module to carry out the calculations in DMESTAR (apsidal) was designed
to facilitate automatic computation of k 2 (apcon) at each model time step. This feature can
be turned off, if necessary, using the new logic variable lapsid, but the code is setup to
always compute k 2 . The only benefit of turning this feature off, in our opinion, is to save
on computation time when computing a large grid of models where k 2 is of little interest.
The value of k 2 at each model time step is printed out in the model track file (unit 37).
The program also keep track of the run of the parameter η(r ), related to the deviation from
sphericity as a function of the radius and density. File unit 77 contains this information.
New Control Variables
lapsid (.true.) – Logical variable controlling whether to compute the internal structure
constant at each time step.
B.3 Convective Overturn Timescale
The convective overturn timescale was added to allow computation of theoretical Rossby
numbers in Feiden & Chaboyer (2012a). A module was created to calculate the convective
turnover time and has been fully integrated into the code. No logic variable was assigned
to turn the feature on or off, it is permanently on. However, disabling this feature can be
achieved by commenting our a single line of code in the primary driver main.f.
Our implementation is similar to the method of Kim & Demarque (1996), who calculate the
“local” convective overturn time at a distance above the base of the convection zone equal to
one half of the mixing-length. This particular location was chosen based on the assumption
that the tachocline (radiative-convective zone interface) is the source region of the stellar
301
magnetic dynamo (Parker 1975). However, using the stellar tachocline as our magnetic field
source location for a fully convective star would be nonsensical. Therefore, in the case
that the star is convective throughout, we utilize the results of Browning (2008) as a first
approximation to the magnetic field source location. Browning found that the magnetic
field strength within a fully convective star was at a maximum at a depth of 85% of the
stellar radius. In accordance with this result, we compute the convective overturn time at
one half the mixing-length above this location. The convective overturn time, in days, is
output to the model track file (unit 37).
B.4 Nuclear Reaction Rates
An updated set of nuclear reaction cross sections was released in 2011 (Adelberger et al.
2011). All of the values recommended were rigorously vetted by the authors and should
therefore represent the most accepted reaction cross sections. Table B.1 provides a comparison between the original Bahcall (1989), NACRE (Angulo et al. 1999), and the latest recommended values (Adelberger et al. 2011). We also quote the ratio between Adelberger et al.
(2011) and Bahcall (1989), which is used as input for DSEP and DMESTAR (sstandard).
B.5 Gough & Taylor Magnetic Inhibition
Magnetic inhibition of convection has been introduced using the magnetic inhibition parameter of Mullan & MacDonald (2001), which is physically motivated by the analytical
work of Gough & Tayler (1966). The radial profile is calculated in mm_mag. See also the more
recent works by Mullan et al. (2007); MacDonald & Mullan (2009); Mullan & MacDonald
(2010); MacDonald & Mullan (2010, 2012, 2013) for updated radial profiles and discussions
of their methods.
302
Table B.1
Updated DSEP nuclear reaction rates.
Reaction
1
H(p, e + ν)D
3
He(3 He, 2p)4 He
3
NACRE
Fusion II
Ratio
4.07E-22
3.94E-22
4.01E-22
0.9853
5150
5180
5210
1.0117
0.54
0.54
0.56
1.0370
C(p, γ)13 N
1.45
1.30
1.34
0.9241
1, 2
2
He( He, γ) Be
13
C(p, γ) N
5.50
7.00
7.6
1.3818
14
N(p, γ)15 O
3.32
3.20
1.66
0.5000
9.40
9.30
10.6
1.1277
0.0243
0.0210
0.0208
0.8560
8.0E-20
2.3E-20
8.6E-20
1.0750
16
7
3
Note
7
4
12
Bachall (1988)
14
17
O(p, γ) F
8
Be(p, γ) B
+
4
He(p, e ν) He
3
(1) NACRE does not publish a 12 C rate. Estimated from their Figure X.
(2) Values in Solar Fusion II (Adelberger et al. 2011) are taken from Solar Fusion I
(Adelberger et al. 1998).
(3) The 3 He + p reaction is not available from NACRE. Fusion I value was quoted instead.
New Control Variables
lgtmag (.false.) – Logical variable controlling whether to modify the adiabatic gradient by a small value, δMM .
B.6 Electron Conduction Opacities
Thermal electron conduction opacities were revised to reflect the improvements presented
by Cassisi et al. (2007). Since the opacities are in tabular form, Alexander Potekchin wrote
the necessary interpolation routines (see http://www.ioffe.ru/astro/conduct/). We cleaned
up his original routines and integrated them into DSEP. Whether or not a given run uses
the new opacities is controlled by a new logic variable. Furthermore, DMESTAR now implements a conduction opacity module thecond with the latest Cassisi et al. (2007) interpolation routines in thecond_int.
303
New Control Variables
lcond07 (.true.) – Logic variable controlling whether to use the updated electron conduction opacities.
B.7 τ=100 Boundary Condition Fit Point
Models of fully convective stars with masses below 0.2M⊙ require boundary conditions
that are prescribed at a deeper optical depth than higher mass models. This is because atmospheres of very-low-mass stars are highly non-adiabatic (Chabrier & Baraffe 1997; Chabrier & Baraffe
2000). Therefore, the specific properties used in mixing length theory are critically important. To avoid an incompatibility between the interior models and the atmosphere, integrating the atmosphere to a level where convection becomes nearly adiabatic is recommended.
DSEP was tested with this type of atmosphere configuration, but lacked a diverse set of
atmospheric boundary condition tables to make it a permanent option. We are presently
compiling the necessary boundary condition tables with points defined at T = Teff and also
where τ = 100, an optical depth of 100. DMESTAR can automatically select which set of
boundary condition tables to use. However, the user can also specify with a new logic variable, lmatm. When set to .true., the code will attach the boundary conditions at τ = 100.
New Control Variables
latm (.false.) – Logic variable controlling whether to use a deeper atmosphere integration, suitable for very-low-mass models.
304
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