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The Host Stars:
To know the Planet you
have to know the Star
Why should we care about the host stars?
1. Stellar properties give hints to planet formation
•
•
•
Abundance: The role of metals in formation
Age of star: Age of planet and evolution
Activity level of star: star-planet interaction, effects
on habitability
Why should we care about the host stars?
2.
Stellar properties are important for accurate planet
parameters
• Doppler measurements for mass
Vobs =
28.4 mp sin i
P1/3 Ms2/3
• Transit light curve gives you k = Rp / Rs
The inner structure of
Kepler-10b and CoRoT-7b
CoRoT-7b
Kepler-10b
r (gm/cm3)
10
7
Earth
Mercury
5
4
3
Venus
Mars
Moon
2
1
0.2
2
0.4
0.6
0.8
1
1.2
1.4
Radius (REarth)
1.6
1.8
The inner structure of Kepler-78b
The limiting factor determining our knowledge of the internal structure
of these planets is the stellar parameters
Why should we care about the host stars?
2.
Stellar properties are important for accurate planet
parameters
• Limb darkening : Affects the “stellar radius” in the
transit modeling
• Temperature of the star : Equilibrium temperature
of the planet
• Distance to the star: Location of planets in our
galaxy and environment
Example: CoRoT-19b
Spectroscopic analysis
Evolutionary models
lightcurve analysis
Up the main
sequence:
1. Higher Mass
2. Higher
temperature
3. Shorter life
2 M‫סּ‬
1 M‫( סּ‬s.m.)
0.1 M‫סּ‬
Sp.T
Mass
(solar)
Radius
(solar)
Teff
(K)
Lum.
(solar)
Lifetime
( years)
B0
17.5
7.4
30000
200 000
3.5 x 106
B5
5.7
5.9
15200
380
1.5 x 108
A0
2.9
2.4
9790
24
1.2 x 109
A5
2.0
1.7
8180
11
1.8 x 109
F0
1.6
1.6
7300
5.1
3.1 x 109
F5
1.24
1.4
6650
2.7
4.6 x 109
G0
1.05
1.05
5940
1.2
8.8 x 109
G5
0.92
0.92
5560
0.73
1.3 x 1010
K0
0.79
0.79
5150
0.38
2.1 x 1010
K5
0.67
0.67
4410
0.15
4.5 x 1010
M0
0.51
0.51
3840
0.080
6.4 x 1010
M5
0.21
0.20
3170
0.011
1.9 x 1011
Fundamental Stellar Parameters: Distance
©Perryman 2011, fig. 8.1
• Accuracy of distances of planet host stars increased substantially
from ground-based (a) to Hipparcos (b).
• GAIA will improve things and give accurate distances, luminosities,
and temperatures of millions of stars out to thousands of parsecs
HR diagram with planet host stars
• HR diagram
of stars
within 25 pc
• based on
Hipparcos
data
Hawley & Reid (2003), Perryman (2011; fig. 8.5)
Fundamental Stellar Parameters: Mass and Radius
Along the main sequence there is a tight
relationship between spectral type and the stellar
radius and mass.
One can get estimates of the stellar radius and
mass from the spectral type. This requires that the
main sequence be calibrated
Calibrating Stellar Masses using Astrometric
Measurements of Binary Stars
W 1062 AB
-0.1
4 5
6
Declination (arcsec)
3
MA = 0.381 ± 0.006
MO
MB = 0.187 ± 0.003
MO
0.0
8
0.1
9
90°(E)
10
11
17
16
15
0.2
2
0° (N )
-0.2
1
14
-0.1
12
0.0
RA (arcsec)
0.1
Calibrating Stellar Masses and Radii using
Eclipsing Binary Stars
The light curve gives you the stellar
radii, Doppler measurements give you
the masses of the stars
Torres et al.
Radius vs. Mass
Radius vs. Effective Temperature
Mass vs. Temperature
Luminosity vs. Mass
Meaurements of Stellar Radii
using Lunar Occultations
R = 45.1 ± 0.1 Rsun
Measurements of Stellar Radii
using Interferometry
A Basic Interferometer
S
s • B = B cos q
q
A2
x2
A1
x1
Delay Line 1
d1
Beam
Combiner
1) Idealized 2-telescope interferometer
Delay Line 2
d2
Interferometry Basics
Adjacent fringe crests projected on the sky are separated
by an angle given by: Ds = l/B, B is the baseline
Every baseline samples a new “frequency”
Nulling Interferometers
Adjusts the optical path length so that the wavefronts
from both telescope destructively interfere at the position of the star
Technological challenges have
prevented nulling interferometry from
being a viable imaging method…for now
Darwin/Terrestrial Path Finder
would have used Nulling
Interferometry
Earth
Venus
Mars
Ground-based European
Nulling Interferometer
Experiment will test
nulling interferometry on
the VLTI
The VLT Interferometer
The Keck Interferometer
Interferometry Basics: The Visibility Function
Michelson Visibility:
Imax –Imin
V=
Imax +Imin
Visibility is measured by changing the path length and
recording minimum and maximum values
Interferometry Basics: Cittert-Zernike theorem
The visibility :
V(k, B) =  da db A(a,b) F(a,b) e −2pi(au+bv)
Cittert-Zernike theorem: The interferometer
response is related to the Fourier transform of the
brightness distribution under certain assumtions
(source incoherence, small-field approximation).
In other words an interferometer is a device that
measures the Fourier transform of the brightness
distribution.
F(a,b) is flux distribution of object on sky
Aperture Synthesis
Aperture Space
V
N
B
E
Spatial Resolution : l/B
Can resolve all angular scales
up to q > l/D, i.e. the diffraction
limit
Fourier Space
U
Frequency Resolution : B/l
Can sample all frequencies out to D/l
One baseline measurement maps into a single location in the (u,v)-plane (i.e. it is
only one frequency measurement of the Fourier transform)
Aperture Synthesis
Consider a 2-telescope array
B2
B1
B4
B3
=
Interferometry
Stellar Angular Diameters:
V(s) = |2J1(pas)/pas|
J1 is the first order Bessel function
Note: as it should be, this is also the diffraction
pattern of a circular aperture.
Diameters of Giant Stars
The Radius of HR 8799
R = 1.44 ± 0.06 RSun
L = 5.05 ± 0.29 LSun
Teff = 7193 ± 87 K
Baines et al.
The Radius of Epsilon Eridani
R = 0.74 ± 0.01 RSun
Teff = 5039 ± 126 K
Stellar Diameters of Giant Stars
Measuring Stellar Atmospheric
Parameters:
Temperature, surface gravity,
abundance of heavy elements
Simple Way: Compare observed spectra with library of
known stars with well determined parameters
• Simple, but limited accuracy
• One still has to determine the stellar parameters for
the library spectra!
Use temperature Sensitive Lines: Sodium
T = 5100 K
T = 7300 K
The strength of Na D decreases with increasing temperature.
Behavior of Hydrogen lines with temperature
B3IV
B9.5V
A0 V
G0V
F0V
The atomic absorption coefficient of hydrogen is temperature sensitive
through the Stark effect. Because of the high excitation of the Balmer series
(10.2 eV) this excitation growth continues to a maximum T = 9000 K
Comparison of Interferometric Temperatures with Spectroscopic
Temperatures for giant stars
L = sT4(4pR2)
Behavior of Spectral Lines: Abundance Dependence
The line strength should also depend on the abundance of the absorber, but the
change in strength (equivalent width) is not a simple proportionality as it depends
on the optical depth.
3 phases:
Weak lines: The line strength increases
proportional to the abundance of the element
Saturation: Line depth (strength) approaches a
constant value
Strong lines: The wings start to grow and the
line strength increases, but not proporinally
The graph specifying the change in equivalent
width with abundance is called the Curve of
Growth
Surface Gravity: GM/R2
Use gravity sensitive lines like Fe II 4508 Å
Stellar Gravity from the Transit Duration
t = 2(R* +Rp)/v
where v is the orbital velocity and i = 90 (transit across disk center)
For circular orbits
v = 2pa/P
From Keplers Law’s:
a = (P2 M*G/4p2)1/3
2R* P (4p2)1/3
t
2p P2/3 M*1/3G1/3
t  R* /M*1/3
This is the „transit gravity“ and is often used to check the value from
spectroscopy.
Best way: Model the spectrum
Set of stellar
parameters:
Teff, logg, [M/H],...
Model atmospheres:
e.g. ATLAS9, MARCS, MAFAGS
Temperature profile
Pressure profile
...
Line formation codes:
e.g. SPECTRUM,
MOOG, LINFOR
Synthetic spectrum
The „Metal“ Abundance of Exoplanet Host Stars
The „Bracket“ [Fe/H]
• e.g. [Fe/H] = –1 → 1/10 the iron abundance of the sun
• unit: „dex“ (contraction of decimal exponent, indicates
decimal logarithmic ratio which is in fact unitless)
• [Fe/H] is often used as an overall metallicity indicator,
other elements then are related to Fe, e.g. [Mg/Fe].
The Star-Metallicity Connection
Astronomer‘s
Metals
More Metals !
Even more Metals !!
Metallicity correlation
• host stars of
planets appear on
average more
metal-rich than
comparison stars
without planets
• probability to find
planet is higher
for metal-rich star
Santos et al. (2005)
The Planet-Metallicity Connection
Valenti & Fischer
There is believed to be a connection between metallicity and planet formation.
Stars with higher metalicity tend to have a higher frequency of planets. This is
often used as evidence in favor of the core accretion theory
Limb Darkening
Temperature
Bottom of photosphere
Temperature profile
of photosphere
10000
8000
6000
4000
q2
q1
z=0
tn =1 surface
Top of photosphere
z
dz
The path length dz is
approximately the same
at all viewing angles, but
at larger the optical depth
of t=1 is reached higher
in the atmosphere
Report that the transit duration
is decreasing with time, i.e. the
inclination is changing:
However, Kepler shows no change
in the inclination!
One solution:
Mislis et al. had to combine
data taken from different
instruments with different
filters, i.e. observed at different
wavelengths. An improper
treatement of limb darkening
will make the transit duration
different.
Limb Darkening Laws used by
Transit Light Curve Modelers
m = cos q, q = 0 at disk center, 90o at limb
Limb darkening for stars other than the Sun use theoretical model
atmospheres
Stellar activity
Sun in Ca II K:
chromosphere!
Sun in white light: photosphere
Stellar Flares
Stix (1991, pp. 352, 355)
Stellar (chromospheric) activity is often meaured using
the Ca II H & K resonance line:
Active Star
Inactive Star
Ca II line
5 Minutes of Stellar Atmospheres:
Optical depth, t : How far you can see in the atmosphere.
t large: You are looking high up in the atmosphere (photons
are readily absorbed so they cannot make it out)
t small: Photons can travel a long way before being absorbed,
i.e. you can see deeper into the atmosphere
Source contribution function: Where the photons are coming from
Sample Contribution Functions
Strong lines
Weak line
On average weaker lines are
formed deeper in the
atmosphere than stronger lines.
For a given line the contribution
to the line center comes from
deeper in the atmosphere from
the wings
Two important rules:
1) Strong spectral lines are formed higher up in the stellar atmosphere than
weak lines.
2) The core of the line is formed higher up than the wings
Temperature profile of a solar like active star
Chromosphere
Dl (Å)
Core formed here
where temperature
is higher
Photosphere
Wing formed here
Strong absorption lines are formed higher up in the stellar atmosphere. The core
of the lines are formed even higher up (wings are formed deeper). Ca II is formed
very high up in the atmospheres of solar type stars.
Measurement: S-Index/R’HK
•measure flux in emission
line f(H) and f(K)
Active star
Inactive star
•absolute flux in H and K of
active star:
•subtract photospheric
contribution Fphot based on
model atmosphere or
inactive standard:
Strassmeier (1997), pp. 249, 250
Measurement principle
Active star
• absolute flux in H and K of
active star:
Inactive star
Strassmeier (1997), pp. 249, 250
Activity-related RV
The RV scatter (“jitter”) is
correlated with the level
of activity
Saar et al. (1998, ApJ 498L, 153)
Star-Planet Interaction
Shkolnik et al. 2003
Determining the Age of a Star
Evolutionary Tracks
Girardi tracks : http://stev.oapd.inaf.it/cgi-bin/param
HR 8799
M = 1.51 ± 0.24 MSun
Interferometric R and T
Age = 30 Million Years
e Eridani
M = 0.82 ± 0.05 MSun
Interferometric R and T
Age = 109 years
Magnetic braking
• convection + rotation are thought to generate magnetic
field via stellar dynamo (Gray, 2005, pp. 490-492)
• stars with convective envelopes form a magnetic field
• stellar wind is coupled to magnetic field lines and thus
to stellar rotation
• therefore, stellar wind takes away angular momentum
and the stellar rotation is braked
Magnetic braking
formation of
convective envelope
Gray (2005), p. 485
Indicators of Stellar Youth
1. Rapid rotation in cool star: The star has not had
enough time to slow down through magnetic
braking
2. High level of activity (Ca II emission): Related to (1)
as stars that rotate rapidly also generate more
magnetic fields
1. Strong lithium line. Lithium is destroyed at
temperatures of a few million K. The star starts with
lithium present, but through time convection brings
it to the bottom of the convection zone where it is
destroyed
Broad lines are
from a young a
star
Narrow lines are
the sun
Dl (Å)
The Lithium Line
The strength of the Lithium line can be calibrated with age, but it is generally not
that good. In a solar type star the presence of Lithium most likely means it is
young. But the processes that affect the strength of lithium are poorly known.
For instance, strong Li is also found in some evolved giant stars!
Skumanich Law
Ca II emission in stars from
several clusters. For older clusters
the emission levels off and it is
difficult to get an accurate age
Asteroseismology: The best way to
get the important stellar parameters
Zonal mode
Sectoral mode
Low degree
modes
High degree
modes
Red: high degree modes have shorter wavelengths and
do not propagate deeper into the sun.
Decreasing density causes
the wave to reflect at the
surface
Increasing density causes
the wave to refract in the
interior.
Blue: low degree modes have longer wavelengths and
propagate deeper into the sun.
The Asymptotic Limit for p-mode Oscillations
For high order modes: n >> l
nn,l = Dn0 (n + l /2 + e)
Dn0 = large spacing
D0 is sensitive to the sound speed near the core
e is sensitive to the surface layers
Modes are separated by odd and even degree (l). The socalled large separation is between sequential odd/even
modes. What we see is one-half the large spacing)
The Solar Power Spectrum
nmax ≈ 3000 mHz
Dn0 ≈ 135 mHz
How do we scale these to other stars?
Large
spacing
Summary of Scaling Relationships
vosc
L/L‫סּ‬
=
(23.4 ± 1.4) cm/sec
M/M‫סּ‬
(
(
dL
L
l
=
(l/ 550 nm)
(M/M‫)סּ‬1/2
Dn0 =
nmax
L/L‫סּ‬
134.9
3/2
(R/R‫)סּ‬
M/M‫סּ‬
=
(R/R‫סּ‬
)2√T
eff/5777K
(4.7 ± 0.3) ppm
(Teff / 5777 K)2
mHz
(M/M‫)סּ‬
The large frequency spacing is
related to the mean density of
the star
3.05 mHz
Asteroseismology of the 16 Cyg A and
16 Cyg B (planet hosting)
Figure 1 from Asteroseismology of the Solar Analogs 16 Cyg A and B from Kepler Observations
T. S. Metcalfe et al. 2012 ApJ 748 L10 doi:10.1088/2041-8205/748/1/L10
Stellar Oscillations in b Gem
Nine nights of RV measurements of b Gem. The solid line represents a 17 sine
component fit. The false alarm probability of these modes is < 1% and most have
FAP < 10–5. The rms scatter about the final fit is 1.9 m s–1
The Oscillation Spectrum of Pollux
The p-mode oscillation spectrum of b Gem based on the 17 frequencies found
via Fourier analysis. The vertical dashed lines represent a grid of evenlyspaced frequencies on an interval of 7.12 mHz
Frequency Spacing
M1/2
Dn0 ≈
Dn0 ≈
135
mHz
R3/2
7.12 mHz
Inteferometric Radius of b Gem = 8.8 R‫סּ‬
For radial modes → M = 1.89 ± 0.09 M‫סּ‬
Evolutionary tracks give M = 1.94 M‫סּ‬