Download Measuring the Gravity in Stars

Spectroscopy – Determination of
Fundamental Parameters
I.  Gravity
II.  Radii
III. Temperature
IV.  Stellar Rotation
Measuring the Gravity in Stars
As we have seen a change in surface gravity or effective temperature can
change the measured abundance. How can we be sure that we have the
correct pressure and temperature. Here we discuss means of measuring the
gravity (pressure) and temperature.
Determination of the gravity of a star is of fundamental importance
to transit detection of planets. The gravity is used to infer the
luminosity class and thus radius of the star which is needed to
derive the radius of the planet.
Measuring the Gravity in Stars
Direct way:
•  Obtain Mass via spectroscopy and the Doppler effect (binary stars for
example)
•  Measure the radius by an independent means (interferometry, lunar
occultations, eclipsing binaries)
GM
g=
R2
In principle this can only be done for very few stars → must rely on
spectroscopic determinations.
Measuring the Gravity in Stars
Continuum measurements
The Balmer jump is the only reliable means of using the
continuum to measure the gravity
D
Recall that the Balmer jump is
due to the change in continuous
opacity across the wavelength
corresponding to the ionization
of the Balmer electron. (Photons
no longer have the energy to
ionize a hydrogen atom, the
opacity decreases and you are
looking deeper into the
atmosphere across the jump).
Measuring the Gravity in Stars
Continuum measurements
The Balmer jump is sensitive to gravity. If D = F+/F- (balmer discontinuity), then the
larger D, the better it is as a gravity indicator. The best sensitivity occurs around
7500 K. This method is not as useful for stars cooler than 6500 K because it is
masked by the large number of metallic lines that appear in cooler stars.
Measuring the Gravity in Stars
Hydrogen lines
One of the more common means of inferring the gravity
Measuring the Gravity in Stars
Other strong lines
Lines like Ca II H & K, Ca I 4227 Å, Na D lines, Mg I b lines show strong
pressure broadened wings in the spectra of cool stars.
Measuring the Gravity in Stars
Other strong lines: Ca I
Measuring the Gravity in Stars
The Gravity-Temperature diagram
Weak lines can be used by comparing the two stages of ionization for
the same element. However, for ionic line strength depends on Pe and
thus indirectly on g. Plus, you really need to know the chemical
composition.
Pressure (gravity) diagnostics are always temperature sensitive.
Therefore one should make simultaneous solutions to the effective
temperature and gravity.
•  Use 2 lines with different response to the variables (g and T), for
example lines with different excitation potentials
•  Each line is computed for a constant elemental abundance
•  Vary surface gravity keeping temperature constant (vice versa) to
recover the observed equivalent width.
•  The crossing point is the solution.
Measuring the Gravity in Stars
The Gravity-Temperature diagram
Measuring the Gravity in Stars
Empirical indicators
There are two empirical indicators of surface gravity:
•  The width of the chromospheric emission reversal of the Ca II H & K lines (WilsonBappu effect)
•  The size of the Doppler broadening called macro-turbulence (next week)
Measuring the Gravity in Stars
Empirical indicators
Visual binaries allow the direct determination of the mass and
thus gravity, if you can measure the stellar radius.
Measuring the Gravity in Stars
If you have measured the surface gravity from spectral fitting, and
the radius by other means, then the mass of the star is given by
M = gR2/G
If you can measure the mass of the star by other means (e.g.
binary systems) as well as the radius you can get a direct
measurement of the surface gravity.
If you can measure the mass of the star by other means and then
the gravity from spectrosctopy, then you have the stellar radius
Simply put, you gravity is related to mass and radius (g, M, R). If you
measure 2 you have the third and there are a several methods to get
the different quantities.
Direct Measurements of Stellar Radii
If one can measure the radius of the star one can get a direct
measurement of the effective temperature and mass (gravity) of
the star. If L is the observed absolute luminosity of the star, then
L = 4πR2 σTeff
…or, if you do not know the radius you can measure the
4
effective temperature via spectroscopy
and derive the
radius. This is the more common way since the direct
measurement of radii can only be done for a few stars.
The absolute luminosity of the star requires an accurate
determination of the stellar distance. Accurate distances of the
brightest stars are available from Hipparcos measurements. In
the near future GAIA will have more accurate stellar distances for
millions of stars.
Stellar Radii
Speckle Photometry
The diffraction limit of a 4m telescope is 0.02 arcsecs. There are a
significant number of stars that have angular diameters greater than
this. However, atmospheric turbulence prevents one from achieving
the diffraction limit
Texereau (1963), Labeyrie (1970) and others realized that the
instantaneous seeing disk in the focal plane of a large telescope
consists of slightly displaced diffraction-modified images of the star.
These „speckles“ indicate that the atmosphere consists of a small
number of refracting elements that redirects portion of the wavefront
entering the telescope aperture.
Stellar Radii
Speckle Photometry
Averaging over a few seconds integration removes all traces of the
individual speckles. One must take rapid (~millisecs) exposures.
Each speckle is the convolution of the telescope instrumental profile
(diffraction and other optical effects) and the true distribution of the
intensity of the stellar disk. Fourier analysis recovers the disk of the
star, and one simply adds the Fourier transforms of the individual
speckle exposures
The instrumental profile of the telescope can be obtained by
observing a star known to be a unresolved. (e.g. distant giant star).
The speckle pattern for Vega taken with the Hale
5m telescope
From space one of course can reach the diffraction limit of the telescope:
Original image
Deconvolved
image
Hot Spot
Stellar Radii
Interferometry
By increasing the size of your telescope, you increase the diffraction limit,
but building large telescopes is difficult and requires and are technically
challenging. Solution: use many small telescopes
Interferometry can increase the angular resolution of your telescope
and thus allow you to measure directly the angular diameter of a star
A Two-telescope Interferometry
S
s • B = B cos θ
A1
x1
θ
B
Delay Line 1
d1
A2
x2
Beam
Combiner
Idealized 2-telescope interferometer
Delay Line 2
d2
Keck: A Two-telescope Interferometry
VLTI: A Multi-Telescope Interferometer
Two Beam Interferometer and 2 Point sources:
Two plane parallel wave fronts:
φ1 ~ eikd1 e-iωt
φ2 ~ eikd2 e-ikŝ ⋅B e-iωt
φTotal = φ1 + φ2 ~ e-iωt (eikd1 + eikd2 eik ŝ ⋅B )
Can show that the total power is:
P = 2(1 + cos k (s⋅B+d1–d2))
s⋅B is the path difference of the light hitting the 2 telescopes
d1 –
d2
is
the
path
difference
caused
by
the
delay
lines
Output of 2 telescope interferometer
Adjacent fringe crests projected on the sky are separated
by an angle given by:
Δs = λ/B
The Visibility Function
Michelson Visibility:
Imax –Imin
V=
Imax +Imin
Visibility is measured by changing the path
length and recording minimum and maximum
values
Cittert-Zernike theorem
•
•
•
•
β
α
Δs ⊥ ŝo
α, β are angles in „x-y“
directions of the source.
Δs = (α,β,0)
in the coordinate system
where ŝo =(0,0,1)
Cittert-Zernike Theorem
V(k, B) = ∫ dα dβ I(α,β) e -2πi(αu+βv)
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.
Footnote: Fourier Transforms
The continous form of the Fourier transform:
F(s) = ∫ f(x) e–ixs dx
f(x) = 1/2π ∫ F(s) eixs ds
eixs = cos(xs) + i sin (xs)
Footnotes: Fourier Transforms
In interferometry it is useful to think of normal space (x,y)
and Fourier space (u,v) where u,v are frequencies
Two important features of Fourier transforms:
a)  The “spatial coordinate” x maps into a “frequency”
coordinate 1/x (= s)
Thus small changes in x map into large changes in s.
A function that is narrow in x is wide in s
I. Background: Fourier Transforms
x
x
ν
I. Background: Fourier Transforms
x
ν
x
ν
I. Background: Fourier Transforms
sinc
x
ν
J1(2πx)
2x
x
ν
Diffraction patterns from the interference of
electromagnetic waves are just Fourier transforms!
Stellar Angular Diameters:
V(s) = |2J1(πas)/πas|
J1 is the first order Bessel function
Note: as it should be, this is also the diffraction
pattern of a circular aperture.
For measuring stellar diameters you only need 2 telescopes. In fact a
common practice is to take a single telescope and use a mask with
two holes in it
Note: Each baseline measures only one point on the visibility function.
Often, you cannot sample all baselines.
Historical Note: Michelson
was the first to measure the
diameter of Betelgeuse. In
1920, he and Francis
Pease mounted a 6 m
interferometer on the front of
the 2.5 metre telescope at
Mount Wilson Observatory.
measured the angular
diameter of α Orionis at
0.047“ = 47 mas. The
current value is 55 mas.
Large uncertainty due to
limb darkening.
Stellar Radii
Lunar Occultations
As the moon occults a star it can be used as
a knife edge to create a diffraction pattern.
The occultation pattern is recorded as a
function of time, and the time coordinate is
converted to angular diameter using the
angular velocity of the moon relative to the
earth:
References:
Williams, J.D. ApJ, 1939, 89, 467
Nather, R.E. & Evans, D.S. 1970, AJ, 75, 575
Schmidtke et al. 1986, AJ, 91, 961.
Lunar occulations are
one of the best ways
to measure stellar
diameters.
Unfortunatley, the
stars have to lie in the
path of the moon!
Stellar Radii
Bolometric Method
4πd2Fν = 4πR2Fν
Fν is the observed flux of the star, Fν is the flux at the stellar surface,
R is the radius of the star, d is the distance to the star
=
(
∫Fνdν
σTeff4
½
(
R
d
Log R = log d – 0.2(mv – BC) – 2log Teff + constant
If we ratio everything to solar units (m
– = ‫סּ‬,
T
= ‫
סּ‬K)
Log R = log d + 0.2BC – 2log Teff – 0.2mv + 7.460
Stellar Radii
Eclipsing Binaries from
CoRoT
Eclipsing Binaries can also yield stellar radii. These are relatively few, but they
can be used to calibrate main sequence radii
Measuring Stellar Masses
Uncertain
•  log g and R (depends on how good gravity and radius is)
•  Spectral type
More certain
•  Dynamical mass (binary)
•  Asteroseismology
Dynamical Masses from Spectroscopic binaries
Measure the Doppler shift of the individual spectral components, apply
Keplers law and viola! You have the ratio of the stellar masses!
Stellar Oscillations: Asteroseismology
Stellar oscillations probe the interior of the star and give you
the mean density
P-mode Oscillations in the Sun
P-mode oscillations (pressure is the restoring force) are equally
spaced in frequency with an interval given by:
M1/2
Δν0 ≈ 135
R3/2
µHz
Stellar Oscillations in β Gem
Above shows the RV measurements of β 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 β Gem based on the 17 frequencies found via
Fourier analysis. The vertical dashed lines represent a grid of evenly-spaced frequencies
on an interval of 7.12 µHz
The Mass of Pollux
Frequency spacing:
Δν0 ≈ 135
M1/2
R3/2
µHz
The radius of β Gem is well
determi
n
ed through
= 8.8
7.12 µHzmeasurements:
(l = 0) → M = R1.89
M‫סּ‬R‫סּ‬
Δνinterferometric
0=
→log g = 2.82
From spectroscopy: log g = 2.70
With enough oscillation frequencies you can derive everything with
asteroseismology!
(M/M1/2)‫סּ‬
Δν0 ≈
νmax
3/2
(R/R )‫סּ‬
=
135
M/M‫סּ‬
(R/R2)
‫√סּ‬Teff/5777K
µHz
3.05 mHz
Stellar Temperatures
Indicators of stellar temperatures:
1.  Slope of Paschen continuum:
∂ log(F4000/F7000)
∂ log Teff
≈ 2.3
If you can measure the continuum slope between 4000 and 7000 Å
to 2.3% then you have the temperature to ≈ 1%
Stellar Temperatures
2. Color indices
Stellar Temperatures
3. The Balmer Jump
Stellar Temperatures: Line Ratios
4. Metallic Lines
Line depth ratios can be used to measure accurately the temperature of a star.
One choses two lines of different temperature sensitivities. These should be very
close in wavelength so as to minimize errors due to continuum placement.
The effects of rotational broadening and macroturbulence should effect both lines
equally.
Stellar Temperatures: Line Ratios
Stellar Temperatures: Line Ratios
To use line depth ratios one must calibrate these against effective temperature
(B –
V):
Line depth ratios can yield effective temperatures accurate to about 10 K. One
can measure temperature variations in one star to about ΔT ≈ 1 K
Note: This method only works for stars of spectral type G–K
The Rotation Profile
v=Ω×R
Doppler shift = yΩx –
xΩy
But there is no x-component
of Ω due to choice of
coordinate system:
vz = xΩsin i
Sign convention: +vz receding (redshift)
–vz
approaching
(blueshift)
Largest velocity occurs at limbs where vL = RΩ sin i = veq sin i.
R is the radius of the star and veq is the equatorial velocity
The Rotation Profile
The flux of the star is still given by:
Fν =
∫I
ν
cos θ dω
But now Iν has to be Doppler shifted by the radial velocity on the star:
dω = dA/R2 , dA is the increment of surface area on the star, the increment on the
apparent disk is dxdy = dAcosθ
Fν =
∫∫
Iν dxdy
R2
The proper way is to calculate Iν at each location on the stellar disk using
a local profile generated by a model atmosphere which takes into account
limb darkening. You the apply the appropriate Doppler shift and integrate
over all points. I will present an analytical solution following Gray.
The apparent disk of the star can be thought of as a series of strips parallel
to the projection axis each having a Doppler shift of xΩsini:
V = –Vrot
x
V=0
V = +Vrot
The Rotation Profile
Define the intrinsic profile at any point on the disk as H(Δλ) = H(v) = Iν/Ic = ratio of
intensity of any point in the spectrum to the continuum intensity. The flux profile
from a rotating star is:
Fν
Fc =
H(v) Iccos θ dω
∫
∫ Iccos θ dω
H(v) depends on the disk position through the Doppler shifts:
Fν =
∫
H(v–vz) Iccos θ dω
Fν =
∫
H(v–vz) Ic dxdy
R2
The Rotation Profile
Integration limits:
x: –R to +R
y: –y1 to +y1
–
x2)½
vz
[
1
–
=R
( veq
2 ½
[(
y1 =
(R2
+R
Fν =
∫
–R
H(v–vz)
+y1
dy dvz
I
c
∫
R vL
–y
1
The Rotation Profile
Define G(Δλ):
y1
∫–y Ic dy/R
1
G(Δλ) = G(vz) =
vL ∫ Iccos θ dω
vz
≤ vL
=0
vz
> vL
1
Normalized profile:
Fν
=
Fc
+∞
∫ H(v–vz) G(vz) dvz
–∞
= H(v) *G(vz)
* is the convolution
Remember that v can also be
replaced with Δλ
Footnote: Convolution
Convolution
∫ f(u)φ(x–u)du = f * φ
f(x):
φ(x):
φ(x-u)
a2
a1
a3
Note:
f*g → F • G
g(x)
a3
a2
Convolution is a smoothing function
a1
In Fourier space the
convolution is a
muliplication of the
individual Fourier
transforms
The Rotation Profile
Important result: The profile of a rotationally broadened spectral line is merely
the convolution of the flux profile from a non-rotating star convolved with the
rotation profile, so long as the shape of the line profile does not change across
the stellar disk. The more rotation, the broader the rotation function G. For
large rotation rates, it dominates the line shape. This means that one does not
have to do a „disk integration“, one merely does a convolution of a simple
profile.
Including limbdarkening:
Ic/Ic0 = (1 – ε) + ε cosθ
Ic0 is the specific intensity at disk center and ε ≈ 0.6 for the sun. θ is the
limb angle. The denominator of G(vz) then becomes
∫
Iccos θ dω = π Ic0(1 –
ε/3)
The Rotation Profile
And the numerator
Use cos θ = [R2 – (x2 + y2)]½/R and the fact that ∫(A2 – y2)½dy = ½[y(A2 – y2)½ +
A2sin–1(y/A)] this becomes
G(λ)
The Rotation Profile
If ε = 0, the second
term is zero and the
function is an ellipse.
If ε =1 the first term
is zero and the
rotation function is a
parabola