Spectroscopy – The Analysis of Spectral
Line Shapes
The detailed analysis of the shapes of spectral lines can give you
information on:
1.  Differential rotation in stars
2.  The convection pattern on the surface of the star
3.  The location of spots on the surface of stars
4.  Stellar oscillations
5.  etc, etc.
The Rotation Profile
The equivalent width of
the line is conserved
under rotational
broadening !!!!
To match the spectrum of a star
that is rotating rapidly, take a
spectrum of a slowly rotating
star with the same spectral type
and convolve with the rotation
function
Rotation in Stars
Note: we only can measure v sin i, the true rotational velocity times the sine
of the inclination axis of the rotating star
Rotation Rates in Stars
Wide range at
earlier spectral
types are due to two
reasons: 1) some
stars rotate slower;
2) We view some
stars at lower
inclinations (from
the pole, sin i is
small)
As one goes to higher
luminosity classes the
rotation break moves to
later spectral types
Rotation in Stars
Fast rotators
The breakup velocity is the speed at which the centrifugal force exceeds
the gravitational force
Rotation in Stars
Gravity darkening
The centrifigal force from rapid rotation provides hydrostatic support to
the atmosphere. Less temperature is needed to maintain hydrostatic
equilibrium, thus the equatorial regions of the star have a cooler
temperature than the polar regions.
Von Zeipel law (1924):
Teff = C g0.25 ,C is a constant
Eg. B-type star
Teff = 20000 K
R = 3 R‫סּ‬
log g = 4.0
vsini = 300 km s–1
log g centrifugal: 3.64
Tequator / Tpole = (5634/10000)0.25
= 17327 K
ΔT ≈ 2700 K
Evidence for Stellar Rotation: The RossiterMcClaughlin Effect
2
1
+v
0
4
3
1
4
2
–v
3
The R-M effect occurs in eclipsing systems when the companion crosses in
front of the star. This creates a distortion in the normal radial velocity of the
star. This occurs at point 2 in the orbit.
The Rossiter-McLaughlin Effect in an
Eclipsing Binary
From Holger Lehmann
The Rossiter-McClaughlin Effect
–v
+v
0
As the companion crosses the star the
observed radial velocity goes from + to –
(as
the
planet
moves
towards
you
the
star
is
moving
away).
The
companion
covers
part
of
the
star
that
is
rotating
towards
you.
You
see
more
possitive
velocities
from
the
receeding
portion
of
the
star)
you
thus
see
a
displacement
to
+
RV.
+v
–v
When the companion covers the
receeding portion of the star, you see
more negatve velocities of the star
rotating
towards
you.
You
thus
see
a
displacement
to
negative
RV.
The effect was discovered in 1924 independently by Rossiter and
McClaughlin
Curves show Radial Velocity after
removing the binary orbital motion
The Rossiter-McClaughlin Effect
What can the RM effect tell you?
1. The inclination of „impact parameter“
–v
–v
+v
+v
Shorter duration
and smaller
amplitude
The Rossiter-McClaughlin Effect
What can the RM effect tell you?
2. Is the companion orbit in the same direction as the rotation of the star?
–v
+v
+v
–v
Note: R-M curves are
schematic, drawn by
hand
The Rossiter-McClaughlin Effect
λ
What can the RM effect tell you?
3. Are the spin axes aligned?
–v
Symmetric R-M
distortion
+v
Orbit axis
–v
+v
Asymmetric R-M
distortion
CoRoT-2b
λ = –7.2 ± 4.5 deg
HAT-P7
λ = 182 deg!
HARPS data : F. Bouchy
Model fit: F. Pont
Lambda ~ 80 deg!
Basic tools for line shape analysis:
1.  The Fourier transform
2.  Line bisectors
Both pioneered by David Gray
To derive reliable information about the line shapes requires high
resolution and high signal-to-noise ratios:
•  R = λ/δλ ≥ 100.000
•  S/N > 200-300
Fourier Transform of the Rotation Profile
David Gray pioneered using the Fourier transform of spectral
lines to derive information from the shapes.
∞
i(f) =
2πiλf dλ
I(λ)e
∫
–∞
Where I(λ) is the intensity profile (absorption line) and
frequency f is in units of cycles/Å or cycles/pixel (detector units)
Because of the inverse relationship between normal and Fourier
space (narrow lines translates into wide features in the Fourier
domain), the Fourier transform is a sensitive measure of subtle
shapes in the line profile. It is also good for measuring rotation
profiles.
The Instrumental Profile
The observed profile is the spectral line profile of the star convolved with the
instrumental profile of the spectrogaph, i(λ)
What is an instrumental profile (IP)?:
Consider a monochromatic beam of light (delta function)
Perfect
spectrograph
A real
spectrograph
If the IP of the instrument is asymmetric, then this can seriously alter the shape
of the observed line profile
No problem with this IP
Problems for line
shape measurements
It is important to measure the IP of an instrument if you are making line shape
measurements
If D(Δλ) is the observed profile (your data) then
D(Δλ) = H(Δλ)*G(Δλ)*I(Δλ)
Where:
D = observed data
H = intrinsic spectral line
G = Broadening function (rotation * macroturbulence)
I = Instrumental profile
* = convolution
In Fourier space:
d(σ) = h(σ)g(σ)i(σ)
You can either include the instrumental response, I, in the modeling, or
deconvolve it from the observed profile.
Fourier Transform of the Rotation Profile
Fourier Transform of the Rotation Profile
The Fourier transform of the rotational profile has zeros which move to
lower frequencies as the rotation rate increases (i.e. wider profile in
wavelength coordinates means narrower profile in frequency space).
Limb Darkening
Limb darkening shifts the zero to higher frequency
Limb Darkening
The limb of the star is darker so
these contribute less to the
observed profile. You thus see
more of regions of the star that
have slower rotation rate. So the
spectral line should look like a
more slowly rotating star, thus the
first zero of the transform should
move to lower frequencies
Limb Darkening
Ic/Ic0 = (1 – ε) + ε cosθ
Effects of Differential Rotation on Line shapes
The sun differentially rotates with equatorial acceleration. The equator
rotational period is about 24 days, for the pole it is about 30 days.
Differential rotation can be quantified by:
ω = ω0 + ω2 sin2φ + ω4sin4φ
α = ω2/(ω0 + ω2)
Solar case α = 0.19
+ → equator rotates faster
–
→ pole rotates faster
φ is the latitude
Differential rotation parameter
Effects of Differential Rotation on Line shapes
Equatorial acceleration →
lines narrower and more ‚Vshaped‘
Polar acceleration → lines
fatter and more ‚U-shaped‘
Effects of Differential Rotation on Line shapes
Equatorial acceleration
Polar acceleration
1.  First zero moves to higher
frequency
1.  First zero moves to lower
frequency
2.  Power in first sidelobe
decreases
2.  Power in first sidelobe
increases
Noise level for low S/N Data (≈10-50)
Noise level for modest S/N Data (≈ 100-150)
Noise level for modest S/N Data (≈ 250-500)
The measurement of differential rotation requires data taken at
high spectral resolution and high signal-to-noise (S/N) ratios.
Ideally one would like to measure the first and second sidelobe,
but that takes data with very high signal-to-noise ratios. Often
this is not possible. The best method is to use the location of
the first zero
The inclination of the star has an
effect on the Fourier transform of the
differential profile. Note: this is for the
same v sin i!
Differential Rotation in A stars
In 1977 Gray looked for
differential rotation in a sample of
A-type star and found none. This
is not surprising since we think
that the presence of a convection
zone is needed for DR and A-type
stars have a radiative envelope.
Differential Rotation in A stars
Gray found two strange stars. γ Boo has a weak first sidelobe and no
second side lobe. γ Her has no sidelobes at all. This may be the effects of
stellar pulsations.
Differential Rotation in F stars
In 1982 Gray looked for differential rotation in a sample of F-type star and concluded
that there was no differential rotation. Spot activity on F-type stars is not seen, but they
do have a convection zone so DR is possible.
Differential Rotation in F stars
ψ Cap
α=0
α = 0.25
However, in 2003 Reiners et al. found evidence for differential rotation in F-stars
What about G-type stars?
For the method to work you need some rotational broadening of the line profile
(several km/s), otherwise differential rotation will not affect the line shapes.
Solar-type stars rotate too slowly.
Velocity Fields in Stars
Early on it was realized that the observed shapes of spectral lines
indicated a velocity broadening in the photosphere termed
„turbulence“ by Rosseland.
A theoretical line profile with thermal broadening alone will not
reproduce the observed spectral line profile. This macroturbulent
velocity broadening is direct evidence of convective motions in the
photospheres of stars
From Velocity to Spectrum
N(v)dv =
1 e–(v/v0)2dv
π½v0
N(v)dv is the fraction of material having velocities in the range v → v + dv
and v is allowed only on stellar radii. The projection of velocities along the
line of sight
= β0 cos θ
β = λ v0cos θ
c
2
Δλ 2
Δλ
1
1
= π½β cosθ exp [–(
β0cosθ )
dΔλ
N(Δλ)dλ = ½ exp [ –(
β )
π β
0
Δλ = λ cos θ
c
[
[
Note that β, the width parameter, is a function of θ, β0 is constant. At
disk center N(λ) reflects N(v) directly, but away from the center the
Doppler distribution becomes narrower. At the limb N(Δλ) is a delta
function.
Including Macroturbulence in Spectra
The observed spectra (ignoring other broadening mechanisms for
now) is the intensity profile convolved with the macroturbulent
profile:
Iν = Iν0 * Θ(Δλ)
Iν0 is the unbroadened profile and Θ(Δλ) is the
macroturbulent velocity distribution.
What do we use for Θ?
The Radial-Tangential Prescription from Gray
We could just use a Gaussian distribution of radial components of
the velocity field (up and down motion), but this is not realistic:
Horizontal motion to lane
Convection zone
Rising hot material
Cool, sinking
intergranule lane
If you included only a distribution of up and down velocities, at the limb these would not
alter the line profile at the limb since the motion would not be in the radial direction. The
horizontal motion would contribute at the limb
Radial motion at disk center → main contrbution at disk center
Tangential motion at disk center → main contribution at limb
The Radial-Tangential Prescription from Gray
Assume that a certain fraction of the stellar surface, AT, has
tangential motion, and the rest, AR, radial motion
Θ(Δλ) = ARΘR(Δλ) + ATΘT(Δλ)
AR
e–(Δλ/ζ cos θ) +
= ½
π ζRcos θ
R
2
AT
–(Δλ/ζ cos θ)
e
π½ζTsin θ
And the observed flux
π/2
∫ ΘR(Δλ)*Iνsin θ cosθ dθ +
Fν = 2πAR
0
π/2
2πAT
∫ ΘT(Δλ)*Iνsin θ cosθ dθ
0
T
2
The Radial-Tangential Prescription from Gray
The R-T prescription produces as slightly different velocity distribution
than an isotropic Gaussian. If you want to get more sophisticated you
can include temperature differences between the radial and tangential
flows.
The Effects of Macroturbulence
Macro
Relative Intensity
10 km/s
5 km/s
2.5 km/s
0 km/s
Pixel shift (1 pixel = 0.015 Å)
Macroturbulence versus Luminosity Class
Macroturbulence increases with luminosity class (decreasing
surface gravity)
Amplitude
Relative Flux
The Effects of Macroturbulence
Pixel (0.015 Å/pixel)
Frequency (c/Å)
There is a trade off between rotation and macroturbulent velocities. You can
compensate a decrease in rotation by increasing the macroturbulent velocity. At
low rotational velocities it is difficult to distinguish the two. Above the red line
represents V = 3 km/s, M = 0 km/s. The blue line represents V=0 km/s, and M = 3
km/s. In wavelength space (left) the differences are barely noticeable. In Fourier
space (right), the differences are larger.
The Effects of Macroturbulence
Pure rotation
Rotation affects the location of the first zero. Macroturbulence affects the size of
the first side lobe and to a lesser extent the main lobe.
For slowly rotating stars one should use Fourier space for measuring accurate
rotational velocities
Sometimes it is very important to measure the rotational velocity accurately.
HD 114762
m sin i = 11 MJup
Most likely vsini is 0-1 km/s. HD
114762 is an F8 star and the
mean rotation of these stars is
about 5 km/s.
The companion could be a more
massive companion, maybe even
a late M-dwarf
0100200300400500.4.6.81
A word of caution about using Fourier transforms
If you want to calculate the Fourier transform of the line you have to „cut out“ the
line.
This is the equivalent of multiplying your data with a box function.
In Fourier space this is a sinc function which gets convolved with your broadening
function. This changes the FT. → need to apply taper function (bell cosine, etc.)
The Funny Shape of the Lines of Vega
A clue may be found in
the slow projected
rotational velocity of
Vega, an A0 V star
Recall Gravity darkening
Von Zeipel law (1924):
equator
Rotation pole
Teff = C g0.25 ,C is a constant
Because of gravity darkening
and centrifugal force, the
equator has lower gravity and
a lower temperature. For a
star viewed pole on this
appears at the limb.
Temperature/gravity sensitive
weak lines will be stronger at
the equator (limb) than at the
poles.
The Power of Spectral Line Bisectors
What is a
bisector?
Curvature
Span
Bisectors as a Measure of Granulation
Hot rising cell
Cool sinking lane
Solar Bisector
Solar bisectors take on a „C“ shape due to more flux and more
area of rising part of convective cells. There is considerable
variations with limb angle due to the change of depth of formation
and the view angle. The line profiles themselves become
shallower and wider towards the limb.
Bisectors as a Measure of Granulation
The measurement of an individual bisector is very noisy. One
should use many lines. These can be from different line strengths
as one can „collapse“ them all into one grand mean. Note: this
cannot be done in hotter stars the weak lines do not mimic the
shape of the top portion of the bisector.
Changes in the Granulation Pattern of Dwarfs
Changes in the Granulation Supergiants
The Granulation and Rotation Boundary
Rapid rotation,
Inverse „C“
bisectors
Slow rotation
„C“ shaped
bisectors
Bisectors as a Measure of Granulation
Can get good results using a 4 stream model (Dravins 1989, A&A, 228,
218). These best reproduce hydrodynamic simulations
1.  Granule center (rising material)
2.  Granules (rising material)
3.  Neutral areas (zero velocity)
4.  Intergranule lanes (cool sinking material)
Each has their own fractional areas An, velocity Vn, and Temperature Tn
Constraints:
1.  A1 + A2 + A3 + A4 = 1
2.  V3 = 0
3.  Mass conservation: A1×V1 + A2 ×V2 = A4×V4
Downflow = upflow
Best way, is to use numerical hydrodynamic simulations
Bisectors as a Measure of Granulation
Examples of 4 component fits
for stars from Dravins (1989)
Rotation amplifies the Bisector span (Gray 1986):
Using Bisectors to Study Variability
The Effects of Stellar Pulsations
Variations of Bisectors with Pulsations
The 51 Peg Controversy
Gray & Hatzes
Gray reported bisector
variations of 51 Peg
with the same period
as the planet. Gray &
Hatzes modeled these
with nonradial
pulsations
A beautiful paper that
was completely wrong.
Hatzes et al.
More and better bisector data for 51 Peg showed that the Gray measurements
were probably wrong. 51 Peg has a planet!
Bisector Variations due to Spots
Spot Pattern
Changes in
Radial
Velocity due
to changing
shapes
Note: this has the same
shape as the R-M effect,
as it should, the spots
can be considered to be
a companion (star or
planet) that blocks flux
from the main star
Star Patches
Bisectors
Bisector span
Star Patches
ΔT = 300 K
Compared to
ΔT = 2000 K for
sunspots
Spots vs. Planets
HD 166435
Radial Velocity
This was reported to be another short period planet with
a period of 4 days until…
Spots vs. Planets
Radial Velocity
Ca II
Brightness
Color
The star was found to vary in Ca II, brightness, and color with the same period
as the presumed planet. This is a spotted star
Correlation of bisector span with radial velocity for HD 166435.
Looking at bisector variations has become a standard way of
confirming planets. The spirit of David Gray continues…
Disk Integration Mechanics
θ
Cell i,j
1.  Divide the star into an x,y grid
2.  At each cell calculate the limb angle θ
3.  Take the appropriate limb angle intrinsic line
profile from model atmospheres, or just
apply limb darkening law to a line profile or
even a Gaussian profile (the poor person‘s
way)
4.  Calculate the radial velocity using the
desired vsini. Include differential rotation if
desired. Doppler shift your line profile
5.  Use a random number generator to
calculate the radial and tangential value of
the macro-turbulent velocity with maximum
value ξ. Apply additional Doppler shift due to
the turbulent velocity
6.  If there is a spot, you can scale the flux. If
there are pulsations you can add velocity
field of star.
7.  Can add convective velocities/fluxes
8.  Take area of cell and multiply it by the
projected area (cos θ)
9.  Go to next i,j cell
10.  Add all profiles from all cells
11.  Normalize by the continuum
12.  Check to make sure line behaves with
vsini macro-turbulence. Make sure
equivalent width is conserved.