spectroscopy – the analysis of spectral line shapes · spectroscopy – the analysis of spectral...

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

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

The equivalent width of the line is conserved under rotational broadening !!!!

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 Rּס

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 Rossiter-McClaughlin Effect

1

1

0

+v

–v

2

3

4

2 3 4

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.

From Holger Lehmann

The Rossiter-McLaughlin Effect in an Eclipsing Binary

The Rossiter-McClaughlin Effect

–v +v

0

As the companion crosses the star the observed radial velocity goes from + to –(astheplanetmovestowardsyouthestarismovingaway).Thecompanioncoverspartofthestarthatisrotatingtowardsyou.Youseemorepossitivevelocitiesfromthereceedingportionofthestar)youthusseeadisplacementto+RV.

–v +v

When the companion covers the receeding portion of the star, you see more negatve velocities of the starrotatingtowardsyou.YouthusseeadisplacementtonegativeRV.

Curves show Radial Velocity after removing the binary orbital motion

The effect was discovered in 1924 independently by Rossiter and McClaughlin


What can the RM effect tell you?

1. The inclination of „impact parameter“

–v +v

–v +v Shorter duration and smaller amplitude



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



3. Are the spin axes aligned?

–v +v

Symmetric R-M distortion

–v +v

Asymmetric R-M distortion

Orbit axis

λ

CoRoT-2b

λ = –7.2 ± 4.5 deg

λ = 182 deg!

HAT-P7

HARPS data : F. Bouchy Model fit: F. Pont Lambda ~ 80 deg!

Basic tools for line shape analysis:

1.  The Fourier transform

2.  Line bisectors

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

Both pioneered by David Gray

Fourier Transform of the Rotation Profile

David Gray pioneered using the Fourier transform of spectral lines to derive information from the shapes.

i(f) = I(λ)e2πiλf ∫ –∞

∞dλ

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.

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 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

Limb Darkening

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

Differential rotation parameter

φ is the latitude


Equatorial acceleration → lines narrower and more ‚V-shaped‘

Polar acceleration → lines fatter and more ‚U-shaped‘


Equatorial acceleration

1.  First zero moves to higher frequency

2.  Power in first sidelobe decreases

Polar acceleration

1.  First zero moves to lower frequency

2.  Power in first sidelobe increases

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

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 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

However, in 2003 Reiners et al. found evidence for differential rotation in F-stars

ψ Cap

α = 0

α = 0.25

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 π½v0

e–(v/v0)2 dv

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

Δλ = λ c cos θ β = λ

c v0cos θ = β0

cos θ

N(Δλ)dλ = 1 π½β

exp [ Δλ –(β )

2 [ 1 π½β0cosθ

exp [ Δλ –(β0cosθ )

2 [

= dΔλ

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:

Rising hot material

Cool, sinking intergranule lane

Horizontal motion to lane

Convection zone

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


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

π½ζRcos θ AT

π½ζTsin θ e –(Δλ/ζRcos θ)2

+ e –(Δλ/ζTcos θ)2

Fν = 2πAR ∫ 0

π/2

ΘR(Δλ)*Iνsin θ cosθ dθ +

2πAT ∫ 0

π/2

ΘT(Δλ)*Iνsin θ cosθ dθ

And the observed flux


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

10 km/s

5 km/s

2.5 km/s

0 km/s

Pixel shift (1 pixel = 0.015 Å)

Rel

ativ

e In

tens

ity

Macroturbulence versus Luminosity Class

Macroturbulence increases with luminosity class (decreasing surface gravity)


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.

Rel

ativ

e Fl

ux

Am

plitu

de

Pixel (0.015 Å/pixel) Frequency (c/Å)

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

Pure rotation

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

Von Zeipel law (1924): Teff = C g0.25 ,C is a constant

Recall Gravity darkening

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.

equator

Rotation pole

Span Curvature

The Power of Spectral Line Bisectors

What is a bisector?

Cool sinking lane Hot rising cell

Bisectors as a Measure of Granulation

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.


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


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


Examples of 4 component fits for stars from Dravins (1989)

Rotation amplifies the Bisector span (Gray 1986):

The Effects of Stellar Pulsations

Using Bisectors to Study Variability

Variations of Bisectors with Pulsations

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.

The 51 Peg Controversy

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

HD 166435

Spots vs. Planets

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

Color

Brightness

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.

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