# Rotational Line Broadening Gray Chapter 18 - PowerPoint PPT Presentation

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## Rotational Line Broadening Gray Chapter 18

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Title: Rotational Line Broadening Gray Chapter 18

1
• Geometry and Doppler Shift
• Profile as a Convolution
• Observed Stellar Rotation
• Other Profile Shaping Processes

2
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3
Doppler Shift of Surface Element
• Assume spherical star with rigid body rotation
• Velocity at any point on visible hemisphere is

4
Doppler Shift of Surface Element
• z component corresponds to radial velocity
• Defined as positive for motion directed away from
us (opposite of sense in diagram)
• Doppler shift is

5
Radial velocity depends only on x
position.Largest at limb, xR.v equatorial
rotational velocity,v sin i projected
rotational velocity
6
Flux Profile
• Observed flux is (R/D)2 F? where
• Angular element for surface element dA
• Projected element
• Expression for flux

7
Assumption profile independent of position on
visible hemisphere
8
Express as a Convolution
9
G(?) for a Linear Limb Darkening Law
• Denominator of G

10
G(?) for a Linear Limb Darkening Law
• Numerator of G

11
G(?) for a Linear Limb Darkening Law
• Analytical solution for second term in
numerator
• Second term is

12
G(?) for a Linear Limb Darkening Law
?ellipse
?parabola
13
Grey atmosphere case e 0.6
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15
v sin i 20 km s-1
v sin i 4.6 km s-1
16
Measurement of Rotation
• Use intrinsically narrow lines
• Possible to calibrate half width with v sin i,
but this will become invalid in very fast
rotators that become oblate and gravity darkened
• Gray shows that G(??) has a distinctive
appearance in the Fourier domain, so that zeros
of FT are related to v sin i
• Rotation period can be determined for stars with
spots and/or active chromospheres by measuring
transit times

17
Rotation in Main Sequence Stars
• massive stars rotate quickly with rapid decline
in F-stars(convection begins)
• low mass stars have early, rapid spin down,
followed by weak breaking due to magnetism and
winds (gyrochronology)

18
L M R v
19
Angular Momentum Mass Relation
• Equilibrium with gravity centripetal
acceleration
• Angular momentum for uniform density
• In terms of angular speed and density
• Density varies slowly along main sequence

20
Rotation in Evolved Stars
• conserve angular momentum, so as R increases, v
decreases
• Magnetic breaking continues (as long as magnetic
field exists)
• Tides in close binary systems lead to synchronous
rotation

21
Fastest Rotators
• Critical rotation
• Closest to critical in the B stars where we find
Be stars (with disks)
• Spun up by Roche lobe overflow from former mass
donor in some cases (? Persei)

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23
Other Processes That Shape Lines
• Macroturbulence and granulationhttp//astro.uwo.c
a/dfgray/Granulation.html

24
Star Spots
Vogt Penrod 1983, ApJ, 275, 661
HR 3831Kochukhov et al. 2004, AA, 424,
935http//www.astro.uu.se/oleg/research.html
25
Stellar Pulsationhttp//staff.not.iac.es/jht/sci
ence/
Vogt Penrod 1983, ApJ, 275, 661
26
Stellar Winds
• Atoms scatter starlight to create P Cygni
shaped profiles
• Observed in stars with strong winds (O stars,
supergiants)
• UV resonance lines (ground state transitions)

http//www.daviddarling.info/encyclopedia/P/P_Cygn
i_profile.html
27
FUSE spectra (Walborn et al. 2002, ApJS, 141,443)
28
To really know a star ... get a spectrum
• If a picture is worth a thousand words, then a
spectrum is worth a thousand pictures.(Prof. Ed
Jenkins, Princeton University)