1. Rotational Line Broadening Gray Chapter 18
Geometry and Doppler Shift
Profile as a Convolution
Rotational Broadening Function
Observed Stellar Rotation
Other Profile Shaping Processes

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)
Radial velocity is
Doppler shift is

5. Radial velocity depends only on x position. Largest at limb, x=R
Radial velocity depends only on x position. Largest at limb, x=R. v = equatorial rotational velocity, v sin i = projected rotational velocity

6. Flux Profile Observed flux is (R/D)2 Fν 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(λ) for a Linear Limb Darkening Law
Denominator of G

10. G(λ) for a Linear Limb Darkening Law
Numerator of G

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

12. G(λ) for a Linear Limb Darkening Law
 ellipse
 parabola

13. Grey atmosphere case: ε = 0.6

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(Δλ) 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 (ϕ Persei)

23. Other Processes That Shape Lines
Macroturbulence and granulation

24. Star Spots Vogt & Penrod 1983, ApJ, 275, 661
HR 3831 Kochukhov et al. 2004, A&A, 424, 935