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 1

2. 2

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

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 4

5. 5 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 F ν where Angular element for surface element dA Projected element Expression for flux 6

7. Assumption: profile independent of position on visible hemisphere 7

8. Express as a Convolution 8

9. G(λ) for a Linear Limb Darkening Law Denominator of G 9

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

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

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

13. 13 Grey atmosphere case: ε = 0.6

14. 14

15. 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 16

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) 17

18. 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 19

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 20

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) 21

22. 22

23. Other Processes That Shape Lines Macroturbulence and granulation http://astro.uwo.ca/~dfgray/Granulation.html 23

24. Star Spots 24 Vogt & Penrod 1983, ApJ, 275, 661 HR 3831 Kochukhov et al. 2004, A&A, 424, 935 http://www.astro.uu.se/~oleg/research.html

25. Stellar Pulsation http://staff.not.iac.es/~jht/science/ 25 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) 26 http://www.daviddarling.info/encyclo pedia/P/P_Cygni_profile.html

27. 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) 28