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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 Doppler Shift of Surface Element • Assume spherical star with rigid body rotation • Velocity at any point on visible hemisphere is v R ^ zsin i ycosi x x y z ^ 0 sin i cosi xcosi y ^ x y z xsin i z ^ ^ ^ 3 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 vR xsin i • Doppler shift is 0 c vR 0 c xsin i 4 Radial velocity depends only on x position. Largest at limb, x=R. L 0 Rsin i 0 c c v = equatorial rotational velocity, v sin i = projected rotational velocity v sin i 5 Flux Profile • Observed flux is (R/D)2 Fν where F I cos d • Angular element for surface element dA d dA 2 R • Projected element dx dy dA cos • Expression for flux I F 2 dx dy R 6 Assumption: profile independent of position on visible hemisphere F H( )I c dx dy /R 2 dy H( ) Ic d R L R y1 y1 R y1 R x 2 2 1/ 2 2 1/ 2 R 1 L 7 Express as a Convolution 1 G() L y1 I c dy /R y1 I R cos d 0 c for L for L F H( ) G( ) H( ) G() Fc R H( ) G( ) 8 G(λ) for a Linear Limb Darkening Law Ic 0 1 cos Ic • Denominator of G I c cos d / 2 2 I cos sin d d c 0 0 2 0 cos 1 I d d 2 I d c 1 c 0 0 1 2 I (1 ) d 2 I 2 3 0 1 0 c 2 0 c I 1 3 0 c 9 G(λ) for a Linear Limb Darkening Law Ic 0 1 cos Ic • Numerator of G y1 dy Ic R 2Ic0 y1 y1 dy 2I 1 cos R 0 0 c y1 y1 dy 0 2I 1 2Ic cos R R 0 0 c y1 1 0 0 2 2 2 2Ic 1 1 2 I R x y dy c 2 0 R L 2 1/ 2 10 G(λ) for a Linear Limb Darkening Law Ic 0 1 cos Ic • Analytical solution for second term in numerator A 2 y 2 1/ 2 1 2 y 2 1/ 2 2 dy y A y A arcsin 2 A • Second term is 2 01 1 2 2 1/ 2 2 2 2Ic y(R x y ) (R x )arcsin 2 2 R Ic0 2 (R x ) R 2 2 0 Ic 1 2 L 2 2 y1 R 2 x 2 y1 y 2 2 R x 0 11 G(λ) for a Linear Limb Darkening Law Ic 0 1 cos Ic 2 21 1 1 2 L L G L 1 3 2 1/ 2 2 1/ 2 2 c1 1 c 2 1 L L ellipse parabola 12 Grey atmosphere case: ε = 0.6 13 14 v sin i = 20 km s-1 v sin i = 4.6 km s-1 15 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 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 L=MRv 18 Angular Momentum – Mass Relation • Equilibrium with gravity = centripetal acceleration GM v 2 GM v 2 2 3 2 2 R R R R • Angular momentum for uniform density 2 2 L I k MR L MRv MR 2 • In terms of angular speed and density R 3 GM 2 GM R 2 1/ 3 L M M 2 / 3 4 / 3 M 5 / 3 1/ 3 M 5 / 3 1/ 6 • Density varies slowly along main sequence L M 5 / 3 19 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 Fastest Rotators • Critical rotation v crit 1/ 2 GM M / M sun 1 437 km s R R /Rsun • 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 Other Processes That Shape Lines • Macroturbulence and granulation http://astro.uwo.ca/~dfgray/Granulation.html 23 Star Spots Vogt & Penrod 1983, ApJ, 275, 661 HR 3831 Kochukhov et al. 2004, A&A, 424, 935 http://www.astro.uu.se/~oleg/research.html 24 Stellar Pulsation http://staff.not.iac.es/~jht/science/ Vogt & Penrod 1983, ApJ, 275, 661 25 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/encyclo pedia/P/P_Cygni_profile.html 26 FUSE spectra (Walborn et al. 2002, ApJS, 141,443) 27 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