Rotational Line Broadening Gray Chapter 18 Geometry and

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

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Doppler Shift of Surface Element • Assume spherical star with rigid body rotation • Velocity at any point on visible hemisphere is 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 • Doppler shift is 4

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

Flux Profile • Observed flux is (R/D)2 Fν where • Angular element for surface element d. A • Projected element • Expression for flux 6

Assumption: profile independent of position on visible hemisphere 7

Express as a Convolution 8

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

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

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

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

Grey atmosphere case: ε = 0. 6 13

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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 • Angular momentum for uniform density • In terms of angular speed and density • Density varies slowly along main sequence 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 • 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