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Rotational Line Broadening Gray Chapter 18

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Title: Rotational Line Broadening Gray Chapter 18


1
Rotational Line BroadeningGray Chapter 18
  • Geometry and Doppler Shift
  • Profile as a Convolution
  • Rotational Broadening Function
  • 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)
  • Radial velocity is
  • 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)
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