Rotational Line Broadening Gray Chapter 18

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Rotational Line BroadeningGray Chapter 18

• Geometry and Doppler Shift
• Profile as a Convolution
• Rotational Broadening Function
• Observed Stellar Rotation
• Other Profile Shaping Processes

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Doppler Shift of Surface Element

• Assume spherical star with rigid body rotation
• Velocity at any point on visible hemisphere is

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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

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Radial velocity depends only on x
position.Largest at limb, xR.v equatorial
rotational velocity,v sin i projected
rotational velocity
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Flux Profile

• Observed flux is (R/D)2 F? where
• Angular element for surface element dA
• Projected element
• Expression for flux

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Assumption profile independent of position on
visible hemisphere
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Express as a Convolution
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G(?) for a Linear Limb Darkening Law

• Denominator of G

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G(?) for a Linear Limb Darkening Law

• Numerator of G

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G(?) for a Linear Limb Darkening Law

• Analytical solution for second term in
  numerator
• Second term is

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G(?) for a Linear Limb Darkening Law
?ellipse
?parabola
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Grey atmosphere case e 0.6
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v sin i 20 km s-1
v sin i 4.6 km s-1
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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

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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)

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L M R v
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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

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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

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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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Other Processes That Shape Lines

• Macroturbulence and granulationhttp//astro.uwo.c
  a/dfgray/Granulation.html

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Star Spots
Vogt Penrod 1983, ApJ, 275, 661
HR 3831Kochukhov et al. 2004, AA, 424,
935http//www.astro.uu.se/oleg/research.html
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Stellar Pulsationhttp//staff.not.iac.es/jht/sci
ence/
Vogt Penrod 1983, ApJ, 275, 661
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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
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FUSE spectra (Walborn et al. 2002, ApJS, 141,443)
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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)