Transcript Stellar Rotation

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
^
^
^
x
y
z
 0 sin i cosi 
x
y
z
^
zsin i  ycosi x
^
xcosi y
^
xsin 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  xsin i
• Doppler shift is
 
0
c
vR 
0
c
xsin i
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Radial velocity depends
only on x position.
Largest at limb, x=R.
L 

0
Rsin i 
0
c
c
v = equatorial
rotational velocity,
v sin i = projected
rotational velocity
v sin i
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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

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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  

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Express as a Convolution


 1
G( )  
L


y1
I
c
dy /R
y1
I
cos  d
0
c
for   L
for   L

F R
  H(  ) G( )   H(    ) G()
Fc R

 H( ) G()
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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
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G(λ) for a Linear
Limb Darkening Law
Ic
0  1     cos 
Ic
• Numerator of G
y1
dy
0
I

2I
 cR c
y1
y1
dy
 2I 1     cos  R
0
0
c
y1
y1
dy
0
 2I 1     2Ic  cos 
R
R
0
0
c
y1
   2 1/ 2
1
0
0
2
2
2


 2Ic 1    1  

2

I
R

x

y
dy

c 
2

0 R
 L  

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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
2Ic
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

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G(λ) for a Linear
Limb Darkening Law
Ic
0  1     cos 
Ic
2 
    

  

21   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
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Grey atmosphere
case: ε = 0.6
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14
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(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
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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=MRv
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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
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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
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)
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Other Processes That Shape Lines
• Macroturbulence and granulation
http://astro.uwo.ca/~dfgray/Granulation.html
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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
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Stellar Pulsation
http://staff.not.iac.es/~jht/science/
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/encyclo
pedia/P/P_Cygni_profile.html 26
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)
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