rotational line broadening gray chapter 18 geometry and doppler shift profile as a convolution...

Rotational Line BroadeningGray Chapter 18

Geometry and Doppler ShiftProfile as a Convolution

Rotational Broadening FunctionObserved Stellar Rotation

Other Profile Shaping Processes

1

Doppler Shift of Surface Element

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

3

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v = Ω × R

=

x^

y^

z^

0 Ωsini Ωcosi

x y z

=

zΩsini − yΩcosi( ) x^

+ xΩcosi( ) y^

+ −xΩsini( ) z^

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€

vR = xΩsin i

€

Δλ =λ0

cvR =

λ 0

cxΩsini( )

5

Radial velocity depends only on x position.Largest at limb, x=R.

v = equatorial rotational velocity,v sin i = projected rotational velocity

€

ΔλL =λ 0

cRΩsini( ) =

λ 0

cv sin i( )

Flux Profile

• Observed flux is (R/D)2 Fν where

• Angular element for surface element dA

• Projected element

• Expression for flux

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Fν = Iν cosθ dω∫

€

dω = dAR2

€

dx dy = dA cosθ

€

Fν =IνR2 dx dy∫∫

Assumption: profile independent of position on visible hemisphere

7

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Fλ = H(λ − Δλ )Ic dx dy /R2∫∫

= H(λ − Δλ ) Icdy

Rd

Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

−y1

+y1

∫−R

+R

∫

y1 = R2 − x 2( )

1/ 2= R 1 −

Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1/ 2

Express as a Convolution

8

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G(Δλ ) =1

Δλ L

Ic−y1

+y1

∫ dy /R

Ic cosθ dω∫for Δλ ≤ Δλ L

0 for Δλ > Δλ L

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Fλ

Fc

= H(λ − Δλ ) G(−R

+R

∫ Δλ ) = H(λ − Δλ ) G(−∞

+∞

∫ Δλ )

= H(λ )∗G(λ )

G(λ) for a Linear Limb Darkening Law

• Denominator of G

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IcIc

0 =1 −ε +ε cosθ

€

Ic cosθ dω =0

π / 2

∫∫ Ic cosθ sinθ dθ dφ μ = cosθ( )0

2π

∫

= −1

0

∫ Icμ dμ dφ = 2π Ic0

1

∫0

2π

∫ μ dμ

= 2π Ic0 (1−ε )μ +εμ 2 dμ =

0

1

∫ 2π Ic0 1−ε

2+ε

3

⎡ ⎣ ⎢

⎤ ⎦ ⎥

= π Ic0 1−

ε

3

⎡ ⎣ ⎢

⎤ ⎦ ⎥


• Numerator of G

10

€

IcIc


€

Icdy

R= 2Ic

0 2Ic0 1−ε( ) +ε cosθ[ ]

0

y1

∫ dy

R−y1

+y1

∫

= 2Ic0 1−ε( )

y1

R+ 2εIc

0 cosθdy

R0

y1

∫

= 2Ic0 1−ε( ) 1−

Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1/ 2

+ 2εIc0 1

R2 R2 − x 2 − y 2 dy0

y1

∫


• Analytical solution for second term in numerator

• Second term is

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€

IcIc


€

2εIc0 1

2

1

R2 y(R2 − x 2 − y 2)1/ 2 + (R2 − x 2)arcsiny

R2 − x 2

⎡

⎣ ⎢

⎤

⎦ ⎥0

y1

=εIc

0

R2 (R2 − x 2)π

2

⎡ ⎣ ⎢

⎤ ⎦ ⎥ y1 = R2 − x 2

( )

=π

2εIc

0 1−Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

A2 − y 2( )

1/ 2dy =

1

2y A2 − y 2( )

1/ 2+ A2 arcsin

y

A

⎡

⎣ ⎢

⎤

⎦ ⎥∫


12

€

IcIc


€

∴G Δλ( ) =

2 1 −ε( ) 1−Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1/ 2

+π

2ε 1 −

Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

πΔλ L 1 −ε

3

⎛

⎝ ⎜

⎞

⎠ ⎟

= c1 1 −Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1/ 2

+ c2 1 −Δλ

Δλ L

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ellipse

parabola

13

Grey atmosphere case: ε = 0.6

15

v sin i = 20 km s-1

v sin i = 4.6 km s-1

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)

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18

L = M R v

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 sequence19

€

GM

R2 =v 2

R⇒

GM

R3 =v 2

R2 =ω 2 ⇒ ω ∝ ρ

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L = MRv = MR2ω L = Iω = k 2MR2ω( )

€

R3 =GM

ω 2 ⇒ R =GM

ω 2

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3

L ∝ M M 2 / 3ω −4 / 3ω = M 5 / 3ω −1/ 3 ∝ M 5 / 3ρ −1/ 6

€

L ∝ M 5 / 3

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

€

vcrit =GM

R= 437

M /Msun

R /Rsun

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

km s−1

Other Processes That Shape Lines• Macroturbulence and granulation

http://astro.uwo.ca/~dfgray/Granulation.html

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Star Spots

24

Vogt & Penrod 1983, ApJ, 275, 661

HR 3831Kochukhov et al. 2004, A&A, 424, 935http://www.astro.uu.se/~oleg/research.html

Stellar Pulsationhttp://staff.not.iac.es/~jht/science/

25Vogt & Penrod 1983, ApJ, 275, 661

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)

26http://www.daviddarling.info/encyclopedia/P/P_Cygni_profile.html

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FUSE spectra (Walborn et al. 2002, ApJS, 141,443)

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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rotational line broadening gray chapter 18 geometry and doppler shift profile as a convolution...

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