Spectral lines analysisRotational velocity and velocity fields
Spring School of Spectroscopic Data Analyses8-12 April 2013
Astronomical Institute of the University of WroclawWroclaw, Poland
Giovanni Catanzaro INAF - Osservatorio Astrofisico di
Catania
๐ฅ
๐ฆ
๐
equator
Projected rotational velocity
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๐ฃ๐๐ sin ๐
Because the Doppler effect we can see only the component of the equatorial velocity parallel to the line of sight
๐ฅ
๐ฆ
๐equator ๐ฃ๐๐
Special case: i=90ยบAll rotational velocity is parallel to line of sight: star appears to rotate with veq
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๐ฅ
๐ฆ
equ ator
๐ฃ๐๐
Special case: i=0ยบAll rotational velocity is perpendicular to line of sight: star appears not rotate
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Rotation shapes line profile
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Rotational profile
๐บ ( ฮ๐ )=๐1[1โ( ฮ๐ฮ ๐๐ฟ)2]
12+๐2[1โ( ฮ๐ฮ ๐๐ฟ)
2] limb darkening
Rotational profile
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Profile fitting for v sin i
Observed spetrum with several synthetics overimposed. Each synthetic spectrum was computed for different value of rotational velocity.
๐ =๐ฮ ๐=57000
Instrumental profile
๐ท (๐ )=๐ผ ( ๐ )โ๐น (๐)
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Importance of Resolution
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Example: FeII 5316.615 ร
log๐ ๐น๐
๐๐๐๐ก=โ4. 48 log
๐ ๐น๐
๐๐๐๐ก=โ4. 3 0
Teff = 7000 KLog g = 4.00HERMESR = 80000
Fourier analysis
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~0,007
๐ (๐ )=๐ (๐ )โ๐(๐ )
Limb darkening
Limb darkening shifts the zero to higher frequency
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The limb of the star is darker so these contribute less to the observed profile. You thus see more of regions of the star that have slower rotation rate. So the spectral line should looks like that of a more slowly rotating star, thus the first zero of the transform move to lower frequencies
๐น(๐
)
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Velocity fields
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Motions of the photospheric gases introduce Doppler shifts that shape the profiles of most spectral lines
Turbulence are non-thermal broadening
We can make two approximations:โข The size of the turbulent elements is large compared to
the unit optical depth Macroturbulent limit
โข The size of the turbulent elements is small compared to the unit optical depth Microturbulent limit
Velocity fields are observed to exist in photospheres oh hot stars as well as cool stars.
MacroturbulenceTurbulent cells are large enough so that photons remain trapped in them from the time they are created until they escape from the starLines are Doppler broadened: each cell produce a complete spectrum that is displaced by the Doppler shift corresponding to the velocity of the cell.
The observed spectra is: In = In0 * Q(Dl)
In0 is the unbroadened profile and Q(Dl) is the macroturbulent velocity distribution.
What do we use for Q?
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Radial-Tangent model
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We could just use a Gaussian (isotropic) distribution of radial components of the velocity field (up and down motion), but this is not realistic:
Rising hot material
Cool sinking material
Horizontal motion
Convection zone
If you included only a distribution of up and down velocities, at the limb these would not alter the line profile since the motion would not be in the radial direction. The horizontal motion would contribute at the limb
Radial motion โ main contribution at disk center
Tangential motion โ main contribution at limb
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Assume that a certain fraction of the stellar surface, AT, has tangential motion, and the rest, AR, radial motion
Q(Dl) = ARQR(Dl) + ATQT(Dl)
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The R-T prescription produces a different velocity distribution than an isotropic Gaussian.
If you want to get more sophisticated you can include temperature differences between the radial and tangential flows.
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Macro
10 km/s
5 km/s
2.5 km/s
0 km/s
Pixel shift (1 pixel = 0.015 ร )
Rel
ativ
e In
tens
ity
Effect of MacroturbulenceIt does not alter the total absorption of the spectral lines, lines broadened by macroturbulence are also made shallower.
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At low rotational velocities it is difficult to distinguish between them: red line is computed for v sini = 3 km/s, x = 0 km/sblue line for v sini = 0 km/s and x = 3 km/s R
elat
ive
Flux
Pixel (0.015 ร /pixel)
Am
plitu
de
Frequency (c/ร )
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There is a tradeoff between rotation and macroturbulent velocities. You can compensate a decrease in rotation by increasing the macroturbulent velocity.
While, In the wavelength space the differences are barely noticeable, in Fourier space (right), the differences are larger.
Example: b Comae (Gray et al., 1996)
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d(s) individual linesh(s) thermal profilei(s) instrumental profile
d(s) averaged and divided by i(s)
Microturbulence
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Contrarly to macroturbulence, we deal with microturbulence when turbulent cells have sizes small compared to the mean free path of a photon.
In this case the velocity distribution of the cells molds the line profile in the same way the particle distribution does.
๐ผ=๐ผ โฒโ๐ (๐ฃ)
Line absorption coefficient without microturbulence
Particles velocity distribution (gaussian)
๐ (๐ฃ )๐๐ฃ= 1
๐12 ๐
๐โ( ๐ฃ๐ )
2
๐๐ฃThe convolution of two gaussian is still a gaussian with a dispersion parameter given by:
๐ฃ2=๐ฃ 02+๐2
ฮ ๐๐ท=๐๐ ( 2๐พ๐
๐ +๐2)12
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Landstreet et al., 2009, A&A, 973
Typical values for x are 1-2 km s-1 , small enough if compared to the other components of the line broadening mechanism.
It is a very hard task to attempt the direct measurement of x by fitting the line profile. Very high resolution (>105), high SNR spectra and slow rotators stars (a few km s-1) are needed.
Blackwell diagrams
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1998, A&A, 338, 1041
FeII
CrII
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Catanzaro & Balona, 2012, MNRAS, 421,1222
Other type of diagram: from a set of spectral lines, we require that the inferred abundance not depend on EW
Example: HD27411, Teff = 7600 ยฑ 150, log g = 4.0 ยฑ 0.1 71 lines FeI
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Thanks for your attention