Spectral lines analysis

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

𝑥

𝑦

𝑖

equator

Projected rotational velocity

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𝑣𝑒𝑞 sin 𝑖

Because the Doppler effect we can see only the component of the equatorial velocity parallel to the line of sight

𝑥

𝑦

𝑖equator 𝑣𝑒𝑞

Special case: i=90ºAll rotational velocity is parallel to line of sight: star appears to rotate with veq


𝑥

𝑦

equ ator

𝑣𝑒𝑞

Special case: i=0ºAll rotational velocity is perpendicular to line of sight: star appears not rotate


Rotation shapes line profile


Rotational profile

𝐺 ( Δ𝜆 )=𝑐1[1−( Δ𝜆Δ 𝜆𝐿)2]

12+𝑐2[1−( Δ𝜆Δ 𝜆𝐿)

2] limb darkening

Rotational profile


Profile fitting for v sin i

Observed spetrum with several synthetics overimposed. Each synthetic spectrum was computed for different value of rotational velocity.

𝑅=𝜆Δ 𝜆=57000

Instrumental profile

𝐷 (𝜆 )=𝐼 ( 𝜆 )∗𝐹 (𝜆)


Importance of Resolution


Example: FeII 5316.615 Å

log𝑁 𝐹𝑒

𝑁𝑇𝑜𝑡=−4. 48 log

𝑁 𝐹𝑒

𝑁𝑇𝑜𝑡=−4. 3 0

Teff = 7000 KLog g = 4.00HERMESR = 80000

Fourier analysis


~0,007

𝑑 (𝜎 )=𝑖 (𝜎 )∗𝑔(𝜎 )

Limb darkening

Limb darkening shifts the zero to higher frequency


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

𝐹(𝜆

)

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Velocity fields


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:• The size of the turbulent elements is large compared to

the unit optical depth Macroturbulent limit

• 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?


Radial-Tangent model


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 → main contribution at disk center

Tangential motion → 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 Å)

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 Å/pixel)

Am

plitu

de

Frequency (c/Å)


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)


d(s) individual linesh(s) thermal profilei(s) instrumental profile

d(s) averaged and divided by i(s)

Microturbulence


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.

𝛼=𝛼 ′∗𝑁 (𝑣)

Line absorption coefficient without microturbulence

Particles velocity distribution (gaussian)

𝑁 (𝑣 )𝑑𝑣= 1

𝜋12 𝜉

𝑒−( 𝑣𝜉 )

2

𝑑𝑣The convolution of two gaussian is still a gaussian with a dispersion parameter given by:

𝑣2=𝑣 02+𝜉2

Δ 𝜆𝐷=𝜆𝑐 ( 2𝐾𝑇

𝑚 +𝜉2)12


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


1998, A&A, 338, 1041

FeII

CrII


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 ± 150, log g = 4.0 ± 0.1 71 lines FeI


Thanks for your attention

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Spectral lines analysis