Radiative transfer and line formation

Adapted from: http://www.astro.uio.no/ita/undervisning/ast2120/xtransp.html

1 Literature

Carroll & Ostlie Sections 9.3 - 9.4 - 9.5 in An introduction to modern astrophysics will be helpful in understanding this computer lab.

2 Introduction

In this computer exercise you will study spectral line formation in a one-dimensional model atmosphere. With the use of a computer program you can vary the parameters that determine the emergent line profile. The Voigt profile (the absorption profile of the spectral line), the total opacity, and the source function are plotted together with the emergent intensity of the line.

Voigt Profile (Damping and Doppler profiles "hand-drawn."

Line source function, Sl. Nl and Nu represent the number of atoms with electrons in the lower energy level and upper energy level, gu and gl are the corresponding statistical weights, ψ(ν) denotes the frequency dependence of the spontaneously emitted radiation, and φ(ν) takes care of the frequency dependence of the absorption and stimulated emission. Sl = Bν(T) in those special cases where equilibrium holds.

Emergent intensity of the line

The idea is to play around with the interactive software and to answer specific questions relating to radiative transfer and line formation.

3 Procedure

The programs are written in the IDL programming language. You do not need to know the language, just to work on a machine with IDL installed.

• Save the IDL program xtransp.sav and s.idlsave (e.g. click on right mouse button and select Save Link Target As...).
• Run the program in the IDL Virtual Machine.
  • Enter:
    idl -vm=xtransp.sav
    in a terminal window (in the directory you have saved the program).
  • Click anywhere in the IDL Virtual Machine splash screen to dismiss the window and run the xtransp.sav file.

The program uses sliders to change the values of the different variables, which are explained very briefly in a window invoked by the Help button and in the next section. You will get messages, % Program caused arithmetic error: Floating underflow, in your terminal window when using xtransp which is normal and can be ignored. [NOTE: I couldn't get the help feature to work. --Ana]

You may want to use the Save button to save the screen output as a Postscript file in your directory. The file will be saved with the file name: xtransp_ + five numbers + .ps. The numbers give the five variables as set when you saved the plot.

To quit the program, click on the Done button.

4 Line formation

The five variables of interest here are: the Voigt damping parameter, a, which determines the shape and maximum value of the Voigt function; the continuous and line opacities relative to the continuum absorption coefficient at the reference wavelength 500 nm (the assumption we worked on in the opacity project); the cosine of the angle between the ray and the normal of the atmosphere, mu, and the plot range on the x-axis in units of the number of Doppler widths from the line center.

The source function button, Sny, enables you to choose between two different source functions, one for LTE and one for the calcium line. You will see that there is a definite difference in the shape of the Ca line when the source function for the line is used.

As a first approximation in spectral analysis, we do assume that the source function is given by the Planck function, B_n (T). If we chose not to, we'd have to calculate the population of every atomic level in each atomic species, as indicated above. These species and their excitation levels affect each other through the radiation field. It becomes exceedingly complex; however, a number of students have tried for their PhD theses.

Begin by playing around to get the feeling for the different parameters. Pay attention to the automatic scales. [Image link.]

1. In particular, take a look at the source function, S_ν compared with different values of log(τ₅₀₀). Correlating the value of the optical depth with relative physical depths in the star, does the x-axis run from high-up in the stellar atmosphere (maybe through part of the chromosphere) to deep in the atmosphere, or from deep-to-high-up?

3. Explore the plot range. How is xmax defined?

4. Explore the mu-dependence of the Ca spectral line.

1. Which parts of the Sun are observed with the different values of mu?
2. How does this explain limb darkening?
3. As mu goes to 0 degrees, the central peak under LTE of the calcium line increases until only the emission core remains. What does this say about the source function?
4. The emission core is due to the effect of the chromosphere. What does this say about the temperature of the chromosphere relative to the temperature of the photosphere?

5. Explore the damping constant.

1. Explain what a Voigt profile is and how its maximum and its shape are determined by the damping constant a.
2. What kind of profile results for small and large values of a?

When you are finished with this section, set the damping constant to log a = -3.

6. Explore the extinction coefficients, a, by working with the line and continuous opacities.

[If you cross-reference Böhm-Vitense, Carroll & Ostlie, and Gray, you will find each has a different definition for extinction coefficient. In these instructions, this is the first time it is called this. Gray states kappa_nu is really an extinction coefficient. We have kappa_nu entering here under the definition of the optical depth, tau. For these questions, however, work with changing the ratio of the line and continuum opacities to the opacity at 500 nm: αl /α500 and αc /α500.)



1. How are the damping profile and the extinction coefficient related? (Also, look at the Voigt parameters besides working with the slider.)
2. What does the extinction coefficient profile look like for small and for large line opacities?
3. Try to explain why the extinction coefficient appears different in the two cases.
4. What happens if you increase the continuous opacity?

Examine different cases by varying the line and continuous opacities. Specifically, compare the effects of having a strong line to a weak line.

Eddington-Barbier relation: For a plane-parallel gray atmosphere in LTE, the (constant) value of the radiative flux is equal to pi times the source function evaluated at an optical depth of 2/3: Frad = π S (τν = 2/3). In other words, the radiative flux received from the surface of the star is determined by the value of the source function at τν = 2/3 (Carroll & Ostlie, 2nd ed., p. 280). In this exercise, be sure that Sny is set to LTE.

1. How does the line opacity and the continuous opacity relate to the emergent profile for strong lines, assuming Eddington-Barbier? What is a strong line?
2. How does the line opacity and the continuous opacity relate to the emergent profile for weak lines, assuming Eddington-Barbier? What is a weak line?

7. Explore the emergent line profile, I_ν, in the LTE case.

In the following, you can keep the damping constant, log(a), the continuous opacity log(α_c / α₅₀₀) = 0 and xmax constant.

1. Explain what you see in the different panels (such as the value of α_tot / α₅₀₀ and the values of τ = τ_tot and τ_c).
2. What happens if you increase the line opacity moderately? Why? What does increasing α_l correspond to physically?
3. Increase the line opacity even more. Do so until an emission peak appears. What is happening? Try to explain the situation with the Eddington-Barbier approximation.
4. When is this approximation valid? What would you expect according to Eddington-Barbier?
5. See what happens if you put τ_tot = 1 (the unit optical depth in the line) in the minimum of the source function. Are there other, more correct ways of explaining it?
6. Can you manipulate the opacities in order to get
• a pure emission line?
• an emission line with absorption in the center?

8. Explore the emergent line profile in the Ca case.

Change the source function to the one for the calcium line. Also in the following, you should let the damping constant and xmax be constant.

Start off by putting log(α_c /α₅₀₀) = 0.

1. Let the line opacity grow. What happens? Give an explanation. Can one observe this phenomenon in reality? For an example, see section 9, and abstracts given at the end of this exercise.
2. Repeat a but for fundamentally different values of the continuous opacity. Describe or save images of at least 3 different cases you found.
3. Can you manipulate the opacities in order to get
• a pure emission line? State the values of the opacities.
• an emission line with absorption in the center? State the values of the opacities.

9. Discuss the source function.

1. Give a qualitative explanation for the LTE source function. Compare it to the source function for the calcium line.
2. What is the difference? Why?



10. Here is an actual spectrum of the calcium K and H lines for the Sun, showing definite emission cores in the 2 lines and very broad wings. Qualitatively, working with all parameters except for mu, reproduce the shape of the wings and size and shape of the central emission peak. Either list your values or save an image of your final result. You should also try toggling Sny between LTE and Ca.




 

11. Summarize what you have learned about how absorption and emission lines are formed.

Use of the calcium emission core in astrophysics:

A&A 401, 997-1007 (2003)
The Wilson-Bappu effect: A tool to determine stellar distances
G. Pace, L. Pasquini and S. Ortolani

Abstract
Wilson & Bappu (1957) have shown the existence of a remarkable correlation between the width of the emission in the core of the K line of Ca II and the absolute visual magnitude of late-type stars.

Here we present a new calibration of the Wilson-Bappu effect based on a sample of 119 nearby stars. We use, for the first time, width measurements based on high resolution and high signal to noise ratio CCD spectra and absolute visual magnitudes from the Hipparcos database.

Our primary goal is to investigate the possibility of using the Wilson-Bappu effect to determine accurate distances to single stars and groups.

The result of our calibration fitting of the Wilson-Bappu relationship is $M_V=33.2{-}18.0 \log W_0$, and the determination seems free of systematic effects. The root mean square error of the fitting is 0.6 mag. This error is mostly accounted for by measurement errors and intrinsic variability of W0, but in addition a possible dependence on the metallicity is found, which becomes clearly noticeable for metallicities below [Fe/H] ~ -0.4$. This detection is possible because in our sample [Fe/H] ranges from -1.5 to 0.4.

The Wilson-Bappu effect can be used confidently for all metallicities not lower than ~ - 0.4, including the LMC. While it does not provide accurate distances to single stars, it is a useful tool to determine accurate distances to clusters and aggregates, where a sufficient number of stars can be observed.

We apply the Wilson-Bappu effect to published data of the open cluster M 67; the retrieved distance modulus is of 9.65 mag, in very good agreement with the best distance estimations for this cluster, based on main sequence fitting.

O. Engvold and Ø. Elgaroøy, "The Wilson-Bappu relationship - a barometric effect" in Cool Stars, Stellar Systems, and the Sun, Vol. 291 (Springer Berlin / Heidelberg, 1987)

Abstract
Optically thick lines in the UV spectra of late type stars obey a Wilson-Bappu type relationship. Optically thin lines reveal no width-luminosity relationship. Consequently, there is no systematic variation in Doppler broadening with stellar luminosity.