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Radiation Field Quantities and Transfer Details

Equation (11) conceals a variety of important issues concerning the treatment of the radiation field and the values which are printed in the various output files produced by xstar. In an effort to clarify this we present here a complete description of the various radiation field quantities which are used internally to xstar, and which are output to the user. In this subsection, all radiation fields are specific luminosity, $L_\varepsilon$, in units erg/s/erg for the continuum, and luminosity, $L_i$, in units erg/s for lines. We distiguish several different radiation fields. First, the radiation field used locally by xstar for the calculation of photoionization rates and heating, we denote $L_{\varepsilon}^{(1)}$. This is calculated during an outward iteration using the transfer equation:


\begin{displaymath}\frac{{\rm d}L_{\varepsilon}^{(1)}}{{\rm d}R}=-\kappa_{cont}(...
...{\varepsilon}^{(1)}
+ 4 \pi R^2 j_{\varepsilon}(R) \eqno{(12)} \end{displaymath}

with the boundary condition that $L_{\varepsilon}^{(1)}=L_{\varepsilon}^{(inc)}$ at the inner radius of the cloud. Here $\kappa_{cont}(\varepsilon)$ and $j_{\varepsilon}$ are the local continuum opacity and emissivity and $L_{\varepsilon}^{(inc)}$ is the incident radiation field at the inner edge of the cloud.

In addition we can define the various radiation fields of interest for use in fitting to observed data. These include the spectrum transmitted by a model, i.e. the radiation which would be observed if the incident radiation field were subject to absorption alone:


\begin{displaymath}L_\varepsilon^{(2)}=L_{\varepsilon}^{(inc)} e^{-\tau_{cont}^{(tot)}(\varepsilon)} \eqno{(13)} \end{displaymath}

where $\tau_{cont}^{(tot)}(\varepsilon)$ is the total optical depth through the model cloud due to continuum photoabsorption,


\begin{displaymath}\tau_{cont}^{(tot)}(\varepsilon)=\int_{R_{inner}}^{R_{outer}} \kappa_{cont}(\varepsilon) dR \eqno{(14)}\end{displaymath}

Also of interest is the total emitted continuum radiation in both the inward and outward directions, which is given by equations similar to (11):


\begin{displaymath}L_\varepsilon^{(3)}=\int_{R_{inner}}^{R_{outer}}{4\pi R^2 j_\...
...}^{(in)}(\varepsilon)}
P_{esc, cont.}^{(in)}(R)}dR \eqno{(15)} \end{displaymath}


\begin{displaymath}L_\varepsilon^{(4)}=\int_{R_{inner}}^{R_{outer}}{4\pi R^2 j_\...
...{(out)}(\varepsilon)}
P_{esc, cont.}^{(out)}(R)dR} \eqno{(16)} \end{displaymath}

where the escape probabilities in the inward and outward directions are $P_{esc, cont.}^{(in)}(R)=(1-C)/2$ and $P_{esc, cont.}^{(out)}(R)=(1+C)/2$, where $C$ is the covering fraction, specified as an input parameter, and $\tau_{cont}^{(out)}(\varepsilon)$ and $\tau_{cont}^{(out)}(\varepsilon)$ are the continuum optical depths in the inward and outward directions.

Line luminosities are calculated separately, one at a time, according to an equation analogous to equation (12):


\begin{displaymath}\frac{d L_i^{(1)}}{dR}=-\kappa_{cont}(\varepsilon) L_i^{(1)}
+ 4 \pi R^2 j_i(R) P_{esc,line}^{(in)} \eqno{(16)} \end{displaymath}


\begin{displaymath}\frac{d L_i^{(2)}}{dR}=-\kappa_{cont}(\varepsilon) L_i^{(2)}
+ 4 \pi R^2 j_i(R) P_{esc,line}^{(out)} \eqno{(17)} \end{displaymath}

where $L_i^{(1)}$ and $L_i^{(2)}$ are the luminosities of individual lines in the inward and outward directions, respectively. The escape probabilities in the inward and outward directions are calculated using $P_{esc., line}(\tau_{line})$ from equations (6)-(8) and $P_{esc., line}^{(in)}=(1-C) P_{esc., line}(\tau_{i}^{(in)})$ and $P_{esc, line}^{(out)}=(1-C) P_{esc., line}(\tau_{i}^{(out)})
+ C P_{esc., line}(\tau_{i}^{(out)}+\tau_{i}^{(in)})/2$, and $\tau_{i}^{(in)}$ and $\tau_{i}^{(out)}$ are the line scattering optical depths in the inward and outward directions:


\begin{displaymath}\tau_{i}^{(in)}(R)=\int_{R_{inner}}^R \kappa_i {\rm dR} \eqno{(18)}\end{displaymath}


\begin{displaymath}\tau_{i}^{(out)}(R)=\int_R^{R_{outer}} \kappa_i {\rm dR} \eqno{(19)}\end{displaymath}

.

and $\kappa_i$ is the line center opacity.

None of the continuum luminosities defined in equations (12)-(16) have the effects of lines included, either in emission or absorption. This is because lines scatter the radiation, while photoionization is true absorption. The effects of lines on the continuum can be added to the continuum for the purposes of comparing with observed spectra by binning the lines, i.e. we can calculate the binned specific luminosity and opacity:


\begin{displaymath}L_{line, \varepsilon}^{(in)}=\Sigma_{i \ni \mid\varepsilon_i ...
... \phi(\varepsilon-\varepsilon_i)}{\Delta \varepsilon} \eqno(20)\end{displaymath}


\begin{displaymath}L_{line, \varepsilon}^{(out)}=\Sigma_{i \ni \mid\varepsilon_i...
...\phi(\varepsilon-\varepsilon_i) }{\Delta \varepsilon} \eqno(21)\end{displaymath}


\begin{displaymath}\kappa_{line}(\varepsilon)=\Sigma_{i \ni \mid\varepsilon_i -\...
...\varepsilon}
\kappa_i \phi(\varepsilon-\varepsilon_i) \eqno(22)\end{displaymath}

where $\varepsilon$ and $\Delta\varepsilon$ are the energy and width, respectively, of the continuum bin closest to line $i$, and $\phi(\varepsilon-\varepsilon_i)$ is the profile function including the effects of broadening due to thermal Doppler motions, natural broadening, and turbulence.

Then we can define the total optical depth of the cloud


\begin{displaymath}\tau^{(tot)}(\varepsilon)=\int_{R_{inner}}^{R_{outer}}
(\kap...
...}(\varepsilon)+\kappa_{line}(\varepsilon)) {\rm dR} \eqno{(23)}\end{displaymath}

and the total transmitted specific luminosity


\begin{displaymath}L_\varepsilon^{(5)}=L_{\varepsilon}^{(inc)} e^{-\tau^{(tot)}(\varepsilon)} \eqno{(24)} \end{displaymath}

and the total emitted specific luminosity in the inward and outward directions:


\begin{displaymath}L_\varepsilon^{(6)}=L_\varepsilon^{(3)}+L_{line, \varepsilon}^{(in)} \eqno{(25)} \end{displaymath}


\begin{displaymath}L_\varepsilon^{(7)}=L_\varepsilon^{(4)}+L_{line, \varepsilon}^{(out)} \eqno{(26)} \end{displaymath}

The quantities $L_\varepsilon^{(inc)}$ $L_\varepsilon^{(5)}$, $L_\varepsilon^{(6)}$ and $L_\varepsilon^{(7)}$ are output in columns 2,3,4,5 of the file xout_spect1.fits. The quantities $L_\varepsilon^{(inc)}$ $L_\varepsilon^{(5)}$, $L_\varepsilon^{(3)}$ and $L_\varepsilon^{(4)}$ are output in columns 2,3,4,5 of the file xout_cont1.fits.

In fact, the lines should be included in the continuum which is responsible for the local ionization and heating of the gas, since they can contribute to these processes. So we define a modified version of equation (12):


\begin{displaymath}\frac{{\rm d}L_{\varepsilon}^{(1')}}{{\rm d}R}=-\kappa_{cont}...
...i(\varepsilon-\varepsilon_i) }{\Delta \varepsilon} \eqno{(27)} \end{displaymath}

$L_{\varepsilon}^{(1')}$ is the quantity which is used by xstar to calculate the local ionizing flux. This is the quantity which is conserved by xstar when it calculates heating=cooling.

The quantities $L_i^{(1)}$, $L_i^{(2)}$, $\tau_i^{(in)}$ and $\tau_i^{(out)}$ are output in columns 6,7,8,9 of the file xout_lines1.fits.

The quantities which contain the continuum only, before the lines are binned and added, are printed out to the log file, xout_step.log, when the print switch lpri is set to 1 or greater. Then they are in a table following the label 'continuum luminosities'. The quantities $L_\varepsilon^{(inc)}$, $L_\varepsilon^{(1')}$, $L_\varepsilon^{(3)}$, $L_\varepsilon^{(4)}$, $\tau_{cont}^{(in)}(\varepsilon)$ and $\tau_{cont}^{(out)}(\varepsilon)$ are output in columns 3,4,5,6,7 and 8. Many other useful quantities are output to the log file when lpri=1. This includes the quantities $L_i^{(1)}$, $L_i^{(2)}$, $\tau_i^{(in)}$ and $\tau_i^{(out)}$ are in columns 2-5 following the label 'line luminosities'


next up previous contents
Next: Energy Conservation Up: Recombination Continuum Emission and Previous: Continuum Transfer   Contents
Tim Kallman 2014-04-04