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Important Notes on Mtables

If you are using mtables with variable abundances, then you will likely get totally unphysical results unless you note the following.

The mtable models are constructed using two kinds of parameters: interpolated parameters and additive parameters. Interpolated parameters are treated as one might expect: if the model is represented by a vector $M_i(x_j)$, corresponding to the model flux at various energies $\varepsilon_i$ and stored at various values of the free (intepolated) parameter $x_j$, then for a value of the parameter not on the tabulated grid xspec calculates the model value as


\begin{displaymath}M_i(x)=\Sigma_j M_i(x_j)\omega_j\end{displaymath}

where $\omega_j$ are suitably chosen weights for, eg. linear interpolation. Ths is in contrast to the treatment of additive parameters: for an mtable, xspec needs the value of the model tabulated at only two values of the free additive parameter $y$, 0 and $y_{max}$. Then the model is calculated for some arbitrary $y$ as


\begin{displaymath}M_i(y)=(M_i(y_{max})-M_i(0)){{y}\over{y_{max}}}+M_i(0)\end{displaymath}

i.e. it is assumed that the model scales linearly with the value of the additive parameter $y$. This formalism was developed for emission models, where emissivities might be expected to scale linearly with elemental abundance. In the case of absorption models, the value of $M$ which xspec uses is a transmission coefficient, and this does not scale linearly with abundance. Rather, the transmission coefficient is related to the optical depth by:


\begin{displaymath}M_i(y)=e^{-\tau_i(y)}\end{displaymath}

and the optical depth $\tau_i(y)$ does scale (approximately) linearly with abundance. For this reason, xspec has incorporated the etable models, in which the model builder supplies optical depths rather than transmission coefficient, linear interpolation is used to calculate the optical depth for a given abundance, and then the transmission coefficient is calculated using the above expression. The results:

(1) If you use an mtable calculated by xstar2xspec using varable abundances (i.e. any additive parameters allowed to vary) in xspec using the `model mtablexout_mtable.fits' command, you will get unphysical results. It is easy to see that if $M_i(0)$ is non-zero (which it often is owing to 0 abundances) then you will never get deep absorption.

(2) If so, you should use the `model etablexout_mtable2.fits' command in xspec instead. Entering parameter, etc., in xspec is the same for etables as for mtables.

(3) BUT, the etable itself is supposed to be filled with optical depths, not transmission coefficients. The tables made by xstar2xspec are transmission coefficients, and they need to be converted. This is a simple transformation: replace the contents of every cell of the model extension of the mtable file ($x_{ij}$) with -ln($x_{ij}$). This can be done various ways. One way is to use the calculator inside of the ftools gui routine xv, one column at a time. This has been done for the various precalculated models on the website, and these are stored with the names xout_mtable2.fits.

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Some Specific Examples, and a Discussion of the Analytic Model warmabs

When fitting to absorption spectra it is important to be aware of the inherent limitations of xstar when applied to the fitting of absorption. There are two distinct reasons for this: spectral resolution effects, and interpolation effects. In order to illustrate these effects, we reproduce here a discussion between TK and P. Oneill of Imperial College in which these questions are raised, and some of the issues are discussed.

Question:

I've been using some of the sample XSPEC models from the current version of XSTAR to model a warm absorber.

I've been using the grid18 models with the abundances fixed at their default values. I found that using the XSPEC model "mtablexout_mtable.fits*powerlaw" gave different results (less absorption) than when I used "etablexout_mtable_2.fits*powerlaw". I thought that these should provide the same results, so long as the abundances are not changed. Am I using these tables correctly, or is there perhaps a problem somewhere?

Answer:

The problem is that variable abundances, they are treated as additive parameters in xspec tables, really can't be treated accurately using either type of table. If you use an mtable, then the transmittivity in a bin, i, at a given ionization parameter and column, is:


\begin{displaymath}
M_i=M_i^0+\Sigma_j x_j M_i^j
\end{displaymath} (7.3)

where $M_i^0$ is the 'zero abundance' version of the model at that ionization parameter and column, $x_j$ is the abundance of element j, and $M_i^j$ is the model calculated with that element set to unity (relative to cosmic). The first problem with this approach is that tranmission doesn't scale this way with abundance. If you double the abundance of an element, you should double the optical depth. Doubling the transmittivity has both the wrong qualitative and quantitative behavior. But an even bigger problem with this is that the whole thing, $M_i$, multiplies your continuum. It is almost guaranteed to bigger than 1 if some of the $x_j$ are non-zero. So this is totally the wrong formulation to use when the abundances vary.

In the case of an etable the transmittivity is:


\begin{displaymath}
M_i=exp(-(E_i^0+\Sigma_j x_j E_i^j))
\end{displaymath} (7.4)

where $E_i^0=-{\rm ln}(M_i^0)$ etc.

This is better, since it predicts the correct dependence on abundance, i.e. that the optical depth scales approximately linearly with abundance.

But it is important to remember that what you are trying to simulate is a 'real' photoionized gas with some abundance set $x_j$ that fits your data. What you are using to fit to your data are $M_i^0$, which in the case of xstar are models calculated with only hydrogen and helium, and the set of $M_i^j$, which are supposed to be calculated with pure element j.

The first problem is that the cooling, and therefore the temperature, depends on the element abundance in the gas, and this is not linear. The temperature couples back into the ionization balance, and that affects the opacity, transmittivity, etc.

The second problem is that, while it is straightforward to calculate a model with no elements heavier than H and He, it is not straightforward to calculate a model with pure iron, or calcium, say. That's because H and He havs nice simple cooling properties at low temperature, and their opacity is simple, etc. Pure iron models at best are likely to be very different than H+He+Fe models, or from a cosmic mix. So what I do is make the $M_i^j$ using H+He+element j in cosmic ratios. But the problem with this is that then the opacity due to H and He are present in every $M_i^j$, and $E_i^j$, particularly at low ionization parameters. So if you sum over cosmic abundances using an etable you are including the H and He opacity with every one of the $E_i^j$. At high ionization parameter (log($\xi)>$1?) this should not be a problem because H and He are ionized sufficiently that they do not contribute to the opacity. But at low ionization parameter, the model spectra calculated using etables will have significantly more low energy opacity due to the multiple counting of H and He than they would in a real xstar model with the corresponding parameters.

The strategy to use at low ionization parameters therefore is to use the etable to find a reasonably good fit to the spectrum with xspec, and then run xstar2xspec and make a very small table, i.e. just 1 column and 2 ionization parameters (xspec needs tables to have at least 2 entries), with constant abundances, set to the values which come from the fit. Constant abundance grids (such as grid19) when used as mtables do not suffer from any of these problems.

I attach 4 figures to illustrate these problems. All use the mtable or etable from grid18. All are plotted between 0.6 and 0.7 keV, power law continuum, index=1, norm=1, log($\xi $)=1. The continuum level at .6 keV is 1.66.

Figure 1: figure 1: cosmic abundances, mtable
\begin{figure}
\epsfxsize =5.6in % narrow the plot\epsfysize =7.0in % shorten the plot
\epsffile{warmabsfig1.epsi}\end{figure}

Figure 2: figure 2: cosmic abundances, etable
\begin{figure}
\epsfxsize =5.6in % narrow the plot\epsfysize =7.0in % shorten the plot
\epsffile{warmabsfig2.epsi}\end{figure}

Comparing these shows the big problem with mtables with variable abundances: they give a transmittivity which is $>$1. In this case, the transmittivity is $>$10.

Figure 3: figure 3: oxygen=1, other metals=0, etable
\begin{figure}
\epsfxsize =5.6in % narrow the plot\epsfysize =7.0in % shorten the plot
\epsffile{warmabsfig3.epsi}\end{figure}

Figure 4: figure 4: oxygen=1, other metals=0, mtable
\begin{figure}
\epsfxsize =5.6in % narrow the plot\epsfysize =7.0in % shorten the plot
\epsffile{warmabsfig4.epsi}\end{figure}

Here again, the problem is that even the pure oxygen mtable has the transmittivity of the 'zero abundance' model, $M_i^0$, which really has pure H and He and therefore has unit transmittivity. The line never can go black. The only model which is nearly correct is 3, but here again you have the problem of multiple counting of the H and He opacity. But at this high ionization parameter that is negligible. Anyway, I hope you understand the problem.

It is straightforward to use xstar2xspec to calculate grids without variable abundances. THEN you can use mtables, and there should be none of these problems.

Question:

I have made some comparisons between vturb=0 models and those with vturb=100. I find that vturb=0 gave very different results (much less absorption) to the vturb=100 grid18 etable. I hadn't expected to see the large difference with a change from vturb=100 to vturb=0. Is this expected behaviour?

Answer:

vturb can make a significant difference in the results of xstar modeling of absorption spectra, and the reason is at least partially due to numerics. In calculating the energy dependent opacity Xstar simply evaluates the profile function for the line at each grid energy. So if the line width is less than, or comparable to, the xstar grid spacing, then maximum depth of a given absorption line in the mtable may be very different from the true depth at line center. Obviously this problem is more severe for small vturb, i.e. narrow lines. The total line equivalent width does not depend on vturb, but the numerics do not reflect that unless the line is broader than the grid resolution. And the limiting resolution is the internal xstar grid spacing, which corresponds to 420 km/s, since this is what is used in constructing the table. This is another flaw in the use of tables for fitting to spectra.

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There is now subroutine which calculates xstar warm absorber (called 'warmabs') and warm emitter (called 'photemis') spectra and which can be called as an xspec `analytic' model. This calculates absorption spectra `on the fly', and so will evaluate the profile function on whatever grid you use in xspec. So you can choose to resolve all the lines if you want, using the 'dummyrsp' command with a very large number of bins. It is available on the xstar web site, look for the link under `XSTAR news'. It also does not suffer from the problems discussed previously about the approximations inherent in variable abundance. It calculates the opacity directly from a stored library of level populations calculated for a generic power law grid.

One drawback of this routine is that it brute-force calculates Voigt profiles and opacity for all the lines and bound-free continua which are above a certain threshold in strength, and so it is time-consuming. Currently calls to this routine can take up to 30 seconds on a slow or busy machine. It will be much faster on a faster machine, or if there are few elements with non-zero abundances, or if only a narrow spectral range is of interest, or if the ionization parameter is frozen.

Another drawback to this routine is that the current version assumes constant ionization throughout a slab of given ionization parameter and column. Real slabs will have lower mean ionization in the deeper, shielded regions, and this lower ionization material will absorb more efficiently. This means that the current warmabs assumption results in an underestimate of the opacity, particularly at low energies, for slabs with large column. This will be remedied in future versions of this routine.

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Yet another issue which deserves mention is that of interpolation and grid spacing in the interpolated parameters as implemented by xspec. Many of the available xstar grids were calculated with models spaced 1 decade apart in column. This has the potential to lead to inaccuracies in the model spectrum calculated by xspec, since it will interpolate between grid points. Sparsely gridded models may miss important details. Grids 19b and 19c provide an example.

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Question:

I have compared the grids with the warmabs analytical model. These are in firgure 5. 19c is the solid line, warmabs is the dashed line, and 19b is the dotted line. The model here is the same model that I used to fit the data using 19c (ie, it has 1.4e20 of wabs also). I hope you can get some info from this. The parameters used for these models were: log$\xi $=1.27, N=5.89e22.

Figure 5: figure 5: Comparison of grid19b, 19c, and warmabs
\begin{figure}
\epsfxsize =5.6in % narrow the plot\epsfysize =7.0in % shorten the plot
\epsffile{warmabsfig5.epsi}\end{figure}

Answer:

OK, so what I see from your plot comparing the models is that:

1) grid19b and grid19c give qualitatively different results near the best fit parameter values in the strength of the absorption near the K edge of oxygen. This seems to me to be clear evidence for the errors introduced by interpolation in a sparsely gridded table in column, as I suggested.

2) grid19c and warmabs agree well at energies above .8 keV, except for the lines. This seems to me to be a comforting check on the validity of warmabs.

3) there is disagreement in the strength of the absorption below 0.8 keV. We decided that the assumption of a constant ionization slab made in warmabs would tend to underestimate the absorption, since the ionization balance would be lower in the shielded parts of a real slab, hence more absorption. This is the sense of the disagreement, and so seems to make sense. Of course, there is still some interpolation involved in the use of grid19c, since your best fit values for N and $/xi$ are not precisely on the grid values. I have compared warmabs and grid19c for the nearest grid values, and the effect of interpolation does not appear to change the sense of the disagreement. So this is a strong argument for fixing up warmabs so that it does not make the uniform slab assumption.

4) There is also disagreement between grid19c and warmabs about the strength of the lines. I think this is due to the effect of energy binning: Your energy grid must be rather coarse, i.e.R=E/Delta(E) 100. If you make the same plot with a grid resolution of R=10000, things look better. I attach these plots: figure 6=warmabs, figure 7=grid19c. But there is still disagreement, and this I think is because the grid19c results have the intrinsic xstar internal resolution, which is much less than 10000, and so information is lost about the line profiles, while warmabs calculates the lines specifically for the resolution requested, and so should be more accurate.

Figure 7: figure 6: warmabs model, same parameters as figure 5, R=10000
\begin{figure}
\epsfxsize =5.6in % narrow the plot\epsfysize =7.0in % shorten the plot
\epsffile{warmabsfig6.epsi}\end{figure}

Figure 8: figure 7: grid19c, same parameters as figure 5, R=10000
\begin{figure}
\epsfxsize =5.6in % narrow the plot\epsfysize =7.0in % shorten the plot
\epsffile{warmabsfig7.epsi}\end{figure}


next up previous contents
Next: Notes on Normalization Up: Running XSTAR2XSPEC Previous: Output   Contents
Tim Kallman 2014-06-30