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Astron. Astrophys. 320, 525-539 (1997)

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Appendix A: spectral fitting simulations

We have performed a large number of spectral fitting simulations in order to investigate the ability of the PSPC to constrain the correct model parameters in the different spectral models, its ability to discern between different spectral models, and the influence of the instrumental sensitivity on the fitting results. The simulations were performed in the following way. First we chose spectral parameters and calculated a model spectrum with a given number of counts, which then was rebinned in the same way as a real spectrum (see section 2). To simulate an observed spectrum, in which the number of counts per bin obeys a Poisson distribution, we replaced the number N of counts in each bin by a random number drawn from a Poisson distribution with mean N. In this way we produced a set of 250 different simulated spectra per model spectrum. Then, each of these spectra was fitted and for the analysis of the resulting fitting parameters we only used the successful fits defined by a statistical acceptance [FORMULA] (see section 3.3).

Recently, similar investigations treating 1T and 2T model spectra have been published by Maggio et al. (1995). Since our results are consistent with theirs, we will focus on our simulations with CED spectra and just briefly summarize the main results of the simulations with 1T and 2T models. The temperature of an isothermal plasma can be reproduced quite well (scatter [FORMULA] for temperatures between [FORMULA] and [FORMULA]) from spectra with about 1000 counts even in the presence of considerable extinction ([FORMULA]). The scatter in the fitted temperatures is lowest for [FORMULA] and [FORMULA] and grows fast for [FORMULA]. This can be explained by the strongly temperature dependent sensitivity of the PSPC, caused by the peaks in the effective area for energies of 0.3 keV and 1.2 keV (see Pfeffermann et al. 1987). The temperature components of a 2T spectrum with at least 1000 counts are reproduced similarly well as long as their amplitudes are not too different ([FORMULA]) and the temperatures are well separated ([FORMULA]).

A.1. Simulations with CED models

In a first set of simulations we investigated how well the spectral parameters of the CED model are reproduced. As an example we show in Fig. 7 the distributions of the fitting parameters for simulated CED spectra with [FORMULA], [FORMULA], about 1000 counts per spectrum and five different values of [FORMULA] (6.0, 6.5, 7.0, 7.5, 8.0). The boxes show the intervals containing 63% of the fitting results (" [FORMULA] -intervals") for [FORMULA] and [FORMULA]. The median of the fitted values for [FORMULA] and [FORMULA] lie at the crossing of the lines within the boxes; the solid dots show the true spectral parameters. The number of successful fits was typically about 240 per set of 250 simulated spectra.

[FIGURE] Fig. 7. Boxplot showing the results of the fits to CED spectra with different values of [FORMULA] (see text for explanation).

We also performed simulations for other values of [FORMULA] and [FORMULA]. Since the results are qualitatively similar we do not present them here. The results of these simulations can be summarized as follows: the 1- [FORMULA] interval of the fitted maximum temperature is within [FORMULA] of the true values for models with [FORMULA]. For very high maximum temperatures ([FORMULA]) the scatter is up to 50%. This means that the maximum temperature can be reproduced equally as well as the temperature of an isothermal plasma spectrum. The 1- [FORMULA] intervals for [FORMULA] are typically [FORMULA]. This proves that the results of the fits with CED models are meaningful.

A.2. 1T- and 2T-fits to CED spectra

In many studies coronal X-ray spectra are fitted with 1T or 2T models, whereas the true temperature distribution might be continuous. Thus it is very interesting to investigate the results of spectral fits with 1T and 2T models to spectra with a continuous temperature distribution.

First of all, we want to investigate whether one can prove the existence of a continuous temperature distribution from the observed spectra. This was done by simulating CED spectra and fitting them with 1T and 2T models. For example, we have simulated CED spectra with [FORMULA] = 7.5 and [FORMULA] resp.  [FORMULA] with 200 resp. 1000 counts and different values of [FORMULA]. For each of these spectral models, the set of 250 simulated spectra was fitted with a 1T model, the spectra with 1000 counts also with a 2T model. In Table 4 we give the number of successful 1T or 2T fits.


[TABLE]

Table 4. Number of successful 1T or 2T fits to sets of 250 simulated CED spectra with [FORMULA].


One can see that 1T models give successful fits only to low S/N CED spectra with rather high extinction. A 2T model, however, is very often successful in fitting the CED spectra even if the extinction is low. Only CED spectra with [FORMULA] and [FORMULA] can usually not be fitted with a 2T model. This means that the PSPC is not able to properly discern between a 2T model and a CED model, given an observed spectrum with about 1000 counts. A successful fit with a 2T model does not prove that there really are two dominant temperature components in the emitting plasma; the real emission measure distribution might be quite different from a 2T plasma. However, it is also clear that a successful fit with the CED model does not prove that there really is a power law temperature distribution.

Another very interesting aspect concerns the temperatures found in these fits, in which the "wrong" 1T or 2T models are used to fit simulated CED spectra. Some examples for the distribution of the fitted temperatures are shown as histograms in Fig. 8. Simulations for other values of [FORMULA] and [FORMULA] yielded qualitatively similar results. One can see that in nearly all cases the fitted temperatures do not exceed the true maximum temperature. This means that the 1T or 2T fits usually do not yield temperatures that are not present in the actual temperature distribution. As long as [FORMULA], the high temperature components found in the 2T fits are very close to the maximum temperatures. However, if [FORMULA], most of the temperatures found in the 1T and 2T fits are considerably lower than the maximum temperature. This can be explained by the decreasing sensitivity of the PSPC to plasma with temperatures [FORMULA]. This is a critical point in the determination of coronal temperatures: the temperatures found in 1T and 2T fits tend to "stay" around [FORMULA] K even if the actual maximum temperature increases to considerably higher values. This means that 1T and 2T fits can underestimate the maximum coronal temperature significantly when there is a continuous distribution of temperatures.

[FIGURE] Fig. 8. Histograms showing the temperatures found in successful 1T (dashed-dotted line) and 2T (solid line) fits to sets of 250 simulated CED spectra with [FORMULA], [FORMULA], 1000 counts per spectrum and different maximum temperatures.

A.3. The influence of abundance variations on the inferred temperatures

As mentioned in section 4, indications of an under-abundance of heavier elements as compared to solar abundances have been found for the coronae of some late type stars. The rather moderate spectral resolution of the PSPC does not allow to derive good constraints on the abundances. However, a wrong assumption on abundances will adulterate the fitted temperatures. For our fits, we have assumed solar abundances, and it is interesting to investigate how strongly abundance variations could affect our fitting results.

We have calculated sets of simulated isothermal spectra with metal (all elements heavier than helium) abundances set to half ([FORMULA]) and 1/10 ([FORMULA]) of the solar value, assuming [FORMULA], 1000 counts per spectrum, and various temperatures. Then we fitted these simulated spectra with isothermal models assuming solar abundances and compared the fitted temperatures to the true temperature. The number of successful fits and the relative deviations of the fitted temperatures (T) from the true temperatures ([FORMULA]) are shown in Fig. 9.

[FIGURE] Fig. 9. Relative deviations of temperatures found in fits with solar abundance models to the true temperatures [FORMULA] of simulated spectra with reduced metal abundances. The solid dots show the median of the fitted temperatures, the error bars show the 1- [FORMULA] intervals. The numbers at the bottom of the plots give the number of successful fits. No data are plotted for model parameters where less than half of the 250 simulated spectra could be fitted successfully.

For [FORMULA] we can find no significant deviations of the fitted temperatures from the true temperatures. The largest systematic deviations occur for [FORMULA], where the temperatures are overestimated by [FORMULA]. However, it should be noted that this does not exceed the typical uncertainties of the fitted temperatures (see last paragraph).

In the case of an extreme metal under-abundance ([FORMULA]), spectra with [FORMULA] cannot be successfully fitted with solar abundance models. For temperatures between [FORMULA] and [FORMULA] a systematic overestimation of the temperature by up to a factor of 2 may occur. However, these deviations are not much larger than the typical uncertainties of the fitted temperatures and restricted to a relatively narrow temperature interval. Furthermore, it should be noted that no significant deviations are found for [FORMULA].

This means that the high maximum temperatures found for many stars in our sample cannot be explained as an artifact caused by assuming wrong abundances. Therefore we conclude that non-solar abundances do not strongly affect the results of our study.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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