## Appendix A: spectral fitting simulationsWe 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 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 for temperatures between and ) from spectra with about 1000 counts even in the presence of considerable extinction (). The scatter in the fitted temperatures is lowest for and and grows fast for . 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 () and the temperatures are well separated (). ## A.1. Simulations with CED modelsIn 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 , , about 1000 counts per spectrum and five different values of (6.0, 6.5, 7.0, 7.5, 8.0). The boxes show the intervals containing 63% of the fitting results (" -intervals") for and . The median of the fitted values for and 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.
We also performed simulations for other values of and . Since the results are qualitatively similar we do not present them here. The results of these simulations can be summarized as follows: the 1- interval of the fitted maximum temperature is within of the true values for models with . For very high maximum temperatures () 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- intervals for are typically . This proves that the results of the fits with CED models are meaningful. ## A.2. 1T- and 2T-fits to CED spectraIn 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 = 7.5 and resp. with 200 resp. 1000 counts and different values of . 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.
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 and
can usually 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 and 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 , the high temperature components found in the 2T fits are very close to the maximum temperatures. However, if , 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 . This is a critical point in the determination of coronal temperatures: the temperatures found in 1T and 2T fits tend to "stay" around 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.
## A.3. The influence of abundance variations on the inferred temperaturesAs 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
() and 1/10 () of the
solar value, assuming , 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
(
For we can find no significant deviations of the fitted temperatures from the true temperatures. The largest systematic deviations occur for , where the temperatures are overestimated by . 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 (), spectra with cannot be successfully fitted with solar abundance models. For temperatures between and 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 . 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. © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |