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

3. Spectral modeling

3.1. Possible spectral models

The simplest spectral model for coronal sources is isothermal plasma emission (1T model). In this model the X-ray flux the satellite receives at energy E is

where D is the distance to the source, is the hydrogen column density along the line of sight, is the extinction cross section of the interstellar matter, EM is the emission measure, T the plasma temperature, and the emissivity of the coronal plasma. This model has three parameters, , EM, and T.

A simple extension of the isothermal model is the two temperature model (2T model), with individual emission measures for the two temperature components, where the X-ray flux is given by

In this study we prefer a model with a continuous distribution of emission measure with temperature (CED model), where we have

and assume a power-law dependence up to a maximum temperature

This model has four parameters, , , , and .

The use of more complicated spectral models seems not to be reasonable, keeping in mind the quite moderate spectral resolution of the PSPC (see also our discussion in the Appendix).

3.2. Selection of a spectral model

Before deciding which spectral model to use, we have tentatively applied all three models to our spectra and obtained the following results: successful 1T fits can be found for only a few spectra, especially those with a low number of counts and rather high extinction. The 2T model gives acceptable fits to nearly all spectra of our sample. Finally, the CED model gives acceptable fits to all spectra. For most sources, the "goodness" of the fit as measured by (see below), is very similar for the 2T model and the CED model. As an example of the fitting success of the three models we show in Fig. 1 the best fits for the spectrum of the Hyades stars VB 40.

Fig. 1. Best fits (solid lines) for the X-ray spectrum (crosses) of the Hyades star VB 40 with different models. The 1T model is clearly inappropriate. The 2T model gives a successful fit with temperatures and (the contributions of these individual temperature components are shown as dotted lines). The CED power-law temperature distribution model yields a successful fit with . Note that the CED model gives a significantly higher maximum temperature and provides a better fit to the spectral region above 1.5 keV than the 2T model, which predicts too little flux in this region.

These fitting results are very similar to those reported by other authors: the 1T model fails to fit nearly all high quality stellar X-ray spectra because it is obviously too simple to adequately describe stellar coronae, that can by no means be expected to be isothermal. On the other hand, most ROSAT spectra of late type stars can successfully be fitted with a 2T model (e.g. Stern et al. 1994; Gagné et al. 1995b). However, it should be noted that at least for some very high quality ROSAT spectra the 2T model seems to be no longer appropriate (Ottmann 1994). Furthermore, observations of the same sources with different instruments have revealed systematic differences in the results of 2T fits, which suggests that the temperature solutions may be partially dependent on the detector (Majer et al. 1986). It is also important to note that a successful 2T fit does not necessarily imply the presence of two physical distinct regions that correspond to the different temperatures (see Appendix).

The CED model has been used by Schmitt et al. (1990), who found that the EINSTEIN spectra of late type stars could be well fitted with this model. Nevertheless, from the fitting results alone it is not obvious whether the 2T model or the CED model is preferable, since in the absence of independent information there is no way to distinguish between two models that fit the data equally well. Furthermore, we have performed spectral fitting simulations (see Appendix) that show that the spectral resolution of the PSPC is not high enough to permit a strict distinction between these models: simulated spectra based on a CED model can very often be successfully fitted with a 2T model. Only observations with considerably higher spectral resolution can reveal the true coronal temperature distribution and decide between both models. In this context, it is important to keep in mind that the 1T and 2T models provide only a very simple parameterization of the coronal temperature structure and can probably not be expected to yield a physically consistent description of coronae. We believe that the CED model, although it is also rather simple, might describe a corona in a more physical manner than the 2T model (see also Schmitt et al. 1990). This is based on the following arguments:

Spatially resolved observations of the solar corona show that the coronal loops exhibit a continuous temperature variation from chromospheric temperatures up to the maximum temperature (e.g. Vaiana & Rosner 1978; Hara et al. 1992). Theoretical loop models (see e.g. Antiochos & Noci 1986; Maggio & Peres 1996) as well as solar observations (see e.g.Dere & Mason 1993; Bruner & McWhirter 1988) yield coronal temperature distributions very similar to the power-law form used in the CED model. Typical parameters inferred from solar extreme-ultraviolet or X-ray observations are from to . In nearly all solar emission measure distributions a rather pronounced minimum is found near K (see Bruner & McWhirter 1988). This lack of material at this temperature can be explained by the maximum of the radiative loss function around K (see Doyle et al. 1985) and is the reason for our choice of the lower boundary of the temperature distribution at K. Since the contribution of material at temperatures below and slightly above K to the X-ray flux in the 0.1 - 2.4 keV ROSAT band can be fully neglected (see Raymond & Smith 1977; Raymond 1988), our choice of K as the lower boundary of the temperature distribution is not critical with respect to the results.

There is also evidence for continuous emission measure distributions in the coronae of late type stars: for nearly all late type stars observed with EUVE, broad continuous temperature distributions are found (e.g. Schrijver et al. 1995), in many cases very similar to power-laws (e.g. Haisch et al. 1994; Drake et al. 1995). In a recent review on EUVE spectroscopy of late type stars Drake (1996) concludes that there are no 2T coronae and a continuous temperature distribution is necessary to explain the spectra.

We are aware that the CED model is also a rather simple description of the coronal temperature structure. While the true temperature distribution is most likely continuous, it will not necessarily be of a power law form since there might be different types of loops with different lengths, temperatures, and pressures. It might even be possible that in some cases (e.g. when there are two dominant families of different loops) the temperature structure is actually more similar to a 2T model than to a power law distribution. Indeed, for some stars, high resolution X-ray and EUV spectra indicate temperature distributions with rather pronounced peaks at a few dominant temperatures (e.g. for AB Dor; Mewe et al. 1996).

However, for most late type stars observed with EUVE there is no evidence for a significant bimodal temperature structure and the common morphology of the temperature distributions is rather similar to the power-law distributions of the CED model (Drake 1996). We think the CED model provides a good compromise between the physical expectations for the actual temperature distribution and the amount of information available in modest resolution X-ray spectra. The use of more complicated models (e.g. a sum of power laws with different maximum temperatures and slopes) is not meaningful, keeping in mind the rather low spectral resolution of our PSPC data.

Furthermore, the CED model has the additional advantage that the parameters and have a clear physical meaning, giving simply the highest temperature present in the corona and the relative amount of hot versus cool plasma. This is not the case for the 2T model, where one has to define some "characteristic" temperature before investigating possible dependences on other stellar quantities. It is not clear, how such a quantity should be obtained from the results of a 2T fit and still have a physical meaning. Indeed, different studies use quite different definitions: e.g. Gagné et al. (1995b) use an emission measure weighted mean of the two temperatures, while Güdel et al. (1996a) consider only the high temperature component.

3.3. Fitting procedure and results

The recent version of the Raymond & Smith model of optically thin thermal plasma emission (see Raymond & Smith 1977; Raymond 1988) was used to calculate the model spectra. For the extinction cross sections we used the Morrison & McCammon (1983) model. Solar abundances were assumed for the plasma and the interstellar matter. For some stars in our sample, information on the optical extinction can be found in the literature and one can calculate the hydrogen column density from the relation (Paresce 1984; Predehl & Schmitt 1995). Nevertheless, we decided to treat as a free fitting parameter, especially since, at least for TTS, the estimation of extinction is rather difficult and uncertain. If optical extinction information was available, we checked the concordance of the fitted extinction with the optical value and found rather good agreement in nearly all cases. It should be noted that a fitted extinction of is merely significant and often is consistent with even no extinction.

In the fits could take values between and , was allowed to vary between and K, and no limits were imposed on and . The best fit model was determined with a fitting routine using the Levenberg-Marquardt method; the resulting quantity is a measure for the quality of the fit (Press et al. 1986). From one can calculate the statistical acceptance Q, which gives the probability that the deviations between the source spectrum and the model spectrum are only due to statistical measurement errors (see Press et al. 1986). We defined fits as "acceptable" if , and could find acceptable fits for all spectra of our sample. To quantify the uncertainties of the fitting parameters we derived confidence intervals for the fitting parameters from grids using the projection method described by Lampton et al. (1976). Since in most aspects of the following analysis we are interested only in , we determined 1- confidence intervals for the case of independent variables.

In Table 3 we summarize the results of our spectral fits. We give the number of the ROSAT data set from which the spectrum was extracted, the number of source counts per spectrum, the "best-fit" parameters together with the limits of the 1- confidence intervals for and , and . We have also calculated the X-ray luminosities in the ROSAT band (0.1 - 2.4 keV) from the spectral parameters. A comparison of our values with the X-ray luminosities derived by other authors (see references to Tab. 1) from the same ROSAT data show generally good agreement within the errors. We are aware that maximum temperatures above K are rather uncertain, since the PSPC is not very sensitive to such high temperatures. Nevertheless, they indicate the presence of very hot plasma.

Table 3. Results of the CED model fits to the X-ray spectra.

Table 3. (continued)

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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