6. Interpretation in the frame of a simple analytic loop model
In this chapter we will use the loop model of Rosner et al. (1978; RTV model in the following) to interpret our results. In this model, scaling laws between the temperature at the loop top [K], the pressure p [ ] at the loop base, the loop semi-length l [cm] and the energy input rate per unit volume [ ] can be derived:
can be found.
According to relation (3), higher temperatures can be obtained by increasing the heating rate or the loop length, or a combination of both. For a further investigation we can derive theoretical relations between and and compare them to our data. We assume the stellar corona to consist of identical loops with constant cross section A. In steady state, each loop radiates the same amount of energy as deposited into it due to the heating: . If N loops are present, the surface "filling factor" gives the fraction of the stellar surface covered by loops. From this we find the surface flux to be . Using the scaling laws (1) and (3) we finally get:
If we assume only the loop length to increase while the heating rate and the filling factor are kept constant, we would expect according to (4). If, on the other hand, we assume only the heating rate to increase while the loop length and the filling factor stay constant, we would expect according to (5). Both relations are inconsistent with the relation found in our data. This means that the high temperatures are probably achieved by a combination of increased heating rate and larger loops.
It is important to note that the RTV model imposes a limit to the loop length: since it assumes a spatially uniform gas pressure, the loop length must not exceed the local pressure scale height, which for a fully ionized coronal plasma is given by
(see Serio et al. 1981). The scaling laws may be extended to loops longer than the pressure scale height (Serio et al. 1981), but we will not take this into account for two reasons. First, this extension introduces a multiplicative term containing the pressure scale height and thus the scaling laws can no longer be solved for and . Second, for the high coronal temperatures found for most star of our sample, the pressure scale height is already very large. For example, a star of solar mass and radius with has . Loops longer than this would exceed several stellar radii and it seems hardly imaginable that such long loops could be stable.
In the following analysis we assume the RTV scaling laws to be valid and the loop length to be smaller than the pressure scale height. Then, we can use this upper limit to the loop length to calculate the minimal pressure necessary to reach the observed maximum temperature according to scaling law (1). Furthermore, since we additionally know the surface X-ray flux, we can also calculate an upper limit to the surface filling factor from relation (5). The stellar masses necessary for calculating were taken from Feigelson et al. (1993) for the TTS in Cha I and estimated from the spectral type for the young main sequence stars in IC2391, the Pleiades, Hyades, and the field stars. We did not try to estimate masses for the other T Tauri stars in our sample.
The results are shown in Fig. 6. In order to allow a comparison with solar values, we note that the total X-ray emission of the solar corona is dominated by active region loops with typical lengths of about cm (Pallavicini et al. 1981), for which our analysis gives and . To be more general, we take into account that the solar corona exhibits a variety of loops with rather different lengths. However, pressures found for quiescent solar loops range only from for extended interconnecting loops to for compact active region loops (Pallavicini et al. 1981; Yoshida & Tsuneta 1996).
The loop pressures found for most of the young stars seem to exceed typical solar values by about one or two orders of magnitude. In principle, the actual pressures could even be higher, since the actual loop length might be smaller than . However, it should be noted that the typical length of solar active region loops is very similar to the pressure scale height (for formula (6) gives cm). Therefore, it seems not too unrealistic to assume and we believe that our results reproduce the correct order of magnitude for p and f. This means that the dominant X-ray emitting regions in the coronae of most young stars are high pressure regions that cover only a small fraction (no more than a few percent) of the stellar surface.
Of course, the validity of the rather simple RTV model may be questioned. Furthermore, these results are based on the assumption that the stellar corona is composed of loops that are all (nearly) equal. However, our findings are consistent with other recent results. In the analysis of density sensitive high-temperature emission lines in the EUVE spectra of active late type stars electron densities considerably higher than typical for solar active regions have been found (see e.g. Schrijver et al. 1995; Drake 1996).
We note that such high pressure loops require quite strong magnetic fields to confine the plasma in the loops. However, the required magnetic field strength scales only with the square root of the pressure. Since the (magnetic) activity of the young stars exceeds the solar level by far, their coronal magnetic fields might also be stronger. In this context, it seems worth noting that strong (photospheric) magnetic fields have been found on active late type stars (e.g. Valenti et al. 1995) as well as on TTS (Basri et al. 1992; Günther 1996). Furthermore, recent results provide some evidence that the strength of the magnetic surface fields may grow with stellar activity (Johns-Krull & Valenti 1996).
An alternative explanation might be suggested by the fact that the high loop pressures found for the young stars are quite similar to pressures in solar flaring loops. Although we have not included stars showing obvious flares during the ROSAT observation into our sample and could not find indications for temporal variability of the spectral parameters, we cannot exclude that the X-ray emission is nevertheless dominated by frequent low-amplitude flares, so-called "microflares". Based on theoretical (e.g. Parker 1988) as well as observational (e.g. Porter et al. 1987) arguments, it is suspected that microflares may be occurring permanently in the solar corona and might be of importance for the coronal heating (Watanabe 1996, Yoshida & Tsuneta 1996). There are also some observational results that might indicate microflaring activity in the coronae of other active stars (e.g. Robinson et al. 1995), but a convincing proof of stellar microflares is still missing (for a recent discussion see Haisch & Schmitt 1996). Thus microflaring is a possible but not yet conclusive interpretation of these results.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998