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Astron. Astrophys. 325, 1115-1124 (1997)

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5. Discussion

5.1. The appropriate magnetic-activity unit

We argued in Sect. 4.1 that in comparing chromospheric and coronal emissions, the normalised emission R provides a better measure of magnetic activity than the surface flux density F, because the relationship between the normalised emission units is valid in the complete colour range  [FORMULA] investigated here, while the relationship between surface flux densities is only valid for stars with [FORMULA]. Rutten et al. (1991) and Schrijver et al. (1992) found a good relationship between surface flux densities, because the sample they used consisted mainly of stars with [FORMULA]. Note, however, that in this paper we do not consider the dependence of the emissions on stellar rotation rate, whereas Basri (1987) and Rutten and Schrijver (1987) based their preference for the surface flux density F as the most appropriate measure of magnetic activity largely on the basis of the relation between chromospheric activity and stellar rotation rate.

Some authors state that relations between normalised emission from the corona and the chromosphere/transition region are colour dependent, in contradiction with our results. Simon and Drake (1989) find, that early F stars lie systematically below the relation between normalised emission of coronal X-rays and of the transition region C IV line. This deviation can be attributed to the minimum flux contribution of the C IV line (Rutten et al. 1991), which the authors did not subtract from the normalised emission. The minimum emission for early F stars is about a factor 200 higher than for K stars. Subtraction of the minimum emission moves the early F stars onto the relationship (Schrijver 1993). Rutten and Schrijver (1987) state that the relationship between normalised emission of X-rays and of the chromospheric Mg II line in excess of the minimum emission is slightly colour dependent, while the relationship between the surface flux densities is not. This statement is based on their Fig. 1, but the colour dependence of the relationship between normalised emission units is not very obvious from this figure, and seems to be based on only three stars with [FORMULA], which lie slightly above the mean relation.

5.2. The slope of the CaII - X-ray relationship

The relationship between the X-ray surface flux density and the excess Ca II H&K line core excess flux density, as derived from the present data, is steeper than the relation previously found from EXOSAT observations (slope  [FORMULA] ; Schrijver et al. 1992), and has about the same slope as the relation found by Rutten et al. (1991; slope 1.9) on the basis of Einstein IPC data. This is probably caused by a combination of two effects. The first one is that Schrijver et al. (1992) derived their relationship giving the same weights to every data point, while here and in Rutten et al. (1991) the points are weighted according to their uncertainties, thereby giving the stars with lowest activity (hence larger uncertainties) less weight. The least active stars - both in the sample used by Schrijver et al. (1992) and in the sample presented here - appear to lie slightly above the relationship so that they lower the slope of the relationship, when given more weight. If we would have derived a relationship with equal uncertainties for all stars in our sample, we would have found a slope of  [FORMULA] for the relation between surface flux densities F ([FORMULA]) and a slope of  [FORMULA] for the relation between normalised flux densities R.

The second cause for the higher slope is that constant factors were assumed in the conversion from count rate to flux for the EXOSAT and Einstein data, which is only correct for stars with coronal temperatures higher than a few MK (see Fig. 3a in Schrijver et al. 1992). Fig. 2 (top) shows that the ROSAT hardness ratio tends to increase with the X-ray surface flux density for main-sequence stars, which points to an increase in the average coronal temperature with increasing activity (also observed for IPC data, e.g., Fig. 13 in Vaiana 1983). The use of the same conversion factor for all stars would lead to a relative overestimation of the X-ray flux density of the least active stars, because from a few MK to 1 MK the countrate-to-flux conversion factor of EXOSAT drops a factor ranging from 2 to 10 (for the 3-Lex filter and the A1/P filter respectively; Fig. 3a in Schrijver et al. 1992). Such an overestimation of the lowest flux densities results in a less steep [FORMULA] - [FORMULA] relationship. Using a constant energy conversion factor in the derivation of the relationship for ROSAT data for our sample, we find a slope of  [FORMULA] for the relation between surface flux densities (equal weights for all stars), consistent with the results of Schrijver et al. (1992), and a slope of  [FORMULA] for the normalised flux densities.

Our assumption of a one-temperature plasma could, in principle, affect the slope of the [FORMULA] - [FORMULA] relationship as well. In calculating the energy conversion factor  [FORMULA], we have assumed that the X-ray emitting plasma is dominated by one temperature, so that the hardness of the spectrum can easily be associated with a temperature and consequently with an energy conversion factor (Sect. 2.2). We investigated the effect of this assumption on the  [FORMULA] corresponding to, e.g., a two-temperature plasma, in the following way. For a grid of temperature combinations between [FORMULA]  K and [FORMULA]  K, we calculated the relative emission measures of the two temperature components for a given hardness ratio, and the corresponding energy conversion factor. We find that even for extreme temperature combinations, this ratio stays close to 1, as long as the hardness ratio is larger than 0.2, which is the case for the large majority of the stars in our sample. For smaller hardness ratios there exist temperature combinations (with temperatures [FORMULA]  MK, [FORMULA]  MK and the ratio of the emission measures [FORMULA]) such that for the same hardness ratio the one-temperature conversion factor is larger than 1.5 times the two-temperature conversion factor. In this case the calculated X-ray surface flux density (assuming one temperature) is underestimated by more than a factor 1.5. If we exclude the 9 stars with hardness ratios smaller than 0.2, the slope of the relationship between surface flux densities becomes  [FORMULA], and the slope of the relationship between normalised flux densities becomes  [FORMULA], which are not significantly different from the slopes derived for the surface flux densities (for [FORMULA]) and the normalised flux densities, respectively.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998