## 4. Relationship between chromospheric and coronal emission## 4.1. Single starsRutten & Schrijver (1987) and Basri (1987) suggested
that the We study the relation between the chromospheric and coronal
emission for each of the units (surface flux density Here and are the logarithmic (excess) emissions. In Figs. 3, 4 and 5 we show logarithmic plots of the ROSAT X-ray emission versus the Ca II H&K line core emission expressed in the three different units, with the best power-law fit (solid line). The results of the fitting procedure are summarised in Table 4, which lists the minimum reduced for every unit and, for , the best-fit parameters. The uncertainties in these best-fit parameters have been determined using a `bootstrap' method. In this method we selected 1000 equally large random sets of data points from our sample, allowing duplications. We then determined the best fit for each set by minimising . The uncertainties in the parameters are given by the standard deviations.
We confirm the findings of Rutten and Schrijver (1987) that no tight relation exists between and . The reduced of the best power-law fit is 3.7, and is therefore unacceptable. Also the best power-law fit for the surface flux densities, with a reduced of 1.6, is unacceptable: if we assume that the data follow a power-law relation and have normally distributed errors, the probability of finding a reduced , for 71 data points and two fit parameters, is 0.11%. However, if we exclude the 19 stars with ,
(spectral types between F0 and F4), we find an acceptable power-law
fit for the surface flux densities, with a reduced
of 1.1; the probability of finding a reduced
(52 data points) is 20%. Since Rutten and
Schrijver (1987) based their results on a sample which contained
very few warm stars, this might explain why they found an acceptable
power-law fit for the relation between surface flux densities. The
slope ( Fig. 6 shows the deviation from the relationship between the
surface flux densities (top) and the normalised flux densities
(bottom) as a function of colour. The rank correlation coefficients
For the 79 stars that where not detected with ROSAT, we have derived upper limit values to the normalised X-ray emission, as described in Piters (1995). These values are shown in Fig. 7 (top panel). Given the number of stars, essentially none of them would be expected to have a higher flux than the given upper limit, so that the upper limits should all lie above the mean relationship. Five stars, however, lie below the mean relationship. We attribute this to the uncertainty in the Ca II flux: for these 79 stars we expect about 16%, or 13 stars, to have a Ca II flux density value that exceeds the actual value by more than its one uncertainty, and 3%, or two stars, with an observed flux density too high by more than . This appears consistent with our data.
## 4.2. BinariesThe flux-flux diagram for visual binaries (with a distance of less
than ) and double-lined spectroscopic binaries
is shown in Fig. 7 (bottom panel). The calculation of the
normalised flux density is the same as for the single stars
(Sect. 3), where we used the stellar parameters of the primary
component. On average, these binaries follow the relationship defined
by the single stars, but the spread is substantially larger than for
single stars (). We attribute this scatter to
contributions from the secondaries, both in X-rays and in
Ca II H&K. In the simple case of two identical stars
with the same magnetic activity level, for instance, we would expect
the binary system, analysed as if it were a single star, to lie a
factor 2 above the relationship, because the ## 4.3. The effect of variabilityStellar magnetic activity is intrinsically variable. The effect of
this variability on the The root mean square of the (logarithmic) differences between the observed X-ray emission and the value expected from the fitted relationship equals 0.43, i.e., somewhat larger than that derived by Schrijver (1983; scatter is 0.35). This is mainly caused by the relatively large uncertainties in the ROSAT All-Sky Survey fluxes as compared to those obtained from the Einstein observations used by Schrijver (1983). We note, however, that in deriving the best fit relationship we minimised , not the above defined scatter. If we include only stars with small observational uncertainties we find that the dispersion around the relationship is reduced (i.e., to 0.30 for the 38 stars with uncertainties in (logarithmic) X-ray and Ca II emission less than 0.3 and 0.15, respectively, and to 0.20 for the 20 stars with uncertainties in X-ray and Ca II emission less than 0.2 and 0.1, respectively). This shows that independent of the observational uncertainties, the dispersion around the average relation is largely accounted for by these uncertainties. © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |