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

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4. Relationship between chromospheric and coronal emission

4.1. Single stars

Rutten & Schrijver (1987) and Basri (1987) suggested that the surface flux density F is the appropriate unit in which to express the radiative emission measuring magnetic activity level, because the activity-rotation relation is tightest when the activity is expressed in F, while it strongly depends on luminosity class when the luminosity [FORMULA] is used as an activity unit, and it seems to be slightly colour dependent when the activity is expressed in units of the normalised emission [FORMULA]. Rutten and Schrijver (1987) further showed that the flux-flux relations are tightest between surface flux densities and between normalised emissions. These findings can only partly be confirmed here: we show below that the relationship expressed in normalised emissions is tightest.

We study the relation between the chromospheric and coronal emission for each of the units (surface flux density F, normalised emission R and luminosity L). For this purpose we selected the 71 stars from our sample, which are single or single-lined spectroscopic binaries. The latter have been included because we expect that both the observed X-ray flux and the Ca II H&K flux originate in the primary component, the secondary probably being to faint to observe. A power-law fit is determined by minimising [FORMULA], defined as the sum of the quadratic distances to the fit, expressed in units of the individual error ellipses (each term is the square of the factor by which the error ellipse must be multiplied to touch the relation [FORMULA]):

[EQUATION]

Here [FORMULA] and [FORMULA] 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 [FORMULA] for every unit and, for [FORMULA], 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 [FORMULA]. The uncertainties in the parameters are given by the standard deviations.

[FIGURE] Fig. 3. The relationship between ROSAT X-ray luminosity and Mt. Wilson Ca II H&K line core excess luminosity for single stars and single-lined spectroscopic binaries. The solid line represents the `best-fit' power law, with [FORMULA].
[FIGURE] Fig. 4. Same as Fig. 3, but for surface flux densities. The solid line represents the best-fit power law relation for stars with [FORMULA] (see Table 4).
[FIGURE] Fig. 5. Same as Fig. 3, but for normalised emissions [FORMULA]. The solid line represents the best-fit power law (see Table 4).

[TABLE]

Table 4. The minimum reduced [FORMULA] for the flux-flux relationship between different units of radiative emission, and the fitting parameters a and b, where [FORMULA] (y being the X-ray emission, and x the Ca II emission).


We confirm the findings of Rutten and Schrijver (1987) that no tight relation exists between [FORMULA] and [FORMULA]. The reduced [FORMULA] 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 [FORMULA] 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 [FORMULA], for 71 data points and two fit parameters, is 0.11%.

However, if we exclude the 19 stars with [FORMULA], (spectral types between F0 and F4), we find an acceptable power-law fit for the surface flux densities, with a reduced [FORMULA] of 1.1; the probability of finding a reduced [FORMULA] (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 (b) of the relation is larger than the one derived by Schrijver et al. (1992) who found a slope of  [FORMULA] for a sample of 20 stars observed with EXOSAT. We discuss this in Sect. 5.2.

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 Kendall's [FORMULA] (Kendall 1975) between the deviation from the relationship and [FORMULA] are 0.42 and 0.07, respectively. The probability that [FORMULA] exceeds 0.07, for 71 non-correlated data points, is 40%, but the probability that it exceeds 0.42 is [FORMULA]. We therefore conclude that the relation between normalised emission units is much less colour dependent than the relation between surface flux densities (which is already suggested by the value of [FORMULA] for normalised emission, which is very close to 1; see Table 4), although there may be a weak dependence on colour for [FORMULA]. Thus, on the basis of our extensive data set, it appears that the normalised emission units provide a better measure of magnetic activity than the surface flux densities, when comparing radiative emission measures in different temperature regimes.

[FIGURE] Fig. 6a and b. Top: Deviation from the best fit power-law relation (for stars with [FORMULA]) between [FORMULA] and [FORMULA] expressed in units of [FORMULA] as a function of [FORMULA]. Bottom: The deviation from the best fit power-law relation (including stars with [FORMULA]) between normalised emissions expressed in units of [FORMULA] as a function of [FORMULA]. Stars with the same [FORMULA] have been separated by an amount 0.0016 in [FORMULA] with respect to each other, in order to be able to distinguish individual stars.

For the 79 stars that where not detected with ROSAT, we have derived [FORMULA] 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 [FORMULA] uncertainty, and 3%, or two stars, with an observed flux density too high by more than [FORMULA]. This appears consistent with our data.

[FIGURE] Fig. 7a and b. Same as Fig. 5, but now for specific groups of stars. The smallest circles repeat the positions of the stars from Fig. 5, which define the best-fit relationship between normalised emissions (solid line). Top: stars with only upper limits in [FORMULA] ; bottom: binary stars.

4.2. Binaries

The flux-flux diagram for visual binaries (with a distance of less than [FORMULA]) 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 ([FORMULA]). 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 S -value is the same as for one star, while the X-ray count rate is twice as high as for one star.

4.3. The effect of variability

Stellar magnetic activity is intrinsically variable. The effect of this variability on the S -value for stars with [FORMULA] can be seen in Fig. 1 (bottom panel). We investigate the effect of the time difference between the observing times in X-ray and Ca II on the scatter about the relationship between the normalised emissions. There is no significant correlation between the distance to the relationship (i.e. [FORMULA]) and the time difference between the X-ray and the Ca II H&K measurements, for stars with [FORMULA] and [FORMULA] between [FORMULA] and [FORMULA] (nor between the distance and the time difference normalised to the rotation period). The X-ray measurements are averages over time intervals of at least two days (Sect. 2.2), so we do not expect to detect variations from the relationship due to variations in activity level on time scales smaller than two days. If we exclude stars which have been observed more than two days apart in X-ray and in Ca II in deriving the relationship between normalised emission units, the reduced [FORMULA] for the new relationship between [FORMULA] and [FORMULA] with 39 stars becomes 0.94, which is not significantly better than the reduced [FORMULA] of 1.04 derived for 71 stars (see Table 4). We conclude from this that deviations from the relationship due to variations in activity level on time scales of a few days are of the same order as or less than the observational uncertainties due to statistical effects and calibrations.

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 [FORMULA], 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.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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