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Astron. Astrophys. 318, 215-230 (1997)

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5. Interpretation and conclusions

5.1. Universality of stellar coronae

As is clear from Table 2 and Fig. 3, for stars classified as late A-type or F-type, the detection rate is complete, for stars classified as G-type, the detection rate is very large. I argue that also for the latter group of stars it is reasonable to assume the existence of coronae, since, first, all cases where survey non-detections were reobserved in the pointing program resulted in detections, and second, my upper limits lie well above the lowest detections. The conclusion then is that coronal formation and X-ray emission are universal for stars in the spectral range A7 to G9. Combining this with the findings obtained by Schmitt et al. (1995) it follows that all cool dwarf stars must be surrounded by X-ray emitting coronae. The existence of truly X-ray dark cool dwarf stars can of course not be entirely excluded, but one can state with confidence that such objects must be very rare. On the other hand, for nearby A-type stars, specifically the prototypical A-star Vega, extremely sensitive upper limits are now available, which demonstrate that coronae around those stars (if at all existent) must be very different from those around cooler stars.

5.2. The Sun in perspective

With my complete sample of solar-like stars I am now in a position to carry out a fair comparison between solar and stellar X-ray emission. Interestingly, the median of the observed X-ray luminosity distribution function at [FORMULA] lies actually somewhat above typical solar maximum emission levels. Therefore one must look at the Sun as a star with an activity below average, however, the observed range (between maximum and minimum) of solar soft X-ray luminosity compares well with the low luminosity part of the observed stellar X-ray luminosity distribution function, and therefore the Sun is certainly not atypical among solar-like stars. The X-ray luminosities (cf., Fig. 3 and Fig. 3 in Schmitt et al. 1995) and mean X-ray surface fluxes (cf., Fig. 8) are very similar for the whole sample studied, and in fact for all cool stars when the F and G type dwarfs discussed in this paper are combined with the K and M dwarfs presented by Schmitt et al. (1995). The distribution functions for X-ray luminosity (cf., Fig. 4 and 5, and Fig. 4 in Schmitt et al. 1995) and X-ray surface flux vary smoothly over the observed range of data values with no sign of any bimodal distribution. The most natural explanation for these findings seems to be to assume that the same heating processes that are operating in the solar corona are also operating in stellar coronae.

5.3. Minimum X-ray surface flux

There appears to be a rather well defined minimum X-ray surface flux below which the observed stellar X-ray emission never drops (cf., Fig. 8); this minimum X-ray surface flux is more or less independent of color. I re-emphasize in this context particularly that the lack of cool stars with mean X-ray surface fluxes below [FORMULA]  erg/cm2 /sec is not a selection effect but real; unlike sensitivity-limited studies no clustering of sources near the sensitivity limit is observed.

It is instructive to relate the here derived minimum X-ray flux for solar-like stars to the "basal" heating claimed by the Utrecht group and collaborators to be operating in the atmospheres of cool stars; for a recent review of basal heating cf., Schrijver (1995) and references therein. When different activity indicators such as broad band soft X-ray emission, CIV, CII, Si IV and Si II emission are considered, one generally finds a correlation of these quantities such that stars with the largest X-ray outputs also tend to have the largest outputs in the other activity indicators. When specifically doubly logarithmic regression analyses of flux-flux correlations are considered, Schrijver (1987) found that the scatter around the regression curves is somewhat reduced if a color-dependent arbitrary function is subtracted from each of the activity indicators, and the regression is performed with the excess fluxes (instead of the observed fluxes); for example, Rutten et al. (1991; Table 4) quote fit improvements between 0% (for Si IV vs. Si II) and 64 % (for X-ray vs. Ca II) between the different regression analyses. The arbitrary function, which perhaps not too surprisingly lies close to the minimum observed fluxes, is then called a "basal" flux and interpreted as the result of acoustic heating, while the "excess" flux is attributed to a magnetic heating component.

Interestingly, for the X-ray emission Rutten et al. do not require such a basal flux component, or putting it differently, all the X-ray emission has to be considered as an "excess" flux. Having found a minimum X-ray flux, which is interpreted as "excess flux" in the basal picture, one can calculate the "minimum" excess fluxes from the regression fit parameters presented by Rutten et al. (1991) and compare those to the basal fluxes in the various activity indicators. In Table 4 I present the result of this calculation for the C and Si lines and a number of representative B-V colors. As is clear from Table 4, these "minimum excess fluxes" do exceed the basal fluxes for the G, K and M-type stars by factors of up to ten or orders of magnitude. Therefore for those stars the existence of a basal flux or the lack thereof seems to be of little practical relevance. The situation may be somewhat different for the F-type stars, which tend to have the largest basal fluxes in all activity indicators. For [FORMULA], the basal fluxes quoted by Schrijver (1995) exceed the "minimum excess fluxes" by factors of 2 and 4 in the CIV and SiII lines respectively. Hence some authors have argued (cf., Mullan & Cheng 1994) that the X-ray fluxes observed especially for early F-type and late A-type stars are basal, with the implication that the underlying heating is acoustic other theoretical arguments notwithstanding. Observationally such an hypothesis cannot be rejected, however, in my opinion none of the observations suggest that the coronal properties of late A-type/early F-type are any different from those of later type stars. The X-ray surface fluxes appear similar (cf., Fig. 8) as well as the spectral distribution of the X-ray emission as far as we can tell from the available PSPC spectra. In addition to the apparent similarity of the X-ray spectra, Schmitt et al. (1996) find - for Procyon - coronal densities consistent with solar active region densities, which strongly argues against the presence of acoustic heating in the corona of that star. As to somewhat cooler atmospheric regions, Walter et al. (1995) studied the CII emission from two late A-type stars, Altair (B-V = 0.22, d = 5 pc, included in our sample) and [FORMULA] Cep (B-V = 0.22, d = 14.7 pc, not included in our sample). Both stars have X-ray surface fluxes near [FORMULA], yet Walter et al. (1995) conclude that the observed CII emission exceeds the basal fluxes by some factors even for stars as early as [FORMULA]. Thus the quest for a purely "basal star", conforming to the acoustic heating picture, has been rather elusive. Such stars are surely not impossible by the laws of physics, but nature need not realize all its options and basal stars may well be a chimaera that nature simply does not provide.


Table 4. Ratio of excess to basal flux for minimum flux coronae

5.4. The meaning of the observed minimum flux

All of observed facts suggest that the corona of the Sun and the coronae of other solar-like stars have common heating mechanisms. By appealing to the solar analogy and the observed similarities between the X-ray properties of low-luminosity solar-like stars and the Sun I argue that this heating ought be magnetic rather than acoustic. This is of course in line with theoretical estimates of acoustic heating; Stepie & Ulmschneider (1989) derived an upper bound of [FORMULA] 30 erg/cm2 /sec for solar-like stars as considered in this paper, i.e., a value more than two orders of magnitude below the lowest observed mean surface X-ray flux. If not acoustic, what is then the meaning of the observed minimum flux ?

It is of some interest in this context to consider the minimum X-ray surface fluxes observed for different structural features found in the solar corona. The X-ray emission from the solar corona is characterized by its enormous degree of spatial inhomogeneity as recently exemplified by many thousands of high spatial resolution images taken by the YOHKOH satellite. Clearly, the regions with the lowest X-ray fluxes are coronal holes. While such regions often appear black in color representations of solar X-ray images, the actual X-ray intensity recorded from such regions is non-zero. A variety of techniques has been employed to measure temperature and densities of the coronal hole plasma (cf., Table 1 in Hara et al. 1994). The most precise temperature measurements appear to be those utilizing UV/EUV lines which typically yield temperatures of [FORMULA] K, on the other hand, the best emission measure measurements appear to be those utilizing soft X-ray images, an approach taken by Maxson & Vaiana (1977), who performed a detailed analysis of Skylab observations of coronal holes, and similarly by Hara et al. (1994), who used the YOHKOH soft X-ray telescope to derive temperature and emission measure of coronal holes. The latter approach is somewhat hampered by the fact that plasma temperatures are not that well constrained by the rather broad band soft X-ray imaging data; furthermore, extreme care has to be taken in correctly accounting for flux scattered into the wings of the instrumental point response function, because otherwise the coronal hole emission measure will be overestimated. Hara et al. (1994) conclude that the model which best describes all the available observations is a two-temperature model with [FORMULA] K and [FORMULA] cm-5 and [FORMULA] K and [FORMULA] cm-5, while Maxson & Vaiana (1977) use a one-temperature model and find that emission measure values between [FORMULA] cm-5 are consistent with the data depending on which temperature in the range between [FORMULA] K is chosen. For the various models consistent with the Skylab data (cf., Maxson & Vaiana 1977, their Fig. 10) and the YOHKOH data (cf., Hara et al. 1994, their Fig. 6) I have computed the energy fluxes emitted in the 0.1 - 2.4 keV, i.e., the PSPC, pass band.

The results of these calculations are shown in Fig. 9, where I plot the X-ray surface flux in the 0.1 - 2.4 keV pass band vs. emission measure; the dotted curve refers to the Skylab data and EM is the emission measure of the single temperature component, the solid and dashed lines refer to the YOHKOH data and EM denotes the emission measure of the low-temperature component assumed by Hara et al. (1994) to be [FORMULA] K (solid line) and [FORMULA] K (dashed line). The upward arrow in Fig. 9 indicates the surface flux corresponding to a coronal hole (single) temperature of [FORMULA] K. From Fig. 9 I conclude that the coronal holes studied with Skylab and YOHKOH produced X-ray surface fluxes between [FORMULA] erg/cm2 /sec in the PSPC band pass; smaller surface fluxes can only be achieved if the mean coronal hole temperatures are assumed to be [FORMULA] K or more, which appears to be inconsistent with other data.

[FIGURE] Fig. 9. X-ray surface flux [FORMULA] (in the 0.1 - 2.4 keV band) vs. emission measure EM. The dotted curve refers to the Skylab data (Maxson & Vaiana 1977), where EM denotes the total emission measure, the solid and dashed lines to YOHKOH data (Hara et al. 1994), where EM denotes the emission measure in the lower temperature component assumed at [FORMULA] K (solid curve) and [FORMULA] K (dashed curve). See text for details.

The same surface flux range as derived for solar coronal holes is also indicated - by dashed lines - in Fig. 8. Clearly, the observed minimum fluxes for solar-like stars are rather close to the X-ray surface fluxes observed from solar coronal holes. Similarly, in Fig. 7 I also plotted a typical data value for a solar coronal hole, using [FORMULA] erg/cm2 /sec and assuming a temperature of 1.3  [FORMULA] K, with the HR-temperature conversion from Haisch et al. (1992); obviously, solar coronal holes would appear as extremely soft X-ray sources when observed with the ROSAT PSPC. These observational findings can be interpreted either to be coincidental or alternatively to suggest - again by employing the solar analogy - that the minimum X-ray flux stars are the ones whose total X-ray luminosity is dominated by coronal holes, whereas the more X-ray luminous stars should have significant contributions from either the stellar analogs of quiet Sun large scale structure or active regions or both. If this latter suggestion is correct, one expects mean surface flux and mean coronal density to be correlated. If it is the case that the lower X-ray surface flux envelope of stars with [FORMULA] erg/cm2 /sec are stars completely covered by coronal holes, and the upper X-ray surface flux envelope of stars with [FORMULA] erg/cm2 /sec are stars completely covered by active regions, the mean coronal density should increase by two orders of magnitude from [FORMULA] cm-3 to [FORMULA] cm-3. The lower density limit is not expected to significantly depend on stellar parameters, while the maximum densities will be higher if the emission scale length drops below the pressure scale height. At present there are unfortunately only a few coronal density measurements for the quiescent emission from solar neighborhood stars available. For the low activity F-star Procyon (cf., Schmitt et al. 1996b) and for the intermediate activity K-star [FORMULA] Eri (cf., Schmitt et al. 1996a) coronal densities with substantial measurement errors but consistent with solar active region densities have been determined; for the active RS CVn like system Capella, Dupree et al. (1993) determine coronal densities from Fe XXI lines in excess of 1011 cm-3, and therefore the existing data are certainly not in conflict with the idea that more active coronae are characterized by significantly larger coronal densities.

5.5. Stellar Maunder minima

The physical properties of the solar atmosphere during the so-called Maunder minimum, i.e., a prolonged state of low or absent activity, have become a subject of considerable scientific interest, ever since Eddy (1976) pointed out in a very convincing fashion that the low sunspot numbers recorded in the period between about 1645 and 1715 should be considered real and not be due to sparse and unreliable observational material. The connection to the Earth's climate was made when variations in the solar output were shown to be related to the solar cycle (Hudson 1988), such that solar maximum activity also corresponds to maximum irradiance. The obvious question then to ask is how typical are Maunder minimum-like episodes for the Sun ?

One approach to answer this question in the context of the solar-stellar connection is to search for Maunder minimum candidates in a sample of solar-like stars. Baliunas & Jastrow (1990) studied the frequency distribution of the mean calcium index [FORMULA] of stars contained in the Mount Wilson HK monitoring project (cf., Baliunas et al. 1995); the observed range in this mean index is [FORMULA], typical solar values are between 0.164 and 0.178. Baliunas & Jastrow (1990) found a bimodal frequency distribution in [FORMULA] with a low-activity peak well separated from solar minimum values. This low-activity peak contains about a third of their sample stars and hence these authors conclude that the "Sun has spent about one-third of its time in the past in several millenia in magnetic minima"; they further suggest that the similar appearance of the Sun in terms of mean sunspot numbers during solar minimum and Maunder minimum "has led to incorrect conclusions about the properties of the Sun in a Maunder minimum" and argue "that the level of magnetic activity is much lower in a Maunder minimum than in a sunspot minimum".

I have investigated the X-ray properties of the stars contained both in my 13 pc sample and in the Mount Wilson HK project; Baliunas et al. (1995) do point out that their sample is not complete with regard to magnitude and distance. Unfortunately only 15 stars are contained in both samples and the median X-ray luminosity of this subsample exceeds the median of the full sample by about a factor of [FORMULA] 2. From the X-ray point of view it is extremely unlikely that up to a third of the solar-like stars are in Maunder minimum-like lulls of activity; the lowest third of the X-ray luminosity distribution function contains stars emitting at solar maximum levels ! Further, in the whole sample not a single star does drop significantly below the observed solar-minimum X-ray surface flux. In consequence, Maunder minimum stars must either be quite rare or they cannot differ significantly from the Sun observed at solar minimum. It would obviously be of interest to establish the cycle properties of stars which are known to have very low X-ray surface flux. Some care ought to be exercised in interpreting the statistical results from the Mount Wilson sample; I suspect that this sample is actually somewhat biased, and it would in my opinion be extremely important to study the [FORMULA] index distribution in a truly complete sample.

5.6. The big picture

I conclude with a few somewhat more speculative considerations. The new ROSAT observations have shown that all cool stars in the solar neighborhood are X-ray sources with a minimum X-ray surface flux typical for solar coronal holes. The question is whether one attaches physical meaning to this finding or not. To me it seems natural to go for the former, although admittedly it is unclear whether coronal holes on the Sun do indeed have a "minimum flux" or not. Interestingly, also the largest surface fluxes observed for stars are close to the typical radiative loss values observed for solar active regions [FORMULA] erg/cm2 /sec; cf., Withbroe & Noyes 1977), with only a few stars exceeding this limit. It is therefore suggestive to interpret the observed stellar variety in terms of the coronal features observed on the Sun, i.e., coronal holes, large scale ("quiet") structure, active regions, and a hot component, which is observed on the Sun only during flares. The differences from star to star arise from the filling factors of the various components. In this picture, the observed minimum (I purposely avoid the word "basal" !) flux would be due to emission from magnetically open regions, and cool (main sequence) stars without magnetic fields do - in this picture - simply not exist at least in the observable universe. I would therefore also take issue with the alleged acoustic origin of the "basal" fluxes discussed by Schrijver (1995) in the sense that it is exceedingly difficult to find stars where the "basal" fluxes provide the dominant component; at least at transition region and coronal levels the non-"basal" fluxes exceed the "basal" ones by at least some factors. I emphasize that I am not arguing that such "basal" stars are not possible by the laws of physics, rather I am arguing that nature does not provide such stars very often and not at all within 13 pc around the Sun.

As one goes from "minimum" stars to more and more active stars, more and more of the magnetic topology becomes closed, leading to larger and larger radiative losses from large scale structures and active regions. For the most active stars soft X-ray observations clearly require a hot corona component with a temperature of [FORMULA] K, which provides the bulk of the observed soft X-ray emission measure (cf., Schmitt et al. 1990). This hot component is not present in the solar corona at least under "quiescent" conditions; it is present during solar flares but always in transient from. In the stellar context, such a hot corona component is always present and seems - somehow - to be associated with the production of relativistic particles, which lose energy via gyrosynchrotron radiation as observed at radio wavelengths leading to the correlation between X-ray and radio luminosity for such active stars (cf., Güdel et al. 1993). On the other hand, non-active stars like the Sun (or Procyon for that matter, cf. Drake et al. 1993) are X-ray overluminous with respect to their microwave emission.

My basic ansatz is by no means new. Already Vaiana & Rosner (1978) considered - prior to the launch of the Einstein Observatory - how the X-ray output of the Sun would change if the filling factors of its various coronal constituents were changed; in particular they showed that any X-ray luminosity between [FORMULA] erg/sec and [FORMULA] erg/sec can be produced by going from a coronal hole dominated Sun to a flare dominated Sun. With the ROSAT observations we have the data at hand to demonstrate that nature also realizes the various options.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998