---
abstract: |
    This work is dedicated to the solar-stellar connection, i.e. the close similarity of the Sun and the late-type stars; in particular this work shows that stellar coronae can be composed of [*X-ray emitting*]{} structures similar to those present in the solar corona. To this end we use a large set of ROSAT/PSPC observations of late-type-stars of all spectral types and activity levels, and a large set of solar X-ray data collected with Yohkoh/SXT.

    Solar data have been analyzed and formatted to study the Sun as an X-ray star; they include observations of the solar corona at various phases of the cycle and data on various kinds of [*X-ray*]{} coronal structures, from flares to the background corona, i.e. the most quiet regions.

    We use the X-ray surface flux ($F_X$) vs. spectral hardness ratio (HR) diagram as a fundamental tool for our study. We find that $F_X$ is strongly correlated to HR in stellar coronae, in the solar corona at all phases of the cycle, and in the individual solar coronal structures; all the above follow the same law. Schmitt found the same correlation in stellar coronae.

    We therefore claim that coronae of late type stars are formed with [*X-ray*]{} structures very similar to the solar ones, since their behavior is identical to that of the solar coronal structures and of the whole solar corona. The spatial location of the [*X-ray*]{} structures on the star, however, can be very different than on the Sun.

    In this scenario, the fraction of the stellar surface covered with active regions and with their bright cores increases with activity; the most active stars are brighter and hotter than if they were entirely covered with active regions so they can be explained only with the additional presence of several flares (or flare-like structures) at any time.

    On the basis of the $F_X$ vs. HR correlation, [*corresponding to $F_X \propto T^6$*]{}, we then derive a set of new laws relating the temperature, pressure, volumetric heating and characteristic loop length of the coronal plasma, on all the late type stars. Also, individual solar coronal structures and the whole solar corona, follow the same laws. These laws also agree with recent findings of higher plasma density at higher temperatures in stellar coronae.

    We claim that the strong correlation between surface flux and temperature and the laws mentioned above are just the effect of more fundamental physical mechanisms driving the coronal structures of all the late-type stars from the emergence of new magnetic structures to their dispersal and dissipation.
author:
- 'G. Peres'
- 'S. Orlando'
- 'F. Reale'
title: 'Are coronae of late type stars made of solar-like structures? The F$_X$-HR diagram and the pressure-temperature correlation. '
---

Introduction
============

Since the seminal work by Vaiana et al. (1981) it has become clear that all late type stars share the same basic coronal characteristics: the presence of thermal plasma at several million degrees, its magnetic confinement, the presence of flares etc. Also the basic plasma physics governing coronae is probably very similar in all these stars. On the other hand, coronae cover a large range of X-ray luminosity (even among stars of the same spectral type and luminosity class) and a few scientists think that the coronae of the most active late-type stars are different from the solar corona, albeit the crucial differences have rarely been stated clearly.

Two otherwise identical stars may have rather different levels of stellar activity, i.e. X-ray luminosity, spottedness, UV luminosity etc., because of very different rotation speed (Pallavicini et al., 1981) and age (for a review on the subject Micela, 2002); the speed probably determines how effectively the stellar magnetic dynamo generates magnetic fields.

One may then expect that many differences among late-type stars coronae are just due to rather different regimes of the dynamo at work inside stars, while the basic plasma phenomena occurring in the outer magnetized coronal plasma are probably the same and only the amount and scale of such phenomena are different. Along this line of thought, this work explores to which extent most of the differences found among the various coronae can be attributed to their different composition in terms of various kinds of coronal structures (ranging from the relatively faint and cool structures of the background corona to the very bright and hot flaring regions) and to the amount of [*X-ray emitting*]{} coronal structures present; on the other hand we explore to which extent the plasma phenomena and the structures involved in the coronae are similar even in stars of very different activity levels. According to this paradigm, the solar corona offers the opportunity to study and understand better the building blocks of any late-type-star corona. To this end we consider the average surface flux ($F_X$) and the spectral hardness ratio (HR) in the ROSAT/PSPC band, of late-type stars with those of solar coronal structures; we also study the strong $F_X$–HR correlation (Schmitt, 1997; henceforth S97).

In the following we do not claim that all the aspects of solar corona are the same in the other stars but that the fundamental features are the same; for instance there is evidence of polar active regions and of large polar flares (e.g. Schmitt and Favata, 1999; Brickhouse, Dupree and Young, 2001;

*see also Schrijver and Title, 2001). We will concentrate on the X-ray emission from stellar coronae and we will compare the solar and the various stellar cases and through analogies and differences we will infer (or just constraint) features of X-ray emitting structures present in stellar coronae.*

A few works along this line of thought have been published, either studying the temperature vs. emission measure per unit area on active stars (Schrijver et al. 1984; Jordan and Montesinos, 1991) or correlating activity parameters (e.g. luminosity in given bands or emission measure) with temperature (Schmitt et al., 1990; Guedel et al., 1997; Preibish, 1997). More in detail, Schrijver et al. (1984) analyzed a sample of 34 late-type stars observed with the Einstein telescope, related the relevant coronal temperature and emission measure per unit area and rotation rate, determining scaling laws among them and relating the first two through loop models. Jordan and Montesinos (1991) studied various scalings among emission measure, coronal temperature, surface gravity and Rossby number. Schmitt et al. (1990) performed an extensive survey of late-type stars observed with the Einstein telescope, studying and relating the coronal features and showing some differences among M stars, F and G stars, giants and RS CVn stars. Guedel et al. (1997) studied the “X-ray Sun in time” with the help of various X-ray observations of solar-like stars of different ages; they found a power law dependence of the total X-ray luminosity with the temperature of the hottest component of a two-temperature spectral fit. Preibish (1997) studied a sample of young late-type stars and found a power law dependence of total X-ray luminosity vs. the temperature of the hottest component.

This work is based on methods and results developed in our program to study the solar-stellar connection in X-rays, mostly based on ROSAT/PSPC and Yohkoh/SXT data. [*Our approach allows us to tie directly, with just little elaboration and interpretation, the vast amount of data collected with the two aforementioned observatories. Also our method and scope is novel with respect to previous works, since we use solar data (made homogeneous to stellar data) as a template and a guide. Our findings are eventually interpreted in terms of coronal loops scaling laws.*]{} ROSAT has provided the largest [*homogeneous*]{} data set of stellar coronal observations so far collected; this data set is an important reference for studies of stellar coronae because: i) the relevant data have been collected with just one instrument, ii) instrumental and observing conditions were relatively stable, iii) a large number of stars of different spectral types and luminosity classes have been observed and iv) the data set has an extended time coverage. For analogous reasons, Yohkoh/SXT data are [*equally*]{} important to study the solar corona. It is, therefore, natural to refer both to ROSAT and to Yohkoh data in order to model a scenario of coronae on late-type stars and to connect it to the solar corona. In the following we also use results of previous works on the solar-stellar connection in X-ray band plus some stellar data already published in different contexts; our scope is to draw a global perspective on coronal structure.

The paper is organized as follows: section 2 describes our method to use the solar corona as a template to study stellar coronae, section 3 presents and discusses some of the related results along with related works on stellar X-ray emission, in section 4 we draw our conclusions.

The “Sun as an X-ray star” program
==================================

The scope of this program has been to study the X-ray emission of the whole Sun as an X-ray star, and the contribution of the various coronal structures to its emission, using Yohkoh/SXT data; we have translated the solar observations into a format and into a context homogeneous to those of stellar observations. Thanks to this approach, we can now easily perform comparative studies among solar and stellar observations.

Yohkoh/SXT data have relatively high time resolution, down to a few seconds, and cover uninterruptedly approximately ten years, from the maximum of the solar cycle in ’91 to the following one. All sorts of phenomena have been observed, ranging from extremely large flares to moments of very low emission, during the cycle minimum. The spatial resolution (typically $\approx 5 "$) allows us to identify and study, with high spatial detail, the various regions origin of the X-ray emission and to discriminate, as well as to isolate, the contribution of different coronal structures to the coronal X-ray emission.

In brief, from the Yohkoh/SXT images of the Sun at a given time, we derive a $512 \times 512$ pixels map of coronal plasma temperature and the corresponding map of emission measure; from them we derive the distributions of coronal emission measure vs. temperature at each observation time; then, summing the optically thin spectrum emitted at each temperature (obtained with the corresponding emission measure value and a spectral synthesis code) we obtain the spectrum of the whole corona; folding it with the known instrumental characteristics (effective area vs. photon energy and photon redistribution matrix), we derive the stellar-like focal plane spectrum of the Sun, as it would be observed with the X-ray telescope of interest (Orlando et al., 2000 - henceforth Paper I; Peres et al., 2000 - henceforth Paper II). In this work we will focus on the ROSAT/PSPC data set and will use the ROSAT standard data analysis also to analyze solar data translated into stellar format and conditions.

The capability to isolate the various kinds of coronal structures has allowed us to study their conditions and their contributions to the coronal spectrum (for the flares, see Reale et al., 2001 - henceforth Paper III; for the background corona, the active regions and the core of active regions, see Orlando et al. 2001 - henceforth Paper IV). The good time coverage of Yohkoh/SXT observations has been exploited to study how the emission measure distribution and the X-ray spectrum of the whole corona changes along the solar cycle, with a regular time sampling and taking care not to include flare events (Paper II and Paper IV).

The F$_X$ - HR diagrams
=======================

The diagrams with stellar data
------------------------------

S97 presented the results of a large set of ROSAT/PSPC observations of late-type stars, within 13 pc from the Sun, in the form of a diagram (Fig. \[fig1\]) of F$_X$ (the average X-ray surface flux of the star) vs. HR (the X-ray spectral Hardness Ratio). The Hardness Ratio is defined as $HR= ~(H-S)/ (H+S)$ - where H is the total flux in the ROSAT/PSPC spectral channels between 0.55 and 1.95 keV, and S the total flux in the channels between 0.13 and 0.40 keV. It is a simple characterization of the relative weight of the hard and soft part of the spectrum [^1].

Both F$_X$ and $HR$ are useful to characterize the strength of coronal activity, albeit averaged over the whole corona. Indeed, F$_X$ gives the coronal radiative losses of the star [*per unit area*]{} and is related to the amount of energy delivered into the corona, i.e. the coronal heating, also [*per unit area*]{}; it can discriminate between two stars of the same X-ray luminosity but of very different radius and surface area: the smaller star will have a much higher flux than the other, hinting at a much higher level of plasma activity; F$_X$ is an activity marker independent of the stellar surface value. HR is a simple tracer, albeit a non-linear one, of the coronal average temperature. It is another index of plasma activity, independent from F$_X$.

Fig. \[fig2\] shows the relationship between the plasma temperature and the HR of the spectrum derived with the Me–Ka–L (Mewe, Lemen, and van den Oord 1986; Kaastra, J.S. 1992; Mewe, Kaastra and Liedahl, 1995 and references therein) spectral synthesis model. Note that HR is little sensitive to temperatures below $10^6$ K; it also saturates and is multi-valued in the range 0.0 - 0.4; therefore HR is also relatively insensitive to very hot plasma. Moreover HR should not go beyond 0.4 at any plasma temperature.

The S97 results show that late type stars cover a diagonal strip, with HR ranging from -1.0 to 0.0 and F$_X$ from $10^4$ to $10^8 ~erg
~cm^{-2} ~s^{-1}$ (Fig. \[fig1\]). The more active a star, the higher are both its F$_X$ and its HR; the highest HR (temperature) and F$_X$ values belong to uncommonly active single stars. The overall trend suggests that the confined plasma in the corona of a late type star, on the average, gets brighter as it gets hotter.

Analogously, Marino et al. (2001) reported the data of 40 late type stars of spectral type between F7 and K2, within 25 pc from the Sun, in an analogous F$_X$ vs. HR diagram, shown in Fig. \[fig3\]. The stars of this sample are, on the average, more luminous and active than those in S97; also, at variance from S97, M and late K stars were not included. There were 70 pointed ROSAT/PSPC observations and some of these stars were observed several times; also some observations, of sufficient signal to noise ratio, were split to allow time-resolved analysis during the observation. Some of these stars moved, in time, along the same diagonal track where most of the stars are located; however some stars evolved along a path in this graph, less steep; we claim, in the following discussion, that this different evolution may be due to the contribution of flares. Also Steltzer (2001) generated preliminary F$_X$ vs. HR diagrams for stars in clusters.

The diagrams with solar data
----------------------------

In Paper II we studied the solar corona at the maximum, at an intermediate phase and at the minimum of the solar cycle, taking care not to include flare events; also the whole solar corona during a large flare event was studied. Paper III was entirely dedicated to study, with the same method, the evolution of eight representative flare events ranging from small to large ones and covering most of the conditions encountered in flares; each flaring region was studied separately from the rest of the corona [*so the surface flux and the HR were derived just for the flaring region and its effect was not added to, and diluted within, the whole corona.*]{}

Paper IV studied the solar corona at several moments of the solar cycle, sampling the cycle better than in Paper II and studying separately the various classes of coronal structures, classified as background corona (very quiet regions), active regions and cores of active regions, according to their surface brightness in Yohkoh/SXT bands. The set of whole the data mentioned above yields a good sampling of the physical conditions in the corona from the background corona to the strongest flares and span the whole solar cycle.

[*For the present work we have taken all the flares in the sample of Paper III because they are well observed so they can be studied and characterized accurately; also, since they span a large range within the flares classification and the various phases of their evolution are well observed, they provide a good sampling of the various conditions encountered in flares, from the largest down to very small events. Most of the attention, in the following will be on the physical conditions of various flare phases. We did not consider micro-flares because they can hardly be discriminated from small AR variability and their contribution in the context of average stellar $F_X$ and HR is, in practice, the same as bright ARs.*]{}

Fig. \[fig4\] shows a $F_X$ - HR graph of the whole solar corona during the cycle plus the evolution of the whole solar corona, near maximum of the cycle, in the course of a X-9 flare; the graph collects some results of Paper II, of Paper IV and of Orlando et al. (2002). The flux of the whole solar corona was derived dividing the total corona luminosity by the whole solar surface area.

The evolution of the whole solar corona along the solar cycle, apart from [**moderate to large**]{} flares, spans the lower part of the diagram, the same covered by low activity stars, i.e. the ones at low F$_X$ and low HR, well below the levels of most active stars [^2]. [**Solar flares span a quite large range, from very faint, barely detectable, to very large ones; the flares included in our work are those that at least triggered the Yohkoh flare mode; in this respect they can be considered to be representative of flares from moderate to large.**]{} If a [**moderate to large**]{} flare happens at maximum of the cycle, it makes the luminosity (and the flux) of the solar corona change at most by a factor $\approx 2$, during large flares. However, the corresponding increase of HR can be larger than the changes occurring along the solar cycle, for the same increase of luminosity. Analogous flares occurring during the solar minimum carry a similar change in HR but a much larger change of luminosity and flux. Thus, under the effect of [**significant**]{} flares, the whole Sun moves rightward in the graph, on a path less steep than that covered by stars of various activity levels.

The occurrence of large flares may explain why the evolution of some stars studied by Marino et al. (2001) follows paths, in this diagram, less steep than the track of all the other stars: albeit they show no evidence of large changes of flux, they may undergo a moderate flaring episode which shows up through the HR change and the different path of the star in the diagram (see also Orlando et al, 2002, 2004).

Fig. \[fig5\] shows F$_X$ vs. HR for four specific classes of structures (i.e. background corona, active regions, cores of active regions and flares). It is worth noting that the flux has been derived by dividing the luminosity of the structure (or of the structures of that class) by the surface covered in Yohkoh/SXT images.

The interpretation of these results is useful for stellar physics: a star entirely covered with solar-like active regions would be in the same part of the F$_X$-HR diagram where solar active regions are, and analogously for all the other kind of structures, since they would yield the same HR (average temperature) and surface flux values. Therefore we can foresee how differently a stellar corona would behave if covered with the various kinds of solar coronal structures. [*More in general, however,*]{} a mixture of two different classes of structures would yield values intermediate between the two, the effective value depending on the relative fractions of stellar surface covered. Interestingly enough, solar structures of different kinds (with the exception of solar flares) span the same region of the graph covered by stellar coronae of different activity levels. [*Quite likely the stellar surface is not entirely covered with X-ray emitting structures and different classes of structures (according to our classification) do coexist on the star, covering different fractions of its surface. While places at the extremes of the graph can most likely be explained with only one kind of structure, a locus between two classes of structures may be explained with an appropriate (non-linear) combination of the two.*]{} [**It is worth noting that, according to the analogy we are drawing here, a star equivalent to a structure, e.g. an active region, would be located in the same place in the diagram if it were fully covered by this type of structure; similarly a star intermediate between two kinds of structure should have its surface close to 100% covered with the two kinds of structures. In the solar case we know from Yohkoh/SXT observations that active regions, their cores and flares (when they occur) take a small fraction of the surface of the Sun, the rest taken by the very faint background corona (as shown by very long exposure images). S97 has shown that very faint coronae have a surface flux comparable to that of very faint solar background corona. We can, therefore, imagine that also on active stars a faint background corona covers the surface not covered by active structures.**]{}

On the basis of the above interpretation, then, we see that the coronae of stars with extremely low surface flux and hardness ratio could be just a large solar coronal hole as, on the other hand, S97 suggested. Continuing this exercise, by increasing the fraction of stellar surface covered with active regions, or with the cores of active regions,[*or a combination thereof,*]{} the corona spans the strip covered by coronae of late-type-stars from low to intermediate activity.

Therefore we can draw a [*hypothesis according to*]{} which the levels of stellar coronal X-ray emission are due to different fractions of the stellar surface covered with active regions or even cores of young active regions. The fraction of more active components increases going towards coronae of higher $F_X$ and higher HR. With one notable exception: the few, very-high-$F_X$ ($> ~10^{6.5} ~erg~cm^{-2}~s^{-1}$) and very-high-HR ($>~~-0.3$), late type stars. The [*average*]{} characteristics of these few stars are beyond the [*peak*]{} values of the solar coronal active regions cores. Yet they are well below [**conditions of solar flares of interest here**]{}. The most active late type stars can, therefore, be explained as being partly covered with active regions, their cores and with flares or flare-like (i.e. very hot and dense) structures [*at all times*]{}. Very active stars may then be continuously subject to several uncorrelated flares whose light curves overlap randomly, canceling most of the evidence of variability. Some authors have already proposed the presence of continuous flaring activity in stellar coronae (Wood et al. 1997; Drake et al. 2000; Guedel et al. 2002, and references therein).

Flares can contribute to coronae of low activity on the strip as long as the flares are small; indeed large flares in low activity stars are very evident and make their $F_X$ - HR point move along a path much less inclined than the strip, bringing them out of the strip itself. There appears to be no match in the solar corona for those super-hot and super-luminous flares detected on very active stars and on RS CVns with the Beppo-SAX and ASCA satellites (Franciosini et al., 2001; Maggio et al. 2000; Favata et al. 2000; Favata and Schmitt, 1999; Schmitt and Favata, 1999).

Global scaling relations
------------------------

The $F_X$ - HR diagram provides information on the average features of the coronal loops forming the structures. The amount of energy emitted from the coronal part of any loop is approximately one half of the entire heating of the loops per unit cross-sectional area (approximately equal to the part of the solar surface covered by the loops footpoints), the other half being emitted from the transition region (Vesecky et al., 1979). Therefore, considering that $F_X$ is proportional to the heating

$F_X = ~K~F_{tot}= ~K~E_H~L$

and, using the scaling laws by Rosner, Tucker and Vaiana (1978), in the following RTV78, i.e.

$$E_H= ~10^5~ p^{7/6}~ L^{-5/6} \label{eq:RTVEH}$$

and $$T= ~1.4 \times 10^3 ~(p~L)^{1/3} \label{eq:RTVT}$$

we find $F_X = ~ K ^ \prime ~p^{7/6}~L^{1/6} =~ K ^ \prime ~p~(pL)^{1/6}= ~K ^{\prime \prime}~p~T^{1/2}$

where $K$, $K ^ \prime $ and $K ^{\prime \prime}$ are proportionality factors, $F_{tot}$ is the total radiated flux, $E_H$ is the heating per unit volume averaged along field lines, p is the coronal plasma pressure, T is the coronal loop [**maximum**]{} temperature and L is the half-length of the magnetic loop. Then, given the very weak dependence on T, $F_X$ is very close to being linearly related to the coronal pressure. A closer look to the above derivation shows that, on one hand, the linear relation between $F_X$ and $F_{tot}$ is appropriate only for a large part (but not all) of the temperature range, since one expects that for very high T (above a few $10^7$ K) or very low T (below $10^6$ K) different fractions of $F_{tot}$ fall outside the ROSAT band. Nonetheless the main point of the above exercise is to show that $F_X$ variations are largely dominated by pressure variations. Of course, HR changes are related to temperature changes.

Using the simple relation derived above and the HR vs. T relation in Fig. \[fig2\], it is straightforward to translate the $F_X$ vs. HR diagram into an “average pressure” vs. “average temperature” diagram, deriving the temperature from the single-valued part ($T<~4 \times
10^6$ K) of the hardness ratio and then the pressure as $p= F_X/(
K^{\prime \prime}~T^{0.5})$. Fig. \[fig6\] shows the pressure-temperature graphs for solar structures, for the whole solar corona, for the F and G stars in S97, and for the G stars (from F7 to K2) in Marino et al. (2001)

The data set by Marino et al. (2001) is the most homogeneous one of all those reported here. We have fitted the relevant data, finding the relation $$p =~ 1.2 \times 10^{-3} ~ {T_6}^{5.2} \label{eq:pT}$$ shown as a dashed line in all the graphs in Fig. \[fig6\] ($T_6$ is the temperature in units of $10^6$ K); more precisely the power index found from the fitting is $5.2 \pm 0.3$. It is quite interesting that all the stars, the solar corona during the cycle and the various coronal structures follow this law, which shows a rather steep dependence of the confined plasma pressure on the plasma temperature, probably governed by the heating mechanism(s) due to the magnetic field dissipation.

[*Flares appear to depart, to some extent, from such a law, more specifically there appear to be a flattening of the pressure - temperature relation with slightly lower pressure than that predicted by the above scaling law.*]{}

The diagonal strip covered by the solar corona at various phases, and by solar coronal structures is a coarse but firm relation between average pressure and temperature of the loops at different evolutionary stages of the solar structures they form, from emergence to decay. There is an analogous, striking, relation for stellar coronae at different activity levels.

Steady loop models cannot be applied to the rise phase of flares. However they can be applied, with caveats, to flares during the slow evolution of the decay phase. Because of this, the pressure values corresponding to solar flares are less precise than all the others and have to be taken with care. Indeed the large spread of solar-flares-related points suggests the limits of using RTV78 scaling laws on flares.

We also note that the pressure values of the solar structures appear slightly lower than [*the values predicted according to eq. (\[eq:pT\]*]{}); we should consider, however, that the surface considered, in the Yohkoh/SXT image, is invariably not uniformly filled with loops. Therefore the local X-ray flux (and the average pressure) is higher than what we measure, even for the solar structures. Determining in detail the area really filled with loops, from each Yohkoh/SXT image, is a formidable task. We note in passing that this choice simply skips on the problem of the surface filling factor by coronal structures, probably present also below the angular resolution (see however Appendix A).

An equivalent, albeit slightly different, form of the same relation is shown in Fig. \[fig7\] where we show the average density vs. temperature, derived with the ideal gas law for a fully ionized hydrogen plasma

$p = ~ 2~n~k_B~T$

where $n$ is the Hydrogen number density and $k_B$ is the Boltzmann constant. In this figure we have put together all the solar and the stellar data; obviously we find $$n = ~4.3 \times 10^6~ {T_6}^{4.2}$$

Using the RTV78 scaling law \[eq:RTVT\] and substituting there the p-T relation found above, it is straightforward to translate these graphs and laws into the analogous ones relating the characteristic structure length (L) vs. temperature (T). Fig. \[fig8\] shows a graph for the same data as above; the L-T scaling is $$L= ~3.0 \times 10^{11} ~{T_6}^{-2.2}.  \label{eq:LT}$$

Analogously, substituting the relations \[eq:pT\] and \[eq:LT\] into the scaling law \[eq:RTVEH\] we obtain the relation $$E_H= ~1.1 \times 10^{-8} ~{T_6}^{7.9} \label{eq:EHT}$$ shown as a dashed line in Fig. \[fig9\] which also shows the same solar and stellar data in a $E_H$ vs. T graph.

[*Flares appear to depart from all the above scaling laws and, in particular, to yield a lower volumetric heating than predicted by the above scaling law, as an implication of the departure from the pressure - temperature scaling law. This may just be an implication of the non-stationarity of the relevant X-ray emitting plasma. However, since many of such data are related to the decay phase, i.e. when the loops are in quasi-steady conditions and RTV scaling laws should be applicable, we should leave open the possibility that there is indeed a difference in the dissipation mechanisms involved in coronal steady loops and flares.*]{}

Since dissipation mechanisms tend to connect volumetric heating with the characteristic size, we are interested to derive a volumetric-heating vs. structure-length law. Combining the above scaling laws we find $$E_H= ~1.7 \times 10^{33}~~L^{-3.6}, \label{eq:EHL}$$ also probably dictated by the dispersal and dissipation mechanisms of the magnetic field.

All the above derivation does not take into account the fraction of the stellar surface covered by the X-ray emitting region, i.e. the surface filling factor, for clarity’s sake. Appendix A contains the generalization of the derivation, considering the surface filling factor.

Discussion
==========

Stellar coronae vs. the solar corona and its building blocks
------------------------------------------------------------

We have described the coronae of late-type stars of different activity levels as covered by correspondingly different fractions of structures similar to those present in the solar corona. We have classified and grouped these structures as background corona, active regions, active regions cores and flares, in order of increasing surface X-ray flux and hardness ratio. Indeed, we find the same correlation of $F_X$ vs. HR for coronae of late-type stars and for coronal structures present on the Sun. The solar coronal structures, from background corona to large flares, fall in the same region of the $F_X$ and HR diagram where the late-type stars coronae, from very inactive to extremely active, are located.

The occurrence of a large flare on a star of medium [*or low*]{} activity level should make the star’s position move, in this diagram, toward the right (i.e. with a significant increase of HR) with relatively moderate increase of $F_X$ (Paper II; Orlando et al., 2002). [*Thus significant flares in these stars should be very prominent for their variability but also because they should make the star move outside the strip covered by coronae.*]{}

The most active late type stars appear to have both $F_X$ and HR significantly higher, even than the active region cores, but one can still explain them assuming a significant presence, at any time, of flares or of flare-like structures. Although one cannot exclude that even moderate activity stars may have a contribution due to flares, most of their behavior can be easily explained with active regions. Flares, in analogy with the solar case, should lead to relatively fast evolution along more horizontal paths slightly departing from the main region of active stars. Many flares may go undetected for a variety of reasons, most likely for the very moderate increase of stellar flux or for the low signal to noise of the observation; undetected small flares may however lead to a spread along the HR direction of the locus of the stars in the $F_X$-HR diagram. The diagram in Marino et al. (2001), containing M stars, indeed shows that the M stars are more spread along the HR direction than the G stars; it is quite fitting that M stars are more variable than G ones on short time scales (Marino et al. 2002).

Only the very late evolutionary phases of active region cores, going toward rather low HR but high surface flux appear to deviate a little from this scenario. However the fraction of surface covered by these very old cores appear to be rather small and so, their emission may be irrelevant with respect to that of the whole stellar corona, as it happens in the solar corona.

The power laws and a unified view of the corona
-----------------------------------------------

The X-ray observations made with Skylab, thanks to their several months coverage, allowed for the first time to study the long term evolution of the X-ray corona (for a review, see Vaiana and Rosner, 1978). It became clear that the magnetically confined coronal plasma evolves from the compact, hot, bright and high pressure loops of emerging active regions to progressively longer, cooler, dimmer and relatively low-pressure loops, first of active regions and then of the background corona. In this context, therefore, one can also consider the $F_X$ vs. HR diagram for solar coronal structures as an evolutionary path. The structures move along the region covered by the various structures starting as very high $F_X$, pressure and temperature and small L and, as they evolve, become of progressively lower $F_X$, pressure, and temperature and get longer.

The various structures of confined plasma, when reported in the pressure-temperature diagram of Fig. \[fig6\] or, equivalently, in the others, in Figs. \[fig7\], \[fig8\] and \[fig9\], are all located along a particular track, probably resulting from the physical effects governing the dispersion and dissipation of the magnetic field as it evolves from emergence to total dispersion into very background corona. Interestingly enough, this track characterizes both individual structures in the solar corona and the solar corona as a whole at different phases of the solar cycle. The global characteristics are the results of a sort of averaging the features of all the structures present in the corona, with a weight due to the surface coverage and to the relative brightness. As a consequence, the average conditions of the solar corona at minimum of the cycle are closer to those of the background corona, even if some active regions may be present, and those of the corona at maximum are closer to active region conditions.

The track also describes the global features of the coronae of solar-type stars. While in the solar case we are able to resolve the entire set of structures at the various evolutionary stages, each star instead just yields one data point in such a diagram, which yields an average over all the structures present in corona. The coronal configuration of the solar-type stars, including the Sun, results from the competing effects of emergence of magnetic field (generated by the dynamo mechanisms inside the star) and the magnetic field dispersal and dissipation in the outer stellar layers; as a consequence, the average values corresponding to each point in the diagram are determined by these competing physical effects. The higher the emergence rate, the more the point characterizing the stellar corona moves towards the high-pressure-high-temperature part of the graph, as we observe on the Sun (Papers II and IV).

In general, stellar coronae should have a complex coronal composition, according to this scenario, with different fractions of their surface covered with compact, hot and bright coronal loops and possibly with interactions among magnetic fields of different AR (Drake et al., 2000). A more active star has a higher number of continuously emerging AR with very effective magnetic field amplification and dissipation mechanisms.

We also claim that each star should have a distribution of structures of confined plasma obeying this sort of universal pressure-temperature relationship with loop plasma pressure higher for higher average loop temperature.

We note that flares appear to behave differently, as for the pressure - temperature relation and the related heating - temperature scaling law. As mentioned, this may depend on the non-stationarity of the relevant X-ray emitting structures but since some of the observations pertain to the decay flares phase when the departure from stationarity is small, we should leave open the possibility that heating mechanisms in flares are indeed different from those of steady structures.

*But, if each of the structures evolve similarly and are equivalent to the solar ones (as we assume) and the various coronae are just the sum of the same structures, one should detect all the coronae just as a time-average of identical coronal structures, thus all with the same characteristics, independently of the number of structures present. So why a higher emergence rate should lead to higher average surface flux and temperature?*

On one hand, a higher emergence rate of magnetic structures brings more ARs and so more X-ray emitting regions to the surface per unit time, leading to higher luminosity and average X-ray flux of the star.

As for the temperature and HR, the emission measure of an AR and of its core are significantly larger around their emergence (Orlando et al., 2004); also flares in an AR are more frequent after emergence. The unresolved corona approximate the time-average of the structures only if the number of structures of various age visible at any time is so large to sample all the evolutionary phases, from the AR emergence to the “final decay”, and including flares. We doubt that a low-activity star has enough ARs to yield a reasonable “average behavior of structures”. Flares make the “poor sampling” effect even worse.

Typically a “low activity” star is caught in a non-flaring phase and, if flares occur, they are easily identified and typically analyzed separately from the rest of the observation. Instead, when we observe a very active star, probably one (or more) of its ARs is in some phase of a flare. Indeed, in order to explain the most active stars we need something brighter and hotter than just active regions, most likely flares, at any time (Drake et al., 2000; Paper III).

Even on the most active stars, however, we may not get a large enough number of ARs to sample well the various flare phases, despite we may observe enough ARs to sample (and average) their non-flaring evolution. The stars in the highest part of the $F_X$ - HR diagram appear to vary somehow from one observation to the other (Micela, private communication). The locus identifying each very active star in the diagram may wander around a bit, probably because of flares; the centroid of the stars’ loci at any time may yield the average star behavior and their spread relative to the centroid may yield the stellar variations, apart from experimental uncertainties and stellar peculiarities.

The invariable presence of flares probably makes a further big difference between less active and very active stars.

For completeness we should consider at least two additional effects: 1) a large number of ARs may subtract substantial amount of surface to the faint, diffuse, structures; so the corona is slightly biased toward higher temperatures and brightness with respect to the average of the structures evolution we would see on the Sun; 2) as mentioned above if the ARs crowding is substantial, ARs may interact triggering even more flares than expected from just a sum of solar structures.

We have shown that we are able to explain the correlation of $F_X$ vs. HR of late-type-stars coronae present in the data of S97, and all the scaling laws we have derived, as due to the plasma structures making the coronae. Now the focus moves to explaining why [*X-ray emitting*]{} solar structures follow that very relationship, a task which probably should be accomplished within solar physics and should address the mechanisms governing magnetic field dispersal and dissipation.

We acknowledge useful suggestions by E. Franciosini, A. Maggio, A. Marino, G. Micela, J. Sanz-Forcada, S. Sciortino and B. Steltzer [*and by an anonimous referee who helped to improve the paper*]{}. This work was partially supported by Agenzia Spaziale Italiana and by Ministero della Istruzione, Università e Ricerca.

Generalization of the global scaling laws to include the filling factor
=======================================================================

While in the text we have derived the stellar X-ray surface flux by dividing the stellar X-ray luminosity $L_X$ by the stellar surface $S_*$, a more correct derivation of the X-ray surface flux at the star should consider that only the area $A$ on the star may be covered by coronal structures. Therefore the whole X-ray luminosity should be $L_X=~ A~E_H~L$ and therefore the surface flux we have derived should include a surface filling factor:

$F_X= ~L_X/S_*=~~A~E_H~L/S_*=~~f~E_H~L$

where $f$, the surface filling factor, is the fraction of the stellar surface covered with coronal structures.

Therefore we would obtain $F_X=~~f~p~T^{1/2}$ and thus, we would have to relate the $f~p$ product with temperature, rather than pressure alone. Thus equation \[eq:pT\] would become\
$f~p =~ 1.2 \times 10^{-3} ~ {T_6}^{5.2}$

and we would obtain

$f~n \propto T^{4.2}$,

$ L/f= ~3.0 \times 10^{11} ~{T_6}^{-2.2}$ instead of equation \[eq:LT\]

and

$f^2 ~ E_H= ~ 1.1 \times 10^{-8} ~{T_6}^{7.9}$ instead of equation \[eq:EHT\].

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[^1]: It is worth noting that we have omitted, from the sample of S97 stars, those with $HR \approx -1$, i.e. $T \approx 10^6$ K since the HR becomes inadequate (see also Fig. \[fig2\]); also small HR errors around that temperature yield large temperature errors.

[^2]: Yohkoh/SXT most likely underestimates the emission of plasma with a temperature around $10^6$ K or lower, as well as the relevant emission measure, while the same plasma would have significant emission in the ROSAT/PSPC band (see Judge, Solomon, Ayres, 2003). The solar data points relative to background corona and/or the very quiet Sun at cycle minimum may have to be shifted at higher flux; this effect is not crucial, however, in the context of this paper.
