EDP Sciences
Free Access
Issue
A&A
Volume 578, June 2015
Article Number A129
Number of page(s) 4
Section Stellar atmospheres
DOI https://doi.org/10.1051/0004-6361/201425283
Published online 16 June 2015

© ESO, 2015

1. Introduction

The X-ray emission properties of low-mass main-sequence stars are strongly dependent on stellar mass and rotation (Vilhu 1984; Pizzolato et al. 2003). Although the basic physical mechanisms responsible for heating coronae to X-ray emitting temperatures are not well understood, it is known empirically that coronal temperatures correlate well with levels of X-ray emission (Vaiana 1983; Schrijver et al. 1984; Schmitt et al. 1990, 1995; Güdel et al. 1997; Schmitt 1997; Telleschi et al. 2005, 2007). For example, Telleschi et al. (2005) analysed high-resolution XMM-Newton spectra of six solar analogues with different levels of magnetic activity and showed that , where is the emission measure weighted average coronal temperature (given by Eq. (1)).

An important question is which measure of X-ray emission correlates best with coronal temperature for stars with a range of surface areas and spectral types; the three obvious candidates are X-ray luminosity, LX, X-ray luminosity normalised by the bolometric luminosity, RX = LX/Lbol, and X-ray surface flux, . Schmitt (1997) compared ROSAT hardness ratio measurements for a sample of main-sequence F and G stars with similar measurements for a sample of main-sequence K and M stars and concluded that all low-mass main-sequence stars have the same relation between surface flux and spectral hardness. Similarly, Preibisch (1997) analysed ROSAT spectra for a sample of stars and argued that FX is a better measure than LX, though they did not consider the parameter RX.

The answer to which of these parameters is best correlated with coronal temperature could contribute to our understanding of the magnetic processes responsible for heating coronal plasma to X-ray emitting temperatures and to our understanding of the coronal properties of saturated stars. For example, if RX is the relevant parameter, then this would indicate that temperature scales somehow with how far a star is below the saturation threshold in rotation, given that saturation happens at a single mass-independent value of RX. It would further imply that all saturated stars have similar coronal temperatures. On the other hand, if either LX or FX are the relevant parameters, then it would indicate that among stars that lie above the saturation threshold, lower mass stars have cooler coronae than their higher mass counterparts. The answer can also important for our understanding of stellar radiation in X-rays since stars with hotter corona will emit more photons at higher energies for a given total X-ray flux. This could be important for our understanding of the influences of stars on the upper atmospheres of planets, given that the influence that high-energy radiation has on a planet is highly wavelength dependent (Tian et al. 2005; Cecchi-Pestellini et al. 2009; Chadney et al. 2015).

In this paper, we analyse coronal temperatures for a large sample of stars with different masses to show that FX is indeed the best indicator of coronal temperature and to derive a general scaling law that can be used to predict for all low-mass main-sequence stars. In Sect. 2, we analyse ROSAT measurements of hardness ratio for a sample of nearby X-ray emitting stars. In Sect. 3, we calculate for a sample of stars based on high-resolution XMM-Newton and Chandra spectra and show how it correlated with FX.

2. ROSAT Hardness ratio measurements

To explore the differences between LX, FX, and RX as indicators of coronal temperature, we collect a large sample of ROSAT X-ray observations from the NEXXUS database (Schmitt & Liefke 2004)1. NEXXUS is a database of X-ray measurements for nearby stars and provides a comprehensive compilation of ROSAT measurements, mostly from the ROSAT All-sky Survey (RASS). We collect X-ray luminosities and hardness ratios reported in the database (specifically the HR1 values) for all stars with masses between ~0.2 M and ~1.1 M with Position Sensitive Proportional Counter (PSPC) measurements (mostly as part of RASS). Since hardness ratio is closely correlated with the temperature of the emitting plasma, they can be used to measure coronal temperature (Schmitt et al. 1995; Schmitt 1997). The stellar masses are estimated by converting (BV) colours from the NEXXUS database using a conversion derived from the stellar evolution models of An et al. (2007).

The correlations between hardness ratio and the three measures of X-ray activity are shown in Fig. 1. The red and blue points are for stars above and below 0.65 M respectively (repeating this analysis with smaller mass bins leads to the same conclusion). Although there is overlap between the two mass bins in the LX and RX distributions, at a given temperature, higher mass stars are clearly likely to have higher values of LX and lower values of RX than lower mass stars. On the other hand, the distributions for the two mass bins in the FX plot overlap excellently, providing a good visual indication that there is one mass-independent relation between FX and coronal temperature.

thumbnail Fig. 1

How different measures of X-ray emission scale with hardness ratio for stars of different masses. The hardness ratios and levels of X-ray emission were measured by ROSAT and collected using the NEXXUS database (Schmitt & Liefke 2004). The blue and red points represent stars with masses below and above 0.65 M respectively. The three quantities of interest are X-ray luminosity (left panel), X-ray surface flux (middle panel), and X-ray luminosity divided by the bolometric luminosity (right panel). In order to calculate FX and RX we estimate the surface areas and the bolometric luminosities by assuming and .

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3. The correlation between X-ray surface flux and coronal temperature

In this section, we show in a more comprehensive way that there exists one universal relation between coronal temperature and FX and derive a simple scaling law between the two quantities. For this purpose, we collect coronal temperature estimates from the literature based on multi-temperature emission measure distribution (EMD) fits to high-resolution Chandra and XMM-Newton spectra. Our sample contains 24 stars in total with masses ranging from ~0.1 M to ~1.2 M, including the Sun. For each EMD, we calculate the coronal average temperature by assuming that the logarithm of the average temperature is the emission measure weighted average of the logarithm of the temperatures of each component (Güdel et al. 2007; Johnstone et al. 2014), i.e. (1)where the sums are over all measured components of the EMD with each component having an emission measure of EMi and a temperature of Ti, and EMtot = ∑ iEMi. Stellar radii, FX values, values, and the references for our sample of stars are given in Table 1.

We make the assumption that the coronal temperature, stellar mass, and LX can be related by (2)where a, b, and c are parameters to be fit to the sample. The interesting parameter here is c since its best fit value is dependent on which measure of emission correlates best with coronal temperature. If LX is the best parameter, then c ≈ 0.0; if FX is the best parameter, then c ≈ −1.6; if RX is the best parameter, then c ≈ −3.9. We define our goodness-of-fit parameter, X, for a given set of values for a, b, and c, as (3)where the sum is over all stars given in Table 1 and γi is a parameter that we use to weight the importance of certain stars in our fit. In order to make the set of low mass stars have the same importance in our fit as the larger set of higher mass stars, we assume γi = 1 for all stars with masses above 0.5 M and γi = 4 for all stars with masses below 0.5 M. To find the value of c, we fit the free parameters in Eq. (2) for each value of c between –5 and 1. In Fig. 2, we show the best fit value of X as a function of c. The best fit value of c is –1.45, which shows that coronal temperature correlates with FX much better than with LX and RX.

Table 1

Properties of our sample of stars shown in Fig. 3.

thumbnail Fig. 2

Best fit parameter, X, against the fit parameter c. For each value of c, we fit Eq. (2) to the sample of stars given in Table 1 and calculate X from Eq. (3). The three vertical dashed lines show the value of c that correspond to the different measure of X-ray emission and the dotted vertical line shows our best fit value of c.

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That FX is better than LX and RX can be easily seen in Fig. 3 and is most clear from the two lowest mass stars in the sample, SCR 1845 and Proxima Centauri. Our best fit line in the upper panel of Fig. 3 is given by (4)where is in MK and FX is in erg s-1 cm-2. Our result is consistent with the relation provided by Telleschi et al. (2005) for just the solar analogues in the sample. Despite the large range of masses, all of our stars fit this relation excellently, providing confirmation of our conclusion that one universal scaling law exists between FX and coronal temperature.

4. Conclusions

It is very clear that the emission measure weighted average coronal temperature scales very closely with X-ray surface flux. Our scaling law between FX and coronal temperature can be used to accurately estimate the coronal average temperature of any low-mass main-sequence star when the value of FX is known. Alternatively, FX can be roughly estimated based on the star’s mass and rotation rate (Pizzolato et al. 2003; Wright et al. 2011; Reiners et al. 2014). The scaling law that we derive could be useful as input into coronal models (e.g. Jardine et al. 2002) and stellar wind models that scale wind temperature with coronal temperature (Holzwarth & Jardine 2007; Johnstone et al. 2015a,b).

thumbnail Fig. 3

Average coronal temperature against FX (upper panel), LX (lower-left panel), and RX (lower-right panel), for a sample of low-mass main-sequence stars. As in Fig. 1, blue and red represent stars with masses below and above 0.65 M respectively. The black line in the upper panel shows the relation , where is in MK and FX is in erg s-1. The two data points connected by the dashed line show the Sun at cycle minimum and cycle maximum. The values and references for these stars are listed in Table 1.

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For a star with a given mass and radius, the level of X-ray emission is determined primarily by its rotation rate, with quickly rotating stars emitting more X-rays than slowly rotating stars until a certain threshold where the X-ray emission saturates (Pallavicini et al. 1981; Vilhu 1984). In the unsaturated regime, a star’s X-ray luminosity scales well with its rotation period approximately as (Reiners et al. 2014), which combined with the assumption that , implies that (5)A similar result can be obtained using the scaling laws of Wright et al. (2011). Given that the saturation threshold appears to approximately be at a single mass-independent value of RX, our result implies that in the saturated regime, low-mass stars have cooler coronae than high-mass stars. Assuming and implies that among saturated stars (6)Assuming a saturation Rossby number (=Prot/τc) of 0.13 (Wright et al. 2011) and a convective turnover time given by τc = 15.49(M/M)-1.08 days (Reiners et al. 2014), the mass dependent satiation threshold is approximately given by (7)where we use Ω = 2.67 × 10-6 rad day-1.

The mass dependence of coronal temperature in the saturated regime is an interesting consequence of the fact that coronal temperature depends on FX and not RX and can have consequences for the influence of stellar high-energy radiation on the upper atmospheres of planets. Given that the location of the habitable zone is to first approximation determined by Lbol, all planets within the habitable zones around saturated stars should be exposed to approximately the same stellar X-ray and extreme-UV fluxes regardless of the central star’s mass. However, the spectra of high-mass saturated stars is likely to be harder (i.e. more photons at higher energies) than the spectra of low-mass stars. The higher photon energies will mean that the radiation is likely to penetrate deeper into a planetary upper atmosphere, which could lead to differences in the atmospheric photochemistry and mass loss rates.


Acknowledgments

The authors thank the referee for providing feedback on our research note and acknowledge the support of the FWF NFN project S11601-N16 “Pathways to Habitability: From Disks to Active Stars, Planets and Life”, and the related FWF NFN subproject S11604-N16 “Radiation & Wind Evolution from T Tauri Phase to ZAMS and Beyond”. This publication is supported by the Austrian Science Fund (FWF).

References

All Tables

Table 1

Properties of our sample of stars shown in Fig. 3.

All Figures

thumbnail Fig. 1

How different measures of X-ray emission scale with hardness ratio for stars of different masses. The hardness ratios and levels of X-ray emission were measured by ROSAT and collected using the NEXXUS database (Schmitt & Liefke 2004). The blue and red points represent stars with masses below and above 0.65 M respectively. The three quantities of interest are X-ray luminosity (left panel), X-ray surface flux (middle panel), and X-ray luminosity divided by the bolometric luminosity (right panel). In order to calculate FX and RX we estimate the surface areas and the bolometric luminosities by assuming and .

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In the text
thumbnail Fig. 2

Best fit parameter, X, against the fit parameter c. For each value of c, we fit Eq. (2) to the sample of stars given in Table 1 and calculate X from Eq. (3). The three vertical dashed lines show the value of c that correspond to the different measure of X-ray emission and the dotted vertical line shows our best fit value of c.

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In the text
thumbnail Fig. 3

Average coronal temperature against FX (upper panel), LX (lower-left panel), and RX (lower-right panel), for a sample of low-mass main-sequence stars. As in Fig. 1, blue and red represent stars with masses below and above 0.65 M respectively. The black line in the upper panel shows the relation , where is in MK and FX is in erg s-1. The two data points connected by the dashed line show the Sun at cycle minimum and cycle maximum. The values and references for these stars are listed in Table 1.

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In the text

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