The effective temperature -- radius relationship of M-dwarfs

TL;DR

This paper compares theoretical models of M-dwarfs with observations, revealing a smooth temperature-radius relationship and identifying an empirical signature of electron degeneracy effects, rather than a discontinuity at 0.2 solar masses.

Contribution

It demonstrates that the observed discontinuity is an artifact of data fitting and links the transition to electron degeneracy, refining understanding of low-mass star models.

Findings

No predicted discontinuity at 0.2 solar masses in models.

Empirical discontinuity is due to data fitting, not stellar structure transition.

Transition to electron degeneracy influences the temperature-radius relationship.

Abstract

M-dwarf stars provide very favourable conditions to find habitable worlds beyond our solar system. The estimation of the fundamental parameters of the transiting exoplanets rely on the accuracy of the theoretical predictions for radius and effective temperature of the host M-dwarf, hence the importance of multiple empirical tests of very low-mass star (VLM) models, the theoretical counterpart of M-dwarfs. Recent determinations of mass, radius and effective temperature of a sample of M-dwarfs of known metallicity have disclosed a supposed discontinuity in the effective temperature-radius diagram corresponding to a stellar mass of about 0.2Mo, that has been ascribed to the transition from partially convective to fully convective stars. In this paper we compare existing VLM models to these observations, and find that theory does not predict any discontinuity at around 0.2Mo, rather a smooth change of slope of the effective temperature-radius relationship around this mass value. The appearance of a discontinuity 5is due to naively fitting the empirical data with linear segments. Also, its origin is unrelated to the transition to fully convective structures. We find that this feature is instead an empirical signature for the transition to a regime where electron degeneracy provides an important contribution to the stellar EOS, and constitutes an additional test of the consistency of the theoretical framework for VLM models.

Full text

1 Introduction

M-dwarf stars may be our biggest opportunity to find habitable worlds beyond our solar system. This class of stars comprise $\approx$70% of all stars in the Milky Way, and small earth-like planets are easier to detect when orbiting small stars via both transit and radial-velocity techniques. Also, the habitable zones are much closer to the host star than the case of Sun-like stars, thus increasing the probability of observing a transit (see, e.g., the review by Shields et al., 2016). About 200 exoplanets have been found around M-dwarfs, many of them in their host star habitable zone (see, e.g. Quintana et al., 2014; Anglada-Escudé et al., 2016). Current and planned missions like NASA’s Transiting Exoplanet Survey Satellite ((TESS –) Ricker et al., 2015), and the ESA’s PLAnetary Transits and Oscillations of stars (PLATO) mission (Rauer et al., 2014) will facilitate the discovery of several more planets hosted by M-dwarfs.

The estimation of the fundamental parameters of a transiting exoplanet, such as its mass and radius, rely on the determination of mass and radius of the host star, while the planet surface temperature and the location of the habitable zone depend on the star radius and effective temperature. These determinations often involve the use very low-mass star (VLM) models –models for stars with mass in the range between about 0.5-0.6 $M_{\odot}$ and the minimum mass that ignites H-burning ($\sim 0.1M_{\odot}$)– that are the theoretical counterpart of M-dwarfs (see, e.g., Chabrier & Baraffe, 2000, for a review).

Comparisons of VLM calculations with empirical determinations of M-dwarf radii, masses and effective temperatures, are therefore crucial to assess the reliability of the models, hence the accuracy of the estimated parameters for planets hosted by M-dwarfs (see, e.g., Torres et al., 2010; Feiden & Chaboyer, 2012; Parsons et al., 2018, and references therein). These empirical benchmarks, for example, have disclosed a small average offset between theoretical and empirical mass-radius relationships for VLM models –theoretical radii being smaller at a given mass– by $\sim$3% on average (Feiden & Chaboyer, 2012; Spada et al., 2013; Hidalgo et al., 2018). This is generally ascribed to the effect of large-scale magnetic fields that are not routinely included in VLM model calculations.

Very recently, Rabus et al. (2019) have combined their own near-infrared long-baseline interferometric measurements obtained with the Very Large Telescope Interferometer, with accurate parallax determinations from Gaia Data Release 2, to estimate mass, linear radius, effective temperature and bolometric luminosity of a sample of M-dwarfs of known metallicity. By implementing their own data set with the much larger sample by Mann et al. (2015), they claimed to have found a discontinuity in the effective temperature-radius diagram around $\sim$3200-3300 K, corresponding to a stellar mass of about 0.23$M_{\odot}$. These authors concluded that the discontinuity is likely due to the transition from partially convective M-dwarfs to the fully convective regime, although no comparison with theoretical models was performed.

The most recent sets of VLM theoretical models actually predict the transition to fully convective stars at masses $\sim$0.35$M_{\odot}$ (see, e.g., Chabrier & Baraffe, 1997; Baraffe et al., 2015; Hidalgo et al., 2018), a value about 0.15$M_{\odot}$ larger than assumed by Rabus et al. (2019) to explain the observed discontinuity. The goal of this paper is therefore to reanalyze the results by Rabus et al. (2019), by performing detailed comparisons with theoretical VLM models in the effective temperature-radius diagram, to assess whether models predict this discontinuity and what is its actual physical origin.

In the next section we analyze the mass, radius and effective temperature data employed by these authors by performing detailed comparisons with theoretical VLM models, and identify the physical reason for the discontinuity in the effective temperature-radius diagram. A summary and conclusions follow in Sect. 3.

2 Analysis

The paper by Rabus et al. (2019) –hereafter R19– presents empirical estimates of mass ($M$), radius ($R$) and effective temperature ($T_{\mathrm{e}ff}$) for 22 low-mass dwarfs with masses between $\sim 0.15$ and $\sim 0.55M_{\odot}$, and [Fe/H] between $\sim-$0.6 and $\sim$ +0.5. Eighteen of these object are in common with Mann et al. (2015) –hereafter M15– who determined $M$, $R$ and $T_{\mathrm{e}ff}$ values for a much larger sample (over 180 objects) of the same class of objects. For the stars in common, the $R$ and $T_{\mathrm{e}ff}$ values determined by R19 are generally in good agreement (within the associated errors) with M15 (see Fig. 2 from R19). After merging their results with the rest of M15 sample, R19 performed simple linear fits to the data in the $T_{\mathrm{e}ff}$ -$R$ diagram –without using theory as a guideline– finding a discontinuity of the slope at $T_{\mathrm{e}ff}\sim$3200-3340 K, corresponding to a radius $R\sim 0.22\leavevmode\nobreak\ R_{\odot}$, and a mass $M\sim 0.23M_{\odot}$.

R19 do not present any comparison with theoretical models, but explain this discontinuity as the signature of the transition from partially to fully convective structures. In the following we compare theoretical VLM models with this data, demonstrating that the existence of a discontinuity is only apparent, due to the way the data are fit, and it is not intrinsic of the models. Also, the physical origin of this apparent discontinuity is completely different from that hypothesised by R19.

2.1 Comparison with stellar evolution models

In our analysis we rely on models from the BaSTI-IAC database (Hidalgo et al., 2018)111Models are publicly available at the following URL: http://basti-iac.oa-abruzzo.inaf.it. We refer to Hidalgo et al. (2018) for details about the model physics inputs and the parameter space covered by the calculations. Here we briefly recall that the radiative Rosseland opacity is calculated employing the OPAL results (Iglesias & Rogers, 1996) for temperatures larger than log($T$)=4.0, and Ferguson et al. (2005) results –including contributions from molecules and grains– for lower temperatures. Both high- and low-temperature opacity tables have been computed for the solar-scaled heavy element distribution determined by Caffau et al. (2011). Electron conduction opacity is calculated using the results by Cassisi et al. (2007), whilst for the equation of state (EOS) we employ the ‘FreeEos’ by A. Irwin (see Cassisi et al., 2003, for a short discussion about the characteristics of this EOS). The EOS tables have been calculated with the option EOS1 in Irwin’s code, that provides the best match to the OPAL EOS Rogers & Nayfonov (2002), and to the EOS by Saumon et al. (1995) in the low-temperature and high-density regime relevant to VLM models.

The temperature gradient in superadiabatic surface convective layers is calculated according to the mixing-length theory (Böhm-Vitense, 1958), using the formalism by Cox & Giuli (1968), with the mixing-length parameter ${\rm\alpha_{ML}}$ set to 2.006 as obtained from the standard solar model calibration (see, Hidalgo et al., 2018, for more details). The outer boundary conditions for the model calculations (pressure and temperature at a Rosseland optical depth $\tau$=100) are obtained from the non-grey PHOENIX model atmosphere library (Allard et al., 2012).

The upper panel of Fig. 1 displays the data by M15 (with error bars) in a $T_{\mathrm{e}ff}$ -$R$ diagram. We consider here only the M15 sample, because these authors provide in tabular form also [Fe/H] values in addition to $M$, $R$ and $T_{\mathrm{e}ff}$. Just considering M15 stars does not alter at all the conclusions by R19. By fitting linear relationships to this $T_{\mathrm{e}ff}$ -$R$ diagram as in R19, the discontinuity claimed by R19 is still clearly visible in Fig. 1 at $R\sim 0.22\leavevmode\nobreak\ R_{\odot}$. A least squares linear fit to the data (also displayed in Fig. 1) provides $R/R_{\odot}=0.763(T_{\mathrm{e}ff}/5777)-0.224$ (with a 1$\sigma$ dispersion equal to 0.01 around this mean relation) if $R/R_{\odot}<$ 0.22, and $R/R_{\odot}=2.583(T_{\mathrm{e}ff}/5777)-1.155$ (with a 1$\sigma$ dispersion equal to 0.05 around this mean relation) if $R/R_{\odot}>$ 0.22 . These values of slope and zero point are consistent with R19 results (see their Eq. 7) within their quoted error bars. Also the values of the dispersion of the observed points around these mean linear relationships are consistent with R19 results.

The lower panel of Fig. 1 displays the theoretical $T_{\mathrm{e}ff}$ -$R$ relationships for an age of 10 Gyr, [Fe/H]=$-$0.60, +0.06 and +0.45, masses between 0.1 and $\sim$ 0.6-0.7$M_{\odot}$, as derived from the BaSTI-IAC models. In the same diagram we show also the $T_{\mathrm{e}ff}$ -$R$ relationship for [Fe/H]=0.06 but an age of 1 Gyr, and the 10 Gyr, [Fe/H]=0.0 relationship derived from the independent Baraffe et al. (2015) calculations. Theoretical models display a smooth and continuous change of slope of the $T_{\mathrm{e}ff}$ -$R$ relationship around $R\sim 0.2R_{\odot}$, at temperatures increasing with decreasing [Fe/H]. Notice that the assumed stellar age does not affect the shape of the $T_{\mathrm{e}ff}$ -$R$ relationship, and also that models from different authors give essentially the same result both qualitatively and – at least at metallicities around solar – quantitatively.

To make a close comparison with M15 data, we have created a synthetic sample of stars with our 10 Gyr BaSTI-IAC grid of models, employing the same mass and [Fe/H] distribution of M15 sample, shown in Fig. 2.

In brief, for each individual star in M15 sample, we have considered the mass $M$ and [Fe/H] values given by these authors, perturbed by a Gaussian random error with the same 1$\sigma$ dispersions given by M15 (typically 0.08 dex for [Fe/H], and about 0.1$M$ for the mass). With these ($M$, [Fe/H]) pairs we then interpolated amongst the model grid to determine the corresponding theoretical $R$ and $T_{\mathrm{e}ff}$ values. We have repeated this procedure several times to create 100 synthetic counterparts of M15 sample. In each case a clear change of slope in the $T_{\mathrm{e}ff}$ -$R$ relation does appear. Figure 3 shows one of our synthetic samples, that is representative of the overall result.

A least squares linear fit to the data provides $R/R_{\odot}=0.552(T_{\mathrm{e}ff}/5777)-0.133$ (with a 1$\sigma$ dispersion equal to 0.01 around this mean relation) if $R/R_{\odot}<$ 0.20 , and $R/R_{\odot}=2.094*(T_{\mathrm{e}ff}/5777)-0.936$ (with a 1$\sigma$ dispersion equal to 0.06 around this mean relation) if $R/R_{\odot}>$ 0.20. These two linear relationships are also shown in Fig. 3. The discontinuity claimed by R19 is retrieved in this synthetic sample, despite the lack of discontinuities in the theoretical models, and it is due to fitting the data in this diagram with linear segments. It simply reflects the smooth change of slope of the model $T_{\mathrm{e}ff}$ -$R$ relation.

The synthetic samples share qualitatively the same properties of the observed one, although quantitatively there are some differences. The change of slope in the $T_{\mathrm{e}ff}$ -$R$ relation appears at slightly lower radii compared to M15. Also, the actual values of slopes and zero points are slightly different from the empirical result. The reason for these differences becomes clear when examining Fig. 4, that displays fractional differences of $R$ and $T_{\mathrm{e}ff}$ between the synthetic sample of Fig. 3 and M15 –calculated as (observations-theory)– as a function of M15 $T_{\mathrm{e}ff}$ estimate.

For observed $T_{\mathrm{e}ff}$ values up to $\sim$3600 K (corresponding to stellar masses $\sim 0.45M_{\odot}$) there are systematic average constant offsets between theory and observations, and no trends with the empirical $T_{\mathrm{e}ff}$. In this range the average difference in radius is 8$\pm$9 % (observed radii being larger) and 3$\pm$3 % in $T_{\mathrm{e}ff}$ (model temperatures being larger). At higher temperatures there are trends with the observed $T_{\mathrm{e}ff}$, in the direction of both model temperatures and radii becoming increasingly larger compared to observations, and an increasing spread of the differences at a given $T_{\mathrm{e}ff}$ .

Varying the age assumed for the stars in M15 sample from 10 Gyr to 1 Gyr does not alter significantly –the change of the model $R$ and $T_{\mathrm{e}ff}$ for masses up to $\sim$0.4$M_{\odot}$ is almost zero– the results for both the slopes in the $T_{\mathrm{e}ff}$ -$R$ diagram, and the differences of $R$ and $T_{\mathrm{e}ff}$ between models and observations, because of the very slow evolution of stars in this mass range.

2.2 Why a change of slope of the $T_{\mathrm{e}ff}$ -$R$ relationship?

Figure 5 displays the theoretical $M$-$R$ and $M$-$T_{\mathrm{e}ff}$ diagrams in the VLM regime, for an age of 10 Gyr (again, the choice of the age is not critical) and a representative metallicity [Fe/H]=0.06. The empirical data are also shown. Around 0.2$M_{\odot}$ both theoretical and empirical $M$-$T_{\mathrm{e}ff}$ relationships display a clear slope change, the effective temperature starting to decrease with mass faster than at higher masses. Figure 5 gives a clearer visual impression of this effect by showing also a comparison between the results of a linear fit to the theoretical relationship in the mass range between 0.4 and 0.25 $M_{\odot}$, and the actual calculations.

The same holds true for the radius, even though the effect is not obvious in the data. Analogous to the case of the $M$-$T_{\mathrm{e}ff}$ relationship, we display in Fig. 5 also a comparison between the linear fit to the theoretical $M$-$R$ relation in the mass range between 0.4 and 0.25 $M_{\odot}$, and the actual results. Clearly, also the theoretical values for the radius start to decrease faster with mass when $M$ decreases below $\sim$0.2$M_{\odot}$. This same behaviour of radius and effective temperature with mass is predicted by Baraffe et al. (2015) models (see also, e.g., Burrows et al., 1989; Baraffe & Chabrier, 1996; Chabrier & Baraffe, 2000). The combination of these two slope changes causes the change of slope of the $T_{\mathrm{e}ff}$ -$R$ relationship.

The physical origin of this phenomenon is certainly not the transition to fully convective stars, that happens at larger masses. The culprit is the electron degeneracy, as shown by Chabrier & Baraffe (2000) and with more details by Chabrier & Baraffe (1997) –see e.g. Figs. 6, 12 and 13 in Chabrier & Baraffe (1997). For masses below $\sim$0.2 $M_{\odot}$ electron degeneracy starts to provide a sizable contribution to the gas EOS. Therefore, at around 0.2 $M_{\odot}$ the model $M$-$R$ relation begins to change slope (faster decrease with $M$) to eventually approach the $M\propto R^{-3}$ relation below the H-burning limit. As an example, when extrapolated at 0.1 $M_{\odot}$, the the $M$-$R$ linear relation in the regime between 0.4 and 0.25 $M_{\odot}$ (see Fig. 5) provides a radius equal to $\sim$0.15$R_{\odot}$. The actual calculations give $R\sim 0.125R_{\odot}$, whilst the zero-temperature degenerate $M$-$R$ relation for solar chemical composition provides $R\sim 0.07R_{\odot}$.

Figure 6 displays the relationships between central temperature ($T_{c}$) and $M$, and central degeneracy parameter ($\psi=k_{B}T/E_{F}$ where $k_{B}$ is the Boltzmann constant and $E_{F}$ the electron Fermi energy) and $M$, in the same mass range as in Fig. 5. It is very clear the steady decrease of $\psi$ and $T_{c}$ with decreasing mass. The rate of decrease of $T_{c}$ also changes around 0.2 $M_{\odot}$. Around this mass, the ever increasing contribution of the electron degeneracy pressure accelerates the decrease of $T_{c}$ with $M$, that in turn causes a faster reduction of the efficiency of the H-burning via the $p-p$ chain. As a consequence, also the surface luminosity decreases faster with decreasing mass below 0.2 $M_{\odot}$. The trend of the luminosity with $M$ is also shown in Fig. 6, together with the empirical data by M15, that follows the trend predicted by the models.

The change of slope of the $M$-$T_{\mathrm{e}ff}$ relationship in Fig. 5 is then a consequence of the change of the trends of $L$ and $R$ with mass at M=0.2$M_{\odot}$, given that $T_{\mathrm{e}ff}=(L/(4\pi\sigma R^{2}))^{1/4}$ (where $\sigma$ is the Stefan-Boltzmann constant, see also Chabrier & Baraffe, 1997).

3 Summary and conclusions

Recent empirical determinations of mass, effective temperature and radius for a large sample of M-dwarfs (M15, R19) have disclosed a supposed discontinuity in the $T_{\mathrm{e}ff}$ -$R$ diagram, corresponding to a mass $\sim$0.2$M_{\odot}$. R19 have hypothesised that the reason for this discontinuity is the transition to fully convective stars, although they did not perform any comparison with theory.

Here we have compared a set of existing theoretical VLM models to these observations, to assess whether this discontinuity is predicted by theory, and its physical origin. Theoretical models do not show any discontinuity in the $T_{\mathrm{e}ff}$ -$R$ diagram, rather a smooth change of slope around $\sim$0.2$M_{\odot}$. The discontinuity found by R19 arises from fitting the empirical data with linear segments. To this purpose, we have created synthetic samples of stars with the same [Fe/H], mass and error distributions of the observations, fitting the resulting $T_{\mathrm{e}ff}$ -$R$ diagram with linear segments, as done by R19. Despite some small offsets between the $T_{\mathrm{e}ff}$ and $R$ scale of the models and the observations, the linear fits show a discontinuity corresponding to a mass $\sim$0.2$M_{\odot}$, where the theoretical models display a smooth change of slope. As discussed by Chabrier & Baraffe (1997), this smooth change of slope of the models reflects the growing contribution of electron degeneracy to the gas EOS below $\sim$0.2$M_{\odot}$, and not the transition to fully convective stars, that is predicted to happen at $\sim$0.35$M_{\odot}$. A stronger electron degeneracy causes a faster decrease of $R$ with decreasing $M$, and a faster decrease of both central temperature and luminosity with decreasing $M$. These, in turn, induce a steeper decrease of $T_{\mathrm{e}ff}$ with $M$. These changes of slope of the $M$-$R$ and $M$-$T_{\mathrm{e}ff}$ produce a corresponding change in the $T_{\mathrm{e}ff}$ -$R$ diagram. The empirical results by M15 and R19 provide therefore a clear signature of the threshold beyond which electron degeneracy provides an important contribution to the stellar EOS. This data are therefore an additional test of the consistency of the theoretical framework for VLM models.

Acknowledgements.

SC acknowledges support from Premiale INAF MITiC, from INFN (Iniziativa specifica TAsP), PLATO ASI-INAF contract n.2015-019-R0 & n.2015-019-R.1-2018, and grant AYA2013-42781P from the Ministry of Economy and Competitiveness of Spain. The authors acknowledge the anonymous referee for pointing out the discussion in Sect. 3.1.1. of Chabrier & Baraffe (1997).