Mass-luminosity relation for FGK main sequence stars: metallicity and age contributions

Abstract

The stellar mass-luminosity relation (MLR) is one of the most famous empirical “laws”, discovered in the beginning of the 20th century. MLR is still used to estimate stellar masses for nearby stars, particularly for those that are not binary systems, hence the mass cannot be derived directly from the observations. It’s well known that the MLR has a statistical dispersion which cannot be explained exclusively due to the observational errors in luminosity (or mass). It is an intrinsic dispersion caused by the differences in age and chemical composition from star to star. In this work we discuss the impact of age and metallicity on the MLR. Using the recent data on mass, luminosity, metallicity, and age for 26 FGK stars (all members of binary systems, with observational mass-errors ≤3 %), including the Sun, we derive the MLR taking into account, separately, mass-luminosity, mass-luminosity-metallicity, and mass-luminosity-metallicity-age. Our results show that the inclusion of age and metallicity in the MLR, for FGK stars, improves the individual mass estimation by 5 % to 15 %.

Appendix: Chemical composition impact on the MLR. Homological approach

According the homology approach applied to the stellar structure equations and assuming ϵ=ϵ ₀ ρ ^λ T ^ν and κ=κ ₀ ρ ⁿ T ^−s for the energy production rate and the opacity, respectively (cf. Cox and Giuli 1968), we can write:

$$ 4d(\ln R) - n d(\ln\rho)+ (4+s)d(\ln T) - d(\ln\emph{L}) = d(\ln M) $$

(7)

On the other hand re-writing the above equation in relation to Sun, we can derive:

$$ \frac{L(r)}{L_\odot(r)} = \biggl(\frac{\epsilon_0}{\epsilon_{\odot0}} \biggr)^{-\alpha} \biggl(\frac{\kappa_0}{\kappa_{\odot0}} \biggr)^{-\beta} \biggl( \frac{\mu_0}{\mu_{\odot0}} \biggr)^{\gamma} \biggl(\frac{M}{M_\odot} \biggr)^{\delta} $$

(8)

where

We want this equation written explicitly as a function of the chemical composition (X, Y and Z). For that we consider that the main source of opacity comes from the bound-free and free-free contributions: κ=κ _bf +κ _ff , where

(9)

(10)

On the other hand assuming the pp-chain

$$\varepsilon(pp) = \frac{2.4\times10^4 \rho X^2}{T_9^{\frac{2}{3}}}e^{\frac{-3.380}{T_9^{\frac{1}{3}}}} $$

and Y≃2Z+Y _p (where Y _p ∼0.25 is the primordial helium value, Tsivilev (2009)) we can finally derive a MLR depending on metal abundance Z:

(11)

Therefore, using this simple approximation, we can conclude that for the solar luminosity a change of the metal abundance (Z) from 0.004 and 0.03 (typical for the metallicity ranges of this work) can have an impact of about 0.25M _⊙. This value is luminosity independent for FGK main sequence stars.