Analytic Models of Brown Dwarfs and the Substellar Mass Limit

Abstract

We present the current status of the analytic theory of brown dwarf evolution and the lower mass limit of the hydrogen burning main sequence stars. In the spirit of a simplified analytic theory we also introduce some modifications to the existing models. We give an exact expression for the pressure of an ideal non-relativistic Fermi gas at a finite temperature, therefore allowing for non-zero values of the degeneracy parameter (ψ = kT µ F , where µ F is the Fermi energy). We review the derivation of surface luminosity using an entropy matching condition and the first-order phase transition between the molecular hydrogen in the outer envelope and the partially-ionized hydrogen in the inner region. We also discuss the results of modern simulations of the plasma phase transition, which illustrate the uncertainties in determining its critical temperature. Based on the existing models and with some simple modification we find the maximum mass for a brown dwarf to be in the range 0.064M ⊙ − 0.087M ⊙ . An analytic formula for the luminosity evolution allows us to estimate the time period of the non-steady state (i.e., non-main sequence) nuclear burning for substellar objects. Standard models also predict that stars that are just above the substellar mass limit can reach an extremely low luminosity main sequence after at least a few million years of evolution, and sometimes much longer. We estimate that ≃ 11% of stars take longer than 10 7 yr to reach the main-sequence, and ≃ 5% of stars take longer than 10 8 yr.

On the Existence of Brown Dwarfs More Massive than the Hydrogen Burning Limit

“…Combining this approximation with a precscribed entropy jump between the core and the surface from Chabrier et al (1992) yields an effective pressure from which one can back out an effective density and temperature. Note that Auddy et al (2016) use several different parameters to describe the jump in entropy, including a case with no change. This case is probably closest to what we observe in the MESA simulations, and is consistent with the modern result that the phase transition as postulated by Chabrier et al (1992) does not occur (Knudson et al 2004;Morales et al 2010;Knudson and Desjarlais 2017), but for simplicity and ease of comparison, we follow the default choice made by Auddy et al (2016)…”

“…The surface luminosity must be balanced by the nuclear luminosity L N in a steady state. According to Burrows and Liebert (1993) and Auddy et al (2016), low-mass stars do not reach sufficient temperatures to burn all the way through the PPI chain of nuclear reactions, in particular because the 3 He nuclei face a much higher Coulomb barrier. The first two reactions in PPI are p + p → d + e + +ν e and p + d → 3 He + γ.…”

“…Recently Auddy et al (2016) updated the classical analytic model for the hydrogen burning limit from Burrows and Liebert (1993). These models are constructed with the goal of explicitly writing down a formula for the hydrogen burning limit.…”

“…For concreteness, we explicitly write the formulae used in deriving the hydrogen burning limit in this way in the appendix. We follow Auddy et al (2016) with some minor updates, acknowledging that particular details and assumptions in this model are at best rough approximations. Nonetheless, the shape of the M (ψ) curve and the qualitative behavior we will discuss in the following sections are insensitive to the details of these approximations.…”

Prospects for Life on Temperate Planets around Brown Dwarfs

“…On the other hand, a brown dwarf will be below the threshold necessary for core hydrogen burning, yet massive enough to ignite deuterium burning (eg. Burrows & Liebert 1993;Burrows et al 2001;Spiegel et al 2011;Luhman 2012;Auddy et al 2016). This places the mass of a brown dwarf (M BD ) between 13M J M BD 90M J .…”