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Analytic Models of Brown Dwarfs and The Substellar Mass Limit

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 ($ψ= \frac{kT}{μ_{F}}$, 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 $0.064M_\odot-0.087M_\odot$. 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 $\simeq 11 \%$ of stars take longer than $10^7$ yr to reach the main-sequence, and $\simeq 5 \%$ of stars take longer than $10^8$ yr.