Analytic Models of Brown Dwarfs and the Substellar Mass Limit

We present the analytic theory of brown dwarf evolution and the lower mass limit of the hydrogen burning main-sequence stars and introduce some modifications to the existing models. We give an exact expression for the pressure of an ideal nonrelativistic Fermi gas at a finite temperature, therefore allowing for nonzero values of the degeneracy parameter. 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 0.064(circle dot)-0.087(circle dot). An analytic formula for the luminosity evolution allows us to estimate the time period of the nonsteady state (i.e., non-main-sequence) nuclear burning for substellar objects. We also calculate the evolution of very low mass stars. We estimate that similar or equal to 11% of stars take longer than 10(7) yr to reach the main sequence, and similar or equal to 5% of stars take longer than 10(8) yr.