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We have been using Keck laser guide star adaptive optics to monitor the orbits of ultracool binaries, providing dynamical masses at lower luminosities and temperatures than previously available and enabling strong tests of theoretical models. We have identified three specific problems with theory: (1) We find that model color-magnitude diagrams cannot be reliably used to infer masses as they do not accurately reproduce the colors of ultracool dwarfs of known mass. (2) Effective temperatures inferred from evolutionary model radii are typically inconsistent with temperatures derived from fitting atmospheric models to observed spectra by 100-300 K. (3) For the only known pair of field brown dwarfs with a precise mass (3%) and age determination (~25%), the measured luminosities are ~2-3x higher than predicted by model cooling rates (i.e., masses inferred from Lbol and age are 20-30% larger than measured). To make progress in understanding the observed discrepancies, more mass measurements spanning a wide range of luminosity, temperature, and age are needed, along with more accurate age determinations (e.g., via asteroseismology) for primary stars with brown dwarf binary companions. Also, resolved optical and infrared spectroscopy are needed to measure lithium depletion and to characterize the atmospheres of binary components in order to better assess model deficiencies.
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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 ($psi = frac{kT}{mu_{F}}$, where $mu_{F}$ 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.
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