Light Element Depletion in Contracting Brown Dwarfs and Pre-Main-Sequence Stars

We present an analytic calculation of the thermonuclear depletion of the light elements lithium, beryllium, and boron in fully convective, low-mass stars. Under the presumption that the pre-main-sequence star is always fully mixed during contraction, we find that the burning of these rare light elements can be computed analytically, even when the star is degenerate. Using the effective temperature as a free parameter, we constrain the properties of low-mass stars from observational data, independently of the uncertainties associated with modeling their atmospheres and convection. Our results are in excellent agreement with the detailed calculations of D'Antona & Mazzitelli and Chabrier, Baraffe, & Plez. Our analytic solution explains the dependence of the age at a given level of elemental depletion on the stellar effective temperature, nuclear cross sections, and chemical composition. These results are also useful as benchmarks to those constructing full stellar models. Most importantly, our results allow observers to translate lithium nondetections in young cluster members into a model-independent minimum age for that cluster. Using this procedure, we have found lower limits to the ages of the Pleiades (100 Myr) and Alpha Persei (60 Myr) clusters. Dating an open cluster using low-mass stars is also independent of techniques that fit upper main-sequence evolution. Comparison of these methods provides crucial information on the amount of convective overshooting (or rotationally induced mixing) that occurs during core hydrogen burning in the 5-10 M_☉ stars typically at the main-sequence turnoff for these clusters. We also discuss beryllium depletion in pre-main-sequence stars. Recent experimental work on the low-energy resonance in the ¹⁰B(p, α)⁷Be reaction has greatly enhanced estimates of the destruction rate of ¹⁰B, making it possible for stars with M ≳ 0.1 M_☉ to deplete both ¹⁰B and ¹¹B before reaching the main sequence. Moreover, there is an interesting range of masses, 0.085 M_☉ ≲ M ≲ 0.13 M_☉, where boron depletion occurs on the main sequence in less than a Hubble time, providing a potential "clock" for dating low-mass stars.