Advances in Astronomy

Volume 2016 (2016), Article ID 5743272, 15 pages

http://dx.doi.org/10.1155/2016/5743272

## Analytic Models of Brown Dwarfs and the Substellar Mass Limit

^{1}Department of Physics and Astronomy, The University of Western Ontario, London, ON, Canada N6A 3K7^{2}King’s University College, The University of Western Ontario, London, ON, Canada N6A 2M3

Received 29 January 2016; Revised 6 June 2016; Accepted 8 June 2016

Academic Editor: Gary Wegner

Copyright © 2016 Sayantan Auddy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 . 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 *≃*11% of stars take longer than yr to reach the main sequence, and *≃*5% of stars take longer than yr.

#### 1. Introduction

One of the most interesting avenues in the study of stellar models lies in understanding the physics of objects at the bottom of and below the hydrogen burning main-sequence stars. The main obstacle in the study of very low mass (VLM) stars and substellar objects is their low luminosity, typically of order , which makes them difficult to detect. There is also a degeneracy between mass and age for these objects, which have a luminosity that decreases with time. This makes the determination of the initial mass function (IMF) difficult in this mass regime. However, in the last two decades, there has been substantial observational evidence that supports the existence of faint substellar objects. Since the first discovery of a brown dwarf [1, 2], several similar objects were identified in young clusters [3] and Galactic fields [4] and have generated great interest among theorists and observational astronomers. The field has matured remarkably in recent years and recent summaries of the observational situation can be found in Luhman et al. [5] and Chabrier et al. [6].

In two consecutive papers, Kumar [7, 8] revolutionized the understanding of low mass objects by studying the Kelvin-Helmholtz time scale and structure of very low mass stars. He successfully estimated that stars below about contract to a radius of about in about years, which was a correction to the earlier estimate of years. The earlier calculation was based on the understanding that low mass stars evolve horizontally in the H-R diagram and thus evolve with a low luminosity for a long period of time. However, Hayashi and Nakano [9] showed that such low mass stars remain fully convective during the pre-main-sequence evolution and are much more luminous than the previously accepted model based on radiative equilibrium. Kumar’s analysis showed that, for a critical mass of , the time scale has a maximum value that decreases on either side. Although this crude model neglected any nuclear reactions, it did give a very close estimate of the time scale. The second paper [8] gave a more detailed insight into the structure of the interior of low mass stars. This model was based on the nonrelativistic degeneracy of electrons in the stellar interior. Kumar’s extensive numerical analysis for a particular abundance of hydrogen, helium, and other chemical compositions yielded a limiting mass below which the central temperature and density are never high enough to fuse hydrogen. A more exact analysis required a detailed understanding of the atmosphere and surface luminosity of such contracting stars.

The next major breakthrough in theoretical understanding came from the work of Hayashi and Nakano [9], who studied the pre-main-sequence evolution of low mass stars in the degenerate limit. Although it was predicted that there exist low mass objects that cannot fuse hydrogen, the internal structure of these objects remained a mystery. A complete theory demanded a better understanding of the physical mechanisms which govern the evolution of these objects. It became essential to develop a complete equation of state (EOS).

D’Antona and Mazzitelli [10] used numerical simulations to study the evolution of VLM stars and brown dwarfs for Population I chemical composition , and different opacities. Their model showed that for the same central condition (nuclear output) an increasing opacity reduces the surface luminosity. Thus, a lower opacity causes a greater surface luminosity and subsequent cooling of the object. Their results implied that the hydrogen burning minimum mass is for opacities considered in their model. Furthermore, they showed that objects with mass close to spend more than a billion years at a luminosity of ~.

Burrows et al. [11] modelled the structure of stars in the mass range . They used a detailed numerical model to study the effects of varying opacity, helium fraction, and the mixing length parameter and compared their results with the existing data. Their important modification was that they considered thermonuclear burning at temperatures and densities relevant for low masses. A detailed analysis of the equation of state was performed in order to study the thermodynamics of the deep interior, which contained a combination of pressure-ionized hydrogen, helium nuclei, and degenerate electrons. This analysis clearly expressed the transition from brown dwarfs to very low mass stars. These two families are connected by a steep luminosity jump of two orders of magnitude for masses in the range of .

Saumon and Chabrier [12] proposed a new EOS for fluid hydrogen that, in particular, connects the low density limit of molecular and atomic hydrogen to the high density fully pressure-ionized plasma. They used the consistent free energy model but with the added prediction of a first-order “plasma phase transition” (PPT) [12] in the intermediate regime of the molecular and the metallic hydrogen. As an application of this EOS, they modelled the evolution of a hydrogen and helium mixture in the interior of Jupiter, Saturn, and a brown dwarf [13, 14]. They adopted a compositional interpolation between the pure hydrogen EOS and a pure helium EOS to obtain a H/He mixed EOS. This was based on the additive volume rule for an extensive variable [15] and allowed calculations of the H/He EOS for any mixing ratio of hydrogen and helium. Their analysis suggested that the cooling of a brown dwarf with a PPT proceeds much more slowly than in previous models [11].

Stevenson [16] presented a detailed theoretical review of brown dwarfs. His simplified EOS related pressure and density for degenerate electrons and for ions in the ideal gas approximation. Although corrections due to Coulomb pressure and exchange pressure are of physical relevance, they together contribute less than in comparison to the other dominant term in the pressure-density relationship for massive brown dwarfs (). The theoretical analysis gave a very good understanding of the behavior of the central temperature as a function of radius and degeneracy parameter . Stevenson [16] discussed the thermal properties of the interior of brown dwarfs and provided an approximate expression for the entropy in the interior and in the atmosphere of a brown dwarf. He also derived an expression for the effective temperature as a function of mass.

A method to use the surface lithium abundance as a test for brown dwarf candidates was proposed by Rebolo et al. [17]. Lithium fusion occurs at a temperature of about , which is easily attainable in the interior of the low mass stars. However, brown dwarfs below the mass of never develop this core temperature. They will then have the same lithium abundance as the interstellar medium independent of their age. However, for objects slightly more massive than , the core temperature can eventually reach . They deplete lithium in the core and the entire lithium content gets exhausted rapidly due to the convection. This causes significant change in the observable photospheric spectra. Thus, lithium can act as a brown dwarf diagnostic [18] as well as a good age detector [19].

Following this, an extensive review on the analytic model of brown dwarfs was presented by Burrows and Liebert [20]. They presented an elaborate discussion on the atmosphere and the interior of brown dwarfs and the lower edge of the hydrogen burning main sequence. Based on the convective nature of these low mass objects, they modelled them as polytropes of order . Once again, the atmospheric model was approximated based on a matching entropy condition of the plasma phase transition between molecular hydrogen at low density and ionized hydrogen at high density. The polytropic approximation enabled the calculation of the nuclear burning luminosity within the core adiabatic density profile [21]. While the luminosity did diminish with time in the substellar limit, the model did show that brown dwarfs can undergo hydrogen burning for a substantial period of time before it eventually ceases. The critical mass deduced from this model did indeed match that obtained from more sophisticated numerical calculations [11].

In this work, we give a general outline of the analytic model of the structure and the evolution of brown dwarfs. We advance some aspects of the existing analytic model by introducing a modification to the equation of state. We also discuss some of the unresolved problems like estimation of the surface temperature and the existence of PPT in the brown dwarf environment. Our paper is organized as follows. In Section 2, we discuss the derivation of a more accurate equation of state for a partially degenerate Fermi gas. We incorporate a finite temperature correction to the expression for the Fermi pressure to give a more general solution to the Fermi integral. In Section 3, we discuss the scaling laws for various thermodynamic quantities for an analytic polytrope model of index . In Section 4, we discuss the derivation of the equations [20] connecting the photospheric (surface) temperature with density, where the entropies at the interior and the exterior are matched using the first-order phase transition. In the spirit of an analytic model, we derive simplified analytic expressions for the specific entropies above and below the PPT. We also highlight the need to seek alternate methods given current concerns about the relevance of the PPT in BD interiors. We discuss the nuclear burning rates for low mass objects in Section 5 and determine the nuclear luminosity [21]. In Section 6, we estimate the range of minimum mass required for stable sustainable nuclear burning. In Section 7, we discuss a cooling model and examine the evolution of photospheric properties over time. In Section 8, we estimate the number fraction of stars that enter the main sequence after more than a million years. In the concluding section, we discuss further possibilities for an improved and generalized theoretical model of brown dwarfs.

#### 2. Equation of State

In main-sequence stars, the thermal pressure due to nuclear burning balances the gravitational pressure and the star can sustain a large radius and nondegenerate interior for a long period of time. However, substellar objects like brown dwarfs fail to have a stable hydrogen burning sequence and instead derive their stability from electron degeneracy pressure. A simple but accurate model needs to have a good equation of state that incorporates the degeneracy effect and the ideal gas behavior at a relative higher temperature. Burrows and Liebert [20] give a pressure law that applies to both extremes but has a poor connection in the intermediate zone. Stevenson [16] also gives an empirical relation for the pressure that does include an approximate correction term to connect the two extremes. Here, in order to obtain a more accurate analytic expression for the pressure, we integrate the Fermi-Dirac integral exactly using the polylogarithm functions . The most general expression for the pressure is[22], where for the Fermi gas and and the other variables are the standard constants. For substellar objects, the electrons are mainly nonrelativistic due to the relatively low temperature and density. In the nonrelativistic limit, that is, , the energy density reduces to . Now, rewriting (1) in terms of the energy density giveswhere , , and we have taken . In the limit , for all , the argument of the exponential is negative and hence the exponential goes to zero as . Thus, the integral reduces to the Fermi pressure at zero temperature. However, in a physical situation at finite temperature, the integral can be solved analytically using the polylogs. The details of the exact derivation for a general Fermi integral are shown in Appendix A. The expression for the pressure of a degenerate Fermi gas at finite temperature isThe above expression for pressure is the most general analytic relation for the pressure of a degenerate Fermi gas at a finite temperature. The first term is the zero temperature pressure and the subsequent terms are the corrections due to the finite temperature of the gas and include , the polylogarithm functions of different orders . The expression is terminated after the fourth term as the polylogs fall off exponentially as the gas becomes more and more degenerate. Equation (3) is a natural extension of the first-order Sommerfeld correction [23].

The central temperature of VLM stars and brown dwarfs is of the same order as the electron Fermi temperature and thus the degeneracy parameter is defined aswhere is the electron Fermi energy in the degenerate limit and is the number of baryons per electron and and are the mass fractions of hydrogen and helium, respectively. Other constants have their standard meaning.

Rewriting (3) in terms of the degeneracy parameter and retaining terms only up to second order, we arrive (for ) atwhere is a constant. However, the interior of a brown dwarf is also composed of ionized hydrogen and helium. The total pressure is a combined effect of both electrons and ions; that is, , where is the Fermi pressure for an ideal nonrelativistic gas at a finite temperature. The pressure due to ions for an ionized hydrogen gas can be approximated as

Therefore, the final equation of state for the combined pressure iswhere and is the mean molecular weight for helium and ionized hydrogen mixture and is expressed aswhere is the ionization fraction of hydrogen. It should be noted that changes as one moves from the core (completely ionized) to the surface which is mainly composed of molecular hydrogen and helium.

There are several corrections to the EOS that can be considered. The Coulomb pressure and the exchange pressure (see in Stevenson [16]) are two important corrections to (7). However, as stated earlier, they are less important for more massive brown dwarfs. Hubbard [24] presents the contribution due to the electron correlation pressure, which depends on the logarithm of , the mean distance between electrons. Stolzmann and Blocker [25] present an analytic formulation of the EOS for fully ionized matter to study the thermodynamic properties of stellar interiors. They show that the inclusion of both electron and the ionic correlation pressure results in a ~10% correction to the EOS. Furthermore, Gericke et al. [26] state that the main volume of the brown dwarfs and the interior of giant gas planets are in a warm dense matter state, where correlation energy, effective ionization energy, and the electron Fermi energy are of the same order of magnitude. Thus, the interiors of these objects effectively form a strongly correlated quantum system. Becker et al. [27] give an EOS for hydrogen and helium covering a wide range of densities and temperatures. They extend their ab initio EOS to the strongly correlated quantum regime and connect it with the data derived using other methods for the neighboring regions of the plane. These simulations are within the framework of density functional theory molecular dynamics (DFT-MD) and give a detailed description of the internal structure of brown dwarfs and giant planets. This leads to a correction in the mass-radius relation.

The study of the EOS of brown dwarfs will help in understanding degenerate bodies in the thermodynamic regime that is not so close to the high pressure limit of a fully degenerate Fermi gas. In this context, the Mie-Grueneisen equation of state is of relevance to test the validity of the assumption that the Grueneisen parameter is independent of the temperature [28] at a constant volume . The brown dwarf regime is in a way more interesting than the white dwarf regime since it is not so close to the limit of a fully degenerate Fermi gas. In Appendix C, we have provided analytic expressions for two parameters that are of particular relevance for the brown dwarfs: the specific heat ( or ) and the Grueneisen parameter.

#### 3. An Analytic Model for Brown Dwarfs

In this section, we derive some of the essential thermodynamic properties of a polytropic gas sphere based on the discussion in Chandrasekhar [29]. As is evident from (7), the relation for a brown dwarf is a polytropewhere the index . is a constant depending on the composition and degeneracy and can be expressed (from (7)) aswhere for a simplified presentation we represent the correction terms asand (on using the values of natural constants, we get cm^{4} g^{−2/3} s^{−2}) = is a constant. The solution to the Lane-Emden equation subject to the zero pressure outer boundary condition can be used to arrive at useful results for , , and for the polytropic equation of state (see (7)). The radius can be expressed as[29]. On substituting (10) for , the radius for a brown dwarf can be expressed as the function of degeneracy and mass:Similarly, the expressions for the central density and central pressure are given by the relations and , where the constant and for the polytrope of [29]. On substituting the expression for (13) in these relations, we get

These are the scaling laws of the density and pressure in the interior core of a brown dwarf. Interestingly, these vary with the degeneracy parameter that is a function of time. Thus, a very simple polytropic model can yield the time evolution of the internal thermodynamical conditions of a brown dwarf. From the definition of the degeneracy parameter in (4) and using (14), the central temperature can be expressed as a function of :The central temperature has a maximum for a certain value of , and it increases for greater values of . Further, using (13), we have shown the variation of central temperature as a function of radius . increases as the object contracts under the influence of gravity. It peaks at a certain and then cools over time. The maximum peak temperature increases for heavier objects and also depends on the extent of ionization of hydrogen and helium. Figure 1 shows the variation of the central temperature as a function of radius for different mass ranges. If the critical temperature for thermonuclear reactions is around , we can roughly estimate the critical mass for the main sequence as ~0.085. This is similar to the estimated critical mass (~0.084) for the main sequence (see Figure in Stevenson [16]). However, it should be noted that the estimate of minimum mass is very sensitive to the mean molecular mass . In Figure 1, we have used for fully neutral gas (similar to for cosmic mixture as used in Stevenson [16]). Depending upon the value of , the minimum mass may vary significantly. For example, if we consider a fully ionized gas, that is, , it yields a minimum mass of .