Analytic Models of Brown Dwarfs and the Substellar Mass Limit Sree Ram Valluri University of Western Ontario, Valluri@Uwo.Ca

Western University Scholarship@Western

Physics and Astronomy Publications Physics and Astronomy Department

6-8-2016 Analytic Models of Brown Dwarfs and the Substellar Mass Limit Sree Ram Valluri University of Western Ontario, [email protected]

Shantanu Basu Western University, [email protected]

Sayantan Auddy Western University, [email protected]

Follow this and additional works at: https://ir.lib.uwo.ca/physicspub Part of the Astrophysics and Astronomy Commons, and the Physics Commons

Citation of this paper: Valluri, Sree Ram; Basu, Shantanu; and Auddy, Sayantan, "Analytic Models of Brown Dwarfs and the Substellar Mass Limit" (2016). Physics and Astronomy Publications. 30. https://ir.lib.uwo.ca/physicspub/30 Hindawi Publishing Corporation Advances in Astronomy Volume 2016, Article ID 5743272, 15 pages http://dx.doi.org/10.1155/2016/5743272

Review Article Analytic Models of Brown Dwarfs and the Substellar Mass Limit

Sayantan Auddy,1 Shantanu Basu,1 and S. R. Valluri1,2

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

Correspondence should be addressed to Sayantan Auddy; [email protected]

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.

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 0.064𝑀⊙–0.087𝑀⊙.Ananalytic 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 7 8 than 10 yr to reach the main sequence, and ≃5% of stars take longer than 10 yr.

1. Introduction Kelvin-Helmholtz time scale and structure of very low mass stars. He successfully estimated that stars below about 0.1𝑀⊙ One of the most interesting avenues in the study of stellar 9 contract to a radius of about 0.1𝑅⊙ in about 10 years, which 11 models lies in understanding the physics of objects at the was a correction to the earlier estimate of 10 years. The bottom of and below the hydrogen burning main-sequence earlier calculation was based on the understanding that low stars. The main obstacle in the study of very low mass (VLM) mass stars evolve horizontally in the H-R diagram and thus starsandsubstellarobjectsistheirlowluminosity,typically 10−4𝐿 evolve with a low luminosity for a long period of time. of order ⊙, which makes them difficult to detect. There However, Hayashi and Nakano [9] showed that such low is also a degeneracy between mass and age for these objects, mass stars remain fully convective during the pre-main- which have a luminosity that decreases with time. This makes sequence evolution and are much more luminous than the the determination of the initial mass function (IMF) difficult previously accepted model based on radiative equilibrium. in this mass regime. However, in the last two decades, there Kumar’s analysis showed that, for a critical mass of 0.09𝑀⊙, has been substantial observational evidence that supports the the time scale has a maximum value that decreases on existence of faint substellar objects. Since the first discovery either side. Although this crude model neglected any nuclear of a brown dwarf [1, 2], several similar objects were iden- reactions,itdidgiveaverycloseestimateofthetimescale. tified in young clusters [3] and Galactic fields [4] and have The second paper [8] gave a more detailed insight into generated great interest among theorists and observational the structure of the interior of low mass stars. This model astronomers. The field has matured remarkably in recent was based on the nonrelativistic degeneracy of electrons years and recent summaries of the observational situation can in the stellar interior. Kumar’s extensive numerical analysis be found in Luhman et al. [5] and Chabrier et al. [6]. foraparticularabundanceofhydrogen,helium,andother Intwoconsecutivepapers,Kumar[7,8]revolutionized chemical compositions yielded a limiting mass below which the understanding of low mass objects by studying the the central temperature and density are never high enough 2 Advances in Astronomy to fuse hydrogen. A more exact analysis required a detailed density for degenerate electrons and for ions in the ideal understanding of the atmosphere and surface luminosity of gas approximation. Although corrections due to Coulomb such contracting stars. pressure and exchange pressure are of physical relevance, The next major breakthrough in theoretical understand- they together contribute less than 10%incomparisonto ing came from the work of Hayashi and Nakano [9], who the other dominant term in the pressure-density relationship studied the pre-main-sequence evolution of low mass stars for massive brown dwarfs (𝑀 ≥ 0.04𝑀⊙). The theoretical in the degenerate limit. Although it was predicted that there analysis gave a very good understanding of the behavior of the exist low mass objects that cannot fuse hydrogen, the internal central temperature 𝑇𝑐 as a function of radius and degeneracy structure of these objects remained a mystery. A complete parameter 𝜓. Stevenson [16] discussed the thermal properties theory demanded a better understanding of the physical of the interior of brown dwarfs and provided an approximate mechanisms which govern the evolution of these objects. expression for the entropy in the interior and in the atmo- It became essential to develop a complete equation of state sphere of a brown dwarf. He also derived an expression for (EOS). theeffectivetemperatureasafunctionofmass. D’Antona and Mazzitelli [10] used numerical simulations A method to use the surface lithium abundance as a to study the evolution of VLM stars and brown dwarfs test for brown dwarf candidates was proposed by Rebolo for Population I chemical composition (𝑌 = 0.25, 𝑍= et al. [17]. Lithium fusion occurs at a temperature of about 6 0.02) and different opacities. Their model showed that for 2.5 × 10 K, which is easily attainable in the interior of the the same central condition (nuclear output) an increasing lowmassstars.However,browndwarfsbelowthemassof opacity reduces the surface luminosity. Thus, a lower opacity 0.065𝑀⊙ never develop this core temperature. They will then causes a greater surface luminosity and subsequent cooling have the same lithium abundance as the interstellar medium of the object. Their results implied that the hydrogen burning independent of their age. However, for objects slightly more minimum mass is 𝑀 = 0.08𝑀⊙ for opacities considered massive than 0.065𝑀⊙, the core temperature can eventually 6 in their model. Furthermore, they showed that objects with reach 3×10 K. They deplete lithium in the core and the entire mass close to 𝑀 = 0.08𝑀⊙ spendmorethanabillionyearsat −5 lithium content gets exhausted rapidly due to the convection. aluminosityof∼10 𝐿⊙. This causes significant change in the observable photospheric Burrows et al. [11] modelled the structure of stars in the spectra. Thus, lithium can act as a brown dwarf diagnostic [18] mass range 0.03𝑀⊙–0.2𝑀⊙. They used a detailed numerical as well as a good age detector [19]. model to study the effects of varying opacity, helium fraction, Following this, an extensive review on the analytic model and the mixing length parameter and compared their results of brown dwarfs was presented by Burrows and Liebert [20]. with the existing data. Their important modification was Theypresentedanelaboratediscussionontheatmosphere that they considered thermonuclear burning at temperatures and the interior of brown dwarfs and the lower edge of the and densities relevant for low masses. A detailed analysis hydrogenburningmainsequence.Basedontheconvective of the equation of state was performed in order to study nature of these low mass objects, they modelled them as the thermodynamics of the deep interior, which contained polytropes of order 𝑛 = 1.5. Once again, the atmospheric a combination of pressure-ionized hydrogen, helium nuclei, model was approximated based on a matching entropy and degenerate electrons. This analysis clearly expressed condition of the plasma phase transition between molecular the transition from brown dwarfs to very low mass stars. hydrogen at low density and ionized hydrogen at high density. These two families are connected by a steep luminosity The polytropic approximation enabled the calculation of the jump of two orders of magnitude for masses in the range of nuclear burning luminosity within the core adiabatic density 0.07𝑀⊙–0.09𝑀⊙. profile [21]. While the luminosity did diminish with timein Saumon and Chabrier [12] proposed a new EOS for fluid thesubstellarlimit,themodeldidshowthatbrowndwarfs hydrogen that, in particular, connects the low density limit can undergo hydrogen burning for a substantial period of of molecular and atomic hydrogen to the high density fully time before it eventually ceases. The critical mass deduced pressure-ionized plasma. They used the consistent free energy from this model did indeed match that obtained from more model but with the added prediction of a first-order “plasma sophisticated numerical calculations [11]. phase transition” (PPT) [12] in the intermediate regime of the Inthiswork,wegiveageneraloutlineoftheanalytic molecular and the metallic hydrogen. As an application of this model of the structure and the evolution of brown dwarfs. EOS, they modelled the evolution of a hydrogen and helium Weadvancesomeaspectsoftheexistinganalyticmodelby mixture in the interior of Jupiter, Saturn, and a brown dwarf introducing a modification to the equation of state. We also [13, 14]. They adopted a compositional interpolation between discuss some of the unresolved problems like estimation of the pure hydrogen EOS and a pure helium EOS to obtain the surface temperature and the existence of PPT in the a H/He mixed EOS. This was based on the additive volume brown dwarf environment. Our paper is organized as follows. rule for an extensive variable [15] and allowed calculations of In Section 2, we discuss the derivation of a more accurate the H/He EOS for any mixing ratio of hydrogen and helium. equation of state for a partially degenerate Fermi gas. We Their analysis suggested that the cooling of a brown dwarf incorporate a finite temperature correction to the expression withaPPTproceedsmuchmoreslowlythaninprevious for the Fermi pressure to give a more general solution to the models [11]. Fermi integral. In Section 3, we discuss the scaling laws for Stevenson [16] presented a detailed theoretical review various thermodynamic quantities for an analytic polytrope of brown dwarfs. His simplified EOS related pressure and model of index 𝑛 = 1.5.InSection4,wediscussthe Advances in Astronomy 3 derivation of the equations [20] connecting the photospheric exact derivation for a general Fermi integral are shown in (surface) temperature with density, where the entropies at the Appendix A. The expression for the pressure of a degenerate interior and the exterior are matched using the first-order Fermi gas at finite temperature is phasetransition.Inthespiritofananalyticmodel,wederive 2 1 simplified analytic expressions for the specific entropies 𝑃 ≃𝑎 𝜇5/2 − 𝑎𝛽−1𝜇3/2 (1 + 𝑒−𝛽𝜇) 𝐹 5 8 ln above and below the PPT. We also highlight the need to (3) seek alternate methods given current concerns about the 3 𝜋2 3 + 𝑎𝛽−2𝜇1/2 + 𝑎𝛽−2𝜇1/2 (−𝑒−𝛽𝜇)⋅⋅⋅ . relevanceofthePPTinBDinteriors.Wediscussthenuclear 2 6 4 Li2 burning rates for low mass objects in Section 5 and determine the nuclear luminosity 𝐿𝑁 [21]. In Section 6, we estimate The above expression for pressure is the most general analytic the range of minimum mass required for stable sustainable relation for the pressure of a degenerate Fermi gas at a finite nuclear burning. In Section 7, we discuss a cooling model and temperature. The first term is the zero temperature pressure examine the evolution of photospheric properties over time. and the subsequent terms are the corrections due to the finite In Section 8, we estimate the number fraction of stars that temperature of the gas and include Li𝑠, the polylogarithm enter the main sequence after more than a million years. In functions of different orders 𝑠. The expression is terminated the concluding section, we discuss further possibilities for an after the fourth term as the polylogs fall off exponentially as improved and generalized theoretical model of brown dwarfs. the gas becomes more and more degenerate. Equation (3) is a natural extension of the first-order Sommerfeld correction 2. Equation of State [23]. The central temperature of VLM stars and brown dwarfs In main-sequence stars, the thermal pressure due to nuclear is of the same order as the electron Fermi temperature and burning balances the gravitational pressure and the star can thus the degeneracy parameter 𝜓 is defined as sustain a large radius and nondegenerate interior for a long period of time. However, substellar objects like brown dwarfs 𝑘 𝑇 2𝑚 𝑘 𝑇 𝜇 2/3 𝜓= 𝐵 = 𝑒 𝐵 [ 𝑒 ] , 2/3 (4) fail to have a stable hydrogen burning sequence and instead 𝜇𝐹 (3𝜋2ℏ3) 𝜌𝑁𝐴 derive their stability from electron degeneracy pressure. A simple but accurate model needs to have a good equation where 𝜇𝐹 is the electron Fermi energy in the degenerate limit of state that incorporates the degeneracy effect and the and 1/𝜇𝑒 =𝑋+𝑌/2isthenumberofbaryonsperelectron ideal gas behavior at a relative higher temperature. Burrows and 𝑋 and 𝑌 are the mass fractions of hydrogen and helium, and Liebert [20] give a pressure law that applies to both respectively. Other constants have their standard meaning. extremes but has a poor connection in the intermediate zone. Rewriting (3) in terms of the degeneracy parameter 𝜓 and Stevenson [16] also gives an empirical relation for the pressure retaining terms only up to second order, we arrive (for 𝜇= that does include an approximate correction term to connect 𝜇𝐹)at the two extremes. Here, in order to obtain a more accurate analytic expression for the pressure, we integrate the Fermi- 2 𝜌 5/3 5 𝑃 = 𝑎𝐴5/2 [ ] [1 − 𝜓 (1 + 𝑒−1/𝜓) Dirac integral exactly using the polylogarithm functions 𝐹 5 𝜇 16 ln (𝑥) 𝑒 Li𝑠 . The most general expression for the pressure is (5) 15 𝜋2 ∞ 2 2 −1/𝜓 4𝜋𝑝 1 1 𝑑𝜖 + 𝜓 { + Li2 (−𝑒 )}] , 𝑃 =𝑔 ∫ 𝑑𝑝 ( )( 𝑝 ) 8 3 𝐹 𝑠 3 𝛽(𝜖−𝜇) (1) 0 (2𝜋ℏ) 𝑒 +𝑏 3 𝑑𝑝 𝐴 = (3𝜋2ℏ3𝑁 )2/3/2𝑚 [22], where 𝑏=1fortheFermigasand𝜖(𝑝) = where 𝐴 𝑒 is a constant. However, the interior of a brown dwarf is also composed of ionized hydro- √ 2 2 2 4 2 𝑝 𝑐 +𝑚 𝑐 −𝑚𝑐 and the other variables are the gen and helium. The total pressure is a combined effect of 𝑃=𝑃+𝑃 𝑃 standard constants. For substellar objects, the electrons are both electrons and ions; that is, 𝐹 ion,where 𝐹 mainly nonrelativistic due to the relatively low temperature is the Fermi pressure for an ideal nonrelativistic gas at a 2 4 2 2 and density. In the nonrelativistic limit, that is, 𝑚 𝑐 ≫𝑝𝑐 , finite temperature. The pressure due to ions for an ionized 2 the energy density reduces to 𝜖(𝑝) ≃𝑝 /2𝑚.Now,rewriting hydrogen gas can be approximated as (1) in terms of the energy density gives 𝑘𝜌𝑇 𝑃 = . ∞ 3/2 ion (6) 𝜖 𝑑𝜖 𝜇1𝑚𝐻 𝑃 =𝑎∫ , (2) 𝐹 𝑒𝛽(𝜖−𝜇) +1 0 Therefore, the final equation of state for the combined 3/2 3 −1 pressure is where 𝑎= (2/3)(4𝜋(2𝑚) /(2𝜋ℏ) ), 𝛽=(𝑘𝐵𝑇) ,andwe have taken 𝑔𝑠 =2.Inthelimit𝑇→0,forall𝜖<𝜇,the 5/3 2 5/2 𝜌 5 −𝛽𝜇 argument of the exponential is negative and hence the 𝑃= 𝑎𝐴 [ ] [1 − 𝜓 (1 + 𝑒 ) 5 𝜇 16 ln exponential goes to zero as 𝛽→∞.Thus,theintegral 𝑒 (7) reduces to the Fermi pressure at zero temperature. However, 15 𝜋2 in a physical situation at finite temperature, the integral can + 𝜓2 { + (−𝑒−1/𝜓)} + 𝛼𝜓] , 8 3 Li2 be solved analytically using the polylogs. The details of the 4 Advances in Astronomy

where 𝛼=5𝜇𝑒/2𝜇1 and 𝜇1 is the mean molecular weight for where the index 𝑛=3/2. 𝐾 is a constant depending on the helium and ionized hydrogen mixture and is expressed as composition and degeneracy and can be expressed (from (7)) as 1 𝑌 =((1+𝑥 + )𝑋+ ), 𝜇 H 4 (8) −5/3 1 𝐾=𝐶𝜇𝑒 (1+𝛾+𝛼𝜓) , (10) where 𝑥H+ is the ionization fraction of hydrogen. It should where for a simplified presentation we represent the correc- be noted that 𝑥H+ changesasonemovesfromthecore tion terms as (completely ionized) to the surface which is mainly composed 2 of molecular hydrogen and helium. 5 −𝛽𝜇 15 2 𝜋 −1/𝜓 𝛾=− 𝜓 ln (1 + 𝑒 )+ 𝜓 { + Li2 (−𝑒 )} (11) ThereareseveralcorrectionstotheEOSthatcanbe 16 8 3 considered. The Coulomb pressure and the exchange pressure (see (13) in Stevenson [16]) are two important corrections and 𝐶 (on using the values of natural constants, we get 𝐶= 13 4 −2/3 −2 5/2 to (7). However, as stated earlier, they are less important 10 cm g s )=(2/5)𝑎𝐴 is a constant. The solution to for more massive brown dwarfs. Hubbard [24] presents the Lane-Emden equation subject to the zero pressure outer the contribution due to the electron correlation pressure, boundary condition can be used to arrive at useful results for which depends on the logarithm of 𝑟𝑒, the mean distance 𝑅, 𝜌𝑐,and𝑃𝑐 for the polytropic equation of state (see (7)). The between electrons. Stolzmann and Blocker [25] present an radius can be expressed as analytic formulation of the EOS for fully ionized matter 𝐾 to study the thermodynamic properties of stellar interiors. 𝑅 = 2.3573 (12) They show that the inclusion of both electron and the ionic 𝐺𝑀1/3 ∼ correlation pressure results in a 10% correction to the EOS. 𝐾 Furthermore, Gericke et al. [26] state that the main volume of [29]. On substituting (10) for ,theradiusforabrowndwarf the brown dwarfs and the interior of giant gas planets are in a can be expressed as the function of degeneracy and mass: warm dense matter state, where correlation energy, effective 1/3 9 𝑀⊙ −5/3 ionization energy, and the electron Fermi energy are of the 𝑅 = 2.80858 × 10 ( ) 𝜇𝑒 (1 + 𝛾 + 𝛼𝜓) cm. (13) same order of magnitude. Thus, the interiors of these objects 𝑀 effectively form a strongly correlated quantum system. Becker Similarly, the expressions for the central density 𝜌𝑐 and central et al. [27] give an EOS for hydrogen and helium covering a 3 pressurearegivenbytherelations𝜌𝑐 =𝛿𝑛(3𝑀/4𝜋𝑅 ) and wide range of densities and temperatures. They extend their 𝑃 =𝑊(𝐺𝑀2/𝑅4) 𝛿 = 5.991 𝑊 = ab initio EOS to the strongly correlated quantum regime and 𝑐 𝑛 ,wheretheconstant 𝑛 and 𝑛 0.77 for the polytrope of 𝑛=1.5[29]. On substituting the connect it with the data derived using other methods for the 𝑅 neighboring regions of the 𝜌-𝑇 plane. These simulations are expression for (13) in these relations, we get within the framework of density functional theory molecular 𝑀 2 𝜇5 dynamics (DFT-MD) and give a detailed description of the 𝜌 = 1.28412 × 105 ( ) 𝑒 / 3, 𝑐 𝑀 3 g cm (14) internal structure of brown dwarfs and giant planets. This ⊙ (1 + 𝛾 + 𝛼𝜓) leads to a 2.5%–5% correction in the mass-radius relation. 𝑀 10/3 𝜇20/3 The study of the EOS of brown dwarfs will help in under- 𝑃 = 3.26763 × 109 ( ) 𝑒 . 𝑐 4 Mbar (15) standing degenerate bodies in the thermodynamic regime 𝑀⊙ (1 + 𝛾 + 𝛼𝜓) that is not so close to the high pressure limit of a fully degenerate Fermi gas. In this context, the Mie-Grueneisen equation These are the scaling laws of the density and pressure of state is of relevance to test the validity of the assumption in the interior core of a brown dwarf. Interestingly, these that the Grueneisen parameter 𝛾=(𝜕log 𝑇/𝜕 log 𝜌)𝑠 is inde- vary with the degeneracy parameter 𝜓 that is a function of pendent of the temperature 𝑇 [28] at a constant volume 𝑉. time. Thus, a very simple polytropic model can yield the The brown dwarf regime is in a way more interesting than time evolution of the internal thermodynamical conditions the white dwarf regime since it is not so close to the limit of a of a brown dwarf. From the definition of the degeneracy fully degenerate Fermi gas. In Appendix C, we have provided parameter in (4) and using (14), the central temperature can analytic expressions for two parameters that are of particular be expressed as a function of 𝜓: relevance for the brown dwarfs: the specific heat (𝐶V or 𝐶𝑝) and the Grueneisen parameter. 𝑀 4/3 𝜓𝜇8/3 𝑇 = 7.68097 × 108 ( ) 𝑒 . 𝑐 K 2 (16) 𝑀⊙ (1+𝛾+𝛼𝜓) 3. An Analytic Model for Brown Dwarfs The central temperature has a maximum for a certain value In this section, we derive some of the essential thermody- of 𝜓, and it increases for greater values of 𝑀.Further,using namicpropertiesofapolytropicgasspherebasedonthe (13),wehaveshownthevariationofcentraltemperature discussion in Chandrasekhar [29]. As is evident from (7), the 𝑇𝑐 as a function of radius 𝑅. 𝑇𝑐 increases as the object 𝑃 𝜌 - relation for a brown dwarf is a polytrope contracts under the influence of gravity. It peaks at a certain 𝑅 andthencoolsovertime.Themaximumpeaktemperature 𝑃=𝐾𝜌(1+1/𝑛), (9) increases for heavier objects and also depends on the extent Advances in Astronomy 5

6 ×10 Tc versus radius is certainly too low for brown dwarfs. Although the pressure 3.5 estimate is relatively well established in these numerical 3.0 simulations, the phase transition temperature is still a matter of continuing investigation [27]. Having noted these caveats, 2.5 we present the existing model for the surface temperature, 2.0 basedonChabrieretal.[14]andBurrowsandLiebert[20].

(K) We also introduce a simpler treatment of the specific entropy. c

T 1.5 The PPT occurs over a narrow range of densities near 3 1.0 / 𝑥 + ∼ 0.5 1.0 g cm from a partially ionized phase ( H )toaneu- −3 tral molecular phase (𝑥H+ <10).Formassivebrown 0.5 dwarfs, the phase transition occurs nearer to the surface. 0 Burrows and Liebert [20] used the following approximate 0.5 1 2 analytic expressions for the specific entropy (equations (2.48) 10 R (cm) ×10 and (2.49) in Burrows and Liebert [20]) for the two phases of the PPT: 0.070M⊙ 0.080M⊙ 0.075M ⊙ 0.085M⊙ 1 𝜎1 = −1.594 ln + 12.43, Figure 1: The variation of 𝑇𝑐 versus radius 𝑅 for different masses. 𝜓 (17) 𝑇 𝜎 = 1.032 ln − 2.438. 2 𝜌0.42 of ionization of hydrogen and helium. Figure 1 shows the variation of the central temperature as a function of radius Similar expressions for the entropy at the interior and the for different mass ranges. If the critical temperature for 6 atmosphere are given in equations (21) and (22) in Stevenson thermonuclear reactions is around 3×10 K, we can roughly [16]. We use a simplified approach to make the origin of the estimatethecriticalmassforthemainsequenceas∼0.085𝑀⊙. above equation clear in the spirit of an analytic model. We This is similar to the estimated critical mass∼ ( 0.084𝑀⊙)for derive analytic expressions for the entropy of the ionized and themainsequence(seeFigure1 in Stevenson [16]). However, the molecular hydrogen separately for the two phases (similar it should be noted that the estimate of minimum mass is very to equations (2.48) and (2.49) in Burrows and Liebert [20]) sensitive to the mean molecular mass 𝜇1.InFigure1,wehave 𝜇 = 1.23 𝐴 ∼ 1.24 andmatchthemviathephasetransition.Itisassumedthat used 1 for fully neutral gas (similar to for thepresenceofheliumdoesnotaffectthehydrogenPPT[14]. cosmic mixture as used in Stevenson [16]). Depending upon 𝜇 The region between the strongly correlated quantum the value of 1, the minimum mass may vary significantly. For regime and the ideal gas limit can be modelled with correc- example, if we consider a fully ionized gas, that is, 𝜇1 = 0.59, 0.12𝑀 tions to the ideal gas equation. Such correction terms can be it yields a minimum mass of ⊙. expressed by virial coefficients (see equation (1) in Becker et al. [27]). For simplicity, we ignore such corrections in our 4. Surface Properties EOS (see (7)) and consider only the contribution of electron pressure (see (5)) and the ion pressure (see (6)) of the partially In this section, we discuss a very simple but crude model ionized hydrogen (of ionization fraction 𝑥H+ )andhelium which is broadly based on the phase transition proposed by mixture. Chabrier et al. [14] and the isentropic nature of the interior The total entropy for our EOS (see (7)) is the sum of of brown dwarfs. The development of a theoretical model the entropies of the atomic/molecular gas and the degenerate for studying the variation of surface luminosity over time for electrons. The internal energy per gram for the monoatomic lowmassstars(LMS)andbrowndwarfsisagreatchallenge. gas particles is There is no stable phase of nuclear burning for brown dwarfs and the luminosity gradually decreases with time. This leads 3 𝑘 𝑁 𝑇 to an age-mass degeneracy in observational determinations. 𝑈= 𝐵 0 . 2 𝜇 (18) Our lack of knowledge in understanding the physics of the 1 interior of brown dwarfs restricts the development of a com- prehensive model. However, extensive simulations were done Ideally, we can consider the total energy as a combination on the molecular-metallic transition of hydrogen for LMS of kinetic energy, radiation energy, and ionization energy and planets [13, 14]. Chabrier et al.’s [14] model predicts a first- (B.3). But as the electron gas is degenerate, the radiation ordertransitionforthemetallizationofhydrogenatapres- pressure is relatively unimportant. Furthermore, as shown in sure of ∼1Mbarandcriticaltemperatureof ∼15300 K. Such Appendix B, we may even neglect the contribution of the pressure and temperature values are appropriate for giant ionization energy as the gas is only partially ionized. Taking planets and brown dwarfs. Modern numerical simulations the partial derivatives of the expression for the internal energy [30, 31] do confirm the existence of such phase transitions at equation(18),wecanexpressthechangeinheat𝑑𝑄 using the same pressure range but predict a much different range of the first law of thermodynamics (see (B.2)). However, as the temperature ∼2000 K–3000 K. This new temperature regime ionization fraction 𝑥H+ (𝜌, 𝑇) is a function of density and 6 Advances in Astronomy temperature, we further use Saha’s ionization equations (B.5) (see Table 1) for the phase transition at each point of the to get coexistence curve of PPT [12] is used to estimate the relation |𝐶1 −𝐶2| between the two constants in (20) and (22). For 𝑑𝑄 3 𝑘 𝑁 𝑑𝑇 𝑘 𝑁 𝑑𝑊 3 2 𝑑𝑇 𝐵 0 𝐵 0 𝑇=𝑇eff and 𝜌2 =𝜌𝑒 in (22), we can use (23) and (22) in (24) =− + +( ) 𝐻𝑘𝐵𝑁0 𝑇 2 𝜇1 𝑇 𝜇1 𝑊 2 𝑇 and the value of |𝐶1 −𝐶2| to obtain a wide range of possible (19) 𝑇 3 𝑑𝑉 values of surface temperature eff in terms of the degeneracy + 𝐻𝑘 𝑁 , parameter and photospheric density 𝜌𝑒: 2 𝐵 0 𝑉 𝑇 =𝑏 ×106𝜌0.4𝜓V . 3 eff 1 𝑒 K (25) where 𝑊=𝑇𝑉, 𝐻=𝑥H+ (1 − 𝑥H+ )/(2 − 𝑥H+ ).Amore 𝑏 V generalized version including the radiation and the ionization The values of the parameters 1 and for different models are 𝑇𝑑𝑆 =𝑑𝑄 showninTable1.AccordingtoChabrier’smodel,thecritical termsisshownin(B.6)inAppendixB.For ,we 4 −3 integrate(19)toexpresstheentropyintheinteriorofthe temperature ∼1.53 × 10 K and critical density ∼0.35 g cm brown dwarf as mark the end of the phase transition with Δ𝜎 = Δ𝑆/𝑘𝐵𝑁= 0. In the following discussion, we briefly summarize the 𝑘 𝑁 3/2 𝐵 0 𝑇 steps from Burrows and Liebert [20]. We replace equation 𝑆1 = ln +𝐶1, (20) 𝜇1mod 𝜌1 (2.50) in Burrows and Liebert [20] by (25) to estimate the surfaceluminosity.Asanexample,weselectaparticular where phase transition point (model D) and show the derivation of

1 1 3 𝑥H+ (1 − 𝑥H+ ) surface luminosity using hydrostatic equilibrium and ideal =( + ) (21) gas approximation. The photosphere of a brown dwarf is 𝜇 𝜇 2 2−𝑥 + 1mod 1 H located at approximately the 𝜏=2/3surface, where ∞ and 𝜇1 is different for each model in this region and is 𝑥 + 𝜏=∫ 𝜅𝑅𝜌𝑑𝑟 (26) calculated using H fromTable1.However,thecontributions 𝑟 to entropy due to radiation and the degenerate electrons (see is the optical depth. Using the general equation for hydro- equation 2–145 in Clayton [32]) are negligible in the range of 2 temperature and density applicable for brown dwarfs. Based static equilibrium, 𝑑𝑃 = −(𝐺𝑀/𝑟 )𝜌𝑑𝑟,and(26),thephoto- on a similar argument, the analytic expression for the entropy spheric pressure can be expressed as of nonionized molecular hydrogen and helium mixture at the 2 𝐺𝑀 𝑃 = , photosphere is expressed as 𝑒 2 (27) 3 𝜅𝑅𝑅 𝑘 𝑁 𝑇5/2 𝐵 0 where 𝜅𝑅 is the Rosseland mean opacity and the other 𝑆2 = ln +𝐶2, (22) 𝜇2 𝜌2 variables have their standard meanings. Furthermore, our EOS (see (7)) in the approximation of negligible degeneracy 1/𝜇 = 𝑋/2 + 𝑌/4 where 2 isthemeanmolecularweight pressure near the photosphere gives the photospheric pres- for the hydrogen and helium mixture. The expression for sure as 𝑆 entropy 2 is derived using the first law of thermodynamics 𝜌 𝑁 𝑘 𝑇 (Appendix B) and the relation of the internal energy of 𝑃 = 𝑒 𝐴 𝐵 eff . 𝑒 𝜇 (28) diatomic molecules (𝑈 = (5/2)(𝑘𝐵𝑁0𝑇/𝜇2)). Here, we have 2 considered only five degrees of freedom as the temperatures Now, using the expression for radius 𝑅 (13) in (27), we can are just sufficient to excite the rotational degrees ofH2 calculate the external pressure 𝑃𝑒 as a function of 𝑀 and 𝜓: but the vibrational degrees remain dormant. It should be 11.2193 𝑀 5/3 𝜇10/3 noted that (20) and (22) are just simplified forms of the 𝑃 = bar ( ) 𝑒 . 𝑒 2 (29) entropy expressions presented in Burrows and Liebert [20] 𝜅𝑅 𝑀⊙ (1+𝛾+𝛼𝜓) and Stevenson [16]. Thus, the entropy in the two phases is dominated by con- Onusing(29)in(28)andsubstituting𝑇eff for model D with tributions from the ionic and molecular gas, respectively. 𝑏1 = 2.00 and V =1.60from Table 1, the effective density 𝜌𝑒 Using the same argument as Burrows and Liebert [20] can be expressed as a function of 𝑀 and 𝜓: that the two regions of different temperature and density 6.89811 𝑀 5/3 𝜇10/3𝜇 3 𝜌1.40 = ( ) 𝑒 2 g/cm . are separated by a phase transition of order one, we can 𝑒 2 (30) estimate the surface temperature. Using the expression for the 𝜅𝑅𝑁𝐴𝑘𝐵 𝑀⊙ (1+𝛾+𝛼𝜓) 𝜓1.58 degeneracy parameter 𝜓 from (4), we can simplify (20) to be Substituting the expression for 𝜌𝑒 from (30) in (25), we derive 3 𝑘 𝑁 the expression for effective temperature for model D asa 𝑆 = 𝐵 0 ( 𝜓 + 12.7065) +𝐶 . 1 ln 1 (23) function of 𝑀 and 𝜓: 2 𝜇1mod 𝑇 Furthermore, the jump of entropy eff 2.57881 × 104 𝑀 0.4764 𝜓1.1456 (31) 𝑆2 −𝑆1 = K ( ) . Δ𝜎 = (24) 𝜅0.2856 𝑀 0.5712 𝑘𝐵𝑁0 𝑅 ⊙ (1 + 𝛾 + 𝛼𝜓) Advances in Astronomy 7

Similarly, the surface temperature can be evaluated for all nuclear burning or settle into a steady state main sequence. the other models. Since the procedure is the same for all The thermonuclear reactions suitable for the brown dwarfs themodelsinTable1,wejustshowonecalculation.For and VLM stars are this range of surface temperatures, the Stefan-Boltzmann law, + 2 4 𝑝+𝑝󳨀→𝑑+𝑒 + ], (33) 𝐿=4𝜋𝑅𝜎𝑇eff ,yieldsasetofpossiblevaluesofthesurface luminosity 𝐿 as a function of the degeneracy parameter 𝜓. 3 𝑝+𝑑󳨀→ He +𝛾. (34) The luminosity for model D using (13) and (31) is As the central temperature is not high enough to over- 0.41470 × 𝐿 𝑀 1.239 𝜓4.5797 3 3 𝐿= ⊙ ( ) , come the Coulomb barrier of the He– He reaction and the 𝜅1.1424 𝑀 0.2848 (32) 4 𝑅 ⊙ (1 + 𝛾 + 𝛼𝜓) p-p chain is truncated, He is not produced. Most of the thermonuclear energy is produced from the burning of the 𝜎 where is the Stefan-Boltzmann constant. Substellar objects primordial deuterium (see (34)). The energy generation rates below the main-sequence mass gradually evolve towards of the above processes are given as complete degeneracy and a state of stable equilibrium as their 2 𝜌𝑋 1/3 luminosity decreases over time. In the following sections, we 6 −33.8/𝑇6 𝜖̇𝑝𝑝 = 2.5 × 10 [ ]𝑒 erg/g ⋅ s, show that the degeneracy parameter 𝜓 is a function of time 𝑇2/3 𝜓=0 6 and it evolves towards over the lifetime of brown (35) dwarfs.Thisgivesusanestimateoftheluminosityatdifferent 1/3 24 𝜌𝑋𝑌𝑑 −37.2/𝑇 𝜖̇ = 1.4 × 10 [ ]𝑒 6 / ⋅ epochs of time. 𝑝𝑑 2/3 erg g s 𝑇6 4.1. Validity of PPT in Brown Dwarfs. There is a distinction [37]. However, one can fit the thermonuclear rates to a power between the temperature-driven PPT with a critical point at law in 𝑇 and 𝜌 in terms of the central temperature (𝑇𝑐)and ∼0.5 Mbar and between 10000 K and 20000 K as predicted by density (𝜌𝑐)asinFowlerandHoyle[21]: the chemical models [33] and the pressure-driven transition 𝑇 𝑠 𝜌 𝑢−1 from an insulating molecular liquid to a metallic liquid with 𝜖̇𝑛 = 𝜖̇𝑐 [ ] [ ] , (36) a critical point below 2000 K at pressures between 1 and 𝑇𝑐 𝜌𝑐 3 Mbar. The latter is predicted, for example, by Lorenzen et al. where 𝑢 ≃ 2.28 and 𝑠 = 6.31 are constants that depend on [34],Mazzolaetal.[35],andMoralesetal.[31]basedontheab the core conditions [20]. To obtain the luminosity due to the initiosimulations.Lorenzenetal.[34]ruleoutthepresenceof 𝐿 =∫𝜖̇ 𝑑𝑚 PPT above 10000 K and give an estimate of the critical points nuclear burning 𝑁 𝑛 ,weusethepowerlawformfor the nuclear burning rate 𝜖𝑛̇ (see (36)), and making the poly- for the transition at 𝑇𝑐 = (1400±100) K, 𝑃𝑐 = 1.32±0.1 Mbar, 3 𝜌=𝜌𝜃𝑛 𝑇/𝑇 =(𝜌/𝜌)2/3 and 𝜌𝑐 = 0.79 ± 0.05 g/cm . Similarly, Morales et al. [31] esti- tropic approximation 𝑐 and setting 𝑐 𝑐 , mated the critical point of the transition at a temperature near we obtain 2000 K and pressure near 1.2 Mbar. Signatures of pressure- 3 𝑛(𝑢+2𝑠/3) 2 𝐿𝑁 = ∫ 𝜖̇𝑛𝑑𝑚 = 4𝜋𝑎 𝜖𝑐̇ 𝜌𝑐 ∫ 𝜃 𝜁 𝑑𝜁, (37) driven PPT in a cold regime below 2000 Kareobtainedby Knudson et al. [36]. Figure 1 in Knudson et al. [36] shows where 𝑟=𝑎𝜁[29]. Inserting (14) and (16) in (37) yields the the melting line (black) as well as the different predictions final expression for luminosity as for the coexistence lines for the first-order transition (green 11.977 6.0316 curves). Brown dwarf interior temperatures are far above 16 𝑀 𝜓 𝐿𝑁 = 7.33 × 10 𝐿⊙ ( ) . (38) these estimates for a first-order transition from the insulating 𝑀 (1+𝛾+𝛼𝜓)16.466 to the metallic system. The same is true for Jupiter. Of course, ⊙ a continuous transition may be possible in Jupiter and brown dwarfs, but a first-order transition may not be possible. 6. Estimate of the Minimum Mass Thus, the determination of the range of temperature of this transition provides a much needed benchmark for the theory In this section, we estimate the minimum main-sequence of the standard models for the internal structures of the gas- mass by comparing the surface luminosity (32) with the 𝐿 giant planets and low mass stars. luminosity ( 𝑁) due to nuclear burning at the core of LMS and brown dwarfs. Instead of just quoting one value as the 5. Nuclear Processes critical mass, we have presented a range of values depending on the various phase transition points listed in Table 1. This VLM stars and brown dwarfs contract during their evolution will give us a range of values for the minimum critical mass due to gravitational collapse. The core temperature increases that is sufficient to ignite hydrogen burning. Model H marks and the contraction is halted by either the degeneracy theendofthephasetransitionandgivesalowerlimitofthe pressure of the electrons or the onset of the nuclear burning, critical mass. However, we calculate the mass limit for model whichever comes first. In the first case, the brown dwarf con- B and model D only. Equating 𝐿𝑁 of (38) with 𝐿 of (32) gives tinues to lose energy through radiation and cools down with us time without any further compression. However, massive 𝑀 0.02440 (1+𝛾+𝛼𝜓)1.507 brown dwarfs or stars at the edge of the main sequence can = =𝐹(𝜓), (39) 𝑀 𝜅0.106 𝜓0.1617 burn hydrogen for a very long time before they either cease ⊙ 𝑅 8 Advances in Astronomy

a Table 1: Effective temperature for different phase transition points .

−3 −3 Model log 𝑇 (K) 𝑃 (Mbar) 𝜌1 (g cm ) 𝜌2 (g cm ) Δ𝜎 H2𝑥 + 𝑏1 V A 3.70 2.14 0.75 0.92 0.62 0.48 2.87 1.58 B 3.78 1.95 0.70 0.88 0.59 0.50 2.70 1.59 C 3.86 1.62 0.64 0.80 0.54 0.50 2.26 1.59 D 3.94 1.39 0.58 0.74 0.51 0.51 2.00 1.60 E 4.02 1.13 0.51 0.65 0.46 0.52 1.68 1.61 F 4.10 0.895 0.43 0.55 0.42 0.50 1.29 1.59 G 4.18 0.631 0.35 0.38 0.14 0.33 0.60 1.44 H 4.185 0.614 0.35 0.35 0.00 0.18 0.40 1.30 a The phase transition points are taken from Chabrier et al. [14]. This gives the possible range of surface temperature depending on the phase transition points. For different values of temperature and density at which the phase transition takes place, the effective surface temperature is calculated using (25).

where 𝛼 = 2.32 for 𝜇𝑒 = 1.143 and 𝜇1 = 1.24,which where 𝐿 is the surface luminosity and 𝜎=𝑆/𝑘𝐵𝑁𝐴.Now, are the number of baryons per electron and the mean mole- replacing 𝑇 in terms of the degeneracy parameter 𝜓 in (4) 5/3 cularweightofneutral(𝑥H+ =0) hydrogen and helium, res- andusingthepolytropicrelation𝑃=𝐾𝜌 , we arrive at pectively. These mass densities are evaluated for hydrogen and helium mass fractions of 𝑋 = 0.75 and 𝑌 = 0.25,res- 𝑑𝜎 𝑁 𝑘 𝜓 𝐴 𝐵 ∫ 𝑃𝑑𝑉=−𝐿. pectively. 2/3 (43) 𝑑𝑡 𝜇𝑒 The right hand side of (39) has a minimum at a certain 𝜓 value of . This gives the lowest mass at which (39) has ∫ 𝑃 𝑑𝑉 = (2/7)(𝐺𝑀2/𝑅) a solution and this corresponds to the boundary of brown Using the standard expression, ,for polytropes of 𝑛 = 1.5, the integral in (42) reduces to dwarfs and VLM stars. The minimum of 𝐹(𝜓) is at 𝜓min = 2 0.042.Substitutingthisin(39)andfor𝜅𝑅 = 0.01 cm /g, the 6.73857 × 1049𝜓𝜇8/3 𝑀 7/3 minimum mass (model D) is ∫ 𝑁 𝑘 𝑇𝑑𝑀= 𝑒 ( ) . 𝐴 𝐵 2 (44) (1+𝛾+𝛼𝜓) 𝑀⊙ 𝑀 = 0.078𝑀⊙. (40) The variation of the entropy with time (42) can be expressed A similar analysis for model B gives the value of minimum as the rate of change of degeneracy over time. As the star mass of 𝑀 = 0.085𝑀⊙ for 𝜓min = 0.042. For the other collapses,thegasintheinteriorbecomesmoreandmore models, the minimum main-sequence mass is in the range degenerate and finally the degeneracy pressure halts further of 0.064–0.087𝑀⊙. contraction. A completely degenerate star (𝜓 = 0) becomes The solution is relatively independent of the mean molec- static and cools with time. Thus, by substituting the time ular weight 𝜇1. For example, using partially ionized gas; that variationofentropy,using(23),thatis, is, 𝜇1 = 0.84 in 𝛼; the minimum stellar mass increases by only ∼ 𝑑𝜎 1.5 1 𝑑𝜓 5%. = , 𝑑𝑡 𝜇 𝜓 𝑑𝑡 (45) 1mod 7. A Cooling Model and (44) into the energy equation (42) and using the lumi- A simple cooling model for a brown dwarf is presented in nosityexpressionformodelD(32),weobtainanevolutionary both Burrows and Liebert [20] and Stevenson [16]. In this equation for 𝜓: section, we review some of these steps using our more exact 𝑑𝜓 9.4486 × 10−18 𝑀 1.094 EOS(7)andrepresenttheevolutionofthebrowndwarfsover = ( ⊙ ) (1 + 𝛾 + 𝛼𝜓)1.715 time.Usingthefirstandthesecondlawofthermodynamics, 𝑑𝑡 𝜅1.1424 𝑀 𝑅 (46) the time varying energy equation for a contracting star is expressed as ⋅𝜓4.5797.

𝑑𝐸 𝑑𝑉 𝑑𝑆 𝜕𝐿 𝜓 +𝑃 =𝑇 = 𝜖−̇ , (41) Thisisanonlineardifferentialequationof for model D, and 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝜕𝑀 an exact solution can only be obtained numerically. However, we use some very simple and physical approximations to solve where 𝑆 is the entropy per unit mass and the other symbols this differential equation to yield a simple relation of 𝜓 as a have their standard meaning. The energy generation term 𝜖̇ is function of time and mass 𝑀. As we are trying to estimate ignored. On integrating over mass, we get the critical mass for hydrogen burning, it is safe to ignore the early evolution of VLM stars and brown dwarfs. Thus, we will 𝑑𝜎 solve the differential equation (46) with the assumption 𝜓≪ [∫ 𝑁𝐴𝑘𝐵𝑇 𝑑𝑀] =−𝐿, (42) 1.715 𝑑𝑡 1. Thus, we can drop the term (1+𝛾+𝛼𝜓) as it is almost Advances in Astronomy 9

Model D However, as our model does not consider any feedback from hydrogen fusion, the curves do not stabilize to a steady state main-sequence regime, in which 𝐿𝑁/𝐿 remains 1 until −3 10 nuclear burning stops. Interestingly, the ratio 𝐿𝑁/𝐿 is close to unity for many objects below the main-sequence transition −1.2 L∼t mass. This suggests that they burn nuclear fuel for a part ⊙ 10−4 of their evolutionary cycle but do not have enough mass to L/L sustain a steady state. Note that the results of Becker et al. [27] discussed in Section 2 would affect the luminosity 𝐿 by less 10−5 than a few percent. For example, if the radius 𝑅 increases (or decreases) by 2.5% for a constant value of mass 𝑀, 𝐾 in (12) 2.5 𝑇 1.5 𝐿 increases by %, and eff (31) decreases by %, therefore 106 107 108 109 decreases by ∼1%. t (years) 0.010M⊙ 0.080M⊙ 7.1. Brown Dwarfs as Clocks. Interestingly, the cooling prop- 0.030M⊙ 0.090M⊙ 0.050M 0.100M ertiesofbrowndwarfs(Figure2)canbecalibratedtoserveas ⊙ ⊙ an astronomical clock. As the electron degeneracy pressure

Figure 2: The variation of 𝐿/𝐿⊙ over time 𝑡 for different masses. puts a lower limit to the size of the dwarf, it cools slowly and radiates its internal energy. The luminosity of a brown dwarf is the most directly accessible observable quantity. As unity in the range 0 < 𝜓 < 0.1 and integrate (46) in the above luminosity is a time variable, one can get important informa- limit to obtain tion on the age of a brown dwarf depending on its mass and the cooling rate. As evident from Figure 2, given mass 1.094 −0.2794 −6 𝑀⊙ 𝑡 and the luminosity, one can roughly identify the age of the 𝜓 = (317.8 + 2.053 ×10 ( ) ) . (47) dwarfs. However, it is still a challenge to estimate the mass of a 𝑀 yr browndwarf.Anessentialpartofthesolutionistofindbrown Similarly, we can solve for 𝜓 forallothermodelsandobtain dwarfs in a binary system where one can get an accurate the evolution of degeneracy over time. We use this expression estimate of the mass and then compare its luminosity against of 𝜓, for model D, and can express luminosity as a function of available models. Newly discovered brown dwarfs in eclips- time 𝑡 and mass 𝑀.Thetimeevolutionofluminosity(model ing binaries [39, 40] can provide a data set of directly meas- D) is represented in Figure 2. It is evident that such low ured mass and radii. This can yield an empirical mass- mass objects continue to have low luminosity for millions of radius relation that also tests the prediction of the theoretical 7 years before they gradually start to cool. For 𝑡>10yr, the models. Furthermore, lithium in brown dwarfs has been used −1.2 luminosity declines as a function of time, 𝐿≃𝑡 ,asshown as a clock to obtain the ages of young open clusters as origi- in Figure 2 (dashed black line). A simplified expression of the nally suggested by Martin et al. [41] and Basri et al. [18] 7 variation of luminosity after 10 yr for model D is and most recently applied to the Pleiades by Dahm [42]. Massive brown dwarfs (𝑀 > 0.065𝑀⊙) deplete their lithium 𝑀 2.63 𝑡 −1.2 on a longer time scale, but VLM stars and objects above 𝐿≃𝐿 ( ) ( ) . ⊙ 7 (48) the hydrogen burning limit fuse lithium on a much shorter 𝑀⊙ 10 yr time scale [43]. A limitation of our model is that it does Our luminosity model is consistent with the simulation not include rotation. But the mechanical equilibrium in our results of the present day stellar evolution code Modules for models may not be significantly affected by this. For example, 107 Experiments in Stellar Astrophysics (MESA) (see Figure 17 in in model D, using (47) in (13), we find the radius of a - 0.075𝑀 ∼8 × 109 Paxton et al. [38]). Paxton et al. [38] use a one-dimensional year-old brown dwarf of mass ⊙ to be cm. stellar evolution module, MESA star, to study evolutionary This implies that for a median observed rotational period phases of brown dwarfs, pre-main-sequence stars, and LMS. (2𝜋/𝜔)ofoneday[44]theratioofmagnitudeoftherotational ∼𝑀𝑅2𝜔2 ∼𝐺𝑀2/𝑅 In Figures 3 and 4, the ratio 𝐿𝑁/𝐿 is plotted against energy ( ) to gravitational energy ( )fora −4 time for different masses in the substellar regime for models 0.075𝑀⊙ brown dwarf is ∼10 . However, convective mixing B and D, respectively. As evident for both models, there and the consequent lithium abundance have a strong con- is a nonsteady state of substantial nuclear burning for mil- nection to the rotation rate of brown dwarfs and pre-main- lionsofyearsforsubstellarobjects.Foracriticalmassof sequence stars [45]. Fast rotators are lithium-rich compared 0.085𝑀⊙ (model B) and 0.078𝑀⊙ (model D), the ratio 𝐿𝑁/𝐿 to their slow rotating counterparts, indicating a connection approaches1inaboutafewbillionyearsandmarksthe between the lithium content and the spin rate of young pre- beginning of main-sequence nuclear burning (note that the main-sequence stars and brown dwarfs [46]. Rapid rotation time to reach the main sequence will increase if we use a reduces the convective mixing, resulting in a higher lithium partially ionized gas; for example, it becomes ≈10 Gyr if we abundance in fast rotating pre-main-sequence stars. Thus, the use 𝜇1 = 0.84). Stars with greater mass reach a steady state rotational evolution of a brown dwarf can potentially be used where the thermal energy balances the gravitational collapse. as a clock as discussed in Scholz et al. [44]. 10 Advances in Astronomy

Model B 0.11 1.2

1.0

0.8 0.10

/L 0.6 N L

0.4 ⊙ 0.09 󳵳

0.2 M/M 󳵳 0.0 107 108 109 1010 0.08 󳵳 t (years)

0.078M⊙ 0.084M⊙ 0.080M⊙ 0.086M⊙ 0.07 󳵳 0.082M⊙ 𝐿 /𝐿 𝑡 Figure 3: The variation of 𝑁 over time for different masses for 7.0 7.5 8.0 8.5 9.0 9.5 10.0 model B in Table 1. log(t) (years)

Model A Model D Model D 1.2 Model B Model F Figure 5: The time needed to reach the main sequence, that is, 1.0 𝐿𝑁/𝐿 = 1, for objects of different masses. Four colored dashed lines represent different models as labeled. The triangles mark the main- 0.8 sequence critical mass for the respective models.

/L 0.6 N L 0.4 extended amount of time to reach the main sequence means that young stellar clusters may contain a significant fraction 0.2 ofobjectsthatarestillinaphaseofdecreasingluminosityand behave like brown dwarfs. These objects will ultimately settle 0.0 107 108 109 1010 into an extremely low luminosity main sequence. Here, we estimate the fraction of stars that take more than a specified t (years) time to reach the main sequence by using the modified log- 0.078M 0.072M⊙ ⊙ normal power law (MLP) probability distribution function of 0.074M 0.080M ⊙ ⊙ Basu et al. [47]. Their cumulative distribution function is 0.076M⊙ 𝐹 (𝑚) Figure 4: The variation of 𝐿𝑁/𝐿 over time 𝑡 for different masses for model D in Table 1. 1 (𝑚) −𝜇 = (−ln 0 ) 2 erfc √ 2𝜎0 (49) 8. Low Mass Stars That Reach 1 𝛼𝜎 (𝑚) −𝜇 − (𝛼𝜇 )𝑚−𝛼 ( 0 − ln 0 ). the Main Sequence 2 exp 0 erfc √ √ 2 2𝜎0 In Figure 5, we plot the time required by low mass stars to reach the main sequence. The curves represent the steady We use the best fit MLP parameters corresponding to state limit where 𝐿𝑁/𝐿 = 1 forfourdifferentmodels.Itis Chabrier’s [48] initial mass function (IMF) as obtained by interesting to note that objects of masses at the critical mass Basu et al. [47], where 𝜇0 = −2.404, 𝜎0 = 1.044,and𝛼= boundary between brown dwarfs and main-sequence stars, 1.396.Wethenusethecumulativefunctiontocalculatethe 107 108 for example, 0.078𝑀⊙ for model D, reach the steady state in fraction of stars taking more than either yr, yr, or 9 about 2.5 Gyr. This suggests that objects just below the critical 10 yr to reach the main sequence. Table 2 contains our results mass undergo nuclear burning for an extended period of time formodelsAtoH.Itturnsoutthatabout0.2%ofstars 9 but fail to enter the main sequence. Furthermore, stars in the take more than 10 yr to reach the main sequence and about 8 mass range 0.078𝑀⊙–0.086𝑀⊙ for model D take more than 4% take longer than 10 yr and about 12% take longer than 8 7 10 yr to reach the main sequence. Depending on the phase 10 yr, for model D. Some of these objects will end up on transition points for different models, these numbers vary but an extremely low luminosity main sequence, and a sample thefactthatstarsclosetotheminimummasslimitcantakean of luminosity values when an object just above the substellar Advances in Astronomy 11

Table 2: Minimum mass and fractions of stars. a b c b c b c Model 𝑀min 𝑀9 𝑁9 𝑀8 𝑁8 𝑀7 𝑁7 d A 0.087 — — 0.090 0.014 0.107 0.085 B 0.085 0.085 0.000 0.089 0.020 0.107 0.095 C 0.081 0.081 0.001 0.087 0.029 0.106 0.108 D 0.078 0.078 0.002 0.086 0.038 0.105 0.118 E 0.073 0.075 0.006 0.085 0.052 0.103 0.127 F 0.069 0.072 0.013 0.084 0.069 0.099 0.129 G 0.064 0.068 0.018 0.082 0.081 0.089 0.109 H 0.064 0.067 0.014 0.080 0.073 0.085 0.093 a 𝑀min is the minimum mass to reach the main sequence. b 9 8 7 𝑀9, 𝑀8,and𝑀7 are the masses up to which the stars take at least 10 yr, 10 yr, and 10 yr, respectively, to reach the main sequence. c 9 8 7 𝑁9, 𝑁8,and𝑁7 are the number fraction of stars reaching the main sequence in more than 10 yr, 10 yr, and 10 yr, respectively. d 9 Notethat,formodelA,lowmassstarsreachthemainsequencein<10 yr.

Table 3: Luminosity at the main sequence. Wearenotawareofwelldefinedanalyticmodelsthathavea 8 a unique way of estimating the surface luminosity apart from 𝑀/𝑀⊙ 10 (yr) 𝐿/𝐿⊙ −5 usingthePPTtechniqueasgiveninBurrowsandLiebert 0.080 3.66 1.90 × 10 [20]. In this work, rather than considering a single value for −5 0.085 1.15 9.12 × 10 the phase transition point, we have used the entire range −4 0.090 0.56 2.33 × 10 of temperatures from the phase transition coexistence table −4 0.095 0.31 4.67 × 10 [14]. These are within the uncertainty range of the critical a The luminosity of low mass stars in model D when they enter the main temperature of the PPT as proposed by the recent simulations sequence. [30]. Thus, considering the large uncertainties involved in such models, this range of values of the minimum mass is much more acceptable than a single distinct transition mass. 𝐿 /𝐿 = 1 limit achieves 𝑁 , that is, reaches the main sequence, However, the next step forward is to develop an analytic is given in Table 3. model for surface temperature that is independent of the PPT. 8 We estimate that ≃5% of stars take more than 10 yr to reach the main sequence, and ≃11% of stars take more than 9. Discussion and Conclusions 7 10 yr to reach the main sequence (Table 2). The stars in these This paper presents a simple analytic model of substellar categories have mass very close to the minimum hydrogen objects. A focus of the paper was to revisit both the develop- burning limit and will eventually settle into an extremely low −5 −4 ment and the shortcomings of the theoretical understanding luminosity main sequence with 𝐿/𝐿⊙ in the range ≈10 –10 . of the physics governing the evolution of low mass stars The very low luminosity non-main-sequence hydrogen burn- andsubstellarobjectsoverthelast50years.Wehave ing in substellar objects and the pre-main-sequence nuclear also made some modifications to the existing models to burning in very low mass stars are very interesting to study better explain the physics using analytic forms. Although further, and our simplified model can certainly be improved observational constraints hinder our understanding, a sim- in its ability to estimate the time evolution. ple analytic model can answer many questions. We have summarized the method of determining the minimum mass Appendix for sustained hydrogen burning. Objects in the mass range 0.064𝑀 0.087𝑀 ⊙– ⊙ mark this critical boundary between A. Pressure Integral brown dwarfs and the main-sequence stars. We have derived a general equation of state using polylog- The general Fermi integral can be written as arithm functions [49] to obtain the 𝑃-𝜌 relation in the interior of brown dwarfs. The inclusion of the finite temperature ∞ 𝜖𝑛𝑑𝜖 correctiongivesusamuchmorecompleteandsophisticated 𝐹 =𝑎∫ , 𝑛 𝑒𝛽(𝜖−𝜇) +1 analytic expression of the Fermi pressure (3). The application 0 of this relation can extend to other branches of physics, espe- 𝜇 𝜖𝑛𝑑𝜖 ∞ 𝜖𝑛𝑑𝜖 =𝑎∫ +𝑎∫ , cially for semiconductor and thermoelectric materials [50]. 𝛽(𝜖−𝜇) 𝛽(𝜖−𝜇) 0 𝑒 +1 𝜇 𝑒 +1 The estimate of the surface luminosity is a challenge given our limitations in understanding the physics inside such low 𝜇 𝜇 𝜇 𝜖𝑛𝑑𝜖 =𝑎∫ 𝜖𝑛𝑑𝜖 −𝑎 ∫ 𝜖𝑛𝑑𝜖 +𝑎 ∫ 𝛽(𝜖−𝜇) mass objects. Also, it is still an open question if a phase 0 0 0 𝑒 +1 transition actually occurs in a brown dwarf. The results of ∞ 𝜖𝑛𝑑𝜖 modern day simulations [30, 31] do raise doubts about the +𝑎∫ , 𝛽(𝜖−𝜇) relevance of phase transitions in the brown dwarf scenario. 𝜇 𝑒 +1 12 Advances in Astronomy

𝜇 𝜇 𝜖𝑛𝑑𝜖 =𝑎∫ 𝜖𝑛𝑑𝜖 −𝑎 ∫ B. Surface Properties −𝛽(𝜖−𝜇) 0 0 𝑒 +1 Thesurfacepropertiesofabrowndwarfcanbeanalyzedby ∞ 𝜖𝑛𝑑𝜖 +𝑎∫ . studying the phase diagram of hydrogen in the interior and 𝛽(𝜖−𝜇) 𝜇 𝑒 +1 the photospheric region. The total pressure inside a stellar or (A.1) substellar object can be represented as 𝑃=𝑃𝑔 +𝑃𝑟, (B.1) On substituting 𝑥=−𝛽(𝜖−𝜇)inthesecondtermand 𝑥=𝛽(𝜖−𝜇)in the third term, we arrive at where 𝑃𝑔 is the gas pressure due to the adiabatic ideal gas and 𝑃𝑟 istheradiationpressure.Atatemperaturecomparableto 𝑛 𝜇 𝑎 𝛽𝜇 (𝜇 − 𝑥/𝛽) that of the envelope surrounding the interior of a substellar 𝐹 =𝑎∫ 𝜖𝑛𝑑𝜖 − ∫ 𝑑𝑥 𝑛 𝑥 0 𝛽 0 𝑒 +1 object, hydrogen is partially ionized and the helium gas is mostly molecular. For a quasistatic change, the first law of 𝑛 (A.2) 𝑎 ∞ (𝜇 + 𝑥/𝛽) thermodynamics yields + ∫ 𝑑𝑥. 𝛽 𝑒𝑥 +1 𝜕𝑈 𝜕𝑈 0 𝑑𝑄 =( ) 𝑑𝑇 +( ) 𝑑𝑉 + 𝑃𝑑𝑉 𝜕𝑇 𝜕𝑉 (B.2) The numerator in the above integrals can be expanded as 𝑉 𝑇 [32]. The most general expression for the internal energy ofa 𝑥 𝑛 𝑥 (𝜇 ± ) ≃𝜇𝑛 ±𝑛𝜇𝑛−1 ( ) monatomic gas is 𝛽 𝛽 3 𝑘𝐵𝑁0𝑇 4 (A.3) 𝑈= +𝑎𝑇𝑉+𝑥𝜒𝑁0, (B.3) 𝑛 (𝑛−1) 𝑥 2 2 𝜇1 + 𝜇𝑛−2 ( ) +⋅⋅⋅. 2 𝛽 where we have considered the energy due to photon radiation and gas ionization. Here, 𝜒 is the ionization energy and 𝑥 is Substituting this in the integral for the pressure, we can the ionization fraction of hydrogen. Taking the partial deriva- proceed as follows: tivesof(B.3)andusingthesecondlawofthermodynamics 𝑇𝑑𝑆 =𝑑𝑄, we arrive at 𝐹 𝑛 3 𝑘 𝑁 𝑑𝑇 𝑘 𝑁 𝑑𝑊 4 𝑑𝑆 =− 𝐵 0 + 𝐵 0 + 𝑎𝑑𝑊 𝜇 𝑎 ∞ 𝑑𝑥 𝛽𝜇 𝑑𝑥 2 𝜇 𝑇 𝜇 𝑊 3 =𝑎∫ 𝜖𝑛𝑑𝜖 + 𝜇𝑛 {∫ − ∫ } 1 1 𝛽 𝑒𝑥 +1 𝑒𝑥 +1 (B.4) 0 0 0 3 𝜕𝑥 𝑑𝑇 𝜕𝑥 𝑑𝑉 +𝑁0 (𝜒 + 𝑘𝐵𝑇) ( + ), 𝑎 ∞ 𝑥𝑑𝑥 𝛽𝜇 𝑥𝑑𝑥 (A.4) 2 𝜕𝑇 𝑇 𝜕𝑉 𝑇 +𝑛 𝜇𝑛−1 {∫ + ∫ } 2 𝑥 𝑥 3 𝛽 0 𝑒 +1 0 𝑒 +1 where 𝑊=𝑇𝑉. The ionization fraction is a function of density and temperature, 𝑥(𝜌,.UsingtheSahaequation, 𝑇) 𝑛 (𝑛−1) 𝑎 𝛽𝜇 𝑥2𝑑𝑥 ∞ 𝑥2𝑑𝑥 − 𝜇𝑛−2 {∫ − ∫ }. we obtain 3 𝑥 𝑥 2 𝛽 0 𝑒 +1 0 𝑒 +1 𝜕𝑥 𝑥 (1−𝑥) 1 3 𝜒 ( ) = ( + ), 𝜕𝑇 2−𝑥 𝑇 2 𝑘 𝑇 Substituting 𝑛=3/2for the Fermi pressure (2) and 𝑉 𝐵 (B.5) evaluating these integrals using the polylogs [49], we arrive 𝜕𝑥 𝑥 (1−𝑥) 1 at a simplified form ( ) = . 𝜕𝑉 𝑇 2−𝑥 𝑉 2 1 𝑃 ≃𝑎 𝜇5/2 − 𝑎𝛽−1𝜇3/2 (1 + 𝑒−𝛽𝜇) Using the above relations, we can simplify (B.4) to be 𝐹 5 8 ln 𝑑𝑆 3 𝑑𝑇 𝑑𝑊 3 2 𝐻𝑑𝑇 3 𝐻𝑑𝑉 2 =− + +( ) + 3 𝜋 −2 1/2 3 −2 1/2 −𝛽𝜇 𝑘 𝑁 2𝜇 𝑇 𝜇 𝑊 2 𝑇 2 𝑉 + 𝑎𝛽 𝜇 + 𝑎𝛽 𝜇 (−𝑒 ) (A.5) 𝐵 0 1 1 2 6 4 Li2 (B.6) 4𝑎𝑑𝑊 𝑑𝑉 𝑑𝑇 𝜒 𝑑𝑇 3 + +𝜒( +3 + ), − 𝑎𝛽−3𝜇−1/2 (−𝑒−𝛽𝜇)⋅⋅⋅ . 3𝑘 𝑁 𝑇𝑉 𝑇2 𝑘 𝑇3 4 Li3 𝐵 0 𝐵 where 𝐻 = 𝑥(1−𝑥)/(2−𝑥). This expression is the same as (19) Similarly, for 𝑛=1/2, the expression for the number density except for the final two terms due to radiation and ionization canbeobtainedas energy, respectively. We can ignore the final term and retain 2 3 terms of linear order in 𝑇. On integrating and simplifying the 𝜌≃𝑎 𝜇3/2 + 𝑎𝛽−1𝜇1/2 (1 + 𝑒−𝛽𝜇) 3 8 ln above expression, we get the entropy for the partially ionized hydrogen and helium gas: 𝜋2 3 + 𝑎𝛽−2𝜇−1/2 + 𝑎𝛽−2𝜇−1/2 (−𝑒−𝛽𝜇) (A.6) 1 3 𝑥 (1−𝑥) 𝑇3/2 4 12 4 Li2 𝑆 =𝑘 𝑁 ( + ) + 𝑎𝑇3𝑉 1 𝐵 0 𝜇 2 2−𝑥 ln 𝜌 3 1 (B.7) 1 −3 −1/2 −𝛽𝜇 + 𝑎𝛽 𝜇 Li3 (−𝑒 )⋅⋅⋅ . 4 +𝐶1. Advances in Astronomy 13

At low temperature (𝑇 < 4000 K) and low pressure, hy- C.4. Ionic Correlation. Ionic correlation is an important con- drogen is predominantly molecular and fluid. Repeating the tribution as considered by Stolzmann and Blocker [25], above derivation for the nonionized diatomic hydrogen gas Becker et al. [27], Hubbard et al. [52], and Gericke et al. with energy 𝑈 = (5/2)(𝑘𝐵𝑇𝑁0/𝜇2) and molecular helium, we [26] to name but a few. Stolzmann and Blocker [25] use the canarriveatanexpressionfortheentropy: method of Pade’s approximations to provide explicit expressions for the fully ionized plasma of the Helmholtz free energy 𝑘 𝑁 5/2 𝐵 0 𝑇 4 3 and pressure. They have considered the nonideal effects of 𝑆2 = ln + 𝑎𝑇 𝑉+𝐶2. (B.8) 𝜇2 𝜌 3 different correlations such as the electron-electron, ion-ion, 𝜌 𝑇 electron-ion, and exchange contribution for a wide range of In the above expressions for entropy, and are the density values of the Coulomb coupling parameter Γ,whichisthe and the temperature, respectively. Other variables are as des- ratio of the Coulomb to thermal energy: cribed in the text. 𝑒2 4𝜋𝑛 1/3 Γ Γ= ( 𝑒 ) = ion . 5/3 (C.5) C. Thermal Properties 𝑘𝐵𝑇 3 ⟨𝑧 ⟩ We discuss some of the more accurate expressions for the 5/3 Here, 𝑛𝑒 stands for the electron density and ⟨𝑧 ⟩ is the important thermal properties of a degenerate system. charge average, given as

5/3 C.1. Fermi Energy. Using (A.6) for the number density, we ∑𝑛𝑍 ⟨𝑧5/3⟩= 𝑖 𝑖 , can write the general expression for the chemical potential 𝜇 (C.6) ∑ 𝑛𝑖 in terms of the Fermi energy 𝜇𝐹 at 𝑇=0[51]. Considering 3/2 𝑖 only the first three terms (A.6) and for 𝜌 = (2/3)𝜇𝐹 ,wefind forionsofdifferentspecies .Thegreatesteffectisinthe relative pressure contribution 𝑃𝑖𝑖/𝑃 of the ionic correlation 2 5 ideal 𝜋 1 3 1 −𝛽𝜇 𝑇=10 ∼− 𝜇≃𝜇 − − (1 + 𝐹 ). term for hydrogen at K, estimated to be 0.1 at 𝐹 12 2 8 𝛽𝜇 ln exp (C.1) 𝜌∼103 / 3 ∼− 𝜌∼10 / 3 (𝛽𝜇𝐹) 𝐹 g cm and a minimum of 0.2 at g cm . The second and the third terms are the correction factor 𝐶 to Competing Interests the zero temperature Fermi energy. For 0.03 < 1/𝜇𝐹𝛽 < 0.20, −4 the correction factor 𝐶 will be in the range ∼8 × 10 <𝐶< The authors declare that they have no competing interests −2 4×10 . regarding the publication of this paper.

C.2. Specific Heat. In the nondegenerate completely ionized Acknowledgments limit, the specific heat 𝐶V ∼3𝑘𝐵𝑁/2. At finite temperatures, The authors thank Dr. Andrea Becker and Dr. Gilles Chabrier the value of the specific heat is less than the limiting value; for their valuable comments and input during the prepara- that is, 𝐶V <3𝑁𝑘𝐵/2. The specific heat of the ideal Fermi gas tion of this article. They also thank Anushrut Sharma for decreases monotonically. At low but finite temperatures, contributing to the initial stages of this project. Shantanu 𝐶 𝜋2 𝑘 𝑇 BasuandS.R.ValluriweresupportedbyDiscoveryGrant V ≃ 𝐵 . 𝑁 2 𝜇 (C.2) from the Natural Sciences and Engineering Research Council 𝐹 (NSERC) of Canada. S. R. Valluri also thanks King’s Uni- Adetailedanalysisinthecalculationofthespecificheat versity College for its continued support for his research shows that the numerical coefficients in the expansion endeavours. approached a limiting value of 2. References

C.3. Grueneisen Parameter. Applying the condition of con- [1] B. R. Oppenheimer, S. R. Kulkarni, K. Matthews, and T. stant entropy to (20) leads to the condition Nakajima,“InfraredspectrumofthecoolbrowndwarfGI 2/3 229B,” Science,vol.270,no.5241,pp.1478–1479,1995. 𝑇=𝐶𝜌 , (C.3) [2] R. Rebolo, M. R. Zapatero Osorio, and E. L. Mart´ın, “Discovery where 𝐶 is a constant. The Grueneisen parameter 𝛾 is given of a brown dwarf in the Pleiades star cluster,” Nature,vol.377, by the expression no. 6545, pp. 129–131, 1995. [3]E.L.Mart´ın, R. Rebolo, and M. R. Zapatero-Osorio, “Spectro- 𝜕 log 𝑇 scopy of new substellar candidates in the pleiades: toward a 𝛾=( ) . (C.4) 𝜕 log 𝜌 𝑠 spectral sequence for young brown dwarfs,” Astrophysical Jour- nal,vol.469,no.2,pp.706–714,1996. Using (C.3) in the above expression, we estimate the value [4]M.T.Ruiz,S.K.Leggett,andF.Allard,“Kelu-1:afree-floating of the Grueneisen parameter 𝛾 to be 2/3.Thisvalueisin brown dwarf in the solar neighborhood,” Astrophysical Journal, approximate accord with Stevenson [16], who indicated that vol. 491, no. 2, pp. L107–L110, 1997. 𝛾 ≃ 0.6 in dense Coulomb plasma when obtained from com- [5] K. L. Luhman, V. Joergens, C. Lada, J. Muzerolle, I. Pascucci, puter simulations. andR.White,“Theformationofbrowndwarfs:observations,” 14 Advances in Astronomy

in Protostars and Planets V,B.Reipurth,D.Jewitt,andK.Keil, [26] O. D. Gericke, K. Wunsch,¨ A. Grinenko, and J. Vorberger, Eds., The University of Arizona Press, 2007. “Structural properties of warm dense matter,” Journal of Physics: [6] G. Chabrier, A. Johansen, M. Janson, and R. Rafikov, “Giant Conference Series, vol. 220, no. 1, Article ID 012001, 2010. planet and brown dwarf formation,” https://arxiv.org/abs/ [27] A. Becker, W. Lorenzen, J. J. Fortney, N. Nettelmann, M. 1401.7559. Schottler,¨ and R. Redmer, “AB initio equations of state for [7] S. S. Kumar, “The structure of stars of very low mass,” The hydrogen (H-REOS.3) and helium (H-REOS.3) and their impli- Astrophysical Journal,vol.137,p.1121,1963. cations for the interior of brown dwarfs,” The Astrophysical [8] S. S. Kumar, “The Helmholtz-Kelvin time scale for stars of very Journal Supplement Series, vol. 215, no. 2, p. 21, 2014. low mass,” The Astrophysical Journal,vol.137,p.1126,1963. [28] O. L. Anderson, “The Gruneisen¨ ratio for the last 30 years,” Geo- [9] C. Hayashi and T. Nakano, “Evolution of stars of small masses physical Journal International,vol.143,no.2,pp.279–294,2000. in the pre-main-sequence stages,” Progress of Theoretical Physics, [29] S. Chandrasekhar, An Introduction to the Study of Stellar Struc- vol. 30, no. 4, pp. 460–474, 1963. ture, The University of Chicago Press, Chicago, Ill, USA, 1939. [10] F. D’Antona and I. Mazzitelli, “Evolution of very low mass stars [30] J. Yang, Ch. L. Tian, F. Sh. Liu, K. H. Yuan, H. M. Zhong, and brown dwarfs. I—the minimum main-sequence mass and and F. Xiao, “A new evidence of first-order phase transition for luminosity,” Astrophysical Journal,vol.296,pp.502–513,1985. hydrogen at 3000 K,” Europhysics Letters,vol.109,no.3,Article [11] A. Burrows, W.B. Hubbard, and J. I. Lunine, “Theoretical mod- ID 36003, 2015. els of very low mass stars and brown dwarfs,” The Astrophysical [31] M. A. Morales, C. Pierleoni, E. Schwegler, and D. M. Ceperley, Journal,vol.345,pp.939–958,1989. “Evidence for a first-order liquid-liquid transition in high- [12] D. Saumon and G. Chabrier, “Fluid hydrogen at high density: pressure hydrogen from ab initio simulations,” Proceedings of the plasma phase transition,” Physical Review Letters,vol.62,no. the National Academy of Sciences of the United States of America, 20, pp. 2397–2400, 1989. vol. 107, no. 29, pp. 12799–12803, 2010. [13] G. Chabrier and H. M. van Horn, “The role of the molecular- [32] D. D. Clayton, Principles of Stellar Evolution and Nucleosynthe- metallic transition of hydrogen in the evolution of Jupiter, sis,McGraw-Hill,NewYork,NY,USA,1968. Saturn, and brown dwarfs,” The Astrophysical Journal,vol.391, [33]D.Saumon,G.Chabrier,andH.M.VanHorn,“Anequationof no. 2, pp. 827–831, 1992. state for low-mass stars and giant planets,” Astrophysical Journal, [14] G. Chabrier, D. Saumon, W. B. Hubbard, and J. I. Lunine, “The Supplement Series,vol.99,no.2,pp.713–741,1995. molecular-metallic transition of hydrogen and the structure of [34] W.Lorenzen, B. Holst, and R. Redmer, “First-order liquid-liquid Jupiter and Saturn,” The Astrophysical Journal,vol.391,no.2,pp. phasetransitionindensehydrogen,”Physical Review B,vol.82, 817–826, 1992. no.19,ArticleID195107,6pages,2010. [15] G. Fontaine, H. C. Graboske, and H. M. van Horn, “Equations [35] G. Mazzola, S. Yunoki, and S. Sorella, “Unexpectedly high of state for stellar partial ionization zones,” The Astrophysical pressure for molecular dissociation in liquid hydrogen by Journal Supplement Series, vol. 35, p. 293, 1977. electronic simulation,” Nature Communications,vol.5,article [16]D.J.Stevenson,“Thesearchforbrowndwarfs,”Annual Review 3487, 2014. of Astronomy and Astrophysics,vol.29,no.1,pp.163–193,1991. [36] M. D. Knudson, M. P.Desjarlais, A. Becker et al., “Direct obser- [17] R. Rebolo, E. L. Martin, and A. Magazzu, “Spectroscopy of vation of an abrupt insulator-to-metal transition in dense liquid a brown dwarf candidate in the Alpha Persei open cluster,” deuterium,” Science,vol.348,no.6242,pp.1455–1460,2015. Astrophysical Journal, vol. 389, pp. L83–L86, 1992. [37] W. A. Fowler, G. R. Caughlan, and B. A. Zimmerman, “Ther- [18] G. Basri, W. G. Marcy, and R. J. Graham, “Lithium in brown monuclear reaction rates, II,” Annual Review of Astronomy & dwarf candidates: the mass and age of the faintest pleiades stars,” Astrophysics,vol.13,pp.69–112,1975. Astrophysical Journal,vol.458,p.600,1996. [38] B. Paxton, L. Bildsten, A. Dotter, F. Herwig, P. Lesaffre, and [19] J. Stauffer, G. Schultz, and D. J. Kirkpatrick, “Keck spectra of F. Timmes, “Modules for Experiments in Stellar Astrophysics pleiades brown dwarf candidates and a precise determination (MESA),” The Astrophysical Journal Supplement,vol.192,no.1, of the lithium depletion edge in the pleiades,” The Astrophysical 3 pages, 2011. Journal, vol. 499, no. 2, p. L199, 1998. [39] K. G. Stassun, R. D. Mathieu, and J. A. Valenti, “Discovery of two [20] A. Burrows and J. Liebert, “The science of brown dwarfs,” young brown dwarfs in an eclipsing binary system,” Nature,vol. Reviews of Modern Physics,vol.65,no.2,pp.301–336,1993. 440,no.7082,pp.311–314,2006. [21] W. A. Fowler and F. Hoyle, “Neutrino processes and pair for- [40] T. J. David, L. A. Hillenbrand, A. M. Cody, J. M. Carpenter, mation in massive stars and supernovae,” Astrophysical Journal and A. W. Howard, “K2 discovery of young eclipsing binaries Supplement,vol.9,p.201,1964. in upper scorpius: direct mass and radius determinations for [22] T. Padmanabhan, TheoreticalAstrophysics,Volume1:Astrophys- the lowest mass stars and initial characterization of an eclipsing ical Processes, Cambridge University Press, Cambridge, UK, brown dwarf binary,” The Astrophysical Journal,vol.816,no.1, 1999. article 21, 27 pages, 2016. [23] A. Sommerfeld, “Zur Elektronentheorie der Metalle auf Grund [41] E. L. Martin, R. Rebolo, and T. Magazzu, “Constraints to the der Fermischen Statistik. I. Teil: Allgemeines, Stromungs-¨ und masses of brown dwarf candidates form the lithium test,” The Austrittsvorgange,”¨ Zeitschriftur f¨ Physik,vol.47,no.1-2,pp.1– Astrophysical Journal,vol.436,no.1,pp.262–269,1994. 32, 1928. [42] E. S. Dahm, “Reexamining the lithium depletion boundary in [24] W. B. Hubbard, Planetary Interiors, Van Nostrand Reinhold, the pleiades and the inferred age of the cluster,” The Astrophysi- New York, NY, USA, 1984. cal Journal,vol.813,no.2,article108,2015. [25] W. Stolzmann and T. Blocker, “Thermodynamical properties of [43] A. Magazzu,` E. L. Martin, and R. Rebolo, “A spectroscopic test stellar matter. I. Equation of state for stellar interiors,” Astron- forsubstellarobjects,”Astrophysical Journal,vol.404,no.1,pp. omy & Astrophysics,vol.314,pp.1024–1040,1996. L17–L20, 1993. Advances in Astronomy 15

[44] A. Scholz, V. Kostov, R. Jayawardhana, and K. Muziˇ c,´ “Rotation periods of young brown dwarfs: k2 survey in upper scorpius,” The Astrophysical Journal, vol. 809, article L29, 2015. [45] E. L. Mart´ın and A. Claret, “Stellar models with rotation: an exploratory application to pre-main sequence lithium depletion,” Astronomy & Astrophysics,vol.306,no.2,pp.408–416, 1996. [46] J. Bouvier, C. A. Lanzafame, L. Venuti et al., “The Gaia-ESO Survey: a lithium-rotation connection at 5 Myr?” Astronomy & Astrophysics,vol.590,articleA78,2016. [47] S. Basu, M. Gil, and S. Auddy, “The MLP distribution: a modified lognormal power-law model for the stellar initial mass function,” Monthly Notices of the Royal Astronomical Society,vol. 449, no. 3, pp. 2413–2420, 2015. [48] G. Chabrier, “The initial mass function: from Salpeter 1955 to 2005,”in The Initial Mass Function 50 Years Later,E.Corbelli,F. Palla,andH.Zinnecker,Eds.,vol.327ofAstrophysics and Space Science Library,pp.41–50,Springer,Dordrecht,Netherlands, 2005. [49] J. Tanguay, M. Gil, D. J. Jeffrey, and S. R. Valluri, “D-dimensional Bose gases and the Lambert W function,” Journal of Mathemat- ical Physics,vol.51,no.12,ArticleID123303,2010. [50] M. Molli, K. Venkataramaniah, and S. R. Valluri, “The polylogarithm and the Lambert W functions in thermoelectrics,” Canadian Journal of Physics, vol. 89, no. 11, pp. 1171–1178, 2011. [51] R. P. Feynman, Statistical Mechanics: A Set of Lectures,Ben- jamin/Cummings Publishing, Reading, Mass, USA, 1972. [52] W. B. Hubbard and H. E. Dewitt, “Statistical mechanics of light elements at high pressure. VII—a perturbative free energy for arbitrary mixtures of H and He,” Astrophysical Journal,vol.290, pp. 388–393, 1985. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Aerodynamics

Journal of Fluids Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Submit your manuscripts at http://www.hindawi.com

International Journal of International Journal of Statistical Mechanics Optics Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Thermodynamics

Journal of Computational Advances in Physics Advances in Methods in Physics High Energy Physics Research International Astronomy Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Journal of Journal of International Journal of Journal of Atomic and Solid State Physics Astrophysics Superconductivity Biophysics Molecular Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014