the science of brown dwarfs

The science of brown dwarfs

Adam Burrows

Departments of Physics and Astronomy, University ofArizona, Tucson, Arizona 85721

James Liebert

Department ofAstronomy and Steward Observatory, University ofArizona,Tucson, Arizona 85721This review summarizes the current status of both brown dwarf theory and the searches for these elusivesubstellar objects. The conceptual continuity between the brown dwarf and the well-studied M dwarfbranches is emphasized throughout. The physics of their atmospheres and interiors is reviewed and ananalytic model of both brown dwarf evolution and the lower edge of the hydrogen main sequence ispresented. An extensive discussion of brown dwarf searches in the field, in binaries, around white dwarfs,in clusters, in the solar neighborhood, in the galactic halo, and using proper-motion catalogs is provided,as is a tentative list of candidates in the field. The theory near and below the edge of the main sequence,while sophisticated, is only now being successfully challenged by optical and infrared observations. Thenear future promises a productive explosion in our knowledge of this problematic galactic population.

CONTENTS

I. IntroductionII. The Physics of Brown Dwarfs

A. The polytropic essentials of brown dwarf theoryB. The equation of state and thermodynamics of hydro-

gen1. The low-density regime2. The high-density regime

C. Brown dwarf and M dwarf atmospheres and spectraD. Nuclear ratesE. An ana1ytic model for brown dwarfsF. Numerical brown dwarf and M dwarf models

1. A fiducial 0.05MO brown dwarf model2. Theoretical brown dwarf and M dwarf properties

and evolution: 0.03MO —0.2MOIII. Searches for Brown Dwarfs

A. Searches for single objects1. Faint stellar objects: Low-mass stars or brown

dwarfs?2. Construction of initial mass functions for field

stars3. Proper-motion surveys4. Optical and infrared color surveys

B. Searches in young clusters and star-forming regions1. The Taurus-Auriga clouds2. The Rho Ophiuchus molecular cloud/cluster3. The Pleiades4. The Hyades

C. Discovery and analysis of substellar companions1. %'ide optical companions with common proper

motion2. Infrared imaging and photometry3. Photographic, astrometric perturbations and orbit

determinations4. Infrared speckle interferometry or imaging5. Radial velocity surveys6. Substellar objects in interacting binary systems

D. Searches for halo brown dwarfsE. gravitational microlensing

IV. The Bottom LineAcknowledgmentsReferences

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As one great furnace flamed, yet from those flamesXo light, but rather darkness visible.

Milton —Paradise Lost

Nothing is secret which shall not be made manifest.

Luke 8 17

I. INTRODUCTION

Between giant planets such as Jupiter ( —10 M&) andstars at the bottom of the hydrogen main sequence( ~0. 1Mo), the galactic census is woefully incomplete.Spanning two orders of magnitude in mass, this brown

dwarf branch (Tarter, 1975, 1986) is all but unexplored,yet may contain the majority of objects in the galacticdisk and halo. That there is a minimum mass for stablehydrogen burning (-0.08M~), and, hence, that there isa lower edge to the stellar branch, has been knowntheoretically since the pioneering work of Kumar (1963).However, no credible arguments have been advanced forwhy the edge of the main sequence should be the edge ofthe "stellar" mass function. Brown dwarfs emit most oftheir energy in the near infrared and can cool quicklybelow the luminosity of —10 I.o at the main-sequenceedge. Nevertheless, the brown dwarf branch should bepopulated, and realistic extrapolations from the adjacentM dwarf branch simply an abundance. However, ifbrown dwarfs are so numerous, why have we no unambi-guous examples? This question cannot yet be answered,but it is now being posed with more and more vigor asnew infrared technology is applied to the search. Paral-leling the increased observational activity has been theincreased theoretical interest. These transition objects,which straddle the realms of planets and stars, are made

Reviews of Modern Physics, Vol. 66, No. 2, April 1993 Copyright 1993 The Americ Physical Society 301

A. Burrows and J. Liebert: The science of brown dwarfs

plcdomlnaIltly of llqUld Q1ctalllc hydlogcn, have I'SXOI'-

thin" atmospheres composed of an exotic soup of mole-cules and grains, and carl achieve central temperaturesadequate to consume their stores of cosmological deuteri-um, while inadequate to ignite sufFicient light hydrogento avoid cooling into obscurity within a Hubble time.

For background material on the general theory of stel-lar evolution and compact objects, the inquisitive willfind the books by Clayton (1983), Shapiro and Teukolsky(1983), Hubbard (1984), and Chandrasekhar (1939)pIofltablc leading. Though wc do not hcI'c attcIDpt to 1n-

cludc any discussion of thc current theories of stal foI'-

mation, the articles by Zinnecker, McCaughrean, andMilking (1992), Shu, Adams, and Lizano (1987), and Boss(1987) will serve as useful introductions to that annoying-ly incomplete subject.

This review is meant to combine the latest theoreticalthinking with the latest observations in a state-of-the-artsnapshot of brown dwarf science. We are not motivatedby any prejudice that brown dwarfs might be the "miss-ing mass" of the galactic disk (if there is such a mass) orof the galactic halo. Rather„we are driven by the lure ofone of the last terrae incognitae in astronomy.

Section II deals with the theory of brown dwarf struc-ture and cooling. It contains not only a useful analyticmodel, but detailed numerical models as well. Browndwarf atmospheres are described and a comprehensiveequation of state is presented, along with an informative,if crude, analytic calculation of the mass at the Inain-sequence edge. Section III surveys the various searchesand classes of searches. Highlighted are investigations inproper-motion catalogs, and the searches in the field, inbinaries, around white dwarfs, in clusters, in the solarneighborhood, and in the galactic halo. The samples inwhich mass, age, and/or distance are known are re-viewed» and a CUITcnt bl own dwal f candlda'tc list ispresented. Throughout, the continuity between the Mdwarf and the brown dwarf branches is emphasized.However, as Sec. III demonstrates, the comparison be-tween a theory that deals with CQ'ective temperatures andbolometric lurninosities, and observations that involvephotometric colors and magnitudes, is handicapped bythe lack of good theoretical color-color and color-rnagnitude relations. Despite this problem, the browndwarf candidate list continues to grow and scientificquestions conccrniIig thc IowcI' edge of thc main se-quence remain as interesting and challenging as ever.

II. THE PHYSICS OF BROWN DWARFS

Despite the small sample of candidate brown dwarfsfrom the optical and infrared surveys conducted to date(Sec. III), the theorists have not been idle [D'Antona andMazzitelli, Nelson, Rappaport, and Joss, 1985, 1986a,1986b, 1986c; Stringfellow, 1986; Burrows, Hubbard, andLunine, 1989; Stringfellow, Black, and Bodenheimer,1990; Burrows, Hubbard, Saumon, and Lunine, 1993j.The brown dwarf branch is the low-mass extension of themuch-studied very-low-mass (VLM); M «0. 3MO) and

low-mass (M «0.7Mo) stellar branches and as suchshould involve merely extrapolations in the equation ofstate (EOS), atmospheric physics, structure, and evolu-tion appropriate for lighter and cooler objects.

BI'own dwarfs Rrc coInpact stars» plcdoInlnantly ofhydrogen, that are not massive enough (M «0.08MO)eventually to ignite hydrogen stably on the hydrogenmain sequence. This is not to say that brown dwarfs donot burn hydrogen during some phase of their lives. Infact, the Inost massive (M ~ 0.07MO ) can burn hydrogenfoI bllllons of ycaIs. HowcvcI' in a brown dwal f thc nu-clear Ares are eventually extinguished as it cools to ob-scurity, and the surface energy losses are never complete-ly compensated by thermonuclear burning in the core.The surface and central temperatures are never stabi-lized.

There is an unspoken prejudice in the cognizant com-munity that "brown dwarfs" are formed (if formed at all)from the interstellar medium (ISM)—as are regularstars —and not in proto-planetary disks —as are giantplanets. Brown dwarfs would then be aborted stars. Thedistinction between giant planets, such as Jupiter, andbl owri dwarfs ls 1SI'gcly semantic» not physical» and attimes objects between 10M& (Mz—=0.001MO) and 80M&are arbitrarily designated as brown dwarfs. Here, weneed not be so narrow-minded, but the reader should beaware that there may be virtue in distinguishing objectswltl1 thc san1c mass, bUt difterent orlglns.

In Scc. II, wc Rddlcss thc scicntiAC ingredients thatmust go into a complete theoretical attack on browndwarfs. The theory that we describe assumes that browndwarfs are formed, as are stars, from the cores of molecu-lar clouds, that they accrete mass for no more than—10 —10 yr, and that they can be treated as spheres.Fortunately, these assumptions are minima, and suchobjects forget their initial conditions quickly ( «10 —10yr). An important assumption that we have made in or-der to focus the presentation is that the metallicity of ourtllcolctlcal Ob)ccts ls solRI' ( 2% by I11ass). Tllls Ilccdnot be the case, and we refer to Burrows et al. (1993) andD Alltolla (1987) fol R dlscllssloll of IlollsolRI' models.For the theory of substellar objects that may accretemass over a good fraction of a Hubble time, we refer thereader to Salpeter (1992).

Section II.A contains a general discussion of the "po-lytropic»» nature and structure of brown dwar'fs, and thisis followed in Sec. II.B by a short primer on the relevant

thermodynamics and equations of state. In Sec. II.C, thesources of atmospheric opacity and the physics of theouter radiative skin are reviewed. Section II.D dealswith the thermonuclear rates and screening correctionsthat RIc important in thc physics both above and jUstbelow thc cdgc of thc main scqucncc. In order to morcclearly highlight the essentials of brown dwarf theory,Sec. II.E provides an analytic model for the time andmass dependences of the effective temperature (T, ) andthe bolometric luminosity (1.) along the entire browndwarf branch. That section also contains an analytic

Rev. Mod. Phys. , Vol. 65, No. 2, April 1993

A. Burrows and J. Liebert: The science of brown dwarfs 303

derivation of the value of the main-sequence transitionmass. Finally, in Sec. II.F we present representative re-sults from detailed numerical simulations, both unpub-lished and published, of brown dwarf properties and evo-lution. Throughout this review, we have relied heavilyupon the insights and numbers to be culled from thebroad literature. However, for practical reasons, we havedrawn on the theory of Burrows, Hubbard, and Lunine(BHL) when the results of the numerical simulationswere required. This habit is not meant to detract in anyway from the important work of Grossman et al. (1974),Vandenberg et aI (1.983), D'Antona and Mazzitelli(1985), Rappaport and Joss (1984), Nelson, Rappaport,and Joss (1986a, 1986b, 1986c), Dorman, Nelson, andChou (1989), or Stringfellow (1986); and the reader isstrongly encouraged to refer to these papers for an alter-native view of the theoretical terrain.

dM(r)Gr

=4~r p, (2.4)

=0 (2.5)

where

gn p p gn+1 (2.5a)

and

g= —,a= (n+1) p"r E''a' 4+G

1/2

(2.5b)

where G is Newton's universal constant of gravitation( G =6.6726 X 10 cgs), can be combined for polytropesto yield the Lane-Emden equation (Chandrasekhar,1939),

A. The polytropic essentialsof brown dwarf theory

Composed predominantly of metallic hydrogen andhelium and with mass densities in the vicinity of 10—1000gm/cm, brown dwarfs are supported in hydrostatic equi-librium by electron degeneracy pressure. It is only for abrief time ( ~ 10 yr), early in their cooling-regulated con-traction, when the central densities are suKciently lowand entropies are su%ciently high, that thermal ideal-gaspressure is important. But no matter the character of thepressure support, since a brown dwarf contracts along aHayashi track, it is at all times convective (but for acaueat, see Nelson, Rappaport, and Joss, 1986a). Thisimplies that the entropy is the same throughout the star(save for in the very thin outer radiative skin) and thatthe P-p relation is approximately a power law:

p ~~ 1+1/n (2.1)

=K'p (n =1.5), (2.2)

dI' GM (r)~r

(2.3)

where P is the pressure, p is the mass density, n is the po-lytropic index, and IC' is a constant that in Eq. (2.2) de-pends only on the composition and the specific entropy.Stars that satisfy Eq. (2.1) are called polytropes of indexn (Note tha. t it is the nonrelativistic nature of the pres-sure support that sets n = 1.5.) They form a sequence ofobjects with self-similar structure and radii (R), centraldensities (p, ), and central pressures (I', ) that depend onone another and on K' and M (the total mass) via simplepower laws. The general equations of hydrostatic equi-librium

The solution of Eq. (2.5) for 8(g) subject to the zero-pressure outer boundary condition solves the problemgiven p„E', and n or M, K', and n. Useful results are

( ) g ~ M2n/(3 —n)3MPc n P n 4

(2.6)

p = W~cvf ~ M2(1+n)/(3 —n)

C n R4 (2.7)

andn /(3 —n)

M(1 —n)/(3 —n)

6 (2.8)

where 6„, 8'„, and y„are functions of n alone. Forn =1.5, 6„=5.991, 8'„=0.770, and y„=2.357 and, im-portantly, p, ~M, P, ~M', and R ~M ' . Forn =1.0, 5„=m /3=3. 29, W„=0.393, and y„=&a/2= 1.253, while p, ~ M, P, ~ M, and R is independent ofM.

At the low-mass end of the brown dwarf branch, forM ~4MJ, the Coulomb corrections to the P-p law com-pete with the degeneracy component and shift theeffective polytropic index from 1.5 [Eq. (2.2)] for themore massive brown dwarfs to 1.0. As stated above,when n =1.0, the radius of the object is independent ofthe mass. Figure 1, which schematically compares Rversus M for cold brown dwarfs, M dwarfs, white dwarfs,and neutron stars, demonstrates that the radii of low-mass brown dwarfs plateau over a broad range of masses.Indeed, even above M =—5MJ when n —1.5 andR ~M ', the dependence of the brown dwarf's radiuson mass is weak. Zapolsky and Salpeter (1969) have pro-vided a useful analytic fit to the radius-mass relation thatincludes both planets and cold brown dwarfs:

R =2.2X10 MoM

1/3

1+0.0032Mo

—1/2 4/3

cm . (2.9)

Rev. Mod. Phys. , Vol. 65, No. 2, April 9 993

A. Burro~s and J. Liebert: The science of brown dwarfs

&0'0—I

Jupiter —=0.5z =A

z=——=1.0A

~lgy ~

Il

Schematic Radius-vs- Mass Plot&06-for Compact Objects

N80tron stall

1050.001 0.0) 0.1

M (Mo)

1.0 10.0

FIG. 1. Radius-mass plots for the variety of compact objects with surfaces. Depicted are the brown dwarf and M dwarf branches,the white dwarf branch (Z/A =0.5), and the unstable branch between white dwarfs and neutron stars. Shown too, is the high-masscontinuation of the cold, pure hydrogen brown dwarf (Z/2 = l.o) sequence. The positions of the Sun and Jupiter are clearlymarked. See text for discussion.

In fact, as the mass ranges from MJ to -80MJ along thefull cold brown dwarf branch, the radius changes by nomore than 50% and is always near R J ( =8.26 X 10cm-0. 12RO). Furthermore, in a structural, but not ancvolut1onary scrlsc, cold br'own dwaI'fs arc at thchydrogen-rich (high Z/A), low-mass end of the whitedwarf family.

Equations (2.6)—(2.8) allows us to derive most of thesalient features of cold brown dwarfs. Their R's are near0. 1RO,' p, ranges from —10 gm/cm for the light to 1000grn/cm for the heavy; and P, can be found above, near,and below 10' Mbar (1 Mbar =10' dyn/cm ). However,the polytropic analysis alone does not illuminate thethermal structur'e and evolution of a brown dwarf. Thetime dependence of brown dwarf characteristics is a cru-cial dimension in brown dwarf theory. T'he CAective pho-tospheric temperatur'e, luminosity, and central tempera-ture (T, ) at a given time require a knowledge of thespecific heat, atmospheric opacities, phases, etc. that can-not be extracted from Eqs. (2.1)—(2.8). Supplementedwith this ancillary knowledge, the polytropic approxima-tion can indeed provide an analytic model of browndwarf theory (versus M and r). Stevenson (1991) andRappaport and Joss (1984), in particular, have developedsuch models, and. we provide our own version in Sec.II.E. However, an accurate theory must incorporate thethermodynamic and atmospheric details. The analyticmodel reveals the systematics, but only the full simula-tions should be used for numbers.

B. The equation of stateand thermodynamics of hydrogen

More than 50 years ago, VA'gncr and Huntington(1935) postulated that under high pressures molecularhydr'ogen would undergo a phase transition to the metal-lic state. This metal-insulator transition has receivedsporadic attention ever since and is now the subject of anintense, but still ambiguous, experimental quest (Maoand Hernley, 1989; Mao, Hemley, and HanQand, 1990;Van Horn, 1991). Nevertheless, the pressure ionizationand metalhzation of hydrogen and hydrogen/he11ummixtures is unavoidable, given suitable pressure (P-3Mbar), and such an alloy must form the bulk of Jupiter(-85%), Saturn ( —50%), and brown dwarfs ( )99.9%).Theoretical treatments of the thermodynamics and ener-getics of such strongly coupled Coulomb plasrnas in as-trophysical contexts have been attempted by many, butfor this synopsis we have been guided by the work ofStevenson (1976, 1978, 1979, 1982, 1983, 1991). DeWittand Hubbard (1976), Slattery, Doolen, and DeWitt(1982), Marley and Hubbard (1988), Magni and Mazzitel-li (1979), Stringfellow, DeWitt, and Slattery (1990),Mestel and Ruderman (1967), and Saurnon and Chabrier(1989), to any and all of which the reader is referred fordetails. Saumon and Chabrier, in particular, have pro-vided a most careful theoretical treatment of pure hydr'o-gen above and below the plasma phase transition (PPT),at h1gh arid low temperatures~ RIld 1rl thc molecular

Rev. Mod. Phys. , VoI. 65, No. 2, April 1993

A. BUrro~s and J. Liebert:

atomic, and ionized state. A revealing temperature-dcIls1ty pIlasc diagram fo1 pul c hydrogcQ 1nsp11 c«i

Saumon and Chabrier (1989; Saumon, 1990) is shown inFlg. 2. Th1s figulc «icplcts most of thc csscnt181 informa-tion concerning the thermodynamics of the entire browndwRI'f an«i giant-planet falrllly. S1ImlaI' phase d1RgIamscan be found in Stevenson (1991)and Hubbard (1984). AJupiter (J) and two 0.05MO brown dwarf adiabats (Bur-rows, Hubbard, and Lunine, 1989) are shown in Fig. 2 asdotted 11ncs. Though thcsc ad1abats assume 8 hc11ummass fraction ( F ) of 25% (for the dwarf) and 22% (forJupiter), their superposition on a pure hydrogen phase di-agram stlH makes foI' 8 fruitful coITlpallson. Thc X s OIl

these lines mark the points interior to which lie 50% ofthc object s 1Ilass. As ls apparent, only 8 s1TlRH mass fx'ac-

tion of these objects inhabits the low-temperature, low-«icIls1ty regime w1111c IIlost of their mass 1s 1ndccd 1Q thcllqu1d mctaHlc stRtc.

p=0. 593 and 2.29 in the two regimes, but it is always ofol «icl unity.

The photosphere of 8 brown dwarf is located approxi-mately Rt thc v=

3 sUlfacc, where T' ls thc optical depthgiven by

(2.13)

Ic& is the Rosseland mean opacity. Using the equation ofhydrostatic equilibrium (2.3) and (2.13), we obtain

(2.14)

where P, is the pressure at the photosphere, g is the sur-face gravity and has been set equal to 10 cm/s, and azhas been set equal to 10 cm /g.

In a region such as the photosphere where it can be as-sumed that g is constant, Eqs. (2.3) and (2.4) can berewritten as

1. The low-density regime

pX~kTP= {2.10)

where X~ is Avogadro's number, k is Boltzmann's con-stant, and p is the mean "molecular" weight. %'hen hy-drogen and helium are completely ionized (ignoring, forihe moment, '*metals" ),

{2.11)

but when they are in the moleculax state,

(2.12)

where X and 1"~ axe the hydrogen and helium mass frac-tions, respectively. For X=0.75 and 7 =0.25,

Referring to the bottom left of Fig. 2, we note that atlow densities (10 gm/cm &p&0. 1 gm/cm ) and lowtemperatures ( T & 4000 K) hydrogen is predominantly inthe moleculax' state and is a Quid. Between -0. 1 and—1.0 gm/cm and far below the critical point (atP„;,=0.614 Mbar, T,n, =1.53X10 K, and p„;,=0.35gm/cm, marked "SC"},hydrogen is a solid. The solid-Auid tlanslt1GQ ls 1Q«ilcRtc«i with 8 dashc«i 11Ilc. It shouldbe emphasized again that the precise phase boundariesand details throughout the p-T plane are stiH a subject ofdebate (cf. Marley and Hubbard, 1988; Robnik andKundt, 1983) and that, though the discussion here re-volves 81GUQd the Saumon Rnd Chabrier cqUatloIl ofstate, the subject is still in a state of productive Aux.

As the temperature increases above —10 K, molecu-lar hydrogen. first dissociates an«i then ionizes. Tempera-ture ionization does not occur via a phase transition. Atlow dcnslt1cs Rnd high tcIIlpcraturcs, hy«ix'ogcn 1s 8 vcI'yweakly coupled gas. Thc pI'cssUI'c law thI'oughout thclow-«icns1ty regime 1s glvcIl approximately by thc 1«ical-

gas law

whcI'c AM 1S thc total mass thRt ovcI'11cs thc shcH Rt pI'cs-sure P and R is the total brown dwax'f ra«iius. SettingP =P„we derive that AMph —10 "Mo. In addition,the polytroplc Eq. (2.7) can be used to give us

hM P=f 4mB"„j

where P, 1s thc ccntlR1 plcssUrc and M 1s thc total stcHarmass. Of course, Eq. (2.16) gives us b,Mp„/M —10Thc I'Rdlatlvc region of 8 blowIl dwaI'f 1s 1Ildccd 8 plo«i1-giously small fx'action of the object.

Equation (2.10) implies that, at a photospheric temper-atux'e of —1500 K, p, —10 gm/cm and puts the pho-tosphere squarely in the gaseous molecular hydrogen x'e-

g1on of Flg. 2. In Rn«i around this low-dcns1ty, low-temperature region, the temperatures are Sufhcient to ex-cite the I'otational levels of the H2 molecule into equipar-tlt1OQ, but Ilot 1ts vlbrat1GIlal lcvcls. This fact 1IIlpllcsthat the adiabatic indices (I,, l"2, 1 3) are all near(=1.4), not 3. Therefore I' ~p'" and T~po~ along anadiabat. The shaHow slopes of the brown dwarf adiabatsin the low-density molecular region of Fig. 2 are 8 conse-qucncc of this 1otatloQ81 softcmng.

2. The high-density regime

Most of a bx'own dwarf can be found on the right-handside of Fig. 2 in the high-density, liquid metallic region.Saumon and Chabrier adopt a "chemical picture" inwhich the actors (H, H2, H, and e ) interact via pairpotcnt181s. They 1dcIltlfy 8 first-order plasma phase tIall-sition (PPT) that occurs over a narrow range of densitiesnear 1.0 gm/cm (shown as the hatched region in Fig. 2)RIld they coIlcludc that molecular dissociation an«i pI'cs-sux'c 1GQ1zatloIl concur. Thc tl'Rnslt1on 1s from 8 IlcUtlRlphase (X, &10 } to a partially ionized phase (X,-0.5)

Rev. Mod. Phys. , Vol. 65„No. 2, April 1993

306 A. Burrows and J. Liebert: The science of brown dwarfs

(K)

10

10 -1 0log p (gm/cm3)

FIG. 2. Phase diagram of temperature (in K) vs density (in gm/cm ) for pure hydrogen. Superposed as dotted curves are two0.05MO, Y =0.25 brown dwarf adiabats from BHL at 5X10' and 2X10' yr and a Jupiter (J) adiabat. The X's on these adiabatsmark the 50% mass point. Lines of constant Coulomb parameter (I ) are provided, as is the g=1 line (see text). Shown clockwiseare the solid H2, liquid and gaseous H&, atomic H, ionized H+, and metallic H states. "SC"marks the critical point as calculated inSaumon and Chabrier (1989), to which one is referred for further details. The hatched region represents schematically the realm ofthe first-order plasma phase transition (PPT). See text for discussion.

and is at a pressure of 1 —3 Mbar. Above the PPT, theionization fraction increases rapidly with p to 1.0.

The characteristics of a strongly coupled Coulombplasma are determined by the dimensionless Coulomb pa-rameter, the ratio of the Coulomb energy per ion to kT:

Z2 2 1/3Z0 227 P Z /

rkT p,(2.17)

1 1 1+X(2.18)

For Y =0.25, p, =1.143. r, is the radius of a spherethat contains one nucleus on average and, along with r„the electron spacing parameter, is the relevant lengthscale. r, and r, are defined as

4vrn,

1/33pe Z

4~%~p(2.19)

1/3

4mn,

3pe

4~%~p(2.20)

where Ze is the ionic charge (Z = 1 for H), T6= T/10K, p is in cgs, and p, is the number of baryons per elec-tron given by

where n, is the electron density (n, =pN„jp, ). Theweakly coupled regime is characterized by I ((1 and iscalled the Debye-Hiickel limit. This regime is encoun-tered in both the Sun and in blood plasma. The transi-tion to the strongly coupled liquid state occurs at I -1.As Fig. 2 implies, brown dwarf matter lies between I -1and 10 in the liquid metal regime. Note that I in abrown dwarf achieves its minimum at its center andgradually grows to its highest value at or near the PPT.

Calculations of the thermodynamic properties ofstrong-coupled Coulomb plasmas have occupied manyworkers over the years and have achieved great sophisti-cation (Hansen, 1973; Ichimaru, 1982; Hubbard andDeWitt, 1985; Ogata and Ichimaru, 1987; Spruch, 1991;Stevenson, 1991). One needs to include not only the sim-

ple Coulomb part of the ion-ion interaction, but also theelectron exchange and correlation terms. Furthermore,electron screening and density-Auctuation corrections arerequired. Some of the various approaches that have beenused in the past include the signer-Seitz approximation,the Thomas-Fermi-Dirac theory, and dielectric functiontheory. The latter, in particular, which focuses on thedielectric response of the plasma to calculate the effect offluctuations in n, on the Coulomb. binding, has been quitefruitful (Hubbard and DeWitt, 1985). Since it will takeus too far afield to describe the details of the physics ofstrongly coupled plasmas, we shall not do so here. How-

Rev. Mod. Phys. , Vol. 65, No. 2, ApriI 1993


ever, some of the essentials that touch directly on browndwarf theory bear mentioning.

The atomic unit of pressure (Po=e /ao, where ao isthe Bohr radius) is 294 Mbar. For pressures significantlyabove Po, the degeneracy pressure of free electrons dom-inates and P-PD where

5/3 ' 5/3= 10 Mbar . (2.21)

cm Pe

3yX~ kTPT=yET= (2.22)

This echoes the polytropic form of Eq. (2.1) with n =1.5.The temperature corrections to P are generally small

along brown dwarf tracks (the electron component is

especially small) and can be approximated by theGriineisen (1926) theory:

cillation be included in the Lindemann analysis. Whenthis is naively done, one finds that hydrogen "melts it-self" at all temperatures for p) 1 gm/cm due to itszero-point motion. Indeed, this is what molecular hydro-gen does at atmospheric pressure in the laboratory. It isa liquid down to 0 K. However, it has been suggested(Ceperly, 1986; Ceperly and Alder, 1986) that a morerigorous quantum-mechanical calculation allows solidmetallic hydrogen/helium mixtures to exist in a restrict-ed region of the p-T plane, far below the I =178 line, atlow temperatures ( T & 10 K) and "low" densities(p & 10 gm/cm ). However, in a recent paper, Chabrier,Ashcroft, and DeWitt (1992) argue that the zero-pointterms in the solid and liquid phases are comparable andthat the critical I is little changed. Nevertheless, wheth-er the solid phase obtains is moot for our purposes, asFig. 2 amply demonstrates.

where y is the Griineisen parameter [0.6~0.5.for Hfrom 1 to 10 Mbar (Hubbard, 1984)], ET is the thermalenergy of the ions in the liquid, and V is the molarvolume. Core brown dwarf temperatures exceed the De-bye temperature, and the Dulong and Petit formula forthe molar specific heat Cz =3R is a satisfactory approxi-mation (Debye, 1912). With the pressure in degenerateelectrons and the specific heat, entropy, and thermal en-

ergy in the ions, the equation of state at high pressuresand densities is conceptually clean. However, at pres-sures at and below Po, the Coulomb interaction is cru-cial. A characteristic length, the Bohr radius, appearsalong with a characteristic density ( -the density of "ter-restrial" matter). Since the central pressure of Jupiter(PCJ ) is —100 Mbar and Pci & Po, Coulomb effects makea qualitative difference for objects near Jupiter's mass.For a pure hydrogen plasma, the density-dependentCoulomb correction to the pressure in Eq. (2.21) is a fac-tor of (1—0.85/p'~ ), which below p-50 gm/cm issignificant. It is at the onset of Coulomb dominance,with its characteristic density and length scale, that theeffective polytropic index slides from n =1.5 throughn = 1.0 to n -0.0 (-incompressible; see Fig. 1).

At a given density, as the temperature is lowered, I in-creases. According to the classic arguments of Lin-demann, the solid~liquid phase transition occurs whenthe thermal oscillation of an ion in the harmonic-oscillator potential of its Wigner-Seitz cell achieves somegiven fraction (e) of r, (e-0. 1, perhaps). Since themean-squared thermal displacement (hX ) of an ion in aharmonic oscillator is -kT/Mco, the condition that(b,X/r, ) =E =(0.1) translates into a condition on I( & 1/e —100). More detailed Monte Carlo calculations(Brush, Sahlin, and Teller, 1966; Slattery, Doolen, andDeWitt, 1982; Stringfellow, DeWitt, and Slattery, 1990)put the transition at I =178, and this line is shown inFig. 2. Since none of the brown dwarf trajectories reachthis line, we conclude that the solid metallic phase is notof astrophysical importance. In fact, the lightness of thehydrogen nucleus (proton) requires that its zero-point os-

C. Brown dwarf and M dwarf atmospheresand spectra

As we have shown, the outer radiative skin of a late Mdwarf or a brown dwarf is exceedingly thin. Witheffective temperatures at and below 3500 K, they bear asuperficial resemblance to red giants and, by their ex-treme red colors, could be confused with them. Howev-er, the surface gravity of our compact protagonists issome six to seven orders of magnitude larger than that ofa giant. Using Eqs. (2.10) and (2.14), we find that thehigh gravities and the similar or lower effective tempera-tures imply that brown dwarf and VLM photosphericpressures and densities exceed those of giants by a similarfactor. Therefore, the atmosphere, instead of being dom-inated by atomic species, is composed mostly of mole-cules. Furthermore, below about 2400 K, grains con-dense out and can contribute significantly to the continu-um opacity. Molecules and grains provide the distin-guishing spectral features at the edge of the main se-quence. In addition to the molecular hydrogen (H2) andneutral helium that dominate the atmosphere, Hz, Hand He can be found. The metals are in their neutral,singly and doubly ionized states deep inside the atmo-sphere at 10 K, but near the photosphere are found ashydrides [CaH, MgH, SiH), oxides (TiO, VO), magnesi-um silicates [e.g., enstatite (Mg2Si206)], carbides (e.g. ,Fe3C), and iron grains. The C/0 ratio determines thechemistry of carbon. For C/0) 1, carbon tends to be in-corporated in CO and graphite grains, but for C/O(1and very low temperatures ( &600 K), it tends to be inCH4. The abundance of Fe3C is also regulated by thecarbon abundance. Oxygen is not only incorporated insilicates and CO, but in H20 as well. "Steam" bandsplay an important role in spectrum formation and may bethe most useful brown dwarf signatures, centered as theyare near the infrared J, H, and IC bands (Berriman andReid, 1987; Leibert and Probst, 1987; Lunine et al. ,1989;Saumon, Lunine, et al. , 1992).

With such a plethora of species to consider, the radia-


A. Burrows and J. Liebert:

tive transfer and spectral synthesis calculations are for-midable. However, the molecular features in VLM andbrown dwarf spectra provide essential and critical diag-nostics that must be mastered before cool stars andbrown dwarfs are to be understood. The major opacitysources come from H, Hz, He, atomic hydrogen tran-sitions, Rayleigh and electron scattering, bound-freetransitions of Mg, Si, and Ca, the electronic bands ofMgH, CaH, and SiH, the rotation-vibration transitions ofTiO, H20, CQ, and CN, and the collisionally induced di-

pole moment of Hz (Lenzuni, Chernoff, and Salpeter,1991). Tllc lattcl' ls pR1 tlculally lIIlpoltallt Rt low IIlctR111-

cities and in Jupiter's atmosphere. The opacity due tograins requires special knowledge —we need the particlesize spectrum (a —1 —10 p, m't), the vertical distribution ofthe grain "clouds, " their patchiness, and their composi-'tloll. Lulllllc et Ql. (1989) made 'tllc fllst detailed Rtfclllptto understand cloud physics by studying grain coales-cence, aggregation, and transport, and the vapor pressurecurves of the relevant solids. However, the problem as itstands is too complicated soon to yield, and we must besatisfied at present with low-order approximations andgucsscs.

Despite these caveats, there has been real theoreticalprogress over the last few years in understanding theemission of cool, dense atmospheres (Lunine et aI. , 1989;Allard, 1991; Ruan, 1991). Allard's (1991) recenttheoretical spectra deserve special mention for their corn-pletcness and the clarity with which she has catalogedthe spectral systematics with T„g, and metallicity. %Cshall briefIy discuss some of her work to provide thereader with a flavor for the theoretical spectra of coolstars. Note, however, that she does not study browndwarf spectra (T, (2000 K), but concentrates on late Mdwarfs (2000 K (T, (3500 K). There are as yet no goodtheoretical models of brown dwarf spectra (however, seeSaumon, Lunine, Hubbard, and Burrows, 1992 and Lun-ine et al. , 1989); so our discussion cannot be ascomprehensive as wc would 11kc.

Flglllc 3, tRkcll fl'olll Allard (1991),dcplcts colltlllull111and band opacities (J 1 ) versus wave number forT, =3000 K, g =10 cmjs, and solar composition at theatmospheric level where ~ at the center of the J band(A, =l.2 pm) is unity. In the infrared, the rotation-vibration bands of H20 and bound-free absorption ofH2, H, and He dominate. In the optical, absorptionby the Inolccular bands of TiQ and the classic bound-freeabsorption of H dominate. The TiO and VO bands aregenerally used to determine spectral type for the Mdwarfs. However, as T, decreases, these bands becomeless and less useful indices of spectral type. Beyond-M6 a better set of features, perhaps the steam featuresin the infrared, shouM be considered to classify the lateM dwarfs (Kirkpatrick, Henry, and McCarthy, 1991).

Figure 4 from Allard (1991) shows a synthetic emer-gent spectrum for T, =3000 K, g =10 cm/s, and solarmetallicity. The standard photometric filter bands are in-dicated. The solid line is the complete spectrum, the dot-

FIG. 3. Opacities (in crn /gm) vs wave number (in p,m ) for

g =10' cm/s, solar composition, and T, =3000 K from Allard(1991). Shown are most of the dominant sources of atmosphericopacity due to H2O, TiO, H, H2, H2, and metals at a depthsuch that w(A, =1.2pm)=1. Notice the prominent role of H2O,H, and TiO.

ted line is the spectrum with the TiO and H2Q bands andthc atomic 11ncs suppressed, alid thc dashed 11nc dcp1cts ablackbody Rt 3000 K. Nc1thc1 M dwaI'f IloI' bI'owIl dwRI'f

spectra are blackbodies, particularly in the optical whereTiO, VQ, and the hydrides severely blanket the emission.Most of the energy emerges in the infrared, and there aredistinctive H2Q and CO features. The systematic altera-tion of the M dwarf spectra wi. th T, from 3250 to 2000 Kis depicted in Fig. 5, also from Allard (1991). The dashedlines are the corresponding blackbodies. It can be seenthat the H20 features emerge inexorably with decreasingT„and the blackbody approximation completely breaksdown. Band absorption causes a systematic shift of thecAcctive peak of thc cmissloI1 to hlghcr fI'cqucnc1cs IclR-

lE

2

ML K, H„JI

, L'

FIG. 4. Complete synthetic emergent spectrum from Allard(1991) for g =10 cm/s, T, =3000 K, and so1ar composition(solid line). The dashed line is a blackbody curve normalized tojust envelop the synthetic spectrum, while the dotted line ig-nores the absorptions due to H20, TiO, and metal lines. Noticethe distinctive H2O and CO features in the infrared and thegrossly unblackbody behavior in the visibl. The regions of thestandard photometric bands are superposed for reference.



I I Ij I I I &

j& I I I

2000 K

02.9

CD

03.5

C4

C

044

2500 K

05.7-

r r I I I

0 2 3&a~ (+m-')

FICx. 5. Same as Fig. 4, but for T, =2000 K to 3250 K (fromAllard, 1991). The dashed curves follow the blackbody func-tion. Theoretical dwarfs are not blackbodies and show a dis-tinctive blueward shift with decreasing T, from the blackbody.At and below T, -2500 K, the color temperature and T, can bedi6'erent by hundreds of K.

cools "like a rock." If the central temperatures achievedbefore the onset of degeneracy are sufficient to igniteenough thermonuclear fuel, the object can become a star.If not, it will become a brown dwarf. Only objects withsufficient mass achieve ignition before degeneracy.Hence there is a minimum "stellar" mass (canonically-0.08MO) above which lies the hydrogen main se-quence. On the main sequence, photospheric energylosses are balanced by burning in the core.

Since thermonuclear rates are functions of temperatureand density, the true ignition condition involves both.Hence the ignition temperature is a function of density.Curiously, one finds that at the edge of the main se-quence, due to the density increase during final contrac-tion, the core temperature of the transition mass actuallybegins to decrease before stabilizing at its main-sequencevalue. Even below the edge of the VLM branch, massivebrown dwarfs ( ~0.07M~) can burn hydrogen for bil-lions of years before their thermonuclear fires are eventu-ally quenched. In fact, down to masses near -0.015Mo.(Grossman, 1970; Grossman, Hays, and Graboske, 1974),primordial deuterium is burned during a brown dwarf'sfirst ten million years, and, down to -0.06Mo, primor-dial lithium is consumed, though over a significantlylonger time.

The main thermonuclear reactions on the VLM andbrown dwarf branches are

tive to the corresponding blackbody peak. This results ina systematic separation of the color temperature and T,with decreasing T„as has been noticed by Berriman andReid (1987); see also Stringfellow, 1991). Brown dwarfsshould continue this trend (Ruan, 1991; Saumon et al.1993).

The Rosseland mean opacities (~z ) are a bit easier toestimate and generally range between 10 and 10cm /gm for solar composition (Tsuji, 1971; Alexander,Johnson, and Rypma, 1983; Lunine, 1986; Lunine et al. ,1989; Carson, 1992; Carson, Luo, and Sharp, 1992). It is~z that determines the relationship between the centralentropy (or the temperature at 10 bars) and T, and,therefore, the rate at which the thin radiative skin of thebrown dwarf releases its bulk internal heat. The atmos-pheric opacity is the valve that controls the evolutionaryhistory of the brown dwarf.

D. Nuclear rates

As a VLM or brown dwarf contracts early in its evolu-tion due to radiation from its surface, its core tempera-ture increases. This is the classic negative specific-heateffect in stars. The contraction is halted either by the on-set of electron degeneracy or by the ignition of ther-monuclear fuel, whichever comes first. With degeneracy,since pressure is then only a weak function of tempera-ture or heat content, further photospheric radiation doesnot lead to significant compression and is accompaniedby a decrease in the core temperature. The object then

and

p +p —+d+ e++v, (Q =1.442),

p+e +p —+d+v, (pep process),

p+d~ He+@ (Q=5.494 MeV) .

(2.23)

(2.24)

(2.25)

aild

—33 8/T=2.5 X 10 [pX /T6 ]egms

(2.26)

—37 2/T r4~ 1024[+++ /7 2/3]&

' 6 gpd (2.27)

The core temperatures achieved in these light objects arenot sufficient to overcome the Coulomb barriers (Z =2)of the He- He and He- He reactions of the complete p-pchain that operates in the sun. Hence the p-p chain istruncated and He is not produced. The strong-interaction process (2.25) is quick, is the process bywhich primordial deuterium is consumed, and providesthe bulk of the thermonuclear energy. The weak-interaction process (2.23) creates the fresh deuterium andis the rate-limiting link in the burning chain [the pep pro-cess (2.24) is of secondary importance]. The efFective Qvalue per proton of combined processes (2.23) and (2.25)is (1.442 —0.262)/2+5. 494=6.084 MeV, instead of the6.901 MeV of the full p-p chain. The 0.262 MeV ac-counts for the average energy of the neutrino (v, ) lost inprocess (2.23).

The unscreened energy generation rates of processes(2.23) and (2.25) are (Fowler, Caughlan, and Zimmerman,1975)



H(o) (2.28)

where Yd is the deuterium mass fraction whose primordi-al value is near 2X10 . (Note that we employ the"dot" to emphasize that this is a rate. ) However, since Iis of order unity, the plasma is strongly coupled, andscreening of the protons and deuterons in the above pro-cesses can significantly affect the reaction rates. In theintermediate and strong screening limit, the enhancementfactor for both processes (2.23) and (2.25) is given byGraboske et al. (1973),

2/3

=3.018X 10Pe

(2.34)

where pF is the electron Fermi energy in the degeneratelimit and the other constants have their standard mean-ings. At high densities and low temperatures (and henceat low entropies), i) »1 and the gas is degenerate, whileat high entropies, i) ((1 and the ideal-gas law (2.10) ob-tains. A pressure law that applies at both extremes, butonly poorly between them, is

where

II(0)=min(0. 977I ', 1.061 ) . (2.29)

5/3 e 5Pe1+—,a='g 2p

(2.35)

=-'T ''En =&C

C

PPC

(2.30)

we can use (2.26) and (2.28) to derive

u =2+ ' H(0)3

(2.31)

and

At the edge of the main sequence, the core S is -2; butlower on the brown dwarf branch, it can be higher.

One can always fit thermonuclear rates to power lawsin T and p, which obtain over only a narrow range inboth, but give a clear indication of the sensitivities. If weset

where p is the mean molecular weight of ionizedhydrogen/helium mixtures. For X=0.75 and Y =0.25,p, =1.143, p=0. 593 and o.'=4. 82. The virtue of Eq.(2.35) is its simplicity. (It is not highly accurate for par-tially degenerate Iluids. ) The crucial point to notice isthat i) is approximately constant along adiabats (see Fig.2). Hence a single value of i) is characteristic of the vastbulk of a brown dwarf of a given mass and composition,at a given time. Brown dwarf evolution proceeds fromhigh to low specific entropies (cr =S/kN~, where S is theentropy per gram) and from low to high i)'s. Knowingeither cr(t) or YI(t) is equivalent to knowing how browndwal fs evolve.

Equations (2.2) and (2.35) imply that the polytropicconstant is

s= ———1.29H(0) .33.8 2

3T1/3 36

(2.32) 1013 1+—5/3 (2.36)

At T, =3X10 K and p, =10characteristic of the core of thederive that u -2.28 and s=6.31.moderately steep dependences andto u =2 and s-4 for the core of the

gm/cm, conditionstransition mass, weThese values imply

should be comparedSun.

The scaling laws for polytropes [Eqs. (2.6) to (2.8)] andEq. (2.34) can now be converted into

K'R =2.3573 GM'"

E. An analytic model for brown dwarfs 05M0=6.05 X 10

' 1/3

1+— cm, (2.37)

Because brown dwarfs are convective and, hence, isen-tropic and their equation of state is polytropic bothabove and below the PPT, all the essentials of browndwarf cooling theory can be analytically obtained. Inthis section, we derive T, (t, M) and L (t,M) using onlythe scaling laws of a polytrope, the entropy matchingcondition at the PPT between metallic liquid hydrogen athigh densities and molecular hydrogen at low densities,and a crude model of the photosphere. Furthermore, weextend this analysis and derive analytically the value ofthe minimum hydrogen main-sequence mass. Despitethe crudity of the assumptions, the derived power lawsare a passable match to the more detailed cooling theory.

When an electron gas is degenerate, the measure of itsdegeneracy is given by

MP 0 05~

2

1+—'9 .

10/3

C

' 2/33.018X 10 K Pc

Pe

P, =3.67x10'0

gm/cm 3 (2.38)

(2.39)

(2.40)

. 4 Mbar,

2/3pF (3~2g3)&~3 pN„'I=kT 2m, kT p,

(2.33) =2.06x10' K0

4/37l

(r)+a)(2.41)


A. Burrows and J. Liebert: The scIence of brown dwarfs

TH18x —106

0.05MO(2.42)

Larger masses achieve larger core temperatures. If themass is sufhcient, the core can reach ignition conditions.Equations (2.37)—(2.41) show that when T, is at peak,R =2R (i)= ~ ), p, = ,'p, (i—l=~), and I', =

—,', P, (g= co).Even after T, =T, ", the "star" undergoes significantcontI"RctioIl.

Thc cncI'gy equation~

dE d v dS . BI.dt dt dt BM '

where, again, S is the entropy per Inass and the otheI'syInbols have their standard meanirgs, determines thepace of cooling and contraction. Ignoring the energygeneration term and integrating over the mass, we find

fK„kTdM = I. , —

where I is, of course, the surface luminosity. Solving Eq.{2.34) for T, inserting into the bracketed expressionabove, and using the polytropic relations I' =Ep / and

(2.45)

where we have set p, =1.143 for definiteness. %'hen g isfinite, the star is thermally expanded; as q increases, thestar contracts. The degenerate-star values of P„p„T„Rnd R Rrc i cached asymptotically Rs g~ Qo .

As is plain from Eq. (2.41), T, has a maximum for agiven mass at g =a:

r 4/3

To derive an equation that connects p, and T„we availourselves of the isentropic condition. Though the atmo-sphere is by its nature radiative, the entropy at the pho-tosphere is not much diferent from that in the vast con-vective regions. Though the entropies are approximatelythe same exterior and interior to the PPT, the depen-dences on T and p are different —we must rnatch them atthe PPT to connect il with T, . [N.B. However, it maywell be a weakly first-order phase transition with Do %0(see Saumon, Hubbard, Chabrier, and Van Horn, 1992).]To do this, we use the approximate analytic expressionsfor the specific entropy radially above (cr, ) and below(o.2) the PPT:

o i= —1.5941ng+12.43 (2.48)

o2=1.032lnT/p ' —2.438,

where o 2 is from Stevenson (1991), 1' has been set equalto 0.25, and o. , is a crude fit from the results of Burrowset al. (1989). Setting o, =o.

2 (isentropic condition),T = Te, RIld p=p~~ wc obtMIl

0.42

This is a useful result and is an approximate measure ofthe thermal energy content of the object. Note the sti6'dependence of M. [Curiously, the —,'power appears byanalogous arguments in the ThoInas-Fern. i model of themany-electron atom (Feynman, Metropolis, and Teller,1949.]

We next turn to an estimate of T, and L,. For this, weemploy Eq. {2.14) and the ideal-gas law:

2 GM3Ra~

7/31

A+ 'gerg . (2.46)

(2.50)

Plugging into Eq. (2.47) and using the polytropic relation

g = — =1.8X10E. ~ Q

5/3

, cm/s(1+u/il )

p, =1.82X10 M0 005mo

.,0.704

10 1 3

(1+a/i) )glii /ciii (2.52)

1.84X10" Kl.088

M0.05MO

which when substituted into Eq. (2.50) gives us- 0.493 .

2. 0.296

10 1

( 1+~/~)0. 59(2.53)

We note in passing that Eqs. (2.47) and (2.51) yield5/3

12.2 bar M{1+~/&)' 0.05MO

the canonical atmospheric pressure.



Setting L =4m.R o T, (here o. is the Stefan-Boltzmann constant) yields

M' ' 005M,

1.305 1.18410 ' (I+~/q)-'36

4.352R

(2.55)

If wc now plug Eqs (2 46) (2 48), Rnd {2.55) I««hc cn«gy equation {2.44), we obtain the evolutionary equation for rithat we seek:

0.05MOg 2 0/x 10 14

M

1.028 . 1.18410 (1+a/Ii)

2. 352(2.56)

If we ignore the early evolution and assume that )I)&1, we can drop the (I+a/Ii) ' term above and integrate. The re-sult is

0.05M .M

. 0.35310 (2.57)

M0.05MO

which, after substituting into Eqs. (2.53) and (2.55), gives us

0.324 0.827 0.08810 y

10(2.58)

1.29710" yr M

0.05MO

0.35

(2.59)

Equations (2.58) and (2.59) encapsulate the analytictheory of brown dwarfs and are surprisingly accurate,given the approximations they embody. (Note that, inthe above, we assume that az is independent of time. ) Itoften will prove useful for the reader to refer to these ex-pressions when brown dwarf systematics is discussed, buta few of their noteworthy features bear mentioning here.Foremost is the fact, that not only are the powers andtrends in Eqs. (2.58) and (2.59) realistic, but the numbersobtained are good to better than 35%, when compared,for example, to detailed model G of BHL, for all but theearliest epochs. The weak dependence on ~z (especiallyfor T, ) encourages us that this most problematic piece ofbrown dwarf theory will not, when completely under-stood, force qualitative changes in our theories. The stiItdependence of I. on M rejects the sti6'dependence of thebinding ci1cl gy [ GM /R M, cf. Eq. {2.46)] oilmass, but is only slightly steeper than the correspondingrelation above the transition region on the VLM branch(L ~M2 2 2 6)

The minimum main-sequence mass (MMSM) is derived

by setting L„(Ji„dm) equal to L [Eq. (2.55)] in Eq.(2.43). If we employ the power-law form [Eq. (2.30)] ofe„, make the polytropic approximation to the structure

(p/p, =&"), and set T/T, ={p/p, ) along the core adi-

abat, we obtain

L„=I~„dm =4Ira e,p, I0"g d g,where b =n(u +—,'s). Setting 0=1—g /6-e ~, which

is a good approximation in a stellar center (Fowler andHoyle, 1964), we find

2.4i,ML, = =0.079' M,

( —', u+s) i (2.61)

where n =1.5, u =2.28 [Eq. (2.31)], and s=6.31 [Eq.(2.32)] were used. Plugging Eqs. (2.38) and (2.41) into Eq.(2.60) yields

11 977 10 1$

(2.62)( ++~ )

16.466

Equating L„and L [Eq. (2.55)] gives us', 0.111 (q+~)'"'

2.70 1M 10

—2 1325 =I(Ii) .

(2.63)

Equation (2.63) is an imphcit equation for Ii(M) along thelowcI' II1Rlil scqucIlcc. Tllc cssciltlRl fRC't of Eq. (2.63) isthat I(II) has a minimum If the m. ass M is too low,there is no solution to Eq. (2.63). The lowest mass forwhich there exists a solution to Eq. (2.63) is at the bound-ary betweeI1 browI1 dwarfs and VLM s. The minimumvalue of the function I(ri) is 2.337 at Ii;„=34.7. Substi-tuting into Eq. (2.63) gives us

M =0.0865MO10 I( ri)

I(Ii;„) (2.64)

and, hence, an MMSM of 0.0865( 10 /KII )

This number is in excellent agreement with that of thefull theory (Kumar, 1963). However, a value of 34.7 forI);„ is too large. Fortunately, I(I)) is a very shallow



function of g and, if we put in a more realistic boundaryvalue of g of 9.4, we obtain for the MMSM a value of-0.1(10 /zz ) '"Mci, still respectably close to the ac-tual numerical result. Equation (2.64) also suggests thatthe dependence of the MMSM on opacity is very weakand inverse. In particular, Eq. (2.64) correctly predictsthat the edge mass of the low-metallicity subdwarf popu-lation is higher (D'Antona, 1987).

The assumptions concerning the equation of state, thedependences of the nuclear rates, and the constancy ofthe atmospheric opacities are too crude to allow us to useEqs. (2.63) and (2.64) to find a reliable g(m) relation and,hence, R (M) and L (M) along the lower main sequence.For this, the numerical simulations are required. Howev-er, all the systematic behavior of the transition mass andbrown dwarf cooling theory are reproduced and summa-rized by the analytical arguments presented herein.

F. Numerical brown dwarf and M dwarf models

To provide a favor of the internal structure (and itsevolution) of a specific brown dwarf, we display in Sec.II.F.1 thermal and structural profiles for a numerical0.05MO brown dwarf model at different epochs after for-mation. Possible cocoon, proto-stellar, T Tauri, and FUOrionis phases that might obtain in the early ( 5 10 yr)stages of a dwarf's life, as well as magnetic activity, rota-tion, and mass loss, have been ignored. In Sec. II.F.2, wepresent a complete suite from 0.03MO to 0.2Mo ofbrown dwarf and VLM models and their evolution. Forspecificity, we focus in both Secs. II.F.1 and II.F.2 onmodel G of BHL, for which Y =0.25 and the mixinglength in standard convection theory is one pressure scaleheight (a =l /H= 1). As was demonstrated in Bur-rows et al. (1989), brown dwarfs are insensitive to awhile M dwarf luminosities and effective temperatures

show only a weak u dependence. The differences be-tween models with o; =1.0 and the inferred solar valueof —1.6 are slight below 0.2MO and we are well withinthe current model uncertainties. Solar heavy-elementabundances were assumed in the opacity calculations.

1. A fiducial 0.05Mo brown dwarf model

The evolution of the temperature-density structure of a0.05MO brown dwarf is depicted in Fig. 6 for times be-tween 5X10 and 2X10' yr, inclusive. The profiles atthe boundary epochs were reproduced in Fig. 2. Thelow-density portion of the curve was truncated slightlyinward of the photosphere. Notice that T, can exceed10 K for billions of years and yet not be sufficient toachieve important light-hydrogen fusion rates. Duringthe entire evolution depicted, the entropy per baryon perBoltzmann's constant (0 ) decreases a paltry 2 units from-9.0 to 7.0. The first few million years (not depicted) in-volved the largest entropy changes and witnesses the de-struction of primordial deuterium (Stringfellow, 1991).All brown dwarfs more massive than -0.015Mo con-sume deuterium during this early epoch.

Figure 7 shows the temperature profile for the base-line 0.05MO brown dwarf. The shallow T-versus-M (en-closed mass) curve and its steady decay are in evidence.Perhaps more interesting is the density profile and itsevolution during the same epochs (Fig. 8). The modelevolves monotonically from the lower curve to the highersolid curve. At 2X 10' yr, p, is 651 gm/cm, and almostthe entire object is found at densities far in excess of the—1.0 gm/cm of the PPT. Closer inspection of Fig. 8 re-veals that the density profile maintains its self-similar,polytropic (n —1.5) behavior from epoch to epoch.

The radius versus interior mass at the two boundaryepochs of 5 X 10 and 2X 10' yr is shown in Fig. 9. No-

~I

~ I ~ I I~ ~ ~ ~ I ~

Temperature vs Density

cg 4

3x10

~&

FIG. 6. Profiles of temperature I,'in K) vs den-

sity (in gm/cm ) for a theoretical 0.05Mobrown dwarf at various times after formation.These lines are adiabats. Model G (BHL) pa-rameters were employed. See text for discus-sion.

2—4

~ ~ I ~ ~ ~ s I ~ ~ ~ ~ I ~ ~ ~ ~ I ~ ~ ~ ~ I ~

—2 -1 0 1

Log, n p(gm/cm )


3/4 J. Liebe«. scie&ce pf bro wn dwarfsA BUrrows &nd

(M 5 M. )p

i.5

4

'~ 4

5~iQio x... . .,

3xi0yPS

2xiO X~

K) » en' massIG 7 Temperaturef a model G

F

ext(in Mo )

S BHI. and the. b own dwarf. SeeO C5~C&l discussiof mo

E

I

~ ~

I4 4II

0.5-

Q.Qi 0.02 Q.Q3

M»s «o)0.04

~ ~

d M dwarf prope leI brow' d(A'arf

M. -P.2MOp p3MO-coincides with the

f

's oint roug y coin

'

Fiusin the relation p=

t the two boun yure rofiles are versus t e rai . 10. Plotte is re

I" . The stellar center ob o d t r" -"h". ~

te the ex-AM=MT and t e p

6 ure serves toThe pres-

AM «MT. This gd in the object.encounter e in

ven showntremes

Mbar) is not evh tmosphereP' "on the grap, y

m a core va ue innitude from

mariz rior structure(2.39)]

mmarize the interior struc u6—10 as a set suFiguresdwarf at 0.05 o aof a representati ive brown w

tion.

ble the results oofa hsandata eof BHL both above

Here p

The pur-simulations

' 'on mass.ers

qn is to provide bene

Table Id 1

p

o of the to-(hfor e

T I. RTp1um1-

il. ) is 1, the object asat various times.he main sequence.rived on t e

-6X109 cm, and T; 2.7XTh ""lu"dedge od of the main sequence. e

ss M = 0.05 Mo)Density vs Mass

800

600

400

Fi . 7, but for density (inenc os

' M. ). The de-enclosed mass (ingpiete d structure isis almost p

de icts se -sim' c-ilar contrac-Panh t t and afterNote that a

structure is no on er

200

Q.oi 0.030.02Mass (Mo)

0.04

I. 65, No. 2, April 1993Rev.. Mod. Phys. , Vol.


I'I

I ~ ~ I I

W

7 Radius vs Mass (M = 0.05 Mo)

e—

U

4

'~ t~f

05lX

FIG. 9. Same as Fig. 7, but for radius (in 10cm) vs enclosed mass (in Mo ). Qnly curves fort =5 X 10 and t =2 X 10' yr are sho~n.

I l I I ~ s ~ ~

O.oi D.OP. 0.03Mass(M )

0.04

opacities, rnetallicities, Y, and the mixing-length param-eter in various ways and are particularly sensitive to Ynear the MMSM. For instance, L varies at 0.08MO by afactor of 5 as Y changes from 0.22 to 0.28. However,since the actual value of the MMSM moves down as Ymoves up (MMSM=0. 075M~ at F =0.28), the value ofL at the rolling MMSM is not as highly variable. Thesti6'dependence on Y of T, near the transition mass isdepicted in Fig. 11. Though for model 8 Y =0.22 andfor model I Y =0.28, they are similar to model G in allother ways. These isochrones at 10, 10, and 10' yearsshow that though the Y dependence is weak at earlytimes, at higher mass, and at lower mass, near the transi-tion mass at 10' years T, can vary by 1000 K.

Figure 12 depicts T, versus time for various massesabove and below the VLM/brown dwarf boundary. Forall masses, the peak central temperature is reachedwithin a few hundred million years, but only for the moremassive objects does T, eventually stabilize at a main-sequence value. At later times, the VLM and browndwarf branches bifurcate, though just below the edge ofthe main sequence thermonuclear burning can proceedfor billions of years. (N.B. The behavior atM =0.075M~. ) Notice that for M-0. 08MO, the main-sequence T, i.s lower than its peak value. Though T, de-creases for a billion years, the increase in p, more thancompensates in the nuclear rate to lift the star onto themain sequence. Figure 13 recapitulates the behavior

~ ~ I ~ I ~ ~ ~ ~ I ~ ~ ~ ~ I ~ ~ ~ ~ I I ~ I ~ I ~ f I ~ I ~ ~ ~ ~8

5$2

9 FIG. 10. Same as Fig. 7, but for pressure (inMbar) vs fractional exterior mass (AM/MT),at t =5X10 and 2X10' yr. The center is atthe left and the PPT is near AM/MT-10See the text for a cursory discussion.

-30.0 -1.0 -2.0 -3.0

Logto(b, M/MT)-5.0 —6.0

~ e a I I I ~ ~ I ~ E ~ I I ~ ~ ~ a I ~ ~ a I I I

-7.0

Rev. Mod. Phys. , Yol. 65, No. 2, April 1993


shown in Fig. 12 for the same object masses. ThoughI.„/L can reach 80%%uo and stay above 50% for many bil-

lions of years, objects just below the MMSM will eventu-ally fade into oblivion. Even if a star reaches the mainsequence, it may take billions of years to do so. Theseshared properties of "stars" just above and below thetransition mass serve to punctuate the fact that what is,or is going to be, a star can be observationally ambiguousand that there is a transition region between the VLM

and brown dwarf branches. However, after VI.M's reachthe main sequence they can burn quietly for 10" ' yr,far longer than a Hubble time.

Figure 14 depicts the luminosity (L) and bolometricmagnitude (M~ ) versus mass at 10', 3X lo', 10, 3 X 10,and 10' years. Superposed are data from Liebert andProbst (1987) and McCarthy et al. (1988). The approxi-mately power-law cooling behavior of brown dwarfs de-rived in Sec. II.E is manifest, as is the steep dependence

TABLE I. Model G (0.03MO &M &0.2Mo).

Mass (MO) I. CLO) T; (X10')

0.2000.1500.125

0.090

0.080

0.070

0.050

0.030

10.00010.0000.1001.000

10.0000.1001.000

10.0000.1001.000

10.0000.1001.000

10.0000.1001.000

10.0000.1001.000

10.00020.0000.1001.000

10.00020.0000.1001.000 .

10.00020.0000.1001.000

10.00020.0000.1001.000

10.00020.0000.1001.000

10.00020.0000.1001.000

10.00020.0000.1001.000

10.00020.000

3500.03340.03260.03230.03230.03200.03150,03150.03160.Q3060.03060.03130.02980.02970.03110.02870.02860.03080.02720.02670.02670.03050.02500.02100.02100.03010.02200.01460.01080.02960.01890.0988.0715.0

2820.01590.0761.0584.0

2630.01400.0645.0508.0

2340.01210.0548.0436.0

1850.0975.0452.0364.0

0.666E —020.358E —020.302E —020.230E —020.230E —020.253E —020.164E—020.164E —020.224E —020.120E —020,120E—020.209E —020.971E—030.957E—030, 194E—020.740E —030.714E—030.179E—020.513E—030.447E —030.446E —030.164E—020.318E—030.134E—030.133E—030.149E—020.173E—030.264E —040.721E—050.132E—020.910E—040.514E—050.135E—050.983E—030.460E —040.190E—050.639E—060.678E —030.290E —040.105E—050.394E —060.379E—030.169E—040.600E —060.236E —060.143E—030.781E—050.309E—060.127E—06

15.4012.4012.0010.7010.7011.409.479.45

11.008.598.57

10.808.148.10

10.607.637.56

10.407.096.886.88

10.106.596.056.059.986.285.625.329.656.195.385.279.166.245.515.438.706.375.725.648.266.565.985.918.126.826.316.23

6.1405.4204.5804.9504.9504.1304.5904.5903.8104.2704.2803.6404.0704.0803.4703.8203.8203.2903.4903.4403.4403.1103.0502.7302.7202.9302.6101.9901.5402.7302.2701.3501.1102.3201.7601.0100.8341.8701.3400.7750.6551.3900.9630.5720.4930.9400.6450.3890.344

144.0206.0188.0267.0267.0190.0332.0334.0191.0401.0404.0192.0447.0453.0192.0510.0525.0193.0597.0653.0654.0193.0697.0894.0895.0194.0748.0

1040.01220.0194.0723.0

1090.01160.0191.0597.0859.0895.0183.0458.0627.0651.0166.0328.0428.0443.0127.0211.0263.0272.0

13.0712.2211.9611.6111.6111.7111.1211.1211.5210.7010.6911.4110.4510.4311.3010.1510.1111.189.779.639.63

11.059.339.858.85

10.918.978.177.62

10.768.777.537.23

10.438.557.337.09

10.058.377.247.029.598.187.156.939.187.977.056.82

1.0001.0000.3230.9971.0000.1780.9901.0000.1090.9821.0000.0830.9681.0000.0610.9381.0000.0440.8580.9971.0000.0310.6500.9971.0000.0210.4180.6500.4470.0130.2670.1810.1550.0050.0530.0240.0150.0010.0050.0020.0010.0000.0000.0000.0000.0000.0000.0000.000



4000

I I I I

Te-vs-MassIsochrones

3000

200010' B

FIG. 11. T, -vs-M isochrones at t =10', 10,and 10' yr for models B and I. These modelsare the same as model G (see text), but havehelium fractions ( F ) of 0.22 and 0.28, respec-tively. Note the sharp dependence on F nearthe transition zone.

1000 — ]0

j010

I I I I I I ~ ~ ~ ~

0.05 O. iO O.i5Mass (Mo)

0.20

at late times of L on M near 0.08MO. In this transitionregion, at 10' years L ~M, though L ~ M justabove 0. 1Mo and L ~M just below 0.06Mo. Fig-ure 15 shows the corresponding T, -versus-mass iso-chrones. Notice that brown dwarfs with masses near orbelow 0.03Mo cool below 1000 K within only 10 yearsand that young brown dwarfs can be mistaken for stars iftheir ages are not known. This age ambiguity is one ofthe central problems in distinguishing brown dwarfs fromVLM's (see Sec. III). Complicating the identification ofbrown dwarfs is a pronounced selection bias for luminousobjects. These caueats alone demand caution when tout-ing putative substellar objects.

Figure 16 compares the radius-versus-mass plot for

model G at 10 and 10' yr with those for various othermodels (B, C, D, E) from Burrows, Hubbard, and Lunine(1989) (BHL) (at 10' yr) and the radius-mass curves ofvarious other theorists (Grossman, Hayes, and Graboske,1974, G74; Vandenberg et al. , 1983, V83; Rappaport andJoss, 1984, RJ84; D'Antona and Mazzitelli, 1985,DM85). Model B has a F of 0.22; models C and D havemixing-length parameters of 0.5 and 0.1, respectively,and a Y of 0.22; and model E, also with a Y of 0.22,has a low opacity as well. All the other properties ofthese models are the same as in model G. Figure 16shows that despite the manifold differences between themodels displayed, theorists believe that the radii can beaccurately predicted. The characteristic n =1.5 depen-

Central Temperature vs Time

0.110

0.1000.095O.DQO

0.085

0.0$0

O.i%5

0.079

0.060

FIG. 12. Central temperature ( T, ) vs time (inyears) for models with masses between0.06MC) and 0.11Mo. There is a clear bifurca-tion between brown dwarfs and VLM's,though the 0.075MO model is hung up bythermonuclear burning for billions of years.Notice that the peak central temperature isreached within only a few hundred millionyears for all masses, but that it can then de-crease before stabilizing even for main-sequence stars.

I I ~ I

8 9 10

gso Ti~e~eam)



L„Lv

0.8

0.4

FIG. 13. Similar to Fig. 12, and for the samemasses in the same order, but L„/L vs time (inyears). L„/L is 1 for a star on the main se-quence. Notice how large this ratio can be foran object just below the main-sequence transi-tion mass.

0.2

9Log&o,Time(Years)

10

dence of R versus M(R -M '~) is clear on the brown

dwarf branch, as is the R ~ M behavior on the VLMbranch. Notice that there is a minimum radius, depend-ing most sensitively on Y and the opacity, that hoversnear 5 X 10 cm ( -7% of Ro ).

The evolution of the central density (p, ) with time forvarious masses is depicted in Fig. 17. These curves showhow much p, can change in the transition region, evenafter 10 yr. A peak p, of —1200 gm/cm can bereached by brown dwarfs just below the MMSM and, atlate times, p, drops precipitously on either side of thispeak. The brown dwarf side of the late isochrone rough-ly reproduces the p, ~M behavior of an n =1.5 po-lytrope. Figure 18, which depicts the surface gravity gversus mass versus time, looks somewhat similar to Fig.17, but involves what is, in principle, an observable. Ei-

ther by a measure of both M and R, or by extraction viadetailed spectral analysis (Ruan, 1991), g is one of thetestable quantities of the theory. Notice that g generallystays below 3.5X10 cm/s and does not Airt with 10cm/s .

Figures 11—18, and Table I summarize the numericaltheory of brown dwarfs, VLM's, and the main-sequencetransition zone. To them should one liberally refer forthe generic characteristics of these objects.

III ~ SEARCHES FOR BROWN DWARFS

As demonstrated in Sec. II, the theory of brown dwarfsand the edge of the hydrogen main sequence is welldeveloped and mature. However, the translation betweenthe theorist's vocabulary and that of the observer is madeawkward by the continuing lack of good theoretical spec-

I I

I.uminosity-vs-MassIsochmnes

1O

0

C)C1Q

10

—510

y010

~ « I I « ~ ~ I ~ ~ ~ ~ I

Model G

LP87, Mc88

Mc88

18

20

FIG. 14. Photospheric luminosity (L, in unitsof Lo) and bolometric magnitude (M&) vsmass from 0.03Mo to 0.2Mo at 10', 3X10',10, 3 X 10, and 10' years. Model G parame-ters are employed. Data from Liebert andProbst (1987; LP87) and McCarthy et al.(1988; Mc88) are superposed with error bars.The brown dwarf branch is to the left and theVLM branch is to the right.

0.05 0.10 0.15Mass (Mo)

O.P,O



4000

I I I

Ye-vs -MassIsochrones

3000

800010'

LT

FIG. 15. Same as Fig. 14, but for T, (in K) vsmass. Only the data from Liebert and Probst(1987) are included.

iooo — ].0'

diplo

Model 0~ LP87

I I I

0.05 D.io O.i5Mass (Mo)

OZ.O

tra below 2500 K. Most of the data involve photometryand, hence, colors and magnitudes, while theorists stillwork with T, and bolometric luminosity. In this section,we employ the observer's vocabulary, not emphasizingthe physics of the model interiors, but the observablesand the astronomical techniques employed to study dimstellar objects.

We divide the searches to discover substellar objectsand to assess their space density into searches for singleobjects of the heterogeneous galactic disk population(Sec. III.A), searches restricted to a young cluster orstar-forming clouds (Sec. III.B), searches for binary com-panions (which might at times be called planets; Sec.III.C), searches for halo brown dwarfs (Sec. III.D), andsearches via gravitational microlensing (Sec. III.E). Thesearches are further subidivided according to the surveytechnique used.

A. Searches for single objects

1. Faint stellar objects: Low-mass starsor brown dwarfs?

Isolated stellar objects in the field present the greatestproblem in answering the question of whether they arestellar or substellar. There is no direct way to measurethe critical parameter —the mass —and their ages maybe estimated only crudely based on the space motions, ro-tation rate, and/or the strength or absence of surface ac-tivity. The only way to distinguish an isolated substellarobject from a low-luminosity star, at present, is to deriveprecise estimates of the effective temperature and theluminosity, and to try to match this position in what as-tronomers call the Hertzsprung-Russell (HR) diagramwith an evolutionary track of unique mass and age.

1e—

14

12V

0)C)

10—

10

6 10'

D

074V83DM85RJ84

0.225

0.200

0.175 gg

0.150 ~

0.125

0.100

FIG. 16. Radius (in 10 cm and Rc)) vs mass(in Mo) at 10' yr {for model G) and 10' yr (formodels G, B, C, D, E and from the calcula-tions of other theorists). G74 is Grossmanet al. (1974), V83 is Vandenberg et al. (1983),DM85 is D'Antona and Mazzitelli (1985), andRJ84 is Rappaport and Joss (1984). ModelsB—E are described in Burrows et al. (1989)and in the text. Radii in the cold brown dwarfregion (bottom left) are generally around0. 1RO. The VLM branch is to the right.

0.05 0.10Mass (MJ

0.15 0.20


A. BUrrows and J. Liebert: The science of brown dwarfs

I t I s I 5

j.800

i000

soo-

600

400

FIG. 17. Evolution of the central density [p,(in gm/cm')] with time along the browndwarf/VI. M mass continuum. Brown dwarfsoccupy the left and peak regions and VLM'soccupy the right. Notice that p, 's above 10gm/cm can be achieved in the transition re-g1on.

200

R l I I l 1 I r

0.10Mass (Mo)

The quandary faced in trying to do this fit is immedi-ately apparent in Fig. 19, adapted from Liebert (1991).The slopes of the locus of points for low-mass stars(called the zero-age main sequence or ZAMS) and thesubstellar evolutionary tracks are quite similar, with thepositions of the latter only 100-300 degrees cooler at aglvcll lumlnoslty. Note that thc formcI I'cIIlaln essential-ly unevolved in position for tens of gigayears until hydro-gen is completely exhausted in a completely convectivestar.

Shown also are some of the faintest known stellar ob-jects, plotted with best estimates of these parameters.Given the impossibility of determining T, to better tha, n

a fcw hulldI'cd dcgI'ccs, lt ls possible, by moving tllc datapoints to modestly higher T„ to fit these data points asVI.M stars of moderate to old age. If the points fall to

the right of the ZAMS locus, they may 6t the contractingtracks of young, substellar objects. The lower the T,value one assumes at a given I, the lower is the mass andthe younger the age.

Hence T, is a critical parameter when the age andmass are otherwise unknown, yet it is not a,ccuratelydetermined by thc comparison of cullcnt obscI'vatloIlswith current atmospheric and interior models. All valuesof T„until very recently, were based on fits of observedspectral energy distributions or colors to blackbcdies, notto synthetic spectra predicted from realistic model atmo-spheres. As we implied in Sec. II.C, the Ph.D. disserta-tions of Allard (1991)and Ruan (1991) represent the firstattempts to apply model atmospheres to some obscIva-tions of very-low-luminosity objects. However, only par-tial success has been achieved. Ruan (1991) concludes

300000

FIG. 18. Same as Fig. 17, but for surface grav-1ty (g„ 1Il cm/s ) vs mass (1n MQ ) at var1oust1mcs bctwccn 3 X 10 and 10 yca's, 1ncluslvc.See text for a brief discussion.

100000

50000

0.05 0.10Mass (Mo)



D0.2

0.15 yCI i,

0.$25'I 4' k k

O. LLCl, 0

Cl

k

VB8 -b~s. 4~

'

VsioCi

0.085

~024k

0.08

VLM StarsL Hyades0 Pleiades~ BZ Candidateso ZAMS Models

G0485B

G20-38B?

El0.075

0.05I I I t f I I I I I t I I I I I I I I I I I I I I'~l I I I

3.6 3.5 3.4 3.3 3.2 3.1 3.0&&g Ten

FIG. 19. Adapted from Liebert (1991),some recent attempts toplace VLM stars and brown candidates onto a true HR dia-gram. Solid circles are VLM stars observed and analyzed byBerriman and Reid (1987). Pleiades candidates from Staufteret al. (1989) are also plotted. Benchmark stars vB8, vB10, andLHS2924 are plotted twice as solid circles (Berriman and Reidestimates) and solid squares (Becklin and Zuckerrnan, 1988, es-timates), illustrating the range of recent temperature determina-tions at the lowest lurninosities. Included also are the resolvedBecklin and Zuckerman source GD165B, and the unresolvedG29-38B(P), assuming the infrared excess of the latter is attri-butable to a companion. The open squares are hydrogen-burning model positions for an age of 10 years, with prior evo-lutionary tracks (shown as dashed lines) at 0. 1MO and below.Shown, too, is a brown dwarf track for 0.05MO with no ter-minus (rightmost dashed line).

from the difficulty in fitting observations that the atmos-pheric stratifications of her models may be incorrect; cer-tainly this implies that the T, determinations remain lessaccurate than the key value of a few hundred degrees re-quired for Fig. 19. Finally, even if one could place thestars perfectly in T, and I., there remain uncertainties as-sociated with the interior models which affect the pre-dicted positions of tracks in the HR diagram, just as theydo the other parameters plotted in Figs. 6—18.

At face value, in Fig. 19 the "best" current estimatesof T, and L for the lowest-luminosity known stellar ob-jects would imply that most of these are brown dwarfs.Given the expectation that most stars in the galactic diskare old rather than young (i.e., less than 1 Csyr or so), onemay argue that the bulk of the points should fit the lowend of the ZAMS instead. This would imply either thatthe empirical fits are yielding values of T, that are toolow, and/or that the theoretical tracks predict ZAMStemperatures that are too high. The latter possibilitymay imply that the radii of the interior models are toosmall, even though one might surmise from Sec. II.F.2and Fig. 16 that the radii are the best theoretically pre-dicted quantities.

2. Construction of initial mass functionsfor field stars

The function that describes the number of stars in eachinterval of luminosity for a stellar population is called the

(3.1)

where a is generally found to vary from 3 to 5 over thestellar mass range above 0.3Mo. The a index varies from2.2 to 2.6 between 0.1 and 0.3Mo. Thus, if we want todetermine the initial mass function (IMF) for low massand VLM stars whose present-day luminosities are closeto their luminosities when they first reached the main se-quence, the simplest procedure (and that used most often)is to use the above M-L relation with the appropriate ex-ponent to convert an entire LF into the IMF.

The appropriate version of Eq. (3.1) can be determinedfrom a sequence of stellar interior models havingdifferent masses, if a uniform chemical composition forthe population can be assumed (see Sec. II). The nearbystars of the galactic disk population —certainly an im-portant sample —will be an unknown mix of metallici-ties. It should be noted that work on M dwarf atmo-spheres (see especially Ruan, 1991) has progressed to thepoint where only rudimentary atmospheric abundance es-tirnates for individual stars can be achieved, perhaps towithin factors of 2 to 3. However, one can also estimatethe M-L, relation empirically using stellar components innoninteracting binary systems where the masses havebeen determined by solving the orbits. Figure 20 is a plotof stars with such determinations from Henry (1991).The correlation coefficient for the power-law fit indicatesthat a simple, linear relation in a log-log diagram,

log(M /Mo ) = —0. 166M+. +0.560, (3.2)

is a good approximation over the range 0. 1 —1MO.Application of either theoretical models or an empiri-

cal M-L relation becomes much more complicated as thelimiting mass near 0.075—0.08MO is approached. This isseen by examination of a similar M-I. plot of the regionof stars of lowest mass, as shown in Fig. 21, also fromHenry 1991; see Secs. II.F and III.C.4). In Fig. 21, theabsolute 2.2 micron E magnitude is used in place of logL,since it is a directly measurable quantity closely relatedto logL. Data points are again VLM stars with as-trometric mass determinations, with large horizontal er-ror bars, and almost negligible vertical error bars; boththe trignonornetric parallaxes and the apparent K magni-tudes of these nearby stars are accurately measurable.Shown also in Fig. 21 are two sets of theoretical tracksshowing the luminosities of stars at a given mass and age,from D'Antona and Mazzitelli [1985 (DM)] and Burrowset al. (1989). At a given mass, the models predict aspread of only about one magnitude in luminosity forages of 0.1 —10 Gyr and masses ~0.12Mo. Below thisvalue, the model luminosities diverge rapidly as the masslimit is approached. However, the small number ofrelevant data points do not show this effect.

The luminosity-mass transformation can be done intwo ways. Since the observed M-I. line does not appear

luminosity function (LF). For the main sequence, theluminosity of a star is normally a power-law function ofthe mass, over a limited interval of mass:



0

Q

8—9—i0-ii—12

~ E

l L

41 'c

Hydrogt nBurrt tngM I II.i mum

Mass{O.O8 M. )

log M/Mo = —0.166 MK + 0.560I

0.4 0.2t t I t t t I l t I I I I I j t t t I 1 t t I t t l I I l I I

0.0 -0.2 -0.4 -0.6 -0.8 -1,0 -1.2 —1.4

Mass/Luminosity Relation: M

jt t t

It t l

It t t

It t t

It I t

It t t

II t t

jt t t

It t t

FICx. 20. Mass-luminosity relation at K. Thesolid line is the weighted linear least-squares fitto the data. The dashed lines represent the fitswith 1o. errors in the slope and interceptadopted. Taken from Henry (1991).

to deviate from the general fj.t near the stellar mass limit,a purely empirical method would be to use Eq. (3.2) asthe ordinate in ordex' to transform the luminosity of a sin-

gle object in the sample into a mass. This approach iswidely used in the literature, as we shall show in the fol-lowing sections. Assuming that the empirical line corre-sponds with theoretical model lines is equivalent to as-suming that the stars of lowest luminosity are mostlyyounger than 1 Gyr, since the empirical line lies aboveand to the right of the 1 Gyr tracks. It is implausible toassume that the majority of M stars in the solar neigh-borhood are younger than 1 Gyr, unless perhaps they aremostly brown dwarfs. If, instead, the VLM star forma-tion rate has been constant to within factors of 2 or 3 forthe history of the galactic disk, then the mean age wouldnot be much ditt'erent from half of its age (half of —10Gyr, or somewhat higher), and only about 10% or less of

low-luminosity stars in a sample might be as young as theempirica1 fit implicitly assumes. Fortunately, also, thetracks for 1 and 10 Gyr stay much closcI' togcthcI' ovcI'the interval of interest. Hence a second way of doing theM-L transformation is to assume that the theoreticaltIacks are correct, that most of the stars have ages of or-der 1 or 10 Gyr, and to use one of these sets or theirmean to assign masses.

Very di6'erent slopes of the IMF near the hydrogen-burning mass limit will be derived from a given LF usingempirical and theoretical transformations. This pointhas been discussed in greater detail in D'Antona andMazzitelli (1986), Reid (1987), Liebert and Probst (1987),and Kroupa, Tout, and Gilmore (1990). In the followingdiscussion, the statement that the derived IMP shows anupturn Ineans that the number density per unit intervalin log mass increases with decreasing mass. Due to the

Mass/Luminosity/Age Diagrami i t

[i i i

Ii i i

Ii l i

GL 6'G 208-44& ~. . ~ ~ "R~V 6&4B .

0BGL 5698

62gB ' Q L.HS 1047BvB 8vB 10

DM y' BHt.

LHS 2924

0.1 Gyr '

1D Gyr' '—+ GD 165B

1.0 GyrHy'(iI OgeD. BUX'I11K1g MRS' L1ZTlit,

I I I I I I I I t I I I I I I I I I I t I I I I kI I I I I I I

O. &8 O. &6 0.14 0.1P. O. 1O O.OB 0.06 0.04 O.O2

FIG. 21. Mass-luminosity-age diagram forvery-low-mass objects. The isochrones ofO'Antona and Mazzitelli (1985, DM, dashedcurves) and Burrows et al. (1989, BHL, dot-dashed curves) are shown for ages of 0.1, 1.0,and 10 Gyr. The seven lowest-luminosity redobjects known with determined masses arewithin the speckle survey, and the four leastmassive are brown dwarf candidates. Otherbrown dwarf candidates are sho~n on the leftat their respectively Mz's. Taken from Henry(1991).



sharp curvature of the theoretical tracks of Fig. 21—incontrast to the linear empirical law —it is possible for theIMF to show an upturn near the stellar mass limit whenthe LF shows a downward slope. This is due to the factthat, as the theoretical curve increases in slope, smallerintervals of mass correspond to the same interval in lumi-nosity; thus a given number of stars in this luminosity in-terval transforms to a larger number for a fixed mass in-terval.

3. Proper-motion surveys

A proper-motion survey is conducted by comparingthe position of stars on photographic plates taken at twodifferent time epochs. The great majority of faint starswill be distant enough to show no perceptible motion andcan therefore serve as a suitable positional referenceframe for the few that do show motion. Such surveyshave been very efficient at culling out nearby stars of thegalactic disk population, as well as high-velocity interlop-ers from the halo population. Only those using largeSchmidt telescopes reach faint enough over large areas ofthe sky so that they might possibly include Inassive,young brown dwarfs of luminosities that are similar tothose of the faintest main-sequence stars.

The most important survey to date was carried out byLuyten (1963) using plates taken with the Palomar 1.2-meter Schmidt. This was the first such survey carriedout with a specially constructed automatic measuringmachine. The combination of using red plates, a veryfaint limiting magnitude (mz —20), and a small motionlimit (the proper motion, p 0. 1"/yr) resulted in thediscovery of very-low-luminosity stellar objects. A cata-log and finding charts have been published for the -3600stars with measured proper motions larger than 0.50"/yr(Luyten, 1979a, 1979b). Dawson (1986) finds this portionof the survey 90% complete for mz & 18 over 65% of thesky. Only a catalog is available for the -50000 starswith 0.2&p&0. 5 arc sec/yr (Luyten, 1979c). A morecomplete discussion of these and other proper-motion ca-talogs is found in Reid (1987) and Liebert and Probst(1987), who discuss their use in constructing the luminos-ity and initial mass functions of VLM stars.

New proper-motion surveys are already under way us-ing the new Palomar Sky Survey conducted since themid-1980s. These plates, utilizing finer-grained emul-sions, will result in improved positional accuracies andlimiting magnitudes.

These luminosity and mass functions are constructedfrom samples selected for particular intervals of propermotion and, hence, are vulnerable to bias based in somecomplicated way on the space motions of the stars.Brown dwarfs detectable at these optical wavelengthsmight be expected to have small mean tangential veloci-ties, since they should be young. (The velocity disper-sions of stars are known to increase with increasing age. )

Despite the possibility that some stars having very lowtangential velocities may have been missed, the proper-

motions surveys have been very efficient at identifyingthe nearest neighbors to the Sun and in finding a largefraction of the intrinsically faintest stars currentlyknown, some of which might be brown dwarfs. The sam-ple of solar neighbors is optimum for study for severalreasons: Their distances and, hence, luminosities may bedirectly measured by trigonometric parallaxes. They arethe brightest objects of a given class for all follow-up ob-servations. Perhaps most importantly, they are also thesample most amenable to the discovery of fainter, closecompanions, a key component of the search for browndwarfs.

The least luminous of the well-studied solar neighborscurrently are two stars cataloged in Luyten (1979a) asLHS 2924 and 2065, each with distances determinedfrom accurate trigonometric parallaxes (Monet et al. ,1992). The inferred absolute visual magnitudes (M~) arefainter than + 19, implying luminosities of a fewX 10 Lo (Fig. 19). Such luminosities indeed place themvery close to the limiting mass for hydrogen burning (seeSec. II.F). As also discussed in Sec. III.A. 1, it is also pos-sible that either or both are brown dwarfs.

4. Optical and infrared color surveys

The red colors of the lowest-mass stellar objects pro-vide a selection method free of kinematical bias. Withthe availability of sensitive emulsions, automated measur-ing machines, and sophisticated computers, very largecolor surveys covering large areas of the sky have beencarried out to produce luminosity and mass functionswhich may also penetrate the substellar regime. Histori-cally, this method has been fraught with danger, sincesmall systematic errors in the calibration of photographiccolors, or in the assignment of absolute magnitudes(luminosities) based on photometric colors, have resultedin large systematic errors in the LFs, the space densitiesof low-luminosity stars, and in the IMFs. More recently,however, the reddest, best candidates for very-low-luminosity objects have been checked with follow-up in-frared and optical photometry and spectrophotometry.

Cxilmore and Reid (1983) and Gilmore et al. (1985)presented the first large samples of stars selected on thebasis of V and I colors. Relatively few very-low-luminosity candidates were found, and they argued thatthe LF and IMFs turn downward at masses below0.3Mo,' but these authors used an empirical M-L calibra-tion (see Sec. III.A.2). Using a theoretical M Lrelation-which featured an inAection point in the luminosity nearthis mass due to opacity effects, Kroupa, Tout, and Gil-more (1990) found that the IMF need not reach a max-imum near 0.3Mo. They argued that the available obser-vations did not rule out either rising or declining slopes,but that the most likely slope was fairly Aat.

Hawkins (1986) showed that one could reach to greaterlimiting volumes by using the same technique with R andI colors. Hawkins and Bessell (1988) and Leggett andHawkins (1988) extended the scope of this work and



featured follow-up (especially infrared) observations ofthe reddest candidates. Their derived IMF's showed up-turns as they approached the stellar mass limit. Presum-ably, some of their reddest candidates could be browndwarfs, and, in any case, the extrapolation of these re-sults to substellar masses would imply a large space den-sity of brown dwarfs. Photometric parallaxes were de-rived based on comparisons of colors and spectra to thoseof the "benchmarks" with trigonometric parallaxes dis-cussed in the previous paragraph. (For a more generaldiscussion of the observational calibrations, see Bessell,1991.) Proper-motion measurements were also reportedfor the color-selected stars in the Hawkins and Bessellpaper. If the motions and assigned distances are correct,the space motions are small, implying that the cooleststars are a relatively young population.

Reid (1987) reviews the apparently discrepant con-clusions of the Gilmore and Hawkins groups and con-cludes that the preponderance of evidence favors an up-turn in the mass function near the stellar mass limit.Earlier, this conclusion had been advanced by D'Antonaand Mazzitelli (1986) using what were then new interiormodels to convert the luminosity to a mass function.However, the most recent analysis to have been pub-lished at the time of this writing is that of Tinncy,Mould, and Reid (1992), which utilizes six fields from thenew Palomar Sky Survey for which follow-up infraredand optical photometry is complete. These authors findstars red enough to reach logI. /Io= —3.5 if they aremain-sequence objects, but the indicated mass function isBat near the substellar limit. It may nonetheless be con-cluded from aO the above studies that large numbers ofbrown dwarfs should exist, unless for some unknown1cason thc staI formation pI'occss tcI minatcs at massesvery near the hydrogen-burning limit. It is unlikely thatthe contribution of brown dwarfs adds substantially tothe local mass density of the galactic disk (Kroupa, Tout,and Gilmore, 1990; Tinney, Mould, and Reid, 1992).

Other color surveys whose conclusions are just beingfinalized are Ph.D. dissertations (Jarrett, 1992; Kirkpa-trick, 1992). The former uses the Ophiuchus and Taurusclouds as opaque screens, in order to search for genuinelyred foreground dwarf stars in the galactic plane. Thelatter utilizes photometric data from J. T. McGraw'sCCD Transit Instrument, a survey of a strip of the sky atfixed declination.

In addition to their red colors, the M dwarf stars showstrong, broad spectral features, recognizable even at lowspectral resolution. Many early surveys therefore usedobjective prisms to identify these stars in large photo-graphic fields, following Vyssotsky (1963). Most recently,a very cool stellar object —PC 0025+0447—has beenfound serendipitously by Schneider et al. (1991)in a deepobjective grism survey with a CCD detector on the Palo-mar 5-meter reAector. The primary goal of this survey isthe discovery of high-redshift quasars. This object showsredder colors and later-type optical and infrared spectra(Cxraham et al. , 1992) than any field object yet found,though not as I'cd as thc companion object GD165B dis-

cussed in Sec. III.C.2. Its most remarkable property,however, is the extraordinary emission line due to Bal-mer alpha. Speculations as to the origin include anunprecedented degree of chromospheric activity at thesurface of the star and some kind of interacting binarysystem. The luminosity of the system remains unknow~,pending measurement of a trigonometric parallax.

Until recently, infrared survey work on the field popu-lation had not been possible, despite the fact that thesestellar objects emitted most radiation longwards of 1 pm.Since the deployment of sensitive infrared array detec-tors, the surveys for field objects have been focused onthe specific fields of young clusters and star-forming asso-ciations; these results will be discussed in Sec. III.B. Sur-veying of large portions of the sky has not been possiblewith existing configurations.

The all-sky survey of the Infrared Astronomy Satellite(IRAS) did not yield any brown dwarfs from 5776sources detected at 12 pm in the Point Source Catalogue(Beichman et al. , 1990). However, at these far-infraredbandpasses, IRAS would have been able to detect browndwarfs of temperatures 1000—3000 K out to distances ofonly a few parsecs.

A deep all-sky survey in the 1 —2 pm bands, on the oth-er hand, could be a powerful way of finding field stellarobjects at low temperatures. For example, a proposal touse dedicated 1-m telescopes with current infrared arraydetectors aims to reach a hmiting magnitude of 14 at 2.2pm (Kleinman et al. 1991). This would permit 2500 Kobjects to be detected out to 50 pc, and 1000 K browndwarfs to be found at 7 pc. The sensitivity of such a sur-vey would be greater than IRAS for even cooler objectswith temperatures above 500 K.

B. Searches in young clustersand star-forming regions

Focusing a survey for substellar objects on a youngstellar aggregation OA'ers some obvious advantages. First,the objects will be young and hence relatively luminous.Secondly, they have at least approximately the same ages,so that it will be possible to assign masses based on esti-mates of the luminosity alone. Furthermore, they all lieat the same known distance. Finally, there are several in-teresting aggregations close enough for observation,ranging from molecular clouds forming stars of the orderof 10 to IO years old to clusters as old as 10 yr. Ex-ploitation has been made possible by the development ofCKcient infrared array detectors.

There are drawbacks as well. Star-forming regionsmay be in the galactic plane, so that heavily reddenedbackground stars may be confused with genuinely coolmembers of an association. Newly formed stars may pos-sess remnant accretion disks, large rotation rates, andcorresponding chromospheric and coronal activity,which might distort their spectra and colors. Further-Inore, there is evidence of considerable variation in theseproperties among stars in the same cluster. In veryyoung systems, the assumption of spherical symmetry



may break down, which greatly complicates the task ofcalculating accurate theoretical tracks and comparingwith observations. Moreover, we cannot necessarily as-sume that a group of objects is coeval. The age spread ina giant molecular cloud may be significant compared tothe mean age. These difficulties are likely to be magnifiedfor the youngest populations, but these are the very casesin which brown dwarfs are expected to be most luminous!

Perhaps the biggest kind of drawback is a set of practi-cal problems related to the faintness of the candidatesand the distance to the cluster; much more informationmay be gained, for example, about field stars in the solarneighborhood. Usually the available information is re-stricted to photometry in a few passbands. These mustbe converted to T, and luminosity based on a calibrationand the estimated distance of the cluster or group. Forexample, several of the groups discussed below used thetemperature scale of Berriman and Reid (1987) to converttheir observed colors. This scale is based on blackbodyfits to energy distributions and infrared spectra of nearbydwarf stars. Despite the recent criticism by Stringfellow(1991), this scale is a valid representation of their as-sumptions and observational data on nearby stars (seeBerriman, Reid, and Leggett, 1992). It nonetheless con-sistently places disk stars to the right of the HR diagrampositions of most theoretical models for ages ~ 10 yr. Anew scale based on an attempt to fit optical and infraredspectrophotometry with synthetic spectra of model atmo-spheres (Kirkpatrick, Kelly, et a/. , 1993) would shift thetemperature scale a few hundred degrees warmer [seealso the Becklin and Zuckerman (1988) data points inFig. 19]. In any case, if the predominantly old disk starsare made to match the theoretical predictions, then thecluster candidates would be shifted towards higher mass.

Finally, even after the cluster/association candidatesare placed in a true HR diagram, there remain a varietyof theoretical calculations from which to choose. Theirdifferent predictions as to L and T, for a given age andmass are attributable to different physical assumptions inthe same group (explored comprehensively in Burrowset al. , 1989), as well as to differences in the codes amongdifferent groups supposedly making similar physical as-sumptions.

1. The Taurus-Auriga clouds

One of the nearest sites of recent star formation is thisloose association. Numerous low-mass, pre-main-sequence objects have been found in optical and x-raysurveys, including many T Tauri and post-T Tauri ob-jects with a characteristic age of 10 years. In an effort todetect lower-luminosity companions to 26 of these knownobjects, Forrest et al. (1989) carried out a 2.2 pm (ICband) imaging survey, covering areas of 25 arc sec squarenear each target. Of the 20 new objects detected atE —13—16, nine were found to have exceptionally redJHE colors; moreover, four of seven objects found to bevisible on first-epoch Palomar Sky Survey plates hadproper motions indicative of membership in the associa-

tion. For the estimated distance of 140 pc, the luminosi-ties of the objects as members would be only of the orderof 10 I.o,' for an age of 10 yr, the inferred masseswould be near 0.01Mo. The large and random separa-tions of the very red objects from the T Tauri imagingtargets indicated that they should generally be single"free Aoaters" rather than binary companions. The in-ferred population of low-mass brown dwarfs in Tauruswould be quite large.

Follow-up spectroscopy, however, has revealed thatmost of these candidates do not have spectra consistentwith their having very low temperatures (Stauffer et al.1991). Instead, they may be background M dwarfs andstars of even earlier spectral type, distant and heavilyreddened. In retrospect, the large and highly variable ex-tinction in the background rendered the interpretation ofthe detected sources very difficult. Thus, while theTaurus-Auriga region remains a fertile potential huntingground, there is as yet no strong evidence for a largebrown dwarf population.

2. The Rho Ophiuchus molecular cloud/cluster

Rho Ophiuchus differs from Taurus-Auriga in havingsuch a high internal extinction ( A, )50 in the core) andstellar density that relatively few background sources arelikely to be detected. This may be a bound cluster in theprocess of formation and includes high-mass stars. Itsclose proximity (160 pc) and relatively young age((3X10 yr) suggest that this cluster will be a fruitfulplace to search for low-mass stellar objects. The questionis how deep one can look into this cloud complex as afunction of the object s luminosity. Rieke, Ashok, andBoyle (1989) surveyed a modest field to a limiting K mag-nitude of —14 and found no new sources over and abovethose found in the previous study of Wilking and Lada(1983) to K —12. Another survey to K —14 over a largerarea detected 35 new sources (Barsony et al. 1989),though there was no indication any of these were substel-lar.

A deeper search (K —15) by Rieke and Rieke (1990)produced several brown dwarf candidates from the pre-liminary analysis. These authors (Comeron, Rieke, Bur-rows, and Rieke, 1993) have now completed the survey of200 square arc min in the cores of the Rho Ophiuchusclouds. Completeness of IC =15.5 and H (1.6 pm) =17 isachieved. They have detected 91 faint sources, all withmultiple measurements usually in both bandpasses, someas faint as K =17.5. Since minimal background contam-ination is expected, it was assumed that each of thesewould fall on a theoretical isochrone for an age of (up to)2X 10 yr, and a unique value of extinction for each ob-ject could be found consistent with this assumption. In-trinsic luminosities were thus derived and masses ap-propriate to each luminosity were found. Perhaps sevenobjects have luminosities low enough for masses below0.05Mo to be inferred. The IMF distribution appears tobe continuous across the substellar boundary with apower-law exponent of roughly —1.3.

Rev. Mod. Phys. , Vot. 65, No. 2, April t993


3. The Pleiades

The most attention in infrared searches for substellarcandidates has been devoted to the Pleiades cluster, withan age, estimated from the main-sequence turnoff, of6—7X10 yr. However, Stauffer (1984) and others havemade the case that some low-mass stars in the clustermay be significantly older, perhaps as much as 10 yr.Moreover, the distance of the cluster has been called intoquestion: Early determinations clustered around 125 pc,while a recent trigonometric parallax study with a newinstrument yielded about 150 pc (Gatewood et al. , 1990).With the smaller, early infrared arrays, the initial effortsagain focused on finding low-luminosity, resolved com-panions. Skrutskie, Forrest, and Shure (1986) and Zuck-erman and Becklin (1987) found no substellar companioncandidates around main-sequence and white dwarfmembers of the cluster, respectively. However, in thiscluster of low internal extinction, the larger-formatCCDs could also be used to search for low-luminositysingle members. Jameson and Skillen (1989) and Staufferet al. (1989) covered some 125 square arc minutes at Rand I and 900 square arc min at V and I, respectively. Inboth cases several very faint, red sources were found inthe part of the color-magnitude diagram expected forbrown dwarfs or VLM stars. Since their LF peaked near0.2MD, Stauffer et al. (1989) did not find evidence thatbrown dwarfs were a major component of the Pleiades.

More extensive and deeper studies of the Pleiades havesince led to different and perhaps contradictory con-clusions. Hambly and Jameson (1991) used R and ISchmidt photographic data to study a three-degree diam-eter region —encompassing the core of the cluster. Thisstudy provides a nice illustration of the complexity of acomprehensive photographic investigation covering mul-tidegree fields using an automatic measuring machine.Some 140000 to 180000 raw stellar images from each oftwo pairs of IIIaF (R) and IV-N (I) plates had to beidentified and matched with counterparts at the same po-sition on the other plates. The procedure had to copewith both extended objects (galaxies), blended images,and plate Aaws due to such things as diffraction patternsaround bright stars. The authors excluded from con-sideration fields around bright stars. The photographicR and I magnitudes had to be calibrated in terms of theR,I photometric systems, a somewhat complicated taskfor objects of red color because of differences between theR passbands of the photographic and photometric detec-tors. Objects corresponding to the locus of the main se-quence and its theoretical extension to substellar massesat the Pleiades age were identified in an R, R -I diagram;this means that the theoretical model loci (I., T, ) had tobe converted to these colors using the relations of Berri-man and Reid (1987) for photometric R,I. The photo-graphic point sources found in a band around the main-sequence and predicted brown dwarf loci were treated aslow-mass Pleiad candidates. The numbers of candidatesin a given I magnitude bin in the core of the cluster were

compared with those measured from an annulus aroundthe cluster, assumed to be representative of the back-ground. The luminosity function was then assembledfrom the difference in these counts per unit surface area.The initial mass function was then constructed from thetheoretical isochrone for 6X10 yr, which predicts amass for a given luminosity. The resulting mass functionwas Aat or rising slightly down to the stellar mass hmit.A lower limit of 30 brown dwarfs was found in this sta-tistical manner, but it is not known from this study alonewhich are actually Pleiades objects and which are back-ground stars. Moreover, a large number of faint objectswere detected only on the I plates, and these could in-clude many more potential brown dwarfs among thegreat majority which will be background stars.

The same group (Hambly, Hawkins, and Jameson,1991) has complemented the above investigation of thePleaides with an astrometric survey, measuring PalomarSky Survey plates from the 1950s as the first epoch. For-tunately, the Pleiades cluster has a quite distinct propermotion, so that contamination from the field does notpose an overwhelming problem. The lower main se-quence is obvious in an I, R-I diagram of the objectswith the cluster motion, and it extends to at leastMr=11.5. Even with fairly reliable criteria for identify-ing the cluster members, problems remain with the con-struction of a luminosity function and an initial massfunction and with their interpretation. These include (1)the fact that even red colors can be affected by intensefiaring activity, a characteristic of low-mass Pleiads, (2)significant uncertainty in the distance of the cluster, (3)doubt as to whether the lower-mass stars could have sub-stantially older ages, (4) significantly different theoreticalmass-luminosity relations for the standard age of6—7X10 yr depending on the theoretical source, and (5)incompleteness for the faintest and most interesting mag-nitude bins, those affecting the IMF below 0. 1MO. Thisgroup's LF appears, nonetheless, to peak at MI-9. 5,similar to that found by Stauffer et al. (1989) and as mayoccur for the LF of the field galactic disk. Whether themass function declines below 0.2MO, and whether thereare any bona fide brown dwarfs among the sample offainter cluster members, remains uncertain.

The deepest survey of the Pleiades thus far is that ofSimons and Becklin (1992), who imaged some 200 squarearc min at both the I and K bands using a CCD and in-frared array. Candidates were thus identified by compar-ison of position in an I, I-K diagram with a theoreticalisochrone for 10 yr. A reference field of 75 square arcminutes was used to estimate the contribution of back-ground stars, and the authors showed that the two fieldshad the same general stellar count rates at the relevantmagnitudes, though the authors calculated that thePleiades field showed an excess of 22+10 objects withinthe designated isochrone band. The range of I magni-tudes corresponded to masses in the range 0.04—0. 1Mo,so that there was little overlap with the photographicstudies discussed previously. However, the correspond-



ing mass functions do not agree at 0. 1MO,' the Simonsand Becklin study predicts a total of -3400 objects inthe above range in the entire cluster.

The Simons and Becklin (1992) survey is potentiallythe most interesting so far in the Pleiades, because thedepth should clearly penetrate well into the brown dwarfregime. The problem is that, for such faint objects, it willbe difficult to garner conclusive information about theparameters of the candidates. The assignment of massesdepends somewhat on which theoretical model set isadopted to construct the relevant isochrone, as discussedat the beginning of Sec. III.C. They also depend on theBerriman and Reid (1987) scale for their conversion of(I K) to T-, . There is some dependence also on the ageand distance assumed for cluster members.

suggestive of higher temperatures. Spectroscopic obser-vations are being attempted to clarify the situation.Reid's (1992, and in preparation) comprehensivecolor/astrometric survey for Hyades members shouldhelp clarify the nature of the faintest of the Leggett andHawkins and Bryja et al. candidates.

Finally, I- and K-band imaging of fields around 49known Hyad stars has resulted in the identification of asequence of VLM/brown dwarf candidates (Macintoshet al. , 1992). Proper-motion measurements should helpdemonstrate whether they are components of wide binarysystems or field members of the cluster.

C. Discovery and analysisof substellar companions

4. The Hyades

This is the oldest (6X10 yr) and closest (-45 pc) ofthe stellar aggregations searched for substellar constitu-ents. Any member brown dwarfs should therefore befairly dim, but well behaved, as measured rotation ratesfor VLM Hyades dwarf stars do not exceed 20 km/sec(Stautt'er, Hartmann, and Latham, 1987); thus, the in-duced coronal and chromospheric activity is modestenough not to affect seriously the behavior of the photo-spheres. Hubbard et al. (1990) and Stringfellow (1991)have noted that the Hyades is sufficiently old that the LFshould separate into distinct lower main-sequence andbrown dwarf components. Due to the size or relativelooseness of the cluster —and, ironically, its proximity tothe Sun —it is difficult to establish membership reliablyfrom proper-motion measurements alone. Moreover, anew astrometric study (Reid, 1992) shows evidence formass segregation in this old cluster: thus the lower-massstellar objects may be less centrally concentrated (andmay also have escaped more easily) than the higher-massstars.

Leggett and Hawkins (1988, 1989) selected field starswith large R Icolors for follow-u-p infrared (JHK) pho-tometry and derived infrared luminosity functions. Apeak in the LF near Mz = +6.7 corresponded to-0.2Mo, near the value often claimed for a low-masspeak in the disk IMF (cf. Scalo, 1986). A comparison oftheoretical isochrones with the observed data sets foundno evidence of a rising mass function near the stellarmass limit or of a bifurcated LF indicative of a browndwarf population (Hubbard et al. , 1990). The dimmestseveral of these candidates might still be substellar, ifHyades members. Unlike their brighter counterparts,however, these are so faint that there is as yet no as-trometric information.

Bryja et al. (1992) identified several faint, red objectson multiple epochs of red Palomar Sky Survey plates andsuggested that they were candidate brown dwarf or VLMstellar members of the cluster based on proper-motionmeasurements. The JHK colors are somewhat incon-clusive, while optical (BVRI) colors are anomalous or

For stellar objects in close binary systems, the massesmay be determined directly by analysis of the binary or-bits. The observational techniques may include directphotography of visual binaries near enough to the Sun(or more distant ones that can be imaged from space), as-trometric perturbation analyses and speckle inter-ferometry for barely resolvable pairs, and/or spectro-scopic measurements of radial velocities for unresolvedand close pairs. By establishing the mass, such studiesoffer the best hope for unambiguous identification of asubstellar companion to a nearby star. Results fromthese and other techniques applicable to more distantcompanions are discussed next.

1. Wide optical companionswith common proper motion

Photography of the fields around nearby stars has beenemployed for decades to discover resolved companionssharing the space motion of the primary. Several early"benchmark" stars of very low luminosity were found ina survey of known solar neighbors by van Biesbroeck(1961);vB8 and vB10 are among the best known of these.The latter, at Mz —+ 18.7, was the least-luminousknown star until the 1980s.

2. infrared imaging and photometry

Continuation of the search for fainter, resolved com-panions in recent years has been carried out at infraredwavelengths, using new generations of wide-format in-frared array detectors. Searches for companions to starsin young clusters and associations were mentioned in Sec.III.B; here we focus on companions to nearby field starsof the galactic disk, closer and easier targets. Jamesonet al. (1983) scanned 21 nearby stars at 1 —2 pm (JIIKbands), but found no resolved companions at least 10 arcsec from the primary star. Skrutskie et al. (1989) con-ducted a more extensive search, but found only onebrown dwarf candidate, a resolved companion toBD+ 16'2708 =Gliese 569 (Forrest, Skrutskie, and



Shure, 1988). However, these authors concluded that Gl5698 could also be a VLM star, though it is significantlymore luminous than v810 and appears to be cooler thanknown VLM stars of similar luminosity (Henry andKirkpatrick, 1990).

G. H. and M. J. Rieke have surveyed all known starswithin 8 pc of the Sun, north of —20 degrees declinationand fainter than Mv =+ 11 (early M spectral type). Thesearch had a detection limit of IC (2.2p ) = 15.5, corre-sponding to at least six magnitudes (a factor of 250 inAux) fainter than vB10 and covering fields of radius) 100 astronomical units (AU) around each star. Noplausible brown dwarf candidates were found.

If the companion cannot be spatially resolved from itsprimary star, there appears to be no simple way to decon-volve the binary components, if both are low-mass stellarobjects (Probst, 1981). However, Probst (1983a, 1983b)realized that if the primary star were a white dwarf —alow-luminosity object of substantially highertemperatures —then a range of low-mass stellar com-panions and even a moderately cool brown dwarf(T ~ 1000 K) might be detected at infrared wavelengths.Using broadband 1 —2 pm ( JHIC) photometry —the tech-nology of the early 1980s—Probst, nonetheless, failed tofind any companions in observations of approximately100 field white dwarfs.

Zuckerman and Becklin (1987a) continued the searchof white dwarf stars for infrared excesses, first concen-trating on those stars in the relatively young Pleiades andHyades clusters where any brown dwarf companionsmight be more luminous than the norm. None of the 14cluster white dwarfs showed an excess. These authorsalso initiated a much broader survey of field whitedwarfs, using infrared array detectors so that they mightfind both unresolved and resolved companions.

Two exciting discoveries resulted from this search.First, Zuckerman and Becklin (1987b) found that the pul-sating white dwarf G29-38 (=ZZ Psc) showed an in-frared excess. Initially, the simplest explanation was thatthe star had an unresolved companion with a color tem-perature of about 1200 K and a luminosity of5 X 10 I.o, these are values much lower than any low-luminosity star. However, Graham, Matthews, et al.(1990) and Tokunaga, Becklin, and Zuckerman (1990)detected ftux at 10 pm in an amount far more than pre-dicted by the thermal energy distribution of the hy-pothesized companion. These authors argue that the in-frared excesses at all wavelengths might more plausiblybe attributed to some kind of dust shell around the whitedwarf (see also Patterson et al. 1991). On the otherhand, Graham, McCarthy, et al. (1990) found that theradial velocity of the star may vary at the 5 —10 kmslevel. Follow-up observations by 8arnbaum and Zucker-man (1992) indicate that the variations are probably realwith a period near 11 months, which leaves open the pos-sibility that G29-38 has a low-mass companion (see alsoZuckerman, 1992).

Second, Becklin and Zuckerman (1988) found aresolved companion to a second white dwarf, GD165. At

a distance corresponding to some 120 AU from the pri-mary star (on the plane of the sky), this component has acolor temperature (2100 K) and luminosity (7.8X 10 )

making it the coolest and dimmest detected brown dwarfcandidate. It also exhibits a very late type and unusualspectrum (Kirkpatrick, Henry, and Liebert, 1993). Evenso, depending on the precise values of these parametersand the theoretical models used in the comparison,GD1658 might still be a stellar object near 0.07—0.08Mo(Burrows et al. 1989; Greenstein, 1989a, 1989b). More-over, plausible brown dwarf tracks of 60—65MJ throughthis position on the HR diagram (Fig. 19) imply that thecompanion is younger than about 10 yr. Yet the 11000K white dwarf has a cooling age of about 6X 10 yr, be-fore allowing for a nuclear-burning lifetime. This con-straint requires the white dwarf to have evolved froma massive progenitor star, with upwards of 5Mo.GD1658 remains one of the best brown dwarf candi-dates, but even its credentials are not unambiguous.

However, with the completion of the survey of approx-imately 200 white dwarfs, Zuckerman and Becklin (1992)find a luminosity function of companions that includesmany low-mass M stars, but very few, if any, below thestellar mass limit of 0.075 —0.08Mo. The primary starsof the binaries in this sample should have had massesnear or above 1MO. A complementary sample is formedby the astrometric and infrared speckle work on stars inthe solar neighborhood, since these are nearly all wellbelow 1MO.

3. Photographic, astrometric perturbationsand orbit determinations

The existence of an invisible, close companion to anearby star may be inferred from the discovery of period-ic perturbations in the position of the primary star, asfirst demonstrated in Bessel's (1845) discovery of thewhite dwarf companion to Sirius. Photographic plate se-

quences of nearby stars spanning many decades havebeen obtained at the Sproul Observatory and more re-cently at the U.S. Naval Observatory. Reviews of theseresults may be found in Lippincott (1978), Harrington(1986), and van de Kamp (1986). While many low-massstellar companions and objects near the limit have beendiscovered in this manner, it has been difficult to estab-lish the reality of many of the perturbations and which ofthe real companions have substellar masses (Heintz,1978). One of the longest-studied cases is Barnard's star,the Sun's nearest neighbor after the Alpha Centauri sys-tem. Van de Kamp's (1977, 1981) suggestion of one ortwo Jovian companions has found no support in the in-

dependent analyses of Gatewood and Eichhorn (1973)and Heintz (1978).

Another interesting astrometric system which di6'ers in

having two components resolved, but quite low in mass,is Wolf 424. Heintz's (1989) analysis suggests that bothare, in fact, substellar, with estimated masses of 0.05 and0.06Mo. The accuracy of this kind of analysis is difficult



to assess, since it depends on the precision with whichthe positions and separations of two faint stellar imagescan be measured on old Sproul Observatory plates span-ning 50 years. A detailed speckle interferometric, pho-tometric, and spectroscopic study by Henry et al. (1992)suggests that the orbital solutions of Heintz (1989) maybe in error. That the objects are substellar is not sup-ported by the infrared spectroscopy of Davidge andBoeshaar (1991).

Considerable improvement can be made in astrometricmeasurements at optical wavelengths by using new tele-scopes and optics designed for this purpose, by locatingthem in modern thermal enclosure sites ("domes" ) withexcellent atmospheric seeing, and by using more accurateand sensitive detectors.

4. Infrared speckle interferometry or imaging

As the size of a telescope aperture increases, the in-stantaneous image of a point source breaks into smallersections, into a distribution of what are called "speckles, "due to the disturbance of the wave front as it passesthrough the Earth's atmosphere. Labeyrie (1970) firstpointed out that individual speckles contain spatial infor-mation on a source to the diffraction limit. The theoryand application of speckle interferometry at opticalwavelengths is discussed by Dainty (1984) and Roddier(1988). McCarthy (1976) pioneered the application ofspeckle interferometry at infrared wavelengths.

The infrared speckle interferometric technique hasbeen used to search for low-luminosity companions to allM dwarfs known within 8 pc of the Sun, north of —25'.M dwarfs have masses up to about 0.4M0. The searchannuli correspond to about 1 —10 AU separations on theplane of the sky. These results are reported in the Ph. D.dissertation of Henry (1991). The remainder of this sec-

tion, with the exception of the comments in the last twoparagraphs, has been extracted from these references.

Six new companions were found, and masses weredetermined which ranged from 0.39 to 0.05MO. Three ofthe new companions —G208-44B, Gliese 623B, and LHS1047B—and one previously known secondary to a surveystar, Ross 614B, are brown dwarf candidates. These andother candidate field brown dwarfs are listed in Table II.However, the error bars on the mass determinations ofall four cases allow the true masses to be above 80MJ(0.08Mo). Infrared photometric observations allow thelurninosities of these objects to be estimated for an empir-ical M Lrelati-on (see Fig. 20, taken from Henry, 1991).Three of the four candidates can be explained theoretical-ly either as substellar objects with ages of the order 0.1

Gyr or as stars above the mass limit with ages of 1 —10Gyr. The dimmest of these objects, G 208-44B atMz —+ 10, cannot be as young as 0.1 Gyr, but might stillbe a marginal brown dwarf. Even so, note that it issignificantly more luminous than the faintest single starslike LHS 2924, discussed in Sec. III.A.3, and a factor of 5more luminous at 2.2 pm than the companion to thewhite dwarf GD165 (Sec. III.C.2).

The luminosity function of the local sample of 99 near-by low-mass stars, including the companions, appearsfairly Rat in a histogram of number versus half magni-tude intervals in Mz (Fig. 22). The numbers declinesharply at Mz =+10.0. This result implies two things:(1) The very faintest known stellar objects in the field arerare enough not to be represented in the known samplewithin 8 pc of the Sun, and (2) companions fainter thanMz —+10, like GD 165B, are relatively rare. There is,however, no reason to assume that the known sample ofsingle stars and companions is complete to 8 pc, and it is,of course, the dimmest cases that are most likely to havebeen missed.

TABLE II. Brown dwarf candidates and lowest-luminosity VLM stars. '

Name

GD165BLHS2065LHS2924GL 569BPC0025+ 0447

G29 —38B?HD114762B

G208 —44BGL 623BLHS 1047BRoss 614B

14.0910.7510.689.65

14.93:

13.45

1.66 11.781.26 10.311.17 10.481.13 9.551.24

(b) Unresolved companions12.70

(c) Companions8.288.858.047.24

with astrometric mass1.06 10.001 ~ 13 9.461.01 9.421.09 9.17

)0.011determinations

0.087+0.0140.079+0.0100.057+0.0790.085+0.030

K J-K M~ Mass (Mo)(a) Field stars and very wide binary companions

Spectraltype

&M9M9M9M8.5M9.5

'Data taken from Henry (1991); Kirkpatrick, Henry, and McCarthy (1991); Latham et al. (1989);Schneider et al. (1991);Becklin and Zuckerman (1988);James Graham (1993,private communication).



FICx. 22. Mass function for all objects in thesurvey, grouped in bins of 0.05MO. Takenfrom Henry (1991).

The empirical, linear relation derived from the data ofFig. 20 and other sources was then used to estimatemasses for all stars in the local sample, including thebinary components (which have large individual errors).The result is the estimated IMF for the VLM's in the so-lar neighborhood (Fig. 22) from the empirical method(Sec. III.A.2). Like many similar IMF determinations fors1Ilglc stcllaI' objects 111 Geld RIld clustcI' populRtloIls, arising number density with decreasing mass is inferreddown to the stellar mass limit. It would rise faster if anM-I. tlansformation from thcoIy ll.Rd bccn Used instead.

5. Radial velocity surveys

A complementary technique to the various imaging ap-pI'oachcs for finding UQI'csolvcd coIIlpan101ls to ncaI'bystars is to search for radial velocity variations due to or-bital motion of a visible component. The stars need to bebright enough for precise, high-resolution line profiles tobc measured.

Thc most comprehensive scaI'cll. to date of MRI'cy Rnd

Benitz (1989) covered 70 low-mass M dwarfs, some 80%of all known single stars later than dM2, brighter thanV=10.5, and accessible to observation from NorthernHermsphere observatories. Since the brightness con-straint generally required the star's to be within 10 pc,there is considerable overlap with the infrared specklesample described previously. While the velocity searchesare sensitive to smaller orbital separations, variations inthe primary out to at least a few AU could be measuredand overlap with McCarthy and Henry might occur ifthe secondary is detectable in the infraIed. Marcy andBellltz (1989) uncovered only one companion that 1s pos-sibly substellar —to the star Gliese 623 (Marcy andMoore, 1989)—and it was independently discovered inthe infrared speckle work. Like several stars in the sam-ple analyzed with the speckle technique, this star has apossible mass range (0.067—0.087MO) that straddles thecutoft mass for hyd. rogen burning. When these and the

results of the complenlentary astrometric searches areconsideIed together, the lack of detections suggests thatbrown dwarfs with masses above about 0.01Mo andwithin about 10 AU of low-mass M stars in the solarneighborhood are rare.

Campbell, Walker, and Yang (1988) employed aspecial-purpose spectrograph able to measure radial ve-locities of bright stars to a relative accuracy of 13 metersper second. Observations spanning six years of 16 F-Kdwarfs and subgiants —stars much closer to the Sun inmass —ylcldcd Ilo 1Ilfcrrcd coIIlp anions w1th Qlasscs 1Q

the 10—80 MJ range for periods ~ 50 year' s. A few starsshowed small, but possibly significant, long-term varia-tions indicative of the possibility of planetary com-panions with masses below 10MJ.

McMillan (1992) is presently carrying out an evenmore accurate radial velocity monitoring program withan 1nstI'UIIlcnt 1nvcntcd by thc late KIzysztof M. Scl-kowski for this purpose (see McMillan et al. , 1985).Standard deviations well below 10 m/sec have beenachieved from multiple observations in an observingseason. Sixteen solar-type stars have been observed rou-tinely for Qlore than four years, but no conclusive resultscan yet be announced.

Thus far, the most exciting discovery of these radialvelocity searches happened in a somewhat circumstantialmanner: Repeated observations of the G dwarf HD114762 were obtained by Latham et al. (1989) in order toestablish this star as an accurate radial velocity standardstar, under a project for the International AstronomicalUnion. Instead, they found that this star sho~s periodicvelocity variations with an amplitude of 0.55 km/s and apcrlod of 84 days. Thc UnsccIl companion has a mass of11MJ /sini, where i is the angle of inclination between theline of sight and the axis of the orbit. As in anynonecl1psing single-hned spectroscopic b1nary, th1s I-clination angle may not be determined. However, for thesystem to exceed the 80MJ "brown dwarf" upper limit,the inclination must be within 8 degrees of pole on.

Rev. Mod. Phys. , Vol. 65, No. 2, April i 993


A new analysis by Cochran, Hatzes, and Hancock(1991) using even more accurate radial velocity measure-ments at McDonald Observatory confirms the con-clusions of Latham et a/. (1989). These authors also findan upper limit to the projected rotational velocity of 1

km/s. If the actual rotation rate is similar to that of oth-er F9V main-sequence stars and its rotation axis is nearlyparallel to the orbital axis, one concludes that the rota-tion axis is within 5.7 degrees of the line of sight. Thisresults in the companion mass exceeding the stellar limit.On the other hand, it is possible that HD 114762 is an in-trinsically slow rotator and that the companion's mass issub stellar.

Some 50 additional stars are being monitored byresearchers at the Harvard-Smithsonian Center for As-trophysics, and it is possible that additional variables arebeing found (Mazeh et a/. , 1990); the target list includes24 nearby M dwarfs.

6. Substellar objects in interacting binary systems

Among other binary pulsars, the ablating companionto PSR1957+20 is also known to have substellar mass(-0.02MO), though this is likely the result of its prox-

imity to an energetic source (Fruchter et a/. 1988). It is

also worth noting, if only for completeness, that less ex-otic interacting binary systems are known in which thesecondary component is likely to be substellar in mass.The evolutionary theory of accreting white dwarfs with

nondegenerate secondaries predicts that the latter havebeen whittled down to substellar masses by the time theorbital period of the system reaches the observedminimum near 80 min (Paczynski and Sienkiewicz, 1981).At this point, the secondary becomes a degenerate hydro-

gen object, and its radius and the orbital period increasewith further decreases in the secondary's mass due tomass transfer. Interacting systems with much shorterperiods exist in which the secondary is a helium degen-

erate of substellar mass (Faulkner, Flannery, and

Warner, 1972).

D. Searches for halo brown dwarfs

Halo brown dwarfs are predicted to have temperaturesbelow 1000 K (see Fig. 11},making direct detection enor-

mously dificult. Nonetheless, the likelihood that thehalo of the galaxy contains a large amount of dark matter(cf. Fich and Tremaine, 1991) has led to widespread in-

terest in the slope of the initial mass function of halostars near the MMSM (minimum main-sequence mass).

Riven that we expect to see only halo objects above thestellar mass limit, one can start by asking if we aredetecting halo stars down to approximately the predictedmass. The answer seems to be yes: Monet et a/. (1992)presented trigonometric parallaxes and estimatedbolometric luminosities for the known sequence of high-

velocity, metal-poor M subdwarfs in the solar neighbor-

hood; the sequence extends to logL/Lo ——3.0. If the

metallicity of these stars is of the order 0.01 of solar([I/H]j= —2), this luminosity corresponds approxi-mately to the position of a 0. 1Mo model of D'Antona(1987) of this metallicity, a mass just above the substellarlimit for this composition. Earlier, Greenstein (1989b)had advanced a similar argument.

The sample of nearby faint halo stars is biased by theirkinematic properties, since these are drawn from catalogsof stars with measured proper motions. Nonetheless, in-

teresting estimates from color-selected samples have been

made possible by the advent of large-format CCD detec-tors. Several globular clusters show main sequences ex-

tending down to near the limit (Richer et a/. , 1991, and

references therein); in the best case of NCxC 6397, the au-

thors estimate that they reach 0.12MO. In none of these

cluster IMFs is there any evidence of a Aattening or turn-

over in the slope at the lowest masses. Some may have

IMF slopes steeper than the Salpeter value long associat-ed with the galactic disk. The authors suggest thatbrown dwarfs likely exist in large numbers in these clus-ters.

Richer and Fahlman (1992}have now estimated a cor-responding IMF for the field halo population. Theyfaced formidable problems in making this estimate thatare not encountered with clusters. First, the much more

numerous galaxies and compact extragalactic objectsmust be eliminated on the basis of having extended im-

ages and/or nonstellar colors in V, R, and I. Then, abso-

lute magnitudes and distances must be assigned to eachof three-dozen halo star candidates, based on a relation

between Mz and the V-I color. M subdwarfs are known

to be subluminous in such a color-magnitude diagram, ly-

ing up to three magnitudes fainter than their PopulationI counterparts at a given V I(see Monte-et a/. , 1992).How subluminous they are is predicted theoretically todepend on the metallicity. Unfortunately, any set of starsin a field halo sample may exhibit a fair range of metalli-

city, rather than the small dispersion usually found in aglobular cluster. This might leave the distance estimatesrather uncertain, and the space densities depend on these

to the third power. Richer and Fahlman (1992) derive afield star IMF that rises faster than a Salpeter slope,down to about 0. 14Mo. Extension of this slope to0.01Mo would allow VLM's and brown dwarfs to dom-

inate the dark matter content of the galaxy. The expect-ed number density of halo brown dwarfs in the solarneighborhood would be 0.5 objects pc

E. Gravitational microlensing

Paczysnki (1986) first proposed the possibility of ob-

serving gravitational lensing events of more distant stars

by compact objects in the foreground of our own galaxy.The general idea was to detect invisible "missing"rnatter, whether in the form of massive black holes orbrown dwarfs. Paczynski suggested specifically the mon-

itoring of the light from millions of stars in the Magellan-ic Clouds in order to detect microlensing events by fore-



ground objects in the galactic halo. The idea of search-ing for MACHOs (massive, compact halo objects-Griest, 1991) is now being actively pursued by severalgroups (cf. Griest et al. 1991). Paczysnki (1991)and Gri-est et al. (1991) have also suggested that stars in thegalactic bulge could be microlensed by similar compactobjects of the foreground galactic disk population.

These groups have argued that detectable light modu-lation should be observable from intervening massesangIng from 10 to 100Mo. The predicted rates are of

the order 10 to 10 events per star per year for an in-tervening lens near the stellar mass limit, each of a dura-tion of the order of months or less. The rates are propor-tional to the lens mass to the —

—,' power, while the dura-

tions are proportional to mass to the +—,'.For lensing stars located at the optimum distance of

about halfway to the galactic center, the size of the Ein-stein ring of a bulge star s light is similar to the separa-tion of Jupiter from the Sun. Hence Mao and Paczynski(1991) and Gould and Loch (1992) have pointed out thatsignatures of Jovian planets in similar orbits around alensing star would be detectable as short-duration eventsin the overall light curve.

IV. THE BOTTOM LINE

What is the net result of the application of all of theseattempts to find brown dwarfs in the galactic disk popu-lation, in young stellar associations, and as companionsto bright and nearby stars? First, it has to be said thatnot a single, unambiguous case of a brown dwarf orplanetary companion to a star other than the Sun hasbeen established, with the exception of the cases thatlikely result from close binary evolution. It should alsobe clear, however, that many brown dwarf candidateshave been found by a variety of methods. The lack ofproven examples is due to the great difhculty of establish-ing accurate masses, temperatures, or ages of the objects.

There is reason to be optimistic that field brown dwarfsmight exist in significant numbers. We list some of thebest such brown dwarf candidates as of this writing inTable II. The disk population —young and oldcomponents —is dominated numerically by low-massstars, and there is growing evidence that the mass func-tion does not decrease with decreasing mass as it reachesthe substellar mass limit. Some of the least luminous ob-jects in various samples may, in fact, be young and sub-stellar, rather than old and above the mass limit. Howev-er, what actually happens to the mass function belowabout 0.08Mo remains unclear in the galactic disk popu-lation. Studies of the halo population, both in globularclusters and the field, suggest that brown dwarfs could bea significant component.

Literally dozens of interesting candidates are beingidentified and studied further in the Pleiades and Hyadesclusters and in the Rho Ophiuchus association. Back-ground problems make membership in the groups some-what uncertain. There are important uncertainties in (1)

the conversion from an observed color to T„and in (2)

the predictions of the theoretical isochrones of different

groups and/or with different physical assumptions.These make it very dificult from the available results tounderstand whether member objects are brown dwarfs orpre-main-sequence stars of very low mass.

From the systematic programs designed to find com-panions of substellar mass, a less optimistic working hy-

pothesis has been developed. This hypothesis holds thatstars in the local disk population on1y rarely have com-panions between about 80 and 10MJ. This conclusion is

advanced from the radial velocity surveys of low-mass Mstars by Marcy and collaborators, from the more precisevelocity measurements of fewer stars of higher mass byCampbell and collaborators, from the infrared specklework on low-mass solar neighbors by McCarthy andHenry, and from the survey of white dwarfs by Zucker-man and Becklin. It is possible that substantial numbersof companions are formed with masses near, but under,the limit. The most likely substellar objects to have beenmissed entirely are old, low-mass, and/or very distantcompanions.

Nevertheless, as should be clear from the extendedtheoretical discussions in Sec. II and the ample observa-tional survey in Sec. III, the science of brown dwarfs isrich not only with promise, but detail. Since newtheoretical spectra and colors will soon be available andinfrared technology and surveys are continuing their ex-plosive advance, brown dwarf studies are rapidly joiningthe other more established branches of stellar astronomy.These substellar objects bridge the realms of planets andstars, may provide the missing mass of the halo, and re-side on one of the last scientific frontiers of the astronom-ical enterprise.

ACKNOWLEDGMENTS

The authors would like to thank F. Allard, and T.Henry, W. Hubbard, J. D. Kirkpatrick, J. Lunine, G.Rieke, D. Saumon for their help during the preparationof this manuscript. A. B. acknowledges the support ofthe NSF and NASA through Grants AST89-14346 andNAGW-2145, respectively. J. L. acknowledges the sup-port of the NSF through Grant AST89-18471.

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