Gravitational Instabilities in Protoplanetary Disks Speaker: Wenhua(Wendy) Ju 11/06/2012 Outline

• Movaon of studying Gravitaonal Instabilites (GI) in protoplanetary disks

• Gravitoturbulence and locality of GI

• Fragmentaon of disks due to GI

• FU Orionis outbursts and models of disk evoluon Motivation – Protoplanetary Disks

• Accreon disks are required: • Observaon(Infrared and sub-mm) • Dynamical role in star formaon: accreon • Site of planet formaon

Credit: C. Burrows and J.Krist(STScI), K.Stapelfeldt(JPL) and NASA Properties of Protoplanetary Disks

-3 -1 • Masses: 10 – 10 Msun in gas • Sizes: 100 – 1000 AU • Lifetimes: 106 – 107 yr • Temperature: cold, ~ several 100 K • !ickness: H/R ~ 0.03 – 0.05 • Opacity: τ ~ 103 – 104 (at 1 AU in optical, mainly due to dust) • Very low ionization fraction: MRI couldn’t work in “dead zones” Motivation – Gravitational Instability(GI)

• Alternave source of angular momentum transport in cold disks (Lin & Pringle 1987; Armitage, Livio & Pringle 2001). • Locally: gravitaonal turbulence à “viscosity” • Globally: bar structures, spiral arm structures • Giant Planet Formaon: • Limit of me: giant planets have to be formed within several million years from the star forming event • Difficulty with current planet formaon theory(i.e. core instability models, (Ida & Lin 2004; Rafikov 2010) • Giant planets might have formed by Gravitaonal Instability(disk fragmentaon, Boss 2000). tion process (a thorough discussion of these aspects can also be found in Lodato 2008). Gravitational instability in a disc is essentially a small modification of the standard Jeans instabil- ity in a three-dimensional homogeneous fluid. In the standard Jeans stability analysis, the natural ten- dency for collapse associated with self-gravity is bal- tion process (a thorough discussion of these aspects anced byTHE pressureASTROPHYSICAL gradients.JOURNAL, 553:174È Now,183, 2001 May pressure 20 gradi- V can also be found in Lodato 2008). ( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A. Gravitational instability in a discents is essentially are particularly a effective at small wavelengths small modification of the standardand Jeans tend instabil- to vanish for very large wavelengths. It ity in a three-dimensional homogeneous fluid. In the is therefore notNONLINEAR surprising OUTCOME that OFa system GRAVITATIONAL becomes INSTABILITY IN COOLING, GASEOUS DISKS standard Jeans stability analysis, the natural ten- dency for collapse associated with self-gravityJeans unstable is bal- if it has a size larger thanC aHARLES funda-F. GAMMIE Center for Theoretical Astrophysics, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801 anced by pressure gradients. Now,mental pressure lengthscale, gradi- called the Jeans lengthReceived 2000 (that May 6; we accepted 2001 January 9 ents are particularly effective at smallhave wavelengths already encountered in Section 2). and tend to vanish for very large wavelengths. It ABSTRACT is therefore not surprising that a systemIn the becomes case ofThin, a rotating Keplerian accretion disc, the disks generically situation become gets gravitationally unstable at large radii. I investigate the nonlinear outcome of such instability in cool disks using razor-thin, local, numerical models. Jeans unstable if it has a size largermore than complicated a funda- not only because of the different Cooling, characterized by a constant cooling timeqc, drives the instability. I show analytically that if the mental lengthscale, called the Jeansgeometry, length (that we butdisk especially can reach a because steady state the in which disc heating rotation by dissipation of turbulence balances cooling, then the have already encountered in Section 2). dimensionless angular momentum Ñux densitya \ [(9/4)c(c [ 1))qc]~1. Numerical experiments show In the case of a rotatingTHE disc,ASTROPHYSICAL theprovidesJOURNAL situation, 553:174È183, another2001 gets May 20 that stabilizing (1) ifqc Z 3)~1 e thenffect, the which disk reaches inV thisa steady, case gravitoturbulent state in which Q D 1 and cooling is ( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.balanced by heating due to dissipation of turbulence; (2) if3 , then the disk fragments, possibly more complicated not only because of the different qc [ )~1 THE ASTROPHYSICAL JOURNAL, 553:174isÈ183, more 2001 May 20 effectiveforming at large planets wavelengths. or stars; (3) in a steady, TheV gravitoturbulent combina- state, surface density structures have a charac- 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A. geometry, but especially( because the disc rotation teristic physical scale 64G / that is independent of the size of the computational domain. NONLINEARtion OUTCOME of a short OF GRAVITATIONAL wavelength INSTABILITY INstabilization COOLING,D & GASEOUS)2 DISKS factor (pres- Fig. 11. Surface density of a self-regulated self-gravitating provides another stabilizing effect, which in this case CHARLES F. GAMMIE HE STROPHYSICAL OURNAL Subject headings: accretion, accretion disks È galaxies: nuclei È solar system: formation T A J ,Center 553:174 forÈ Theoretical183, 2001 Astrophysics, May 20 University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801 V disc. In this case the mass ratio between the disc and the is more effective at( 2001. largeNONLINEAR The American wavelengths. Astronomical OUTCOME Society.sure) All rights The OF reserved. GRAVITATIONALcombina- and Printed in U.S.A. aReceived largeOn 2000 INSTABILITY May-line 6; wavelength accepted material: 2001 IN January COOLING, 9color Ð one GASEOUSgures (rotation) DISKS makes star was Mdisc/M =0.1. From Rice et al. (2005). tion of a short wavelength stabilizationit possible, factor (pres-CHARLES underF. GFig.ABSTRACTAMMIE appropriate 11. Surface density conditions, of a self-regulated to self-gravitat rendering Center for TheoreticalThin, Astrophysics, Keplerian accretion University disks of Illinois generically at Urbana-Champaign, become gravitationally 1002 WestINTRODUCTION Green unstable Street, at Urbana, large radii. IL 61801 I investigate from the central star (although there is some evidence for sure) and a large wavelength onethe nonlinear (rotation) outcomeReceived makes of such 2000 May instability 6; accepteddisc. in 2001 cool JanuaryIn disks this1 9 using. case razor-thin, the mass local, numerical ratio between models. the disc and the NONLINEARCooling, OUTCOMEthe characterized disc OF by GRAVITATIONAL a constantstable coolingstar timeat INSTABILITYqc was, all drivesM the wavelengths.disc instability./M IN=0 COOLING, I show.1. analytically From GASEOUS Rice that This if et the al. DISKS (2005). is clearlysuch low temperatures in the solar nebula; e.g., Owen et al. it possible, under appropriate conditions,disk can reach a steady to render state in whichABSTRACT heating by dissipation of turbulence balances cooling, then the It hasCHARLES longF. been GAMMIE realized that the outer reaches of accre- 1999). If the e†ective value of a is small and heating is Thin, Kepleriandimensionless accretion angular disks generically momentum become Ñux density gravitationallya \ [(9/4)c( unstablec [ 1))qc] at~1. large Numerical radii. I experiments investigate show Center for Theoreticalseen Astrophysics, when University one of Illinois considers at Urbana-Champaign, 1002 the West Green dispersion Street, Urbana, IL 61801 relation for As we have seen above, disc masses of this order of the disc stable at allthe wavelengths. nonlinearthat outcome (1) if Thisqc ofZ 3 such)~1 is then instability clearly thetion disk in reaches disks cool a disks steady,around using gravitoturbulent active razor-thin, galactic state local, in which numerical nucleiQ D 1 and models. (AGN) cooling is and young conÐned to surface layers, however, as in the layered accre- balanced by heating due toReceived dissipation 2000 May of turbulence; 6; accepted (2) 2001 ifq January[ 3)~1 9 , then the disk fragments, possibly Cooling, characterized by a constant coolingstellar time objectsqc, drives the(YSOs) instability.c may I show be analyticallygravitationally that if the unstable (for a tion model of Gammie (1996a), then instability can occur at seen when one considersdisk can the reach dispersionforming a steadytightly planets state relation or in stars; which wound (3) heating in for a steady, by gravitoturbulent dissipation axi-symmetricAs we of have turbulence state, surface seen balances density above, disturbances structures cooling, disc have then massesa the charac- of in this an order in- of magnitude might not be uncommon and it is there- teristic physical scale D64Greview&/)2 that is see, independent for AGN: of the size Shlosman of the computational et al. domain. 1990; YSOs: Adams dimensionless angular momentum Ñux densitya \ABSTRACT[(9/4)c(c [ 1))q ]~1. Numerical experiments show much higher temperatures. tightly wound axi-symmetric disturbancesSubject headings: accretion, in an accretion in- disks Èmagnitude galaxies: nucleic È might solar system: not formation be uncommon and it is there- thatThin, (1) ifq Keplerianc Z 3)~1finitesimally then accretion the disk disks reaches generically& a Lin steady, thin, 1993). become gravitoturbulent Instability gravitationally rotating state in in unstable which a Keplerian discQ atD 1 large (cf.and radii.cooling disk Toomre I sets isinvestigate in where 1964 the in AGN diskfore heating likely is typically that youngdominated circumstellar by illumination discs are gravi- balanced by heatingOn-line due material: to dissipationcolor Ðgures of turbulence; (2) if3 , then the disk fragments, possibly finitesimally thin, rotatingthe nonlinear disc (cf. outcome Toomre of such 1964 instabilitysound in speed in coolforecq disksc,[ likely the) using~1 rotation that razor-thin, young frequency local, circumstellar numerical), and models. the discs surface are gravi-from a central source. The temperature then depends on the formingWhen does GI work? Cooling, planets characterized or stars; (3) by in a a constant steady, gravitoturbulent cooling timeq , state, drivess surface the instability. density structures I show have analytically a charac- that if the the case of a stellar disc,teristic see physical also scale BinneytheD641G. &INTRODUCTION case&/)2 Tremainethat is ofdensity independent a stellar ofsatisfytationally thec size disc, offrom the the computational unstable. central see star also (although domain. Binney there is some evidence & Tremaine for tationally unstable. disk can reach a steady state in which heating& by dissipationsuch low of temperatures turbulence in balances the solar nebula; cooling, e.g., then Owen the et al. shape of the disk. If the disk is Ñat or shadowed, however, Subjectdimensionless headings:It has long angularaccretion, been realized momentum accretion that the disks outer Ñux È reaches density galaxies: of accre-a nuclei\ [(9/4) È1999).c solar(c [ Ifsystem:1))q the] e†ective~1 formation. Numerical value of a experimentsis small and heating show is 1987 and Bertin 2000 for• Dispersion relaon for density waves in gas disks in WKB thetion disks fluid around1987 case): active galactic and nuclei Bertin (AGN) and young 2000Having for establishedc the fluid that case): self-gravitating discs areand transportHaving is dominated established by internal torques, that self-gravitating one can discs are Onthat-line (1) material: ifq Zcolor3)~1Ð thengures the disk reaches a steady, gravitoturbulentconÐned toc surface state) layers, in which however,Q D as1 in and the coolinglayered accre- is stellarc objects (YSOs) may be gravitationally unstable (forlinearly a tionQ model unstable,4 ofs Gammie\ (1996a),weQ should then^ 1 instability now can occur ask at what (1) is theapply equation (3). For example, in the nucleus of NGC 2 2 2 balancedlimit(2review byBinney see, heating for AGN: & due ShlosmanTremaine to dissipation et al. 1990; 1987): of YSOs: turbulence; Adams (2) ifmuchqc [ higher3)~1 temperatures., then the disk fragments, possibly ω = c k 2πGΣ k +formingκ &, Lin planets1. 1993).INTRODUCTIONInstability or stars; in (3) a Keplerian in a steady,(78) disk sets gravitoturbulent in wherefrom the the state, central surface starnG (although& density there structurescrit is some have evidence a charac- for 4258 (Miyoshilinearly et al. 1995) unstable, the accretion we rate should may be as large now ask what is the s 2 2 2 suchnon-linear low temperaturesAGN disk evolution heating2 in the is solar typically of nebula; the dominated e.g., instability Owen by illumination et al. and whether − | | teristic sound physical speed scalec , the rotation64G / frequencythat is), independent and the surface of thefrom size a of central the source. computational The temperature domain. then depends on the It has long been realized thatωs theD= outer&c reaches)s2 k of accre-2πG1999).Σ Ifk the+ e†ectiveκ value, of is small and heating is (78)as 10~2 M yr~1 (Lasota et al. 1996; Gammie et al. 1999). density & satisfy (Toomre 1964;shape Goldreich of the disk. If & the diska Lynden-Bell is Ñat or shadowed, 1965). however, Here non-linear_ evolution of the instability and whether where, as above,tionω disksandSubject aroundk are headings: active the galactic perturbationaccretion, nuclei accretion(AGN) and fre- disks− young È galaxies:conitÐ canned nuclei| to lead| surface È solar to layers, asystem: sustained however, formation as in the transport layered accre- of angular mo-Equation (3) then implies that instability sets in where stellar objects• Similar to Jeans Instability: (YSOs) may be gravitationally unstable (for a 2 and transport is dominated byQ = 1 internal torques, one can On-line material: color Ðgurescs ) Q \ 1ωfortion a model ““apply razor-thin of Gammie equation (1996a), (3). ÏÏ For (two-dimensional) then example, instability in the can nucleus occur of Ñuid at NGC disk, review see, for AGN: Shlosman etQ 4 al. 1990;\ Q YSOs:^ 1crit Adams mentum. (1) The stability criterion, Eq. (79) offersT a \ 104(ita/10 can~2) K. lead If the to disk a is sustained illumination dominated, transport of angular mo- quency and wavenumber• and κ iswhere, the epicyclicnG& ascrit and fre- above,Q \much0.676ω higherand4258 fortemperatures. (Miyoshik a etÐare al.nite-thickness 1995) the accretion perturbation rate isothermal may be as large disk fre- & Lin 1993). Instability Gravity in a Keplerian Pressure + disk sets in whereFcentri the AGNQ > 1 diskas 10 heatingM isyr typically(Lasota dominated et al. 1996; by Gammie illumination et al. 1999). however, then Q Ñuctuates with the luminosity of the (Toomre1 1964;INTRODUCTION Goldreich & Lynden-Bell 1965).crit Herenaturalfrom the way~2 central_ to~1 describe star (although this. there Indeed, is some evidence we should for note quency. In Eq. (78),sound the speed firstc , the term rotation. on frequency the right), and thehand(Goldreich surface from & aLynden-Bell centralEquation source. (3) The then 1965). temperature implies The that then instability instability depends sets on in the where condition s Q 1 for a ““ razor-thin ÏÏ (two-dimensional) Ñuid disk, mentum. The stability criterion, Eq. (79) offers a density & satisfy• Small wavelengths: P \ quency and wavenumbersuchT low10 temperatures( /10 and) K. If the inκ the diskis solar is illumination the nebula; epicyclic dominated,e.g., Owen et al. fre-central source. It has long beenandcrit Q realized0.676 that for the a Ð outernite-thickness reaches(1) can isothermal of accre- beshape rewritten,that disk of the the\ disk. stability for4 a If thea~2 disk disk parameter is with Ñat or shadowed,scaleQ heightis however, proportionalH ^ c /), to the side corresponds to the stabilizing\ effect of pressure, 1999).however, If the then e†ectiveQ Ñuctuates value with of a theis luminosity small and of heating the s is (Goldreichcrit & Lynden-Bell 1965). The instability conditionand transport is dominated by internal torques, one can The fate of a gravitationally unstable YSO or AGN disk tion disks around• Long wavelengths: activec ) galactic nuclei (AGN)Fcentriaround and young a centralcon objectcentralÐned to source. surfaceof mass layers,M however,, as in the layered accre- natural way to describe this. Indeed, we should note (1) canQ bes rewritten,Q for a1 disk with scale height (1) H applycsound/ , equation speed (3). For (and example, hence in the to nucleus temperature), of NGC so that the second is thestellar de-stabilizing objects (YSOs)4 e mayffectquency.\ be gravitationally of^ self-gravity, In unstable Eq. (for (78),^ a s ) theThe fate first of a gravitationally term* unstable on YSO the or AGN right disk handdepends on how it arrived in an unstable state. To under- around an centralG& objectcrit of mass M , Ω42582 (Miyoshition model et al. of 1995) Gammie the accretion (1996a), rate then may instability be as large can occur at review see, for• AGN:Intermediate: GI Shlosman et al. 1990;* YSOs: Adamsascolder 10 muchMdepends discs higheryr on(Lasota aretemperatures. how it more et arrived al. 1996; in unstable. an Gammie unstable et state. al. Now, 1999). To under- let us considerstand why, consider an analogy with convective instability while the third is(Toomre the stabilizing 1964; Goldreich e &ffect Lynden-Bell of rotation. 1965). Here ~2 stand_ why,~1 consider anH analogy with convective instability that the stability parameter Q is proportional to the & Lin 1993). Instability inside a Keplerian corresponds diskH sets in where theEquation to theAGN (3) then disk stabilizing impliesheating that is typically instability dominated eff setsect in whereby of illumination pressure, Q \ 1 for a ““ razor-thin ÏÏ (two-dimensional)M Z M Ñuid, disk, (2) in stellarM structureZ theory.M The evolution, of stellar models (2) in stellar structure theory. The evolution of stellar models sound speed• GI occurs when: c , the rotation frequencydisk r), and* the surfaceTadiscwhichisinitiallyhot,sothat\ 104(a/10~2) K. If the disk is illumination* dominated, Q 1and Eq. (78) is a simpleandcrit Q quadratic\ 0.676s for aequationÐnite-thickness in isothermalk and it disk fromwith a central highlydisk unstable source.r The radial temperatureQ < 1 entropy proÐ thenles are depends of little on the sound speed (and hence to temperature), so that density & satisfy however, then Q Ñuctuates with the luminosity of the % with highly unstable radial entropy proÐles are of little (Goldreichcrit & Lynden-Bellwhere M2 4 1965).thenr2&. The second instability condition is thethe de-stabilizingshape discinterest, of is the because stable. disk. Ifconvection the In disk e preventstheff isect Ñat absence or such shadowed, of models self-gravity, of from however, other trans- is straightforward to see that• 1. ωdisk< 0ifandonlyif: central source. (1) can be rewritten,In fora steady a disk state with disk scale whose height heatingH ^ isc / dominated), by andbeing transport realized is in dominatedan astrophysically by internal plausible setting. torques, Simi- one can interest, because convection prevents such models from c ) wheres M 4Then fater2larly,& of. a highly gravitationally unstable disks unstable may be YSO irrelevant or AGN because disk the colder discs are more unstable. Now, let us consider around a centralinterior objectQ of turbulent massswhileM dissipationQ, 1 the and whose third evolution (1)disk isport iscon- apply the mechanisms equation stabilizing (3). For (and example, neglecting eff inect the nucleus of stellar rotation.of NGC heating), • trolled2. by4 internal\ transport * ^ of angular(Toomre momentum,In Parameter) a steady thedepends acc- state onaction how diskof it the arrived instability whose in an would unstable heating prevent state. one is from To dominated under- ever arriv- by being realized in an astrophysically plausible setting. Simi- csκ nG& crit 4258 (Miyoshi et al. 1995) the accretion rate2 may be as large retion rateM 3 c / , where I have used thestand why,ing consider in such an a analogy state. Gravitational with convective instabilityk instability must be Q = < 1. 0 H\ na s2 & ) interior(79) turbulentthea as disc 10 dissipation coolsM yr down(Lasota and due et whose al. 1996; to radiativeevolution Gammie et al. is cooling 1999). con- untillarly, highlyadiscwhichisinitiallyhot,sothat unstable disks may be irrelevant because the Q 1and πGΣ (Toomre 1964;• formalism3. GoldreichM ofZ ShakuraEq. &M & Lynden-Bell, Sunyaev (78) (1973). is 1965). Then a (2) simpleHerein stellar““ structure turnedquadratic~2 on_ theory. ÏÏ in~1 a natural The way.evolution equation of stellar models in k and it disk r * EquationAn initially (3) then stable implies Keplerian that disk caninstability be driven sets unstable in where action of the instability would prevent one from ever arriv- % Q \ 1 for a ““ razor-thin ÏÏ (two-dimensional)trolled Ñuid disk,by internalwitheventually highly transport unstableQ radial1. of entropy Atangular this pro momentum,Ðstage,les are of the little disc the acc-develops a crit • 4. 3acs3 cs 3 interest,T \ becauseby10 an4( increasea/10 convection~2 in) K. surface If prevents the density2 disk (e.g., such is illumination Sellwood models & from Carlberg dominated, the disc is stable. In the absence of other trans- whereandQM 40.676nr2&. M for0 Z a isÐ\nite-thickness7.1 straightforward] 10~4a isothermalretionM_ yr diskrate~1 (3)M to1984)3 see orc by cooling.that≈/ , In where anω-disk< model, I0ifandonlyif: have the surface used density the ing in such a state. Gravitational instability must be Thus, the dimensionlessdisk\ parameterG Q playsA1 an km s im-~1B beinggravitational0however, realized\ na in then ans2 astrophysically& instabilityQ)Ñuctuatesa plausible with in the the setting. form luminosity Simi- of a of spiral the a struc- (GoldreichIn a steadycrit & state Lynden-Bell disk whose 1965). heating The is instability dominated condition by changes on the accretion timescale D(r/H)2(a))~1 (more ““ turned on ÏÏ in a natural way. interior turbulentPre-assumpon for 4: No external torques, no heang from external illuminaon . dissipation and whose evolutionformalism is con- oflarly, Shakuracentral highly unstable source. & Sunyaev disks may be(1973). irrelevant Then because the portant role in determining(1) can be rewritten,implies whether gravitational for a disk a with instability gaseous scale (e.g., height Shlosman discH ^ etc al./), 1990).ture,rapid that variation leads of surface to angular density could momentum be obtained by transport, port mechanisms (and neglecting stellar heating), trolled by internalThis transport does not of apply angular to disks momentum, dominatedc κ the by external acc- s torquesaction ofThe thedumping instabilityfate ofmaterial a gravitationally would onto theprevent disk or one unstable by fromapplication ever YSO arriv- of or a direct AGN disk An initially stable Keplerian disk can be driven unstable around a central object of mass M , s ing in such a state. Gravitational instability must be is gravitationallyretion unstable rateM0 \(e.g., or3na ac not.s2 magnetohydrodynamic&/),Q We where= can* I have now [MHD] used ask the< wind)a1 or. disksand3ac3depends ultimatelymagnetic on torque). how it The to arrived temperature, energy inc an by unstable dissipation contrast,3 state. changes To on and under- heating.(79)by an increasethe indisc surface cools density down (e.g., Sellwood due & to Carlberg radiative cooling until formalism of Shakuraheated & mainly Sunyaev by external (1973). illumination. Then M0 Z““ turnedsstand\ onthe ÏÏ7.1 why, incooling a] natural consider time10~4 way.D(a)a an)~1 analogy. This suggestss with convectivethatM in coolyr disks instability~1 (3) how massive should the discFor be a young, in order solar-massH to star be accreting grav- from a disk withAn initially(r/H ? stable1), cooling Keplerian is the disk dominant can be driver driven of unstable gravitational_ 1984) or by cooling. In an a-disk model, the surface density M Z M π,GΣ (2) InG turn,in stellar the structure stability theory.A1 kmcondition The evolution s~1B works of stellar like models a kind of 3acs3 a \ 10~2 atdisk 10~6 Mr_cyrs *~1 equation3 (3) implies that insta-by an increaseinstability. in surface density (e.g., Sellwood & Carlberg changeseventually on the accretionQ timescale1. At( / this) ( stage,) (more the disc develops a itationally unstable.M0 ThisZ is\bility7.1 easily occurs] 10~4 where deriveda the temperature fromM yr Eq.~1 drops(3) below 17 K. withOnce highly gravitational unstable radial instability entropy sets in, pro theÐles disk are can of little D r H 2 a) ~1 G A1 km s~1B _ 1984)‘thermostat’, or by cooling. In so an a-disk that model, heating the surface turns density on only if the ≈ where M 4Disksnr2&. may notThus, be this cold if the the starimplies is dimensionlesslocated in gravitational a warmchangesinterest, onattempt the becauseinstability accretion to regain parameter stabilityconvection timescale (e.g., by rearrangingD prevents Shlosman(r/H)2(Qa) its) such mass~1plays(more to et models reduce al. 1990). from an im-rapid variationgravitational of surface density instability could be in obtained the form by of a spiral struc- (79) in the case whereimpliesIn a gravitational steadydiskMdiscmolecular state instability diskM cloud: whose where (e.g., heatingShlosman the ambient is et dominatedtemperature al. 1990). is by greaterrapiddiscbeing variation is& colderor realized by of heating surface thanin an itself densityastrophysically a through given could dissipation temperature. be plausible obtained of turbulence. setting. by Simi-If the ‘ther- than 17 K, or if the disk is bathed in scatteredThis does infrared not light applyDissipation to disks can occur dominated directly through by shocks external or indirectly torques dumping material onto the disk or by application of a direct Thisinterior does turbulent not apply" to dissipation disks dominated and whose by external evolution torques is con-dumpinglarly, material highly onto unstable the disk disks or by may application be irrelevant of a direct because the (e.g., a magnetohydrodynamicportant [MHD] wind) role(e.g., or disks ain magnetohydrodynamic determiningmagneticmostat’174action torque). of works, the The instability temperature, we whether should would [MHD] by contrast, prevent then changesone wind)a expect fromgaseous on ever or the arriv- disks instabil- discmagneticture, torque). that The temperature, leads to by contrast, angular changes momentum on transport, csΩ cs trolledGM by internalH transportM of angular momentum, the acc- Q = heatedretion mainly rate byM0 external\ 3nac2 illumination.&/), where, I haveheated(80) used the mainlyatheity cooling bytoing externalself-regulate in time suchD(a) a illumination.)~1 state.. This in Gravitational suggests such that a way in instability cool that disks must the be disc is al-the cooling time D(a))~1. This suggests that in cool disks For a young,2 solar-masss staris accreting gravitationally from a disk with (r/H ? unstable1), cooling is the dominantor not. driver We of gravitational can now ask ≈ πGΣ ΩRformalismπGΣR of Shakura≈ R &M Sunyaevdisc(R (1973).) Then For a young,““ turned solar-mass on ÏÏ in a natural star accreting way. from a disk with (r/H ? 1),and cooling ultimately is the dominant to driver energy of gravitational dissipation and heating. a \ 10~2 at 10~6 M_ yr~1 equation (3) implies that insta- instability.waysAn kept initially close stable to Keplerian marginal disk stability, can be driven and unstable thus a self- bility occurs3 wherec the temperature dropsc belowa \ 1710 K.~2 atOnce 10by~6 angravitationalM increaseyr~1 in instability surfaceequation density sets (3) in, (e.g., the implies Sellwood disk can that & Carlberg insta- instability. Disks may nota bes3 this cold ifhow the star is massives located3 in a warm shouldattempt to the regain_ stability disc by rearrangingbe in its order mass to reduce to be grav-¯ In turn, the stability condition works like a kind of which then shows that,M0 Z as anticipated,\ 7.1 ] 10~4a in orderMbility_ foryr~1 occurs(3) gravitating where1984) or by the cooling. disc temperature will In an evolvea-disk model, drops to a the state below surface where density 17 K.Q Q, Once gravitational instability sets in, the disk can molecular cloudG where the ambientA1 temperature km s~1B is greater & or by heating itself through dissipation of turbulence. ≈ Q 1, we shouldthan require 17 K, or if that the disk (M is batheddiscitationally/M in scattered) (H/R infraredDisks). unstable. light mayDissipationwhere notchanges beQ This¯ thiscanis on occur cold a the constant directly is accretion if easily the through star timescale of shocks order is derived locatedD or(r indirectly/ unityH)2(a) in)~1 (physicalfrom a warm(more Eq. pro-attempt to regain stability by rearranging its mass to reduce implies gravitational instability (e.g., Shlosman et al. 1990). rapid variation of surface density could be obtained by ‘thermostat’, so that heating turns on only if the ∼ ∼ molecular174 cloud where the ambient temperature is greater & or by heating itself through dissipation of turbulence. This does not apply to disks(79) dominated in by the external case torques wheredumpingM materialdisc onto the diskM or: by application of a direct disc is colder than a given temperature. If the ‘ther- (e.g., a magnetohydrodynamic [MHD] wind)than or 1726 disks K, or ifmagnetic the disk torque). is bathed The" temperature, in scattered by contrast, infrared changes light on Dissipation can occur directly through shocks or indirectly heated mainly by external illumination. the cooling time D(a))~1. This suggests that in cool disks mostat’ works, we should then expect the instabil- For a young, solar-mass star accretingc Ω from a disk withc (r/GMH ? 1), cooling is theH dominantM driver of gravitational 174 a \ 10~2 at 10~6 M_ yr~1 equation (3)s implies that insta-s instability. bility occurs where the temperatureQ drops below= 17 K. Once gravitational2 instability sets in, the, disk can (80) ity to self-regulate in such a way that the disc is al- Disks may not be this cold if the≈ starπ isG locatedΣ in a warmΩR πattemptGΣ toR regain≈ stabilityR byM rearrangingdisc(R its mass) to reduce molecular cloud where the ambient temperature is greater & or by heating itself through dissipation of turbulence. ways kept close to marginal stability, and thus a self- than 17 K, or if the disk is bathed in scattered infrared light Dissipation can occur directly through shocks or indirectly which then shows174 that, as anticipated, in order for gravitating disc will evolve to a state where Q Q¯, Q 1, we should require that (M /M ) (H/R). where Q¯ is a constant of order unity (physical≈ pro- ∼ disc ∼ 26 THE ASTROPHYSICAL JOURNAL, 553:174È183, 2001 May 20 V ( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

NONLINEAR OUTCOME OF GRAVITATIONAL INSTABILITY IN COOLING, GASEOUS DISKS CHARLES F. GAMMIE Center for Theoretical Astrophysics, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801 Received 2000 May 6; accepted 2001 January 9

ABSTRACT Thin, Keplerian accretion disks generically become gravitationally unstable at large radii. I investigate the nonlinear outcome of such instability in cool disks using razor-thin, local, numerical models. Cooling, characterized by a constant cooling timeqc, drives the instability. I show analytically that if the disk can reach a steady state in which heating by dissipation of turbulence balances cooling, then the dimensionless angular momentum Ñux densitya \ [(9/4)c(c [ 1))qc]~1. Numerical experiments show that (1) ifqc Z 3)~1 then the disk reaches a steady, gravitoturbulent state in which Q D 1 and cooling is balanced by heating due to dissipation of turbulence; (2) ifqc [ 3)~1, then the disk fragments, possibly forming planets or stars; (3) in a steady, gravitoturbulent state, surface density structures have a charac- teristic physical scale D64G&/)2 that is independent of the size of the computational domain. Subject headings: accretion, accretion disks È galaxies: nuclei È solar system: formation On-line material: color Ðgures

1. INTRODUCTION from the central star (although there is some evidence for such low temperatures in the solar nebula; e.g., Owen et al. It has long been realized that the outer reaches of accre- 1999). If the e†ective value of a is small and heating is tion disks around active galactic nuclei (AGN) and young conÐned to surface layers, however, as in the layered accre- stellar objects (YSOs) may be gravitationally unstable (for a tion model of Gammie (1996a), then instability can occur at review see, for AGN: Shlosman et al. 1990; YSOs: Adams much higher temperatures. & Lin 1993). Instability in a Keplerian disk sets in where the AGN disk heating is typically dominated by illumination soundThermal processes are important to GI speedcs, the rotation frequency ), and the surface from a central source. The temperature then depends on the density & satisfy shape of the disk. If the disk is Ñat or shadowed, however, and transport is dominated by internal torques, one can c ) Q 4 s \ Q ^ 1 (1) apply equation (3). For example, in the nucleus of NGC nG& crit 4258 (Miyoshi et al. 1995) the accretion rate may be as large as 10 M yr (Lasota et al. 1996; Gammie et al. 1999). (Toomre• Timescales: 1964; Goldreich & Lynden-Bell 1965). Here ~2 _ ~1 Equation (3) then implies that instability sets in where Q \•1 Change of for a ““ razor-thinΣ: accreon mescale ~ (r/H) ÏÏ (two-dimensional)2(αΩ)-1 Ñuid disk, crit T \ 104(a/10~2) K. If the disk is illumination dominated, andQ 0.676 for a Ðnite-thickness isothermal-1 disk • Cooling or heang: thermal mescale ~(α\ Ω) however, then Q Ñuctuates with the luminosity of the (Goldreichcrit & Lynden-Bell 1965). The instability condition • r/H >> 1 central source. (1) can be rewritten, for a disk with scale height H c / , ^ s ) The fate of a gravitationally unstable YSO or AGN disk around a central object of mass M , * depends on how it arrived in an unstable state. To under- H stand why, consider an analogy with convective instability M Z M , (2) in stellar structure theory. The evolution of stellar models * disk r with highly unstable radial entropy proÐles are of little where M 4 nr2&. interest, because convection prevents such models from In a steadydisk state disk whose heating is dominated by being realized in an astrophysically plausible setting. Simi- interior turbulent dissipation and whose evolution is con- larly, highly unstable disks may be irrelevant because the trolled by internal transport of angular momentum, the acc- action of the instability would prevent one from ever arriv- ing in such a state. Gravitational instability must be retion rateM0 \ 3nacs2 &/), where I have used the a formalism of Shakura & Sunyaev (1973). Then ““ turned on ÏÏ in a natural way. An initially stable Keplerian disk can be driven unstable 3ac3 c 3 by an increase in surface density (e.g., Sellwood & Carlberg M0 Z s \ 7.1 ] 10~4a s M yr~1 (3) G A1 km s~1B _ 1984) or by cooling. In an a-disk model, the surface density changes on the accretion timescale D(r/H)2(a))~1 (more implies gravitational instability (e.g., Shlosman et al. 1990). rapid variation of surface density could be obtained by This does not apply to disks dominated by external torques dumping material onto the disk or by application of a direct (e.g., a magnetohydrodynamic [MHD] wind) or disks magnetic torque). The temperature, by contrast, changes on heated mainly by external illumination. the cooling time D(a))~1. This suggests that in cool disks For a young, solar-mass star accreting from a disk with (r/H ? 1), cooling is the dominant driver of gravitational a \ 10~2 at 10~6 M_ yr~1 equation (3) implies that insta- instability. bility occurs where the temperature drops below 17 K. Once gravitational instability sets in, the disk can Disks may not be this cold if the star is located in a warm attempt to regain stability by rearranging its mass to reduce molecular cloud where the ambient temperature is greater & or by heating itself through dissipation of turbulence. than 17 K, or if the disk is bathed in scattered infrared light Dissipation can occur directly through shocks or indirectly 174 Final Fate for GI Unstable Disks

• Quasi-steady, long-lived state: • Gravitaonal turbulences transport angular momentum out

• Marginally stable Q~Q0. • Self-regulated state cooling ~ heang by GI

• Rapid fragmentaon: • If cooling is much more efficient than heang • Formaon of massive planets or substellar objects (Boss 1997).

• Bursts of accreon (FU Ori bursts): • If temperature becomes high enough to trigger MRI 1.GI as a specific form of “viscosity”: Controversy about locality of GI 70 Accretion in binary systems

Re > 1014. (4.33) 4.6 Viscousmol torques 65

Even in regions where extreme conditions hold, it is hard to reduce Remol very much below this estimate. Hence, molecular viscosity is far too weak to bring about the viscous dissipation and angular momentum transport we require. Curiously, a clue to the right kind of viscosityB is provided by the very large estimate (4.33); it is known from laboratory experiments that, if the (molecular) Reynolds number of a fluid flow is gradually increased,v~ therevφ ( isR a +criticalλ/2)Reynolds number at which turbulence sets in: i.e. the fluid velocity v suddenly begins to exhibit largeR and+λ chaotic variations on an arbitrarily short time and lengthscale. For most laboratory ~ 3 fluids, theLet us critical then summarise Reynolds what have number we learned is so ofv the order 10–10 . Because of the size of far. The radial component of the equation of mo- the estimatetion generally (4.33), takes the we form may of centrifugal conjecture balance, that the gas flow in accretion discs is also turbulent,leading although to there is, as yet, no proof thatvφ ( thisR− isλ so./2) If thisR is the case, 1/2 A the flow willGM be characterizedc 2 by the size λturb and turnover velocity vturb of the v = 1 β s , (37) φ R − v largest turbulent! " eddies# K $ (see% e.g. Landau & Lifshitz (1959) and Frisch (1995)). Since the turbulentthus indicating motion that rotation is completely is Keplerian, with chaotic some about the mean gas velocity, our simple correction terms due to pressure. The vertical equa- viscositytion calculations of motion takes the apply: form of hydrostatic there is balance, a turbulentviscosityνturb Rλturb−vλturb. Although so that (in the non-self-gravitating case) the vertical ∼ this resultdensity has profile a is neat Gaussian appearance, with thickness it is here that our troubles with viscosity really 4.6 Viscous torques 65 begin.H Turbulencec is one of the major uncharted areas of classical physics and we do Fig. 3. Disc viscosity from kinetic theory. Sketch of the angu- = s , (38) lar momentum flux from emission point E to referenceFig.not sur- understand 4.8.R vExchangeK the onsetof fluid of blobs turbulence, in differentially still less rotating the physical flow. mechanisms involved face S as seen in a reference frame corotating with E.Even which shows than in order for the disc to be thin B if particles are emitted isotropically at E,duetoCoriolisandLocality how they determine the lengthscale, λturb, and turnover velocity vturb.Themost force, their path is not straight and they tend to intersect the we require cs vK.Finally,conservationofangular ~ vφ (R+λ/2) " v R+λ reference surface S more normally if they are emittedwe in a canmomentum do with and present the continuity knowledge equation for is a Keple- to place plausible limits on λturb and vturb. prograde direction and more tangentially if they are emitted rian disc lead to the following diffusive evolutionary v~ in a retrograde direction, thus enhancing the contributionseeofFirst, that• equationEvoluon of viscous disks: the typical kinematicfor the disc size surface viscosityof density the largestΣ due turbulent to either eddieshydrodynamic cannot exceed turbulence the disc or thickness molecular prograde particles to the flux at S. R A vφ (R−λ/2) transportH,soλ∂Σturb in3< a∂ H simple.Second,itisunlikelythattheturnovervelocity∂ shearing motion is vturb is supersonic, = R1/2 νΣR1/2 . (39) ∂t R∼∂R ∂R The resolution of this difficulty was givenfor, by in this case,& the turbulent' () motions would probably beR− thermalizedλ by shocks (see Clarke and Pringle (2004). The point is that even ν λv,˜ Section4.6. 3.8).∼Time-dependent Thus, we solutions can writeExchange of fluid blobs in differenally rotang if the random distribution of particles is isotropic Fig. 4.8. Exchange of fluid blobs in differentially rotating flow. at the location where they are emitted, it is not • α-prescripon: flow(Frank et al. 2002). isotropic at the position where they collide andwith re- theClearly, appropriate the equationsinterpretations derived above have a lim- of λ andv ˜.Inthecaseofmolecular transport, ited applicability(Shakura if we do not and specifyseeSunyaev that thewhat kinematic, 1973 ) the vis- viscosity due to either hydrodynamic turbulence or molecular lease their angular momentum, and relatively more ν = αcsH transport in a simple shearing motion is (4.34) λ andcosityv ˜ are is, the and in mean particular free whether path it depends and thermal or speed of the molecules respectively. In particles move on prograde orbits than retrograde ν λv,˜ ones. Thus, even if an annulus at a smaller radius Lin & Pringle, 1987: not on other disc properties, such as∼ surfaceBalbus den- & Papaloizou, 1999: the casesity, of temperature, turbulent< radius, motions, etc. Aswith will theλ appropriate beis discussed theinterpretations spatial of λ scaleandv ˜.Inthecaseof or characteristicmolecular transport, wavelength of the has on average asmallerspecificangularmomen-and expect α 1. This is theλ and famousv ˜ are the mean freeα- pathprescription and thermal speed of theof molecules Shakura respectively. and In Sunyaev (1973). Self-gravity could be treated as below, in some cases, the relationshiplocal, betweenSelf-gravitang disks could only be vis- tum than an annulus at a larger radius, it isturbulence, pref- and∼ v ˜ is the typicalthe case ofvelocity turbulent motions,described as local process ofλ is the the spatial eddies. scale or characteristicwhen the wavelength of the erentially those particles that happen to haveIt an isgravitoturbulent importantcosity and surface to process, and thus realize density might thatturbulence, even (4.34) changeandv ˜ is the is the typical merely velocity of the a eddies. parametrization: all our ignorance Cancould be described in a di weffusive use character the above of equationmodified arguments (35).Can we use the above topaern speed arguments obtain to obtain an anΩ estimate estimate matches the local for the viscosity for in the the case viscosity in the case excess angular momentum that collide at a largerabout the viscosity mechanismof a diff haserentially been rotating fluid isolated and for an accretion inp α discrather in particular? than After all, we in ν and apart from Nevertheless, we can still gainmay some reason insight that as long into as λ R, the processes responsible for the stresses are local, radius, releasing their angular momentum and so viscous α framework. angular velocity " Ω. Otherwise, the of a differentially rotating fluidand a small and neighbourhood for of an an arbitrary accretion point will look disc very similar in to particular? a simple After all, we transporting it (correctly) from the inside out.the The expectationdisc dynamicsα by< considering1 we have a couple gained ofenergy equaon sim- nothing. Therecannot is be put into the certainly no cogent reason, ple cases, which were initiallyshearing discussed motion. by It turns von out that this case is more complex than it appears at first picture is even clearer if one considers the dynamicsmay reason that as∼ long as λsight, althoughR, as thewe shallform of a see, processes the finaldiffusion equaon answer can responsible be obtained by a simple as required argument. for the stresses are< local, in a reference frame co-rotating with the emissionfor example,Weizs¨acker for (1948) believing and L¨ust (1952).αWithto" the be application constant to accretion discs throughout in mind, we consider the a situation disc shown structure; in even α 1 location, E (see Fig. 3). In the (non-inertial) frame Fig. 4.8, where the gasin local viscous scenario. flow lies between the planes z = 0 and z = H of cylindrical andmight a smallbe violated neighbourhood in small regionspolar of coordinates an of arbitrary (R, φ the,z). We disc assume azimuthalpoint where symmetry, will some with look gas physical flowing very in the similar input continually to a simple∼ of E,particlesareemittedisotropically.However, 4.6.1. The spreading ring φ direction with angular velocity Ω(R)abouttheorigin. their path is not straight, due to the effect of Cori-shearingfeeds theConsider, motion. supersonic for example, It turns turbulence. the case outLet in which us that consider Nonetheless,ν twoisthis sim- neighbouring case rings is the of width moreαλ -prescriptionon both complex sides of a surface thanR = has it proved appears a useful at first const. Because of the chaotic motions, gas elements such as A and B are constantly olis force. As seen in Figure 3, particles emitted in ply a constant, independent ofexchanged radius andacrossΣ this,and surface, with speeds v˜.Atypicalelementtravelsadistance sight, although as we shall see, the final answer∼ can be obtained by a simple argument. the prograde direction (that have an excess angularparametrizationthe initial condition of our for the ignorance surfaceλ before density interacting and is an on in- has the other encouraged side of the surface R =const.Astheseelements a semi-empirical approach to ∼ momentum with respect to E)tendtointersecta finitesimally thin ring of mass mof,whoseshapeisa fluid are exchanged, they carry slightly different amounts of angular momentum: Withthe viscosity the application problem, to which accretionelements seeks such as discstoA will estimateon average in mind,carry an the angular we magnitude momentum consider corresponding of the toα situationby a comparison shown in reference surface S more normally, while retrograde δ function centered at some radiusthe locationR0: R λ/2, while elements such as B will be representative of a radial − particles tend to intersect S on more tangentialFig.of theory 4.8, where and observation. them gas flow There lies is,between for example, the planes somez reason= 0 and to believez = H thatof cylindricalα 0.1 paths. Thus the contribution of prograde particles Σ(R, t =0)= δ(R R0)(40) 2πR − ∼ to the flux at S is larger than that of retrogradepolar onesin accretion coordinates discs in (0R, cataclysmicφ,z). We variables, assume azimuthal at least some symmetry, of the time with (see gas Section flowing 5.8). in the (see Matsuda and Hayashi 2004), thus removing the The solution to equation (35) in this case was found inconsistency. φMoreover,directionby Lynden-Bell the withα-prescription angular and Pringle (1974): velocity can beΩ(R formulated)abouttheorigin. in an identical fashion for viscosities due to the winding-up of chaotic magnetic fields in the disc; here, too, one expects Let10 us consider two neighbouring rings of width λ on both sides of a surface R = const.α < 1 on Because dimensional of the grounds chaotic (see motions, Section gas 4.8 elements below). such as A and B are constantly exchanged∼ across this surface, with speeds v˜.Atypicalelementtravelsadistance ∼ λ before interacting on the other side of the surface R =const.Astheseelements ∼ of fluid are exchanged, they carry slightly different amounts of angular momentum: elements such as A will on average carry an angular momentum corresponding to the location R λ/2, while elements such as B will be representative of a radial − cesses of this kind also operate in the the dynamics tion 8.3 above). In this case, the thermal timescale of spiral galaxies, Bertin and Lin 1996). Analytical in the disc is closely related to the viscosity param- models of such self-regulated discs have been con- eter α through the requirement of thermal equilib- AA49CH06-Armitage ARI 5 AugustLocality 2011 12:22 structed and described by Bertin (1997), Bertin and rium (Eq. (59)). In thermal equilibrium the thermal Lodato (1999) and Lodato and Bertin (2003). The • Gravitoturbulencetimescale is equal to: the cooling timescale tcool,and effectiveness of such a self-regulation mechanism has • weLocal Thermal Equilibrium: therefore have: been shown (at least for relatively light discs, with 4 1 Mdisc/M ! 0.1) by Lodato and Rice (2004, 2005), α = . (81) 9γ(γ 1) tcoolΩ who have performed global hydrodynamicalwhere τ, the optical simu- depth between the− disk surface and midplane, is given in terms of the midplane lations of self-gravitating discs inopacity the presence by τ (1 of/2)"κ.The Writing strengthκ κ ofT theβ , one torque finds induced that α can by begravitational expressed as an explicit = 634• Assuming opacity to be power G. Lodato =and0W.cK. M. Rice cooling. These simulations indeedfunction show that of just after the surfaceinstabilities density and angular has been velocity measured (Levin from 2007), numerical sim- (Gammie, 2001) an initial transient the disc settles down in a quasi- law of ulationsmidplane in several temperature: papers (Gammie, 2001; Lodato 4 β steady state, characterised by the presence of a spi- and Rice,σ µ 2004,mp − 2005; Mejia6 2β et4 al.,2β 2005;2β 7 Boley et al., α (Q0πG) − " − ) − . (15) ral structure (to provide a source of transport and 2006)∼ κ andγ k the relation described in Eq. (81) above 0 ! B " heating), and which only evolves secularly, on the has been confirmed both in cases where t Ω is a Here, we have omitted numerical(Levin 2007) prefactors, the exact values of which varycool in the literature de- viscous timescale. Fig. 11 shows a snapshot of one constant (Gammie, 2001; Lodato and Rice, 2004) pending upon the assumed• vertical structure (which has not been accurately determined from such simulation, where the spiral structure is clearly Locality is proved when: and in cases where it is not (Boley et al., 2006). This simulations) and upon whether the sound speed used is isothermal or adiabatic. Analogous ex- visible. In such quasi-steady state, the radial profile • isM also / M a confirmation < 0.25 (Lodato&Rice that, at 2004) least in the cases con- pressions can be derived for opticallydisk * thin disks or for disks with more general opacity laws that of Q is indeed found to be remarkably constant and sidered in these papers (with some exceptions, see are a function of density• as M well as / M temperature = 0.1, departure is 10% (Cossins, Lodato & Clarke 2010). close to unity. below),disk the* transport induced by gravitational in- The simplicity of Equation 15 results (Cossin from, theLodato assumption & Clarke, 2009) that angular momentum transport What about the transport induced by gravita- stabilities can be relatively well described in terms due to self-gravity can be treated as a local, gravitoturbulent process. Because gravity is a long- tional instabilities? Here the very important point to • ofM adisk ‘local’ / M process.* < 0.5 range force, the general validity of the local approximation is questionable, and it is easy to remember is that this is strongly dependent on the From(including the discussionradiave transfer, above, it would then ap- (Lodato & Rice, 2004) effective cooling rate. This is a simpleenvisage consequence circumstances inpear which that it Forgan would in principle et al. 2010) fail entirely. it can A be disk possible that develops to produce a strong bar-like of the self-regulation process describedstructure, above. for example, For couldthrough have regions gravitational in which instabilities gravitational a torques stress fromwith the an bar remove agivencoolingrate,thediscwilltryandprovideangular momentum and driveeffective accretionα much without larger the than attendant unity. energy This would dissipation be the that occurs in a local model. More formally, Balbus & Papaloizou (1999) show that a self-gravitating disk can (through the development of gravitational instabil- value predicted by equation (81) for tcool tdyn = − # ities) a balancing heating term, otherwiseonly be described the disc by a localΩ α1.Actually,itturnsoutthatthisisnotthecase.model if, at all radii, the pattern speed of the waves )p matches the local angular velocity ). If, on the other hand, ) is very different from −)1, the radial energy would cool down further and become more unsta- Gammie (2001) has shownp that if tcool ! 3Ω ,the ble, thus increasing the amplitudeflux of associated the perturba- with wavesdisc, in the rather disk will than violate achieving the assumption the of above-mentioned locality and mandate a global tion and the resulting stress and energytreatment dissipation. of the disk dynamics.self-regulated state, undergoes fragmentation. Rice Thus, once in a self-regulated state,The not locality only does of angularet momentum al. (2005) transport (see also by Clarke disk self-gravity et al. 2007) has been have tested later numerically. the temperature of the disc settleLodato to a &value Rice such (2004, 2005)shown studied that angular the momentum actual fragmentation transport in a model boundary system is composed of that Q 1, but also the amplitudean isolated of the disk spiral that cools everywheredependent on on a the timescale ratio of that specific is a fixed heats multipleγ and of ranges the local dynamical ≈ −1 −1 structure induced by gravitationaltime. instability They found has to that thebetween analytic estimatetcool =6 forΩ theand stress,tcool derived=13 fromΩ .Alsothis the assumption of local provide a stress large enough to balancethermal the balance, external was a goodbehaviourFigur matche 1. toSurf theace isdensity numericaleasilystructure understood. resultsat the end of providedthe simulations Indeed, thatfor the the(upper disk growthleft) massq = 0.05, was(upper notright) tooq = 0.1 and (bottom) q = 0.25. cooling rate. Of course, in order tohigh, reproduceMdisk/M such< 0.25. Arate more quantitativeof the instability analysis was is of carried the out order by Cossins, of the Lodato dy- & Clarke ∗ behaviour in numerical simulations,(2009), as who one explicitly might evaluatednamical the timescale, pattern speed and of therefore the waves generated for cooling by times self-gravity within momentum of the disc is conserved to within a few per cent through- The resulting radial profiles of α are shown in Fig. 5 for the three expect, one should go beyond a simplea disk described isothermal using a similarlyshorter simple than prescription that, the disc for the has cooling no time time. to For developMdisk/M 0.1, out all the simulations. cases q = 0.05, 0.1∗ and= 0.25. The upper panels show separately equation of state, as done in the pioneeringthey bounded work departures of When fromthelow v instabilityalues localof behaviorthe artifi andcial atviscosity reach the 10%are thermalused, level.particle Broadly equilibrium,in- similarthe hydrodynamic and results alsoand gra holdvitational contributions to α, while the Laughlin and Bodenheimer (1994).for The more importance realistic disks,terpenetration incollapse whichmight the becomeslead coolingto a poor inevitable.representation timescale, setof Thestrong by dependency theshocks radiativebottom physics, ofpanels the sho isw athe strongsum of the two. The plots show the time of thermodynamics in the development of gravita- in SPH.fragmentationThis is not a serious issue thresholdin our case, onbecauseγ foundin our sim- by Riceaverage etof al.α at the end of the simulation, once the disc has reached function of radius. Boleyulations et al.only (2006)mildly supersonic used a radiationshocks are hydrodynamicsinvolved. Based on the code toa studyquasi-steady the evolutionstate. The time-a of veraging interval is 500 time units, tional instabilities was firstby Princeton University Library on 10/03/12. For personal use only. noted ina moderately the simulations massivedensity disk(2005)contrast (Mdisk/inMthe offspiralers0.14)arms, another aroundwe estimate interpretation a low-massthe Mach number star. of They thei.e. found0.6 sameorbital that,times althoughat the outer disc edge. ∗ = by Pickett et al. (1998) and Pickettsome et nonlocal al. (2000). transportof the didshocksphenomenon. occur,to be M the! predictions1.5. ItAt turnsthese lo ofw out,v aalues local thatof M model, fora value any based value uponWe can of thenoγw coolinguse the general timeresults of viscous disc theory outlined of β SPH ≈ 0.2 is already sufficient to stop particle interpenetration in Section 2 to test the locality of transport in our simulations. In fact, However, it was not until Gammiewere (2001) consistent that withthe (Bate the1995). simulationtheThis fragmentationis con acrossfirmed aby substantialthe boundarywell-defined regionspiral alwaysstructure of the occurs disk.equation Likewise, at the(9) giv simulationses us firm expectations for the value of α needed to connection described above was made explicit. Ac- that wesameobtain (which valuewould of αhave been0.06.smeared It thusout if appearsignificant thatbalance gravita-the imposed cooling, if energy dissipation can be treated in

Annu. Rev. Astro. Astrophys. 2011.49:195-236. Downloaded from www.annualreviews.org including radiative transfer by Forgan et al. (2010) showed significant nonlocal effects only when tually, we can also make a further step forward if particletionalinterpenetration instabilitieswas indeed≈ present), in a steadyconsistent statewith the cannotre- a viscous provideframework, i.e. by using equation (7). In particular, in our Mdisk/M > 0.5. sults of previous simulations that used the standard SPH viscosity simulations t $ = β = 7.5, γ = 5/3 and, because our discs are we assume that the transport induced by∗ gravita- astresslargerthanα 0.06. If a larger stress is re-cool The conditions forand fragmentationhigher values for the ofviscosity a low-mass,coefficients≈ self-gravitating(Rice et al. 2003a,b). disk cannearly alsoKeplerian, be written,d ln $/ todln aR ≈−3/2. Inserting these numbers tional instabilities can be describedfirst in approximation, terms of a as a functionquired of in local order disk to conditions. reach thermal Gammie equilibrium, (2001),in usingequation then local(9) w numericalould give us an expected value of α ≈ 0.05. It ‘local’ viscosity (i.e. if we neglect possible non-local the disc will fragment. is important to note that the fact that the expected α turns out to simulations, found that4.2 Angular if the coolingmomentum couldtransport be parameterizedand energy dissipation as be nearly independent on radius is a result of choosing the cooling energy transport, that might in principle occur, see The torque producedThe naturalby gravitational questioninstabilities atin thisthe disc stageis give isn whethertime to be thesimply proportional to the dynamical time-scale. In the 1 general case, of course, the resulting α need not be constant. The Balbus and Papaloizou 1999 and discussion in sec- by the sumcoolingof the tw timeo terms intdescribedcool real discsβ)in equationsK− is, fast(10) enoughand (11). to allow frag- (16) After averaging the stress tensor azimuthally= and radially, over a dotted line in Fig. 5 shows the expected value of α. Our results are then for β 3 fragmentationsmall region "R followed= 0.1, we compute promptlythe corresponding as soon asvalue theof diskα cooledin fairly togood lowagreementQ.Shortwith the expectations of viscous transport ≤ (see27equation 4): theory. cooling times allow the disk to radiate away the energy provided by shocks or turbulentIt is also interesting dissipation,to compare the dissipation rate D(R) that −1 grav Reyn dln $ TRφ + TRφ would result if the transport process were viscous, i.e. computing such that neither pressureα(R) = support nor shear suffices. to avert gravitational(14) collapse (Shlosman & ! dln R ! " #&c2" # D(R) based on equation (7), with the actual dissipation rate. Because Begelman 1987). The thermal! condition! s for fragmentation in this simple form applies strictly ! ! ! ! only in the limit where the cooling time in the disk is locally constant. Using a more realistic #C 2004 RAS, MNRAS 351, 630–642

www.annualreviews.org Dynamics of Protoplanetary Disks 203 • 2. Fragmentaon due to GI Disk Fragmentation 2D hydrodynamic simulaon by Gammie 2001: Disks will fragment when -1 tcool ≤ 3 Ω

-1 -1 tcool = 50 Ω , no fragmentaon tcool = 2 Ω , fragmentaon Disk Fragmentation 3D hydrodynamic simulaon by Rice et al. 2003: Disks will fragment when -1 tcool ≤ 5 Ω

-1 -1 tcool = 5 Ω , no fragmentaon tcool = 3 Ω , fragmentaon THE ASTROPHYSICAL JOURNAL, 553:174È183, 2001 May 20 V ( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

NONLINEAR OUTCOME OF GRAVITATIONAL INSTABILITY IN COOLING, GASEOUS DISKS CHARLES F. GAMMIE Center for Theoretical Astrophysics, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801 Received 2000 May 6; accepted 2001 January 9

ABSTRACT Thin, Keplerian accretion disks generically become gravitationally unstable at large radii. I investigate the nonlinear outcome of such instability in cool disks using razor-thin, local, numerical models. Cooling, characterized by a constant cooling timeqc, drives the instability. I show analytically that if the disk can reach a steady state in which heating by dissipation of turbulence balances cooling, then the dimensionless angular momentum Ñux densitya \ [(9/4)c(c [ 1))qc]~1. Numerical experiments show that (1) ifqc Z 3)~1 then the disk reaches a steady, gravitoturbulent state in which Q D 1 and cooling is balanced by heating due to dissipation of turbulence; (2) ifqc [ 3)~1, then the disk fragments, possibly forming planets or stars; (3) in a steady, gravitoturbulent state, surface density structures have a charac- teristic physical scale D64G&/)2 that is independent of the size of the computational domain. Subject headings: accretion, accretion disks È galaxies: nuclei È solar system: formation On-line material: color Ðgures

1. INTRODUCTION from the central star (although there is some evidence for such low temperatures in the solar nebula; e.g., Owen et al. It has long been realized that the outer reaches of accre- 1999). If the e†ective value of a is small and heating is tion disks around active galactic nuclei (AGN) and young conÐned to surface layers, however, as in the layered accre- stellar objects (YSOs) may be gravitationally unstable (for a tion model of Gammie (1996a), then instability can occur at review see, for AGN: Shlosman et al. 1990; YSOs: Adams much higher temperatures. & Lin 1993). Instability in a Keplerian disk sets in where the AGN disk heating is typically dominated by illumination sound speedcs, the rotation frequency ), and the surface from a central source. The temperature then depends on the density & satisfy shape of the disk. If the disk is Ñat or shadowed, however, and transport is dominated by internal torques, one can c ) Q 4 s \ Q ^ 1 (1) apply equation (3). For example, in the nucleus of NGC nG& crit 4258 (Miyoshi et al. 1995) the accretion rate may be as large as 10 M yr (Lasota et al. 1996; Gammie et al. 1999). (Toomre 1964; Goldreich & Lynden-Bell 1965). Here ~2 _ ~1 Equation (3) then implies that instability sets in where Q 1 for a ““ razor-thin ÏÏ (two-dimensional) Ñuid disk, \ T 10 ( /10 ) K. If the disk is illumination dominated, andcrit Q 0.676 for a Ðnite-thickness isothermal disk \ 4 a ~2 \ however, then Q Ñuctuates with the luminosity of the (Goldreichcrit & Lynden-Bell 1965). The instability condition central source. (1) can be rewritten, for a disk with scale height H c / , ^ s ) The fate of a gravitationally unstable YSO or AGN disk around a central object of mass M , * depends on how it arrived in an unstable state. To under- H stand why, consider an analogy with convective instability M M , (2) in stellar structure theory. The evolution of stellar models Z * disk r with highly unstable radial entropy proÐles are of little where M 4 nr2&. interest, because convection prevents such models from cesses of this kind also operate in the the dynamics tiondisk 8.3 above). In this case, the thermal timescale of spiral galaxies, Bertin and Lin 1996). AnalyticalIn ain the steady disc is closely state related disk to whose the viscosity heating param- is dominated by being realized in an astrophysically plausible setting. Simi- models of such self-regulated discs have been con-interioreter turbulentα through the dissipation requirement of thermal and whose equilib- evolution is con- larly, highly unstable disks may be irrelevant because the structed and described by Bertin (1997), Bertin and rium (Eq. (59)). In thermal equilibrium the thermal action of the instability would prevent one from ever arriv- Lodato (1999) and Lodato and Bertin (2003). Thetrolledtimescale by internal is equal to transport the cooling timescale of angulart ,and momentum, the acc- cool ing in such a state. Gravitational instability must be effectiveness of such a self-regulation mechanism hasretionwe rate thereforeM0 have:\ 3nacs2 &/), where I have used the a been shown (at least for relatively light discs, with 4 1 ““ turned on ÏÏ in a natural way. M /M ! 0.1) by Lodato and Rice (2004, 2005),formalismα = of Shakura. & Sunyaev (1973).(81) Then disc 9γ(γ 1) t Ω Disk Fragmentation who have performed global hydrodynamical simu- cool An initially stable Keplerian disk can be driven unstable AA49CH06-Armitage ARI 5 August 2011 12:22 − lations• If: of self-gravitating discs in the presence of The strength3acs3 of the torque induced by gravitationalcs 3 by an increase in surface density (e.g., Sellwood & Carlberg cooling. These simulations indeed show that after Minstabilities0 Z has\ been7.1 measured] 10 from~4a numerical sim- M_ yr~1 (3) 1984) or by cooling. In an a-disk model, the surface density an initial• there’s transientno irradiaon the disc settlesfrom central star; down in a quasi- ulations inG several papers (Gammie,A 2001;1 km Lodato s~1B steady• angular momentum transport is totally dominated by state, characterised by the presence of a spi- and Rice, 2004, 2005; Mejia et al., 2005; Boley et al., changes on the accretion timescale D(r/H)2(a))~1 (more ral structure (to provide a source of transport and 2006) and the relation described in Eq. (81) above gravitoturbulence 10 –5 implies gravitational instability (e.g., Shlosman et al. 1990). rapid variation of surface density could be obtained by heating), and which only evolves secularly, on the has been confirmed both in cases where tcoolΩ is a viscous• Then: timescale. Fig. 11 shows a snapshot of oneThis doesconstant not (Gammie,FFragmentation applyragmentatio 2001;n to disks Lodato dominated and Rice, 2004) by external torques dumping material onto the disk or by application of a direct such simulation, where the spiral structure is clearly and in cases wherepossiblepossibl ite is not (Boley et al., 2006). This 10 –6 (e.g., a magnetohydrodynamic [MHD] wind) or disks magnetic torque). The temperature, by contrast, changes on visible. In such quasi-steady state,Self-gravity the radial suppressed profile is also a confirmation that, at least in the cases con- by hydrodynamic angular of Q is indeed found to be remarkablymomentum constant transport andheatedsidered mainly in these by papers external (with illumination. some exceptions, see the cooling time D(a))~1. This suggests that in cool disks ) –7 close to unity. –1 10 Forbelow), a young, the transport solar-mass induced by star gravitational accreting in- from a disk with (r/H ? 1), cooling is the dominant driver of gravitational What about the transport induced by gravita- stabilities can be relatively well described in terms year 10 at 10 yr equation (3) implies that insta- instability. tional instabilities? Here the very important pointa to\ of~2 a ‘local’ process.~6 M ~1 α(self _

(M –8 -grav) = 10 10 –2 remember is that this is stronglyM dependent onbility the occursFrom the where discussion the above, temperature it would then ap- drops below 17 K. Once gravitational instability sets in, the disk can effective cooling rate. This is a simple consequence pear that in principle it can be possible to produce K (no irradiation) Disks may not be this cold if the star is located in a warm attempt to regain stability by rearranging its mass to reduce through gravitational instabilities a stress with an of the self-regulation process–9 described = 20 above. For 10 T eff agivencoolingrate,thediscwilltryandprovidemoleculareffective cloudαα(semuchlf-grav) larger where= 1 than the unity. ambient This would be temperature the is greater & or by heating itself through dissipation of turbulence. 0 –3 = 10 K (no irradiation) (through the development of gravitationalff instabil- value predicted by equation (81) for tcool tdyn = T e than 17 K, or if the disk is bathed in scattered infrared light Dissipation can occur directly through shocks or indirectly −1 # ities) a balancing heating10 term,–10 otherwise the disc Ω .Actually,itturnsoutthatthisisnotthecase. 10 1 10 2 −1 would cool down further and become more unsta- r (AU)Gammie (2001) has shown that if tcool ! 3Ω ,the 174 ble, thus increasing the amplitude of the perturba- disc, rather(Armitage than, 2011) achieving the above-mentioned Figure 3 tion and theThe resulting structure stress of a steady-state, and energy self-gravitating dissipation. disk around a 1self-regulated M star, calculated state, assuming undergoes that self-gravity fragmentation. Rice ! Thus, once inacts a as self-regulated a local angular momentum state, transport not only process does whose efficiencyet al. is (2005) set by the (see requirement also Clarke of thermal et al. 2007) have later the temperatureequilibrium of the (Clarke disc 2009, settle Rafikov to 2009). a value The opacity such includesshown contributions that from the water actual ice, amorphous fragmentation boundary is that Q 1,carbon, but silicates,also the and amplitude graphite (Z. Zhu, of private the spiral communication).dependent If self-gravity on is the sole ratio source of of specific angular heats γ and ranges ≈ momentum transport, the effective αgrav generated by the self-gravitating turbulence is an−1 increasing −1 structure inducedfunction by of radius, gravitational and there is instability a maximum radius has (shown to as thebetweenpurple areat)cool beyond=6 whichΩ fragmentationand tcool is=13Ω .Alsothis provide a stresspredicted large to enough occur. The to blue balance curves show the radii external where α behaviour10 2 and 10 3 is. If easily other sources understood. of angular Indeed, the growth grav = − − cooling rate.momentum Of course, transport in order coexist to with reproduce self-gravity and such generate stressesrate of of this the magnitude, instability these radii is denote of the the order of the dy- approximate inner edge of the self-gravitating disk. The green lines show where the effective temperature of behaviour inthe numerical luminous disk simulations, equals 10 K or 20 as K; oneif an external might irradiationnamical field heated timescale, the disk, it would and affect therefore the for cooling times expect, one shoulddynamics to go the beyond right of the a green simple lines. isothermal shorter than that, the disc has no time to develop equation of state, as done in the pioneering work of the instability and reach thermal equilibrium, and Laughlin andRice, Bodenheimer Mayo & Armitage (1994). 2010). The There importance are, however, twocollapse caveats. becomes First, the inevitable. effective tempera- The dependency of the of thermodynamicsture of an in isolated the development self-gravitating disk of gravita- at the innermostfragmentation fragmenting radius threshold is quite low, on typicallyγ found by Rice et al. tional instabilitiesaround was 10–20 first K (Matzner noted in & Levinthe simulations 2005). This implies(2005) that the dynamics offers another of self-gravitating interpretation disks of the same by Pickett etin al. the (1998)potentially and fragmenting Pickett region et al. are (2000). sensitive to modestphenomenon. levels of external It turns irradiation, out, that which for any value of γ However, it waswill tend not to until weaken Gammie gravitational (2001) instabilities that the (Cai et al.the 2008, fragmentation Stamatellos & boundaryWhitworth 2008, always occurs at the connectionby Princeton University Library on 10/03/12. For personal use only. describedVorobyov & above Basu 2010). was madeSecond, explicit. the innermost Ac- fragmentingsame radius value is of calculatedα 0.06. for a It given thus opac- appear that gravita- ity and will vary with the metallicity of the disk and as a result of particle coagulation≈ (Cai et al. tually, we can also make a further step forward if tional instabilities2 in a steady state cannot provide 2006). For an opacity appropriate for ice grains, for example, κ κ0T (Semenov et al. 2003), we assume that the transport1/3 induced by gravita- astresslargerthan= α 0.06. If a larger stress is re- Annu. Rev. Astro. Astrophys. 2011.49:195-236. Downloaded from www.annualreviews.org and #frag κ0− (Matzner & Levin 2005, Rafikov 2009). Reduced values of the≈ opacity would tional instabilities can∝ be described in terms of a quired in order to reach thermal equilibrium, then therefore push the fragmentation boundary interior to the fiducial radius of 50–102 AU. ‘local’ viscosityImmediately (i.e. if we neglectinterior to possible the radius non-local where # # the,thestrengthofself-gravitatingtransport disc will fragment. K = frag energy transport,is close that to the might maximum in principle value, α occur,0.1. At smaller see radii, theThe disk natural becomes question increasingly at opticallythis stage is whether the ≈ Balbus and Papaloizouthick, and the 1999longer and cooling discussion time results in in sec- lower valuescooling of α (Equation time in 13). real This discs means is fast that enough the to allow frag- surface density in self-gravitating disks at small radii is high and that even very weak transport by other mechanisms will suffice to render the disk stable27 against self-gravity. Curves corresponding 2 3 to a self-gravitating α 10− and 10− are shown in Figure 3,fromwhichitisclearthatitis = hard to sustain a strongly self-gravitating disk much interior to 10 AU. This implies that, although shocks in a strongly self-gravitating disk are a candidate mechanism for chondrule formation (Boss & Durisen 2005), it is hard to construct self-consistent models in which strong enough shocks would be present at small radii (Boley & Durisen 2008).

www.annualreviews.org Dynamics of Protoplanetary Disks 205 • Disk Fragmentation

• Two caveats: • Background viscosity: if background viscosity dominates, then the disk might be gravitaonally stable; • Stellar irradiaon: external heang source keeps the disk gravitaonally stable Disk Fragmentation No. 1, 2009 PROPERTIES OF GRAVITOTURBULENT ACCRETION DISKS 287 expressions (B1)inAppendixB.Unshadedregioncorresponds • Background viscosity: to a self-luminous, gravitoturbulent disk. A constant M disk ˙ • αν = 0.003 corresponds to a straight vertical line in Figure 1 cutting through different regimes. • Strong stellar irradiaon : The numerical values on the left and lower axes of this figure • T = 35K due to irradiaon, correspond to a particular choice of αν 0.003, T /Tf 3 0 = 0 = spaally constant (T0 35 K), and the opacity in form (7). As described in Section= 3.5,theverylowM 10 7 M yr 1 disk is always ˙ ! − − • Opacity: viscous and gravitationally stable, even at# very large distances • κ = κ T2 from the central object. Above this value of M the disk must 0 ˙ 7 be gravitoturbulent within some range of distances. For 10− 1 5 1 M yr− ! M ! 4 10− M yr− ,thegravitoturbulentregion # ˙ × 10 # 9 1 extends from Ω 10− –10− s− all the way to very large distances (where ∼Ω 0), limited only by conditions (49)or → 5 1 (51). However, for M above M " 4 10− M yr− ,the ˙ ˙ × # 10 1 gravitoturbulent disk fragments at Ω 1.4 10− s− (in the optically thick case with opacity characterized≈ × by β 2 = fragmentation occurs exactly at Ωf ,seeSection3.4)sothata constant M disk can be maintained only interior to this point. ˙ For any M 10 7 M yr 1, the gravitoturbulent angular ˙ " − − momentum transport becomes# weak at large enough Ω so that the background viscosity starts to determine disk properties at Figure 1. Plot of different possible states in which an(Rofikov accretion, 2009) disk can be 9 1 small radii (the upper part of the plot, at Ω > 10− s− in this found as a function of mass accretion rate M and angular frequency (the ˙ Ω corresponding dimensionless quantities m M/M and ω / are particular case). ˙ ˙ f Ω Ωf shown on the upper and right axes). The˙ disk= has a non-zero= background As an example, let us use Figure 1 to figure out where the viscosity αν 0.003 and is externally irradiated at a spatially constant tem- transitions between different regimes occur in a disk with M = ˙ perature T0 3Tf 35 K, corresponding to condition (57). Shading indi- 5 1 = = ≈ 10− M yr− , αν 0.003, T0/Tf 3andopacityintheform cates viscous regions (slant solid shading), irradiated regions (horizontal dotted # 9 =1 = shading), and region where the disk must fragment (vertical solid shading). (7). At Ω > 10− s− , the disk is gravitationally stable and self- The gravitoturbulent, self-luminous region is unshaded. Different curves sepa- luminous, angular momentum is transported through the disk 10 1 9 1 rate regions with distinct physical conditions and are marked on the plot (see by background viscosity. For 2 10− s− < Ω < 10− s− , the text for more details). Coordinates of the points where these curves cross in the disk is gravitoturbulent and× self-luminous. For Ω < 2 the (m, ω) coordinates are given by expressions (B1). 10 1 × ˙ 10− s− , the disk is gravitoturbulent and externally irradiated. 11 1 4.1. Separation of Different Regimes The disk is optically thick for all Ω " 10− s− and optically thin for smaller Ω. This example clearly shows the complexity We consider a particular case when external irradiation at of possible states in which a given accretion disk can be found constant temperature T0 is strong, at different distances from the central object. Figure 1 exhibits four invariant properties of accretion disks: T0/Tf > 1, (57) (1) the disk is dominated by background viscosity for all Ω 3/2 at very low M (for m<αν (T /Tf ) ,seeAppendixB) and in which case the phase space of possible regimes can be ˙ ˙ 0 non-zero T ;(2)forallM the disk is dominated by background represented by Figure 1.Giventhatinthecaseofopacity 0 ˙ viscosity at high enough Ω;(3)atintermediatevaluesofM dominated by cold dust Tf is just slightly higher than 10 K 3/2 3/2 ˙ (see Equation (21)) it is clear that condition (57)shouldapply (for αν (T0/Tf ) < m<(T0/Tf ) ), the disk possesses a gravitoturbulent, externally˙ irradiated region that extends to to virtually all accretion disks in the universe as even the lowest 3/2 large distances; (4) at large values of M (for m>(T /Tf ) ), measured temperatures encountered in some dense molecular ˙ ˙ 0 cores are not very different from 10 K (Di Francesco et al. the disk has a gravitoturbulent region within a finite range of 2007). For that reason in the following we consider only disks distances but must inevitably fragment at some large distance. for which condition (57)isfulfilled. Similar qualitative conclusions hold also for opacity behaviors In Figure 1,onecanexaminedifferentregimes(indicatedby different from (7). shading) which are relevant for a given M.Curves3 separating We should note here that calculations presented in this section ˙ different regimes correspond to various critical transitions: α rely on our use of opacity (6)and(7)throughoutthewhole region of M, phase space that we consider. In reality, at α (dotted lines) implies a transition from viscous (above and= ˙ Ω ν high M and the disk temperature should exceed 102 Kat to the left of this curve, slant solid shading) to gravitoturbulent ˙ Ω state, α 1 (solid lines) describes the onset of fragmentation which point icy grains sublimate leaving metal-silicate grains (below and= to the right of this curve, vertical solid shading), as a source of opacity. This latter opacity source while still T T is a (long-dashed) line below which external irradiation being in form (6)ischaracterizedbysmallervaluesofκ at the = 0 same temperature and β 1/2. This is likely to quantitatively starts to dominate disk structure (region with horizontal dotted ≈ shading), τ 1curve(short-dashed)showsatransitionfrom (but not qualitatively) affect results presented in Figure 1 at high M and .Tonotcomplicatethingsfurtherherewedo an optically= thick (above this curve) to an optically thin disk. ˙ Ω Coordinates (in phase space m, ω,upperandrightaxes)of not attempt to self-consistently describe transitions between the points A, B, C, D where these˙ curves cross are given by different opacity regimes but rather display a qualitative picture for a single opacity law. 3 Equations for these curves can be found in Sections 3.1–3.5 and Appendix A. 3. Disk Evoluon & FU Orionis Bursts July 24, 1996 14:50 Annual Reviews HARTTEX1 AR12-06

July 24, 1996 14:50 Annual Reviews HARTTEX1 AR12-06

FU ORIONIS PHENOMENON 211 208 HARTMANN & KENYON kilometers per second, is typically observed in the Balmer lines, especially in accreted in FU Ori outbursts reinforce the notion that disk accretion plays a ↵ H . The Na I resonance lines also show broad blueshifted absorption, some-major role in the formation of stars and not just their associated planetary sys- times in distinct velocityPhenomena components or “shells.” The emission component in tems. FU Ori outbursts also demonstrate that accretion rates through such disks the P Cygni H↵ profile is often absent; when present, this emission extendscan be highly time variable and unexpectedly large at times, with implications to much smaller velocities redward than the blueshifted absorption. Infraredfor disk physics and grain processing. Finally, the powerful winds of FU Ori spectra of FU Oris show strong CO absorption at 2.2 µm and water vapor bandsobjects have important implications for understanding the production of bipolar in the near-infrared ( 1–2 µm) region. The near-infrared spectral character- ⇠ outflows and jets. istics are inconsistent• withLuminosity problem of the optical spectra, if interpreted in terms ofprotostars stellar Figure 1 summarizes(Kenyon et al. 1990): the current picture of a typical FU Ori object. A young, 7 photospheric emission; the infrared features are best matched with K–M giant-low-mass (T Tauri) star is surrounded by a disk normally accreting at 10 M 1 ⇠ supergiant atmospheresTypical bolometric luminosity (effective temperatures of 2000–3000 K). FGKMyr .<< This slowinfall accretion rate is punctuated by occasional, brief FU Ori outbursts, in ⇠ 4 1 supergiants are rare in any case and are not commonly found in star formationwhich the inner disk erupts, resulting in an accretion rate 10 M yr . The regions; thus, optical and near-infrared spectra serve to identify FU Ori systemsdisk becomes hot enough to radiate most of its energy at⇠ optical wavelengths, uniquely. • FU Orions outbursts(Hartmann & Kenyon 1996) and it dumps as much as 0.01 M onto the central star during the century-long by Princeton University Library on 10/31/12. For personal use only. by Princeton University Library on 10/31/12. For personal use only. Annu. Rev. Astro. Astrophys. 1996.34:207-240. Downloaded from www.annualreviews.org Annu. Rev. Astro. Astrophys. 1996.34:207-240. Downloaded from www.annualreviews.org

Figure 1 Schematic picture of FU Ori objects. FU Ori outbursts are caused by disk accretion 7 1 4 1 2 Figure 3 Optical (B) photometry of outbursts in three FU Ori objects. The FU Ori photometryincreasing from 10 M yr to 10 M yr , adding 10 M to the central T Tauri ⇠ ⇠ ⇠ is taken from Herbig (1977), Kolotilov & Petrov (1985), and Kenyon et al (1988); the V1057star during the event. Mass is fed into the disk by the remanant collapsing protostellar envelope < 5 1 Cyg photometric references are contained in Kenyon & Hartmann (1991); and the V1515 Cygwith an infall rate 10 M yr ; the disk ejects roughly 10% of the accreted material in a ⇠ photometry is taken from Landolt (1975, 1977), Herbig (1977), Gottlieb & Liller (1978), Tsvetkovahigh-velocity wind. (1982), Kolotilov & Petrov (1983), and Kenyon et al (1991b). Models of Disk Evolution

• Rapid maer addion to the disk • Perturbaons from companions.(Bonnell & Basen 1992; Forgan & Rice 2010)

• Inspiral of clumps formed in outer self-gravitang region of the disk.(Vorobyov & Basu 2005, 2010)

(Forgan & Rice 2010) (Vorobyov & Basu 2010)

Models of Disk Evolution

• Thermal Insabilies (TI, Bell & Lin 1994): • Opacity of the disk changes rapidly at T ~ 104 K due to ionizaon AA49CH06-Armitageof hydrogen ARI 5 August 2011. 12:22 • Problem: only works if angular momentum is extremely inefficient(α~10-3 – 10-4).

Opacity κ (T) Local disk vertical structure

10 2 Outburst Teff (K)

1 10 4

) 10 –1 g 2 0 10 H ionization (cm

κ 10 –1 2,000 K

10 –2 103 10 4 10 5 Quiescence T (K)

10 5 10 6 10 7 Σ (0.1 AU) (g cm–2) (Armitage 2011)

Figure 10 A classical disk thermal instability is a possibility if the accretion rate through the inner disk is such that the midplane temperature approaches T 104 K, at which point hydrogen becomes ionized and the opacity κ(T ) rises rapidly. Under these conditions, there can ∼ be multiple solutions (an S-curve) for the local equilibrium vertical structure at fixed r and ", with the stable possibilities being a quiescent state of low accretion rate or an outburst state with a much higher temperature and accretion rate. Provided that these states are sufficiently well separated, it is then possible to set up a global limit cycle in which the inner disk as a whole cycles between quiescent and outburst behavior.

the requirement that the inner disk be spontaneously unstable due to the slow axisymmetric accumulation of matter. Instead, it is proposed that some other process is able to dump matter rapidly into the inner disk, thereby triggering an outburst (Clarke, Lin & Pringle 1990). Potential triggers include perturbations from companions (Bonnell & Bastien 1992, Pfalzner 2008, Forgan & Rice 2010) or the inspiral of clumps formed in an outer self-gravitating region of the disk (Vorobyov & Basu 2005, 2010; Boley et al. 2010; Nayakshin 2010). External perturbations to the inner disk almost certainly do occur, but it is not known whether they are strong and frequent enough to account for FU Orionis events. An alternate class of models retains the self-regulated character of the classical thermal instability (Bell & Lin 1994)

by Princeton University Library on 10/03/12. For personal use only. and solves the timescale problem by associating the instability with MRI physics (and specifically the expected change in transport properties at T 103 K) at larger radii, where the viscous c ∼ timescale is longer. The basic idea is to assume that a dead zone exists at r 1AUandthatthe ∼

Annu. Rev. Astro. Astrophys. 2011.49:195-236. Downloaded from www.annualreviews.org strength of any residual transport in the midplane layer is low (this is an important assumption, because a significant residual viscosity would allow the disk to reach a steady state; Terquem 2008). Such a dead zone is massive and potentially unstable, because if it could once be heated above T 103 K the onset of the MRI would subsequently be able to maintain a high dissipation state c ∼ until much of the mass in the inner disk was accreted. Self-gravity in the dead zone, caused by the slow accumulation of mass there, could provide the heating source necessary to realize the latent instability (Gammie 1999; Armitage, Livio & Pringle 2001). These early ideas of how a dead zone might give rise to outbursts have recently been studied in detail, using both two-dimensional hydrodynamic simulations (Zhu et al. 2009) and one- dimensional time-dependent disk models (Zhu et al. 2010; Zhu, Hartmann & Gammie 2010). Zhu et al. (2010) find that protostellar accretion is liable to FU Orionis–type events for infall rates of 6 1 the order of M˙ infall 10− M year− , with dead zone instabilities being triggered at radii between ∼ #

www.annualreviews.org Dynamics of Protoplanetary Disks 229 • Models of Disk Evolution

• Associaon with MRI: • Dead zone at r ~ 1 AU, where angular momentum transport is low in the midplane, and where materials pile up and heat up;

624 ZHU ET AL. Vol. 701 • MRI triggered at TM ~ 1200K (Zhu et al. 2009), rapid accreon

α= α= 0.1 0.025

Lasts 100yr at -4 2x10 Msun/yr

(Zhu et al. 2009) Figure 1. Disk’s temperature color contours at four stages during the outburst in R–Z coordinate. The black curves are the α contours which also outline different regions in the disk. The vertical-dashed line in Panel (a) shows αQ 0.025: (a) is the stage before the outburst, (b) represents the stage when the MRI is triggered at 2 AU, (c) represents the moment when the TI is triggered at the innermost= region, and (d) is the stage when the TI is fully active within 0.3 AU. (A color version of this figure is available in the online journal.)

We apply outflow boundary conditions (Stone & Norman infalling envelope) causing matter to pile up at the radius where 1992) to all boundaries. In some simulations, the disk is as- GI becomes ineffective yet again, leading to another outburst. sumed to be symmetric about the midplane to reduce computa- Figures 1–5 show the disk temperature and density structure tional cost; for these half-disk simulations, we use a reflecting at four consecutive stages of the outburst for our fiducial model. boundary condition (Stone & Norman 1992)atthemidplane. Figure 1 is shown in R–Z coordinate while Figures 2 and 3 are The inner boundary is set to either 0.06, 0.1, or 0.3 AU to shown in log10 R–θ coordinates (R from 0.06 to 40 AU, θ from examine how different inner radii affect disk evolution. These 1.1 to π/2). Since the GI, MRI and TI are taking place at quite parameters are listed in Table 1. different radii, most of our plots are in log10 R–θ coordinates We follow the evolution of the disk for 105 yr. Typically so that all of them can be seen in one figure, although it is less several outbursts occur during the simulation. After the first intuitive than R–Z coordinates. The black solid curves plotted outburst, the subsequent outbursts are similar and do not depend on top of the color contours delimit regions of differing α. on the initial condition. Thus, we use the disk density and Regions above the top solid curve limit the low-ρ photosphere 3 temperature distribution after the first two outbursts as the new with τ < 10− to α 0. Between the top and next lower = 1 initial condition to study the third outburst for various one- solid curve lies the MRI-active layer with Σ+ < 50 gcm− and dimensional and two-dimensional models. The disk mass for α αM 0.1. Regions below the two solid curves constitute = = this initial condition is 0.5 M . the “dead zone” with only αQ if any. The vertical-dashed lines ∼ " at large radii represent αQ 0.025. 3.2. Outbursts Panel (a) of Figures 1–3 =shows the disk temperature structure In the protostellar phase, the disk is unlikely to transport mass before the outburst. The disk is too cool to sustain any MRI steadily from 100 AU all the way to the star at an accretion rate except in the active layer. The contours in Figures 1–3 show the ∼ 6 4 1 outer disk (R>1 AU) is geometrically thick, with a ratio of matching the mass infall rate 10− to 10− M yr− from the " vertical scale height H to radius H/R 0.2, and massive, while envelope to the outer disk. The outburst sequence we find here ∼ is qualitatively similar to that found by Armitage et al. (2001), the inner disk is thin with a low surface density. The inner disk Gammie (1999), and Book & Hartmann (2005). Mass added to is stable against GI and cannot transport mass effectively. The the outer disk moves inward due to GI, but piles up in the inner mass coming from the GI driven outer disk piles up at several disk as GI becomes less effective at small R.Eventually,thelarge AU. The GI thus gradually becomes stronger at several AU and Σ and energy dissipation leads to enough thermal ionization to the disk heats up. trigger the MRI. Then the inner disk accretes at a much higher The disk midplane temperature at 2 AU eventually reaches the MRI trigger temperature (TM 1200 K) and MRI driven angular mass accretion rate, which resembles FU Orionis-type outbursts. ∼ After the inner disk has drained out and becomes too cold to momentum transport turns on. Panel (b) of Figures 1–3 shows sustain the MRI, the disk returns to the low state. During the low- the stage when the thermal MRI is triggered at the midplane state mass continuously accretes from large radius (or from an around 2 AU. The thermal MRI active region are outlined by

July 24, 1996 14:50 Annual Reviews HARTTEX1 AR12-06 628 ZHU ET AL. Vol. 701

FU ORIONIS PHENOMENON 211

kilometers per second, is typically observed in the Balmer lines, especially in H↵. The Na I resonance lines also show broad blueshifted absorption, some- times in distinct velocity components or “shells.” The emission component in the P Cygni H↵ profile is often absent; when present, this emission extends to much smaller velocities redward than the blueshifted absorption. Infrared spectra of FU Oris show strong CO absorption at 2.2 µm and water vapor bands in the near-infrared ( 1–2 µm) region. The near-infrared spectral character- istics are inconsistent⇠ with the optical spectra, if interpreted in terms of stellar Models of Disk Evolution photospheric emission; the infrared features are best matched with K–M giant- supergiant atmospheres (effective temperatures of 2000–3000 K). FGKM supergiants are rare in any case and are not commonly⇠ found in star formation regions; thus, optical and near-infrared spectra serve to identify FU Ori systems uniquely. 2D simulaon (Zhu et al. 2009): Observed FU Ori (Hartmann & Kenyon 1996): by Princeton University Library on 10/31/12. For personal use only. Annu. Rev. Astro. Astrophys. 1996.34:207-240. Downloaded from www.annualreviews.org

Figure 7. First three panels show the disk’s mass accretion rate for half-disk models with different inner radius (0.3 to 0.06 AU). The lower right panel shows the Figure 3 Optical (B) photometry of outbursts in three FU Ori objects. The FU Ori photometry disk’s mass accretion rate for a full disk with the inner radius 0.06 AU. The red lines are the mass accretion rates after being smoothed.is taken from Herbig (1977), Kolotilov & Petrov (1985), and Kenyon et al (1988); the V1057 Cyg photometric references are contained in Kenyon & Hartmann (1991); and the V1515 Cyg (A color version of this figure is available in the online journal.) photometry is taken from Landolt (1975, 1977), Herbig (1977), Gottlieb & Liller (1978), Tsvetkova (1982), Kolotilov & Petrov (1983), and Kenyon et al (1991b). 4. VERTICAL STRUCTURE AND CONVECTION ρ, T,andp are functions of Z.Equation(29)canbeintegrated toward the midplane, given Teff T (τ 2/3) as the boundary Although our two-dimensional simulations exhibit complex = = condition. The integration is halted at zf ,wherethetotalviscous time-dependent behavior, the vertical structure of the simulated 4 heating is equal to σ T .Ingeneral,zf 0, so we change disk can be well fitted by a simple analytical calculation plus eff #= the initial conditions and repeat the integration until zf 0, some basic assumptions. in which case we have the self-consistent vertical structure= Assuming local energy dissipation and with a given effective solution. This procedure is identical to that described in Zhu temperature and viscosity parameter α,thediskstructurecanbe et al. (2009); we will call these local vertical multizone (LVMZ) defined by the following equations. Assuming all the viscosity- models. generated energy radiates vertically, we have Figure 9 shows the vertical structure at 0.1 AU from the 9 2 d two-dimensional simulation and that from the LVMZ with νV ρΩ (F ) , (29) 4 = dz the same central temperature. Although the two-dimensional model cannot resolve the region where T 104 K, the two- where the vertical flux F is dimensional model otherwise agrees with∼ the LVMZ very 4 well, which means that at 0.1 AU the local treatment is a 4σ dTτ F , (30) good approximation even during the TI high state, and that = 3 dτ radial energy transport is not important. We find that, in the viscosity is general, the simulation follows the “S” curve calculated by the 2 νV αc /Ω , (31) LVMZ very well during the whole TI activation process (see = s and Appendix B). dτ ρκ(T,p)dz. (32) In the lower left panel of Figure 9, the pressure plateau around = − τ 10 is caused by the rapid increase in opacity with increasing With hydrostatic equilibrium, temperature∼ near 104 KasHionizes(seetheappendixin dp GM ρz Zhu et al. 2009). This∼ means that the optical depth can change ∗ , (33) dz = − R3 significantly, while the pressure remains nearly the same. The large vertical temperature gradient causes this region and the equation of state, to become convectively unstable. As shown in the lower right kB panel, the dense gas is on top of the thin gas at τ < 100, p ρT, (34) which is clearly unstable against convection or Rayleigh-Taylor = µ instability. Although this convective region is only at the surface 1146 ZHU, HARTMANN, & GAMMIE Vol. 713 Long-term evolution Table 1 1D2Z Models at 0.3 and 1 Myr

a b c Cloud ΩΣA αM RD 0.3/1 Myr Min Mo Mt 2 (g cm− )(AU)(M )(M )(M ) " " " 2 10 14 100 0.01 40/13 0.29/0.19 0.18/0.14 0.47/0.33 × − • Layered accreon with 2 10 14 10 0.01infall 127/66 (Zhu, Hartmann & 0.42/0.33 0.05/0.09Gammie 0.48/0.42 , 2010): × − 2 10 14 00.01 0.47/0.41 0.47/0.41 × − ··· ··· 1 10 14 100 0.01 24/7.6 0.17/0.1 0.07/0.06 0.24/0.16 × − 1148 ZHU, HARTMANN, & GAMMIE Vol. 713 1 10 14 10 0.01 47/34 0.32/0.23 0.008/0.022 0.32/0.26 × − 3 10 15 100 0.01 4.6/2.4 0.05/0.04 0.006/0.007 0.057/0.049 × − 3 10 15 10 0.01 5.5/4.6 0.06/0.06 0.0004/0.001 0.063/0.063 × − 3 10 15 00.01 0.064/0.064 0.064/0.064 × − ··· ··· Notes. a The disk mass within RD. b The disk mass beyond RD. c The total disk mass. The disk structure during the outburst stage is illustrated in Figure 2,shownatthetimewhenthecentrifugalradius(Rc) is 10 AU (labeled by the triangle). Although the infall ends at Rc,thediskextendsfarbeyondRc due to the outward mass transfer by the active layer. With the continuous infall, the disk is marginally gravitationally stable within Rc,andtheenergy dissipation due to the GI has significantly heated the disk so that the outburst will be triggered at 2 AU (the upper right panel shows that Tc is approaching 1500 K). At the time when infall ends, the region between 1AUand 20 AU is still marginally gravitationally unstable, although∼ the∼ disk within 1 AU has been depleted by the previous outburst (Figure 3). The innermost disk (R < 0.1 AU) is purely MRI active due Figure 5. Unstable regions in the R–M plane for an accretion disk around the to the high temperature there produced by viscous heating and ˙ 1 M central star. The shaded region in the upper left shows the region subject Figureirradiation. 4. Disk mass accretion Due to rates the and efficient the mass of mass the central transport star and by disk the with MRI, to classical$ thermal instability. The shaded region in the middle shows where time forthis different inner core region rotations. can The be upper depleted two panels rapidly, show butthe simulation is limited with by the the central temperature of steady GI models exceeds an assumed MRI trigger 14 1 15 1 Solves the luminosity problem! Ωc accretion2 10− rad from s− ,whiletheloweroneswith the outer active layer.Ω Eventually,c 3 10− arad balance s− . is temperature of 1400 K, thus subject to MRI–GI instability. The lines labeled The= solid× curves in the right panels show the mass of the= central× star with time achieved, and the inner disk behaves like a constant-α disk RM and RQ are the maximum MRI and minimum GI steady accretion radius while the dotted curves show the mass of the disk (dotted curve in the lower (the boundaries of the shaded region, detailed in Zhu et al. 2009b). The shaded right panelwith is mass too close accretion to 0 and hardrate to equal be seen). to the mass accretion rate of the region on the right shows where the GI disk will fragment (see Appendix B for innermost active layer (e.g., Gammie 1996). details). The fragmentation limit agrees with Cossins et al. (2010). describedIn in the the standard previous layered section (see accretion also Paper model, I). In the the disk case mass of ouraccretion fiducial rate model, is set R ascmax the active12 AU;layer this accretion places rate the at disk the dead Figure 1. Disk mass accretion rate (upper panel) and the mass of the central star zone inner radius where the∼ MRI becomes thermally activated 14 1 well beyond the region where the mismatch between GI and and disk (lower panel) with time for our fiducial model (Ωc 10 rad s ). = − − MRI(e.g., occurs, Gammie resulting1996 in outbursts.). We use theThe modification fiducial model of accretes the accretion The upper curve in the lower panel represents the central star mass while the 4 quasi-steadilyrate including for the irradiation first 8 derived10 yr; at by this Hartmann point R etc al.1AU. (2006;see lower one represents the disk mass. × ∼ For thealso rapidly Appendix rotatingA), case, Rcmax 50 AU; very little mass infalls to radii within the MRI-steady∼ region, and most of the can become hot enough to sustain the MRI thermally. As infall 1/2 mass then gets accreted duringR outbursts.M − Becauseα the mass proceeds, thermal activation of the MRI cannot be sustained, M 6.9 10 9 c M transport is dominated− by the GI at large radii, a relatively2 and GI takes over as the main mechanism of mass transfer. ˙ = × 0.2AU 1M 10− ! "! " " However, the GI is inefficient at these radii and so matter massive disk results. Note that the outer1/4 regions1#/4 of the$ disk reside near the limitΣ whereA gravitationalL fragmentationf rather1 piles up. Eventually, the disk becomes opaque enough that the ∗ M yr , (20) 2 − than simple transport× 100 g might cm− occurL (Appendix0.1B;Section" 4.4). radiative trapping of the energy dissipated by the GI raises ! "! " " ! " the central disk temperature sufficiently to thermally activate the Thus, the range of rotations chosen here (given our assumed where f is as defined in Equation (4). At the sublimation radius, MRI, and an outburst of accretion occurs. As shown in Paper I, model for the initial cloud core) approximately span the ex- Figure 6. Disk surface density with radii for different rotation cores as shown f may be significantly larger than 0.1. With our numerical inner the result is cycles of outbursts transferring material inward; the tremes of disk evolution; at smaller ΩC ,theaccretionisalmost in Figure 4 at 0.3 (solid curves) and 1 Myr (dashed curves). boundary 0.2 AU, the simulation shows the disk has a mass GI moves material in slowly until enough mass is accumulated spherical and occurs quasi-steadily, while for larger values of accretion rate of 10 8M yr 1 in the quiescent state, which with enough viscous dissipation to trigger the thermal MRI Ωc gravitational fragmentation− is likely− to occur, considerably agrees with the above estimate." it is, most of the disk mass can be at small radii, rather than outbursts. Finally, after infall stops, the disk is not massive modifying disk evolution. As discussed in Paper I, we find unstable behavior at the the large radii probed by current millimeter and submillimeter enough to sustain GI, and it accretes only from the MRI active The disk mass and surface density distribution are also dead zone inner radius because of the rapid change in α . observations (in comparison with constant-α models). layer. Since the outer disk has a surface density smaller than the significantly affected by the initial core rotation. With Ωc M This14 accounts1 for the modest but significant variability= seen maximum surface density that can be ionized by cosmic and/or 2 10− rad s− (left panel in Figure 6), although the dead zone× during shrinks the from quasi-steady 35 AU at the state. end6 This of the instability infall phase cannot (0.3 beMyr) reliably X-rays, it becomes fully viscous due to the MRI; this leads to to 10 AU at the T Tauri phase (1 Myr), the disk is very massive continued expansion of the outer disk, with a slow draining of 3.3. Different Disk Configurations (0.3 M6 Although)andstillgravitationallyunstablewithin10AUduring we avoided this instability at the T Tauri phase by choosing the material into the inner disk and central star. During the T Tauri inner$ numerical boundary at 0.2 AU, this instability is present in our 15 the T Tauri phase. However, for smaller rotation Ωc 3 10− The ionization structure of disk surface layers depends on phase, the layered accretion itself may also trigger outbursts, 5 rad ssimulation1 (right in panel), the first 10 theyears dead (short zone variations is significantly in Figure=1)whenthedead× smaller the dust size spectrum and the flux of ionizing radiation and depending on the detailed layered structure (see below). zone− inner radius is larger than 0.2 AU. (2 AU) and has less mass (0.05 M ). cosmic rays, and is therefore poorly constrained. Angular $ When the initial core rotates slowly and Rcmax is small, momentum transport in magnetically decoupled disks is also the dead zone size is determined by Rcmax.However,ifthe poorly understood, although recent work by Lesur & Papaloizou initial core rotates fast and Rcmax is large, the dead zone size (2009), following Petersen et al. (2007), strongly suggests that is constrained by the radius where the active layer becomes anonlinearinstabilitydrivenbybaroclinicityandradiative gravitationally unstable, as discussed further in Appendices. diffusion may give rise to hydrodynamic turbulence and angular One important point is that even though the dead zone is momentum transport. Therefore, we have investigated disks small <50 AU, it can contain a significant fraction (or most) of with a reduced active layer (Section 3.3.1)andaresidualdead the disk mass. Figure 7 shows that, no matter what the infall zone viscosity (Section 3.3.2). Conclusions

• Gravitaonal instability could be common and so important to protoplanetary disks • Locality of gravitaonal instability is plausible in low-mass disks • Gravitaonal instability results in two outcomes depending on the cooling me: • Gravitoturbulence (steady angular momentum transport) • Fragmentaon • Irradiaon and background viscosity constrain regimes where GI works: • Irradiaon stabilizes disks against fragmentaon at low mass accreon rates • Background viscosity could dominate angular momentum transport and stabilize the disk • Heang up due to GI could trigger MRI and produce bursts of accreon, which explains FU Ori outbursts. References

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