A theoretical perspective on the formation and fragmentation of protostellar discs
aa r X i v : . [ a s t r o - ph . S R ] M a y Publications of the Astronomical Society of Australia (PASA)c (cid:13)
Astronomical Society of Australia 2018; published by Cambridge University Press.doi: 10.1017/pas.2018.xxx.
A theoretical perspective on the formation andfragmentation of protostellar discs
A Whitworth ∗ and O Lomax School of Physics & Astronomy, Cardiff University, Cardiff CF24 3AA, Wales, UK
Abstract
We discuss the factors influencing the formation and gravitational fragmentation of protostellar discs. Westart with a review of how observations of prestellar cores can be analysed statistically to yield plausibleinitial conditions for simulations of their subsequent collapse. Simulations based on these initial conditionsshow that, despite the low levels of turbulence in prestellar cores, they deliver primary protostars and asso-ciated discs which are routinely subject to stochastic impulsive perturbations; consequently misalignmentof the spins and orbits of protostars are common. Also, the simulations produce protostars that collectivelyhave a mass function and binary statistics matching those observed in nearby star formation regions, butonly if a significant fraction of the turbulent energy in the core is solenoidal, and accretion onto the primaryprotostar is episodic with a duty cycle > ∼ there is a sweet spot for disc fragmentation at radii 70 AU < ∼ R < ∼
100 AU and temperatures10 K < ∼ T < ∼
20 K, and this might explain the Brown Dwarf Desert.
Keywords: low-mass star formation – disc fragmentation – prestellar cores – radiative feedback – solenoidalturbulence – the Brown Dwarf Desert
Discs play a critical role in the formation of stars andplanets and moons. They act as repositories for the an-gular momentum that must be lost by the matter con-densing into a star or planet or moon (e.g. Zhu et al.2009; Li et al. 2014), but they also constitute reservoirsof material from which further stars and/or planetsand/or moons can form (e.g. Whitworth & Stamatellos2006; Stamatellos et al. 2007a; Chabrier et al. 2014).This paper is concerned with the formation of discsaround newly-formed (primary) stars , and the gravi-tational fragmentation of such discs to produce addi-tional (secondary) stars . These stars can have massesequal to those of planets , but here they are termed stars on the grounds that they form early in the life-time of the disc, by gravitational instability (as op-posed to core accretion), on a relatively short dynamicaltimescale, and – in the first instance – with an approxi-mately uniform elemental composition (presumably re-flecting the composition prevailing in the local inter-stellar medium). To put the physics of disc formationand fragmentation in context, we start by discussing the ∗ email: [email protected] prestellar cores out of which stars and their attendantdiscs are presumed to condense (e.g. di Francesco et al.2007; Ward-Thompson et al. 2007); it is core propertiesthat determine the initial and boundary conditions fordisc formation.In Section 2 we discuss procedures for extractingstatistical information about the intrinsic struc-tures of cores from observations of an ensem-ble of cores. In Section 3 we present the results ofsimulations that follow the evolution of an ensemble ofcores having properties constrained, as closely as pos-sible, by observations of cores in the nearby Ophiuchusstar formation region; we discuss the results, both fromthe viewpoint of the statistics of the stars and multiplesystems formed, and from the viewpoint of the highlychaotic environment influencing the subsequent forma-tion and evolution of the associated circumstellar discs.In Section 4 we discuss the statistical relationship be-tween core properties and the stars that they spawn, inorder to emphasise further the dynamical and disorgan-ised environments in which circumstellar discs are likelyto form and evolve. In Section 5 we present the basictheory of the gravitational fragmentation of a disc. InSections 6 we calculate how fast a proto-fragment in a1
A Whitworth et al. circumstellar disc must lose thermal energy in order tocondense out of the disc, and in Section 7 we convert thisrequirement into an explanation for the brown dwarfdesert. In Section 8 we explore how cool a circumstel-lar disc has to be for proto-fragments to condense outby gravitational instability, and conclude that discs areonly sufficiently cool if radiative feedback from the cen-tral primary star is episodic. In Section 9 we calculatehow fast a proto-fragment in a circumstellar disc mustlose angular momentum in order to condense out of thedisc, and explain why the critical dynamical timescalefor condensation of a proto-fragment in a disc is of or-der the orbital period of the proto-fragment. Our mainconclusions are summarised in Section 10.
In well observed, nearby star formation regions likeOphiuchus, one can identify – rather unambiguously– the small fraction of material destined to form thenext generation of stars. It is concentrated in dense,low-mass, self-gravitating prestellar cores, which appearto be relatively isolated from one another (Motte et al.1998; Andr´e et al. 2007). Thus, although an individualcore may continue to grow by accreting from the sur-rounding lower-density gas, it is likely to be finishedwith forming stars before it interacts significantly withanother neighbouring prestellar core. Therefore it is in-teresting to explore the intrinsic properties of prestel-lar cores; to use them as the basis for detailed numeri-cal simulations of core collapse and fragmentation (seeSection 3); and to analyse how the properties of coresrelate, statistically, to the properties of the stars theyspawn (see Section 4).On the basis of far-infrared and submillimetre contin-uum intensity maps, one can estimate the dust column-density through, and mean dust temperature along, dif-ferent lines of sight through a prestellar core. Knowingthe distance to the core, one then obtains the core mass,and its projected physical dimensions (i.e. projectedarea and aspect ratio). On the basis of molecular-lineobservations, one can also evaluate both the mean gas-kinetic temperature and the non-thermal component ofthe radial velocity dispersion. Inevitably, one does notknow the distributions of density, temperature and ra-dial velocity dispersion along the line of sight, and onehas no information on the velocity dispersion perpen-dicular to the line of sight. Therefore, it is impossible toconstruct a detailed three-dimensional model of a par-ticular core without making a large number of extremely ad hoc assumptions.Since the processes forming cores are chaotic, analternative procedure is to consider a large ensem-ble of observed cores. With a sufficiently large num- ber of observed cores, and assuming they are ran-domly oriented, one can straightforwardly deduce thedistributions of mass, size, aspect ratio, temperature,and three-dimensional velocity dispersion, plus anysignificant correlations between these global param-eters (e.g. Lomax et al. 2013). A synthetic core (orsuite of cores), representing this ensemble, can thenbe generated by picking values from these distribu-tions and correlations. The large-scale density profileof the synthetic core can be approximated with a criti-cal Bonnor-Ebert profile, or a simple Plummer-like form(Whitworth & Ward-Thompson 2001) ρ ( r ) = ρ O n r/r O ) o − . (1)The only remaining issue is to define the internal sub-structure of the synthetic core. This is critical, since itdetermines how the core subsequently fragments.The procedure we have adopted is to assume thatthe internal non-thermal velocity field is turbulent, withpower spectrum P k ∝ k − n . We have adopted n = 4, butthis is not very critical, as long as 3 < ∼ n < ∼
5, sincemost of the power is then concentrated at short wave-numbers. In addition, we must specify the ratio ofsolenoidal to compressive turbulent energy, and this is critical, since it strongly influences the properties of thestars that the core spawns, as we discuss in Section 3.We also adjust the phases of the shortest wave-numbersolenoidal and compressive modes, i.e. those on the scaleof the core, so that they are centred on the core. Inother words, these modes are obliged to represent – re-spectively – ordered rotation and ordered radial excur-sions of the core; all other modes have random phases(e.g. Lomax et al. 2015a). Finally we introduce fractaldensity perturbations, correlated with the velocity per-turbations. The fractal dimension is D ≃ . PASA (2018)doi:10.1017/pas.2018.xxx isc formation and fragmentation Figure 1.
Grey-scale column-density image of a synthetic core
We have simulated the collapse and fragmentation of anensemble of 100 prestellar cores constructed in this way,i.e. 100 prestellar cores that, if placed at the distance ofOphiuchus and observed in dust continuum and molec-ular line radiation, would be indistinguishable statisti-cally from the actual cores in Ophiuchus (Lomax et al.2015b). Whether they are a good representation of thecores in Ophiuchus depends on whether the assump-tions we have made in the preceding section are valid.The assumptions that turn out to be critical are (i) thatthe cores are randomly oriented (Lomax et al. 2013),(ii) the treatment of radiative feedback (Lomax et al.2014, 2015b, see below), and (iii) the ratio of solenoidalto compressive turbulence (Lomax et al. 2015a). Ourassumptions about the density profile, the fractal di-mension, and the density- and velocity-scaling expo-nents appear to be much less critical, in the sense thatthey do not greatly influence the properties of the starsthat form in the simulations. We note that the observednon-thermal kinetic energies in the Ophiuchus cores –here interpreted as isotropic turbulence – are typicallymuch smaller than their gravitational potential ener-gies, specifically α NONTHERM = E NONTHERM | E GRAV | < ∼ . . (2)The associated non-thermal velocities are typicallytrans-sonic. The simulations use the seren ∇ h -SPH code(Hubber et al. 2011), with η = 1 . ∼
56 neighbours). Gravitational forces arecomputed using a tree, and the Morris & Monaghan(1997) formulation of time-dependent artificial viscosityis invoked. In all simulations, an SPH particle has mass m SPH = 10 − M ⊙ , so the opacity limit ( ∼ × − M ⊙ )is resolved with ∼
300 particles. Gravitationally boundregions with density higher than ρ SINK = 10 − g cm − are replaced with sink particles (Hubber et al. 2013).Sink particles have radius r SINK ≃ . ρ SINK . The equation of state and the en-ergy equation are treated with the algorithm describedin Stamatellos et al. (2007b), which captures approxi-mately the effects of radiation transport.Radiative feedback from protostars is also included,using two distinct prescriptions. In the first prescrip-tion, labelled continuous feedback , accretion onto a pro-tostar is presumed to occur at the same rate at whichmatter is assimilated by the associated sink, so the stel-lar luminosity is L ⋆ ∼ GM SINK ˙ M SINK R ⊙ . (3)Here M SINK is the mass of the sink, ˙ M SINK is therate at which the sink is growing due to accre-tion, and 3 R ⊙ is the approximate radius of a low-mass protostar (Palla & Stahler 1993). In the sec-ond prescription, labelled episodic feedback , we usethe phenomenological “sub-sink” accretion model de-scribed in Stamatellos et al. (2011) – and also used inStamatellos et al. (2012) and Lomax et al. (2014). Inthis phenomenological model, which is based on the the-ory developed by Zhu et al. (2009), highly luminous,short-lived accretion bursts are separated by long in-tervals of low accretion and low luminosity. During thelong intervals, matter collects in the inner accretion discinside a sink, but the rate of accretion onto the centralprimary protostar is low – basically because there is noeffective mechanism for redistributing angular momen-tum: the matter in the inner accretion disc is too hot forgravitational structures to develop and exert torques,and too cool to be thermally ionised and couple to themagnetic field. Consequently the luminosity of the cen-tral primary protostar is also low, and the disc coolsdown and may become cool enough to fragment. Oncesufficient matter has collected in the inner accretiondisc, it becomes hot enough for thermal ionisation tobe significant; the Magneto-Rotational Instability thencuts in, delivering efficient outward angular momentumtransport; matter is dumped onto the protostar givingan accretion outburst and an associated peak in the lu-minosity, which heats the disc and stabilises it againstfragmentation. PASA (2018)doi:10.1017/pas.2018.xxx
A Whitworth et al.
Each of the 100 cores in the ensemble has beenevolved numerically with both continuous and episodicradiative feedback, and with different proportions ofsolenoidal and compressive turbulence (Lomax et al.2014, 2015b,a). The simulations reproduce very well theobserved mass function of young stars, and their multi-plicity properties, including the incidence of high-orderhierarchical multiples, but only if at least ∼
30% of theturbulent kinetic energy is in solenoidal modes, and onlyif the radiative feedback from protostars is episodic.Some solenoidal turbulent energy is required, to pro-mote the formation of massive extended circumstellardiscs around the first protostars to form, and episodicfeedback is required to allow these discs to periodicallybecome sufficiently cool to fragment gravitationally andspawn additional protostars, in particular brown-dwarfprotostars.If there is no solenoidal turbulence, the circumstellardiscs around the first protostars to form (which by theend of the simulation are usually the most massive pro-tostars) are compact and unable to fragment. Low-massprotostars still form, by fragmentation of the filamentsthat feed material towards these more massive proto-stars, but very few of these low-mass protostars end upbelow the hydrogen-burning limit, and so in the endthere are far too few brown-dwarf protostars to matchobservations . As the amount of solenoidal turbulenceincreases, the mean stellar mass decreases, and the frac-tion of brown dwarfs increases.Even if there is solenoidal turbulence (and so ex-tended massive circumstellar discs do form around thefirst protostars), as long as there is continuous feed-back, the discs are warm and disc fragmentation is ef-fectively suppressed. Consequently, the first protostarstend to accrete all the mass in their circumstellar discsand become very massive. Again, the upshot is that themean stellar mass is high and there are very few browndwarfs. Moreover, the few brown dwarfs that do formare very close to the hydrogen-burning limit, and thereare none close to the deuterium-burning limit. Episodicaccretion allows circumstellar discs to cool down, and– provided that the duty-cycle for episodic accretion issufficiently long, i.e comparable with or greater than thedynamical timescale of the outer parts of the disc, say > ∼ ,
000 yrs – the discs are cool enough for long enoughto fragment, thereby spawning brown dwarfs and low-mass hydrogen-burning stars. Recent estimates of theduty-cycle for episodic accretion indicate that it mayindeed be this long (e.g. Scholz et al. 2013).Disc fragmentation is also conducive to the formationof hierarchical (and therefore stable) higher-order mul-tiples, and our simulations with solenoidal turbulenceand episodic feedback deliver distributions of binaries The fraction of stars below the hydrogen-burning limit is esti-mated to be 0 . . and higher-order multiples that are in good agreementwith observation, including systems with up to six com-ponents. Figure 2 illustrates an hierarchical sextuplesystem formed in one of our simulations. If there istoo little solenoidal turbulence, and/or more continu-ous radiative feedback, far too few multiple systems areformed.With continuous radiative feedback, an average corespawns between 1 and 2 protostars; with episodic feed-back this increases to between 4 and 5 protostars. Thisis a controversial result, in the sense that it has usuallybeen assumed that a core typically forms one or two pro-tostars, with the majority of the core mass then beingdispersed. However, there are good statistical reasonsto suggest that a typical core may well spawn between4 and 5 protostars (see Section 4 and Holman et al.2013). There are also recent observations implying thatthe number of protostars in a core has been underes-timated. Additionally, the fact that young populationsinclude significant numbers of high-order multiples sug-gests that cores must, at least occasionally, spawn manymore than two protostars (e.g. Kraus et al. 2011).A key factor promoting the formation and grav-itational fragmentation of discs in simulations withsolenoidal turbulence is that the infall onto a circum-stellar disc is seldom smooth. The discs form in a verydynamic environment, in which new material is oftendelivered to a disc quite impulsively, i.e. at an abruptlyincreased rate, and/or along a filament whose directionand interception point bear little relation to the struc-ture and orientation of the existing disc. Discs are alsooccasionally perturbed tidally by close passages of otherprotostars that have formed in the disc, an occurrencethat is more frequent if cores spawn between 4 and 5protostars. As well as promoting disc fragmentation, theseperturbations represent a stochastic and im-pulsive input of angular momentum. This hasthe consequence that discs can be quite poorlyaligned with the spins of their central stars, asshown by Bate et al. (2010), and that the spinsof stars in multiple systems are often ratherpoorly aligned with one another and with theirmutual orbits (e.g. Lomax et al. 2015b).4 THE STATISTICS OF COREFRAGMENTATION
It has frequently been noted that the prestellar coremass function (hereafter CMF) is similar in shape tothe stellar initial mass function (hereafter IMF) – ba-sically log-normal with a power-law tail at high masses– but shifted to higher masses by a factor of 4 ± PASA (2018)doi:10.1017/pas.2018.xxx isc formation and fragmentation Figure 2.
Montage of false-colour column-density maps of thecentral regions of a collapsed core which has formed an hierarchi-cal sextuplet. The protostars are marked with black dots from the CMF to the IMF must be statistically self-similar. In other words, for a core of any mass M CORE ,the mean fraction of the core’s mass that ends up inprotostars ( η O ), the mean number of protostars formedfrom the core ( N O ), the distribution of relative stellarmasses (which we here characterise as a log-normal withstandard deviation σ O ), and the predisposition of theseprotostars to end up in binaries or higher-order multi-ples (which we here characterise with an exponent α O such that the relative probability for a protostar of mass M ⋆ to end up in a binary is proportional to M α O ⋆ ) areall universal. For example, the probability that a corewith mass 0 . ⊙ spawns two protostars with massesbetween 0 . ⊙ and 0 . ⊙ is the same as the proba-bility that a core with mass 5 M ⊙ spawns two protostarswith masses between 1 M ⊙ and 2 M ⊙ .Holman et al. (2013) have used a Monte CarloMarkov Chain analysis to determine which parameter Thus large η O means that a large fraction of the core massends up in protostars, large N O means that a core typicallyspawns many protostars, a large σ O means that the protostarsspawned by a core have a wide range of masses, and a large α O means that there is a strong tendency for the two most mas-sive protostars in a core to be the ones that end up in a binary.The results are not changed significantly if the distribution ofrelative masses is not lognormal but is characterised by a pa-rameter σ O that measures the width of the distribution, or ifthe relative probability of ending up in a binary depends onmass in a different way but involves a parameter α O such thatlarge α O favours higher-mass protostars, i.e. dynamical biasing(McDonald & Clarke 1993, 1995). values deliver an acceptable mapping between the ob-served distribution of core mass and the observed distri-butions of stellar mass, binary frequency (as a functionof primary mass) and binary mass ratio (as a functionof primary mass). This analysis predicts – very ‘force-fully’ – that the preferred value of η O is η O ≃ . ± . η O > N O is N O ≃ . ± . N O /η O ≃ . ± . N O ∼ N O ∼ σ O is σ O ≃ . ± .
03, so theinterquartile range for the protostars spawned by a sin-gle core spans a factor of order 2.Finally, the preferred value of α O is α O ≃ . ± . N -body simulations (McDonald & Clarke 1995) wouldsuggest. This implies that the protostars in a core donot form first and then pair up. Rather, the identityof a future companion protostar is determined duringformation. For example, our simulations of Ophiuchus-like cores suggest that companions to the more massiveprotostars often form by disc fragmentation.These aspects of protostar formation, and those dis-cussed in Section 3, all indicate that the notion of an iso-lated primary protostar with a symmetric infalling en-velope and a symmetric circumstellar disc may be mis-leading. Circumstellar (and circumbinary) discs proba-bly form in chaotic environments, where impulsive per-turbations due to anisotropic lumpy accretion streamsand nearby passing protostars are the norm. Consider an equilibrium circumstellar disc with surfacedensity Σ( R ), isothermal sound-speed c S ( R ), angularspeed Ω( R ) and epicyclic frequency κ ( R ). As long asthe overall mass of the disc is less than the mass of thecentral protostar, we can put κ ( R ) → Ω( R ) ∼ (cid:18) GM ⋆ R (cid:19) / , (4) PASA (2018)doi:10.1017/pas.2018.xxx
A Whitworth et al. M u l t i p li c i t y F r a c t i on , f Mass [M ¯ ] Figure 3.
The dashed line gives the binary frequency as a function of primary mass for the best-fitting values of η O ≃ . , N O ≃ . , σ O ≃ . and α O ≃ . identified by the Monte Carlo Markov Chain analysis . The boxes represent the observational estimatesof multiplicity frequency in different primary-mass intervals, due to Close et al. (2003); Basri & Reiners (2006); Fischer & Marcy (1992);Duquennoy & Mayor (1991); Preibisch et al. (1999); Mason et al. (1998). The error bars represent the observational uncertainties. (and henceforth the variable κ will be used exclusivelyfor opacity). In the sequel we shall not normally includethe R -dependence of Σ( R ), c S ( R ) and Ω( R ) explicitly.Now suppose that a small circular patch of the disc atradius R , having radius r ≪ R (i.e. a proto-fragment),becomes unstable and starts to condense out. The radialexcursions of this proto-fragment are dictated by thebalance between gravitational, pressure and centrifugalaccelerations,¨ r ∼ − πG Σ + c S r + Ω r . (5)Condensation requires ¨ r <
0, and so the range of unsta-ble proto-fragment radii, ( r MIN , r
MAX ), is given by r MIN , MAX ∼ πG Σ ∓ (cid:8) ( πG Σ) − ( c S Ω) (cid:9) / Ω . (6)There are real roots, and therefore there are only un-stable proto-fragments at radius R , if πG Σ > c S Ω, i.e.Σ > Σ MIN ∼ c S Ω πG . (7)This is the Toomre criterion for gravitational instabilityof an equilibrium disc (Toomre 1964).We note that proto-fragments with r < r MIN are un-able to condense out because their pressure supportis stronger than their self-gravity. In contrast, proto-fragments with r > r
MAX are unable to condense out because their rotational support is stronger than theirself-gravity.The timescale on which a proto-fragment condensesout is t COND ∼ (cid:18) r ¨ r (cid:19) / ∼ (cid:26) πG Σ r − c S r − Ω (cid:27) − / , (8)and the fastest condensing fragment has r FAST ∼ c S πG Σ ,t FAST ∼ (cid:26) ( πG Σ) c S − Ω (cid:27) − / (9) ∼ t ORB / π { (Σ / Σ MIN ) − } / , (10)where t ORB = 2 π Ω (11)is the orbital period at radius R . Thus a proto-fragmentcan condense out in one orbital period ifΣΣ MIN > ∼ π ) ∼ .
025 ; (12)
PASA (2018)doi:10.1017/pas.2018.xxx isc formation and fragmentation
In what follows, we shall substitute for the temperaturein terms of the isothermal sound-speed, c S , viz. T −→ ¯ m c S k B , (13)where ¯ m is the mean gas-particle mass (which for molec-ular gas is ¯ m ∼ × − g) and k B is Boltzmann’s con-stant. The flux from a blackbody is then F BB = σ B T = 2 π ¯ m c S c L h , (14)where σ B is the Stefan-Boltzmann constant, and c L isthe speed of light.A proto-fragment in a disc can only condense out if itcan cool radiatively, on a dynamical timescale. Other-wise it will undergo an adiabatic bounce, and be shearedapart or merge with another proto-fragment. We shallassume that cooling is dominated by thermal emissionfrom dust.The frequency-averaged mass-opacity for a standardmixture of interstellar gas and dust (at the long wave-lengths of concern here) can by approximated by¯ κ R ∼ κ T T ∼ κ c c S , (15)with κ T ≃ − cm g − K − and κ c = κ T ( ¯ m/k B ) ≃ . × − s cm − g − . The uncertainty on these co-efficients is the same as on, say, the mass-opacity thatis routinely used to estimate column-densities from850 µ m intensities, i.e. a factor of order two. For sim-plicity, we do not distinguish between the Rosseland-and Planck-mean opacities.At the temperatures with which we are concernedhere, T < ∼
100 K, the rotational and vibrational degreesof freedom of a hydrogen molecule are not significantlyexcited, and therefore the gas is effectively monatomic,with specific internal energy u = 3 c S /
2. If we assumethat the dynamical timescale for a condensing proto-fragment in a disc is the same as its orbital period,then the requirement that a proto-fragment be able tocool on a dynamical timescale becomes3Σ c S / t ORB < (cid:26) F BB ¯ τ , ¯ τ < , F BB ¯ τ − , ¯ τ > , (16)¯ τ = Σ¯ κ , (17)which is equivalent to the Gammie Criterion (Gammie2001). The Gammie Criterion has been confirmed numerically by various groups, usually using aparametrised cooling law of the form d ln[ u ] /dt = − π/βt ORB , e.g. Rice et al. (2003), although there isstill some debate as to the validity of the Gam-mie Criterion (see, for example, Meru & Bate 2011;Rice et al. 2014). The Gammie Criterion is basicallythe same as the Opacity Limit (e.g. Rees 1976;Low & Lynden-Bell 1976), but formulated for the caseof two-dimensional fragmentation of a disc (Toomreinstability), rather than three-dimensional fragmenta-tion of an extended 3D medium (Jeans instability). Wewill return to the reason why the dynamical timescaleof a proto-fragment might be the same as its orbitaltimescale in Section 9.
If we adopt the first option in Eqn. (16), i.e. that theproto-fragment is optically thin to its own cooling ra-diation, and substitute from Eqns. (11), (14), (15) and(17), we obtain 45 c L h Ω8 π ¯ m κ c c S < ∼ . (18)Using Eqn, (4) to replace Ω, this reduces to R > ∼ R OPT . THINMIN ∼ (cid:18) c L ¯ mGM ⋆ c S κ c (cid:19) / h c L π ¯ m c S ∼
70 AU (cid:18) M ⋆ M ⊙ (cid:19) / (cid:16) c S . − (cid:17) − / , (19)which might seem to imply that proto-fragments cancondense out at any radius where the disc is opticallythin, provided the gas is hot enough (sufficiently large c S ).However, if the optically thin limit is to obtain, thesurface-density must not be too large,Σ < ∼ Σ τ =1 ∼ κ c c S ∼
10 g cm − (cid:16) c S . − (cid:17) − . (20)Consequently the initial radii and masses of proto-fragments must satisfy r FRAG > ∼ r τ =1 ∼ κ c c S πG ∼
14 AU (cid:16) c S . − (cid:17) , (21) m FRAG > ∼ m τ =1 ∼ κ c c S πG ∼ .
001 M ⊙ (cid:16) c S . − (cid:17) . (22) PASA (2018)doi:10.1017/pas.2018.xxx
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A proto-fragment with radius and mass obeying theseconstraints (Eqns. 21 and 22) marginally , can only avoidbeing disrupted by the tidal field of the central protostarif it is formed at R > ∼ R TIDAL τ =1 ∼ (cid:18) κ c M ⋆ c S (4 π ) G (cid:19) / ∼
70 AU (cid:18) M ⋆ M ⊙ (cid:19) / (cid:16) c S . − (cid:17) / ; (23)and, if it obeys them more conservatively , it has to formeven further out from the central protostar.Evidently, it is hard for optically thin proto-fragmentsto condense out if the gas in the disc is much hotter than10 K ( c S ∼ . − ). For example, a modest tem-perature increase to 20 K ( c S ∼ . − ) increasesthe critical tidal radius (Eqn. 23) to R TIDAL τ =1 ∼
200 AU,and the minimum proto-fragment radius (Eqn. 21) to r OPT . THIN τ =1 ∼
110 AU. Even setting aside the assumptionimplicit in our analysis, that a proto-fragment be muchsmaller than the distance to the central primary proto-star, this would require the disc to extend to ∼
300 AU,with surface-density of order 10 g cm − , and hence arather large disc mass M DISC > ∼ M ⊙ . Any further tem-perature increase would make fragmentation of an op-tically thin part of the disc very unlikely.In other words, at radii where a disc has sufficientlylow surface-density and/or sufficiently low temperatureto be optically thin, proto-fragments are so extended,and hence so susceptible to tidal disruption by the cen-tral protostar, that they are unlikely to survive – unlessthey are far from the central protostar, i.e. at radii sat-isfying Eqn. (23). Eqn. (23) defines the sweet spot forfragmentation of optically thin regions in a disc. Con-sideration of the extents of observed discs ( R < ∼
300 AU)then implies that fragmentation is only likely if the tem-perature is low, T ∼
10 to 20 K ( c S ∼ . . − )at radii R > ∼
70 AU.
Conversely, if we adopt the second option in Eqn. (16),i.e. that the proto-fragment is optically thick to its owncooling radiation, then substituting from Eqns. (11),(14), (15) and (17), we obtainΣ < ∼ Σ OPT . THICKMAX ∼ (cid:18) h κ c Ω (cid:19) / π ¯ m c S c L . (24)In order to satisfy Eqns. (7) and (24) simultaneously,we require Σ MIN < Σ OPT . THICKMAX , and, since both Σ
MIN andΣ
OPT . THICKMAX are linear in c S , we are left withΩ < ∼ (cid:18) π G ¯ m h c L κ c (cid:19) / . (25) This upper limit on Ω translates into a lower limit onthe radii at which an optically-thick disc can fragment, R > ∼ R OPT . THICKMIN ∼ (cid:18) h c L κ c M ⋆ π G ¯ m (cid:19) / ∼
70 AU (cid:18) M ⋆ M ⊙ (cid:19) / , (26)a result that was first derived by Matzner & Levin(2005).If the optically thick limit is to obtain, the surface-density must be sufficiently large, Σ > ∼ Σ τ =1 (see Eqn.20), and the initial radii and masses of proto-fragmentsmust satisfy r FRAG < ∼ r τ =1 (see Eqn. 21) and m FRAG < ∼ m τ =1 (see Eqn. 22). We note that all three inequal-ities are reversed here, i.e. we have a lower limit onΣ and upper limits on r FRAG and m FRAG . This meansthat a proto-fragment with radius and mass obeyingthese inequalities marginally can only avoid being dis-rupted by the tidal field of the central protostar if it isformed at R > ∼ R τ =1 (see Eqn. 23). However, if a proto-fragment obeys the inequalities more conservatively , i.e.its surface-density significantly exceeds Σ τ =1 , it canform closer to the central protostar and avoid beingtidally disrupted. Specifically, it can form at R OPT . THICKMIN (see Eqn. 26), provided the surface-density exceeds10 g cm − (cid:16) c S . − (cid:17) . Eqn. (26) is not affected by changing c S , so fragmen-tation of an optically thick disc is not possible closerto the central protostar than R OPT . THICKMIN under any cir-cumstance. Thus, for example, if the temperature is in-creased to 20 K ( c S ∼ . − ), fragmentation is pos-sible at R OPT . THICKMIN provided the surface-density exceeds ∼
20 g cm − . This implies a very massive disc, but isprobably just credible.In other words, if a disc has sufficiently high column-density and/or sufficiently high temperature at radiiexceeding R OPT . THICKMIN to be optically thick, proto-fragments can cool fast enough to condense out at anytemperature. However, as the temperature increases,the surface-density required for a proto-fragment toavoid tidal disruption by the central protostar alsoincreases, linearly with the temperature, and quicklyreaches implausible values. Thus Eqn. (26) defines thesweet spot for fragmentation of optically thick regionsin a disc. Consideration of the surface-densities in theoutskirts of observed discs (Σ < ∼
20 g cm − ) then im-plies that fragmentation is again only likely at lowtemperatures, T ∼
10 to 20 K ( c S ∼ . . − ),since fragmentation at these low temperatures requiresrelatively modest surface-densities. PASA (2018)doi:10.1017/pas.2018.xxx isc formation and fragmentation Irrespective of whether a disc is optically thin or thick,it seems that fragmentation is only likely to occur atradii > ∼
70 AU, and even then only if the disc is cold, T < ∼
20 K. At smaller radii and higher temperatures,fragments are either unable to cool fast enough, or theyare susceptible to tidal disruption by the central proto-star. This would seem to offer a simple and attractiveexplanation for the Brown Dwarf Desert, i.e. the appar-ent paucity of brown dwarfs in close orbits round Sun-like stars (Marcy & Butler 2000). Disc fragmentation ismost likely to occur in the region R ∼
70 to 100 AU, andonly if the temperature is below 20 K ( c S < ∼ . − ). We assume that the luminosity of the central protostaris dominated by accretion, and hence its luminosity isgiven by L ⋆ ∼ GM ⋆ ˙ M ⋆ ⊙ . (27)If accretion is steady, we can define an accretiontimescale, t ⋆ = M ⋆ ˙ M ⋆ , (28)where, for example, t ⋆ = 0 . M ⋆ = 1 M ⊙ protostar accreting at a rate of ˙ M ⋆ =10 − M ⊙ yr − . The luminosity can then be written as L ⋆ ∼ GM ⋆ ⊙ t ⋆ ∼
100 L ⊙ (cid:18) M ⋆ M ⊙ (cid:19) (cid:18) t ⋆ . (cid:19) − . (29)The temperature structure in very young accretiondiscs can be complicated, but near the midplane – wherethe density is highest, and proto-fragments are likely tooriginate – the gas and dust are thermally coupled, andan approximate fit to the observed continuum emissionfrom discs, as a function of the primary protostar’s lu-minosity (e.g. Osterloh & Beckwith 1995) suggests that c S ( R ) ∼ . − (cid:18) L ⋆
100 L ⊙ (cid:19) / (cid:18) R
70 AU (cid:19) − / ∼ . − (cid:18) M ⋆ M ⊙ (cid:19) / (cid:18) t ⋆ . (cid:19) − / (cid:18) R (cid:19) − / . (30)The results derived in Section 7 indicate that suchdiscs will have great difficulty fragmenting, becausethey are too hot, and indeed, in simulations withsteady accretion, and hence steady radiative feed-back (e.g. Offner et al. 2009; Stamatellos et al. 2011, 2012; Lomax et al. 2014, 2015b,a), disc fragmentationis strongly suppressed.However, if accretion is episodic, a large fraction ofthe final mass of a protostar is accreted during short-lived bursts, and in between there are long periods oflow accretion. This means long periods of large t ⋆ andlow luminosity, during which the disc cools down, anddisc fragmentation can occur quite routinely. Specifi-cally, the low-accretion period must be longer than thetime it takes for proto-fragments to condense out of thedisc, which is of order the local orbital period, i.e. t ORB ∼ π Ω ∼ Myr (cid:18) M ⋆ M ⊙ (cid:19) − / (cid:18) R
70 AU (cid:19) / . (31)Observations suggest that accretion onto young proto-stars is indeed episodic (e.g. Kenyon et al. 1990), andthat the low-accretion periods could be as long as10 yr (Scholz et al. 2013). Simulations of protostar for-mation in prestellar cores that invoke episodic accre-tion, using a phenomenological model based on the the-ory of Zhu et al. (2009), find that disc fragmentationis a regular occurrence and makes the main contribu-tion to forming brown dwarfs and low-mass H-burningstars in the numbers observed (Stamatellos et al. 2011,2012; Lomax et al. 2014, 2015b,a). Most of these browndwarfs and low-mass H-burning stars are formed by discfragmentation during low accretion episodes when thediscs can cool down. A proto-fragment can also only condense out if it is ableto lose angular momentum, on a dynamical timescale;otherwise it is likely to undergo a rotational bounce, andbe sheared apart or merge with another proto-fragment.Our simulations of core collapse and fragmentation (seeSection 3) suggest that discs routinely experience im-pulsive perturbations. Local patches with higher thanaverage surface-density and/or lower than average spinare created stochastically by the interaction of densitywaves in the disc, or the delivery onto the disc of freshmaterial by an anisotropic accretion stream. The tidaleffect of the central protostar will then first extrude,and secondly torque, a proto-fragment, resulting in anexchange of angular momentum between the spin andthe orbit of the proto-fragment.To evaluate the timescale on which this happens, con-sider a proto-fragment with radius r and mass m , spin-ning at angular speed Ω (see Eqn. 4). The tidal accelera-tion due to the primary protostar, which acts to extrudethe proto-fragment, is¨ r TIDAL ∼ GM ⋆ rR . (32) PASA (2018)doi:10.1017/pas.2018.xxx A Whitworth et al.
However, because the proto-fragment is spinning, thistidal acceleration only acts coherently (i.e. in approxi-mately the same direction in a frame rotating with theproto-fragment, thereby extruding the proto-fragment)for a time t RADIAN ∼ Ω − . During t RADIAN , the primaryprotostar moves through one radian, as seen in a framerotating with the proto-fragment. Consequently, after t RADIAN the tidal acceleration due to the primary proto-star ceases to be even approximately aligned with theelongation it has caused, and the elongation saturates.Thus, the elongation is of order∆ r ∼ ¨ r TIDAL t RADIAN ∼ r , (33)and the fragment has an aspect ratio ∼ τ SPIN DOWN ∼ GM ⋆ m ∆ r R ∼ mr Ω , (34)and since the moment of inertia of the proto-fragmentis I FRAG ∼ mr , the time it takes to significantly reducethe spin of a proto-fragment is t SPIN DOWN ∼ I FRAG Ω τ SPIN DOWN ∼ t RADIAN . (35)The net time taken to first extrude and then spin downthe proto-fragment is therefore of order t ANG . MOM . LOSS ∼ t RADIAN , (36)i.e. about one third of an orbital period. If we allowthat our calculation has probably overestimated some-what the efficiency of the processes extruding and thentorquing the proto-fragment, this is of order one orbitalperiod.This approximate calculation suggests that, in thesituation where a proto-fragment initially spins at thesame angular speed as it orbits the primary protostar(i.e. when a proto-fragment condenses out of an ap-proximately Keplerian disc), the tide of the primaryprotostar can, in about one orbital period, induce anelongation in the proto-fragment, and then torque theelongated proto-fragment, thereby reducing its spin suf-ficiently to enable it to overcome rotational supportand start to condense out. This timescale is essentiallythe period of epicyclic pulsations of the proto-fragment,and defines the maximum time available for the proto-fragment to also lose some of its thermal energy – asassumed in Section 6.
10 CONCLUSIONS
We have introduced and discussed the factors and pro-cesses that may influence the formation and gravita-tional fragmentation of accretion discs around newly-formed protostars forming in typical prestellar cores: • We have outlined procedures for inferring, in a sta-tistical sense, the intrinsic three-dimensional struc-tures of prestellar cores, from dust-continuum andmolecular-line observations; and how these can beused to construct initial conditions for simulationsof core collapse and fragmentation. • We have presented the results of simulations ofcore collapse using initial conditions constructedin this way on the basis of detailed observations ofOphiuchus. These simulations suggest that the for-mation of brown dwarfs and low-mass H-burningstars requires disc fragmentation, which in turnrequires (i) that a significant fraction ( > ∼ .
3) ofthe turbulent energy in a core be in solenoidalmodes, and (ii) that accretion onto the primaryprotostar at the centre of the disc – and hence alsoits radiative feedback – be episodic, with a duty-cycle > ∼ ,
000 yrs. Typically, in the simulations us-ing Ophiuchus-like cores, each core spawns 4 or 5stars. • We have stressed that, even though the nonther-mal energy (interpreted here as turbulence) in thecores in Ophiuchus is low, typically trans-sonic,the flows delivering matter into the central regionwhere the protostars form are very irregular. Thishas the consequence that discs are often not wellaligned with the spins of the protostars they sur-round, and discs are subject to perturbations dueto lumpy, anisotropic inflows. • We have shown that, if the mapping from coresinto protostars is statistically self-similar – whichit must be if the shape of the Initial Mass Functionis to be inherited from the shape of the Core MassFunction – then a typical core must spawn between4 and 5 protostars – in excellent agreement withthe predictions of our simulations of Ophiuchus-like cores. Additionally, these protostars shouldhave a relatively broad range of masses (interquar-tile mass-ratio ∼ • We have presented a simple derivation of theToomre Criterion for gravitational instability in adisc, and formulated the associated conditions onthe speed with which a proto-fragment must losethermal energy and angular momentum to con-dense out. • We have shown that the need for a proto-fragmentto lose thermal energy on a dynamical timescale(the Gammie Criterion) converts into a lower limiton the radius at which a proto-fragment can con-dense out, which might explain the Brown DwarfDesert. • We have explained why episodic feedback, with aduty-cycle > ∼ ,
000 yr, is required if a disc is to
PASA (2018)doi:10.1017/pas.2018.xxx isc formation and fragmentation • We have shown that there is a sweet spot atwhich disc fragmentation, and hence the formationof brown dwarfs and low-mass hydrogen-burningstars, is most likely to occur, namely radii 70 AU < ∼ R < ∼
100 AU, and temperatures 10 K < ∼ T < ∼
20 K. • We have shown that tidal interactions between aproto-fragment and the primary protostar at thecentre of the disc define a timescale (essentiallythe epicyclic period) which is the maximum timeavailable for the proto-fragment to lose thermalenergy.
11 ACKNOWLEDGEMENTS
APW and OL both gratefully acknowledges the support ofa consolidated grant (ST/K00926/1) from the UK’s Scienceand Technology Funding Council (STFC).
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