When a self-gravitating disc is subject to irradiation, its propensity to fragmentation will be affected. The strength of self-gravitating disc stresses is expected to dictate disc fragmentation: as the strength of these torques typically decrease with increasing sound speed, it is reasonable to assume, to first-order, that disc fragmentation is suppressed when compared to the non-irradiated case, although previous work has shown that the details are complicated by the source of the irradiation. We expand on previous analysis of the Jeans mass inside spiral structures in self-gravitating discs, incorporating the effects of stellar irradiation and background irradiation. If irradiation is present, fragmentation is suppressed for marginally unstable discs at low accretion rates (compared to the no-irradiation case), but these lower accretion rates correspond to higher mass discs. Fragmentation can still occur for high accretion rates, but is consequently suppressed at lower disc surface densities, and the subsequent Jeans mass is boosted. These results further bolster the consensus that, without subsequent fragment disruption or mass loss, the gravitational instability is more likely to form brown dwarfs and low-mass stars than gas giant planets. Discs around both supermassive black holes. Young stars are thought to undergo phases in which they are self-gravitating. It is possible that if these discs are sufficiently unstable while they are self-gravitating, they may fragment into bound objects, providing a mode of planet formation in protostellar discs (Kuiper, 1951; Cameron, 1978; Boss, 1997) or star formation in AGN discs (Levin & Beloborodov, 2003; Nayakshin & Sunyaev, 2005).

The conditions under which a self-gravitating disc is expected to fragment have been extensively studied. The most important criterion is the Toomre parameter (Toomre, 1964)

Q=csκeπGΣ<1.5-1.7,ݑ„subscriptݑݑ subscriptݜ…ݑ’ݜ‹ݐºΣ1.51.7Q=\fracc_s\kappa_e\pi G\Sigma<1.5-1.7,italic_Q = divide start_ARG italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT italic_κ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT end_ARG start_ARG italic_π italic_G roman_Σ end_ARG <1.5 - 1.7 , (1)

where cssubscriptݑݑ c_sitalic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT is the local sound speed, κesubscriptݜ…ݑ’\kappa_eitalic_κ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT is the local epicyclic frequency (equal to the angular velocity ΩΩ\Omegaroman_Ω in Keplerian discs) and ΣΣ\Sigmaroman_Σ is the disc surface density. This is a linear stability criterion - the critical values given above apply to non-axisymmetric perturbations (see e.g. Durisen et al. 2007).

Once the disc becomes gravitationally unstable, spiral waves are excited in the disc, producing stresses and providing heating through shocks. This stress can be described as a pseudo-viscosity in some cases (Shakura & Sunyaev, 1973; Balbus & Papaloizou, 1999; Lodato & Rice, 2004; Forgan et al., 2011), allowing a simple page_content of angular momentum transport in the disc. The stress produced by the spiral structure can therefore be described by a Shakura-Sunyaev turbulent viscosity parameter αݛ¼\alphaitalic_α, and semi-analytic models of self-gravitating discs can be constructed at low computational cost (e.g. Rice & Armitage 2009; Clarke 2009). This pseudo-viscous approximation has been shown to fail if the disc is too massive or geometrically thick (Lodato & Rice, 2005; Forgan et al., 2011), but it remains useful for moderately massive, geometrically thin discs.

A balance can be struck between the shock heating from the spiral arms and the local radiative cooling. Discs which achieve this balance are described as being marginally stable (Paczynski, 1978). This allows a relationship between the local cooling time normalised to the local angular frequency (βc=tcoolΩsubscriptݛ½ݑsubscriptݑ¡coolΩ\beta_c=t_\rm cool\Omegaitalic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT roman_cool end_POSTSUBSCRIPT roman_Ω), and αݛ¼\alphaitalic_α:

α=49γ(γ-1)βc.ݛ¼49ݛ¾ݛ¾1subscriptݛ½ݑ\alpha=\frac49\gamma(\gamma-1)\beta_c.italic_α = divide start_ARG 4 end_ARG start_ARG 9 italic_γ ( italic_γ - 1 ) italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG . (2)

For fragmentation to be successful, a density perturbation produced by a spiral arm should be able to grow until it is gravitationally bound. Therefore, the fragment should be able to cool efficiently to reduce pressure support due to thermal energy, and continue collapsing. As cooling becomes more efficient in a marginally unstable disc (i.e. as βcsubscriptݛ½ݑ\beta_citalic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT decreases), αݛ¼\alphaitalic_α must increase to redress the balance. A consensus has developed in recent years that there is a maximum value of αݛ¼\alphaitalic_α that the disc can sustain before quasi-steady self-gravitating torques saturate: αcrit∼0.06similar-tosubscriptݛ¼ݑݑŸݑ–ݑ¡0.06\alpha_crit\sim 0.06italic_α start_POSTSUBSCRIPT italic_c italic_r italic_i italic_t end_POSTSUBSCRIPT ∼ 0.06 (Gammie, 2001; Rice et al., 2005). This is supported by numerical experiments (Cossins et al., 2009) that confirm an inverse relationship between βcsubscriptݛ½ݑ\beta_citalic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT and surface density perturbations:

<ΔΣrmsΣ>∝1βc∝α.proportional-toexpectationΔsubscriptΣrmsΣ1subscriptݛ½ݑproportional-toݛ¼\propto\frac1\sqrt\beta_c% \propto\sqrt\alpha.< divide start_ARG roman_Δ roman_Σ start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT end_ARG start_ARG roman_Σ end_ARG >∝ divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG end_ARG ∝ square-root start_ARG italic_α end_ARG . (3)

Once the disc reaches a state such that αݛ¼\alphaitalic_α cannot increase to maintain thermal equilibrium, ΔΣ/ΣΔΣΣ\Delta\Sigma/\Sigmaroman_Δ roman_Σ / roman_Σ is typically of order unity (i.e. density perturbations become non-linear) and fragmentation can occur. Rapid cooling also helps to prevent destructive fragment-fragment collisions, as the initial fragment spacing will be such that collisions occur within a few orbital periods (Shlosman & Begelman, 1989). Rapid cooling can then ensure that colliding fragments do not become unbound as a result of the collision. The combination of these criteria assure fragmentation can only occur in the outer regions of protostellar discs, typically at radii greater than r=30-40ݑŸ3040r=30-40italic_r = 30 - 40 AU (Rafikov, 2005; Matzner & Levin, 2005; Whitworth & Stamatellos, 2006; Boley et al., 2006; Stamatellos & Whitworth, 2008; Forgan et al., 2009; Clarke, 2009; Vorobyov & Basu, 2010; Forgan & Rice, 2011). In the case of AGN discs, this critical radius is approximately r=0.1pcݑŸ0.1ݑݑr=0.1pcitalic_r = 0.1 italic_p italic_c (Levin & Beloborodov, 2003; Nayakshin et al., 2007; Levin, 2007; Alexander et al., 2008).

This page_content, however, is incomplete. As fragmentation is consigned to the outer regions of self-gravitating discs, the local temperature is unlikely to be determined purely by gravitational stresses, and is likely to be governed by irradiation, either from the central object or from an external bath (e.g. envelope irradiation in embedded protostellar discs). In this regime, there is no longer a uniquely determined relation between αݛ¼\alphaitalic_α and βcsubscriptݛ½ݑ\beta_citalic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, as irradiation provides an extra term to the disc’s energy budget.

It is somewhat intuitive to assume that adding extra heating will push the disc away from marginal instability by increasing the sound speed, weakening self-gravity and therefore suppressing fragmentation. Matzner & Levin (2005) demonstrate this using an analytic prescription for protostellar disc formation and evolution from Bonnor-Ebert Spheres, showing that fragmentation is typically inhibited for orbital periods lower than 20,000 years. Rafikov (2009) also show that fragmentation is inhibited at large orbital periods when the disc accretion rate is low.

Cai et al. (2008) studied the effect of envelope irradiation in grid based hydrodynamic simulations with radiative transfer, showing that envelope irradiation suppresses the higher mݑšmitalic_m spiral modes of the gravitational instability (a result to some extent predicted by Boss 2002, although they disagree on the role of convection in resisting this suppresion). Stamatellos & Whitworth (2008) compare stellar. Background irradiation using smoothed particle hydrodynamics (SPH) simulations. The background irradiation behaviour is similar to that of Cai et al. (2008) - stellar irradiation allows the outer disc regions (r>30ݑŸ30r>30italic_r >30 AU) to cool sufficiently rapidly to fragment according to the minimum cooling time criterion, but they no longer satisfy the Toomre Qݑ„Qitalic_Q criterion.

More recently, Kratter & Murray-Clay (2011) have noted that weakening self-gravity is not generally an impediment to disc fragmentation if the full effects of mass infall are considered. As was shown by Stamatellos & Whitworth (2008), the cooling time criterion can be satisfied quite easily in irradiated discs (although maintaining a low Qݑ„Qitalic_Q is difficult), and that low mass discs in the irradiation-dominated regime may be made more susceptible to fragmentation given the correct infall rate. This is consistent with Cai et al. 2008)’s finding that mild irradiation increases gravitational torques. Therefore increases the equilibrium mass infall rate).

These results should be compared with local shearing sheet simulations of irradiated discs (Rice et al., 2011). Under irradiation, there is no longer a fixed fragmentation boundary using either βcsubscriptݛ½ݑ\beta_citalic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT or αݛ¼\alphaitalic_α. Comparing to the non-irradiated case, irradiated discs can fragment for lower values of αݛ¼\alphaitalic_α, but still requires rapid cooling.

With the straightforward criteria for fragmentation in the non-irradiated regime becoming less clear when irradiation is incorporated, it is instructive to consider other, more generalised fragmentation criteria. We have developed such a criterion, based on the local Jeans mass inside a spiral wave perturbation (Forgan & Rice, 2011). If a fragment is to collapse and become bound, it must have a mass greater than the local Jeans mass. By measuring how the Jeans mass evolves with time, we can ascertain whether regions of a self-gravitating disc are becoming more or less susceptible to fragmentation. From this criterion, we are able to estimate not only when a disc fragments, but what the initial mass of the fragment should be, an important initial condition to models such as the “tidal downsizing” hypothesis of terrestrial and giant planet formation (Boley et al., 2010; Michael et al., 2011; Nayakshin, 2011, 2010a, 2010b).

This is somewhat similar to other work which has focused on the relationship between spiral arm width and the local Hill radius (Rogers & Wadsley, 2011). These criteria reflect different competing influences: the Jeans criterion compares self-gravity to local pressure forces, whereas the Hill criterion compares self-gravity to local shear. The advantage of both these criteria is that they are easily generalised to cases difficult to describe or explain by traditional minimum cooling time/maximum stress criteria.

In this paper, we apply the Jeans criterion to 1D self-gravitating disc models where the effects of irradiation are accounted for. We compare these to models where irradiation is not present, to assess whether irradiation promotes or inhibits fragmentation. In section 2, we outline the Jeans criterion and how the 1D disc models are constructed. In section 3 we describe and discuss the results from the models, and in section 4 we summarise the work.

2 Method

2.1 Calculating the Jeans Mass in irradiated spiral arms

To calculate the Jeans mass inside a spiral density perturbation, we adopt the same procedure as described in Forgan & Rice (2011). We assume the Jeans mass is spherical:

MJ=43π(πcs2Gρpert)3/2ρpert=43π5/2cs3G3/2ρpert1/2.subscriptݑ€ݐ½43ݜ‹superscriptݜ‹subscriptsuperscriptݑ2ݑ ݐºsubscriptݜŒݑݑ’ݑŸݑ¡32subscriptݜŒݑݑ’ݑŸݑ¡43superscriptݜ‹52subscriptsuperscriptݑ3ݑ superscriptݐº32subscriptsuperscriptݜŒ12ݑݑ’ݑŸݑ¡M_J=\frac43\pi\left(\frac\pi c^2_sG\rho_pert\right)^3/2\rho_% pert=\frac43\pi^5/2\fracc^3_sG^3/2\rho^1/2_pert.italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG 3 end_ARG italic_π ( divide start_ARG italic_π italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG start_ARG italic_G italic_ρ start_POSTSUBSCRIPT italic_p italic_e italic_r italic_t end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_p italic_e italic_r italic_t end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG 3 end_ARG italic_π start_POSTSUPERSCRIPT 5 / 2 end_POSTSUPERSCRIPT divide start_ARG italic_c start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG start_ARG italic_G start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p italic_e italic_r italic_t end_POSTSUBSCRIPT end_ARG . (4)

Under the thin disc approximation

ρpert=Σpert/2H=Σ(1+ΔΣΣ)/2H.subscriptݜŒݑݑ’ݑŸݑ¡subscriptΣݑݑ’ݑŸݑ¡2ݐ»Σ1ΔΣΣ2ݐ»\rho_pert=\Sigma_pert/2H=\Sigma\left(1+\frac\Delta\Sigma\Sigma\right)/% 2H.italic_ρ start_POSTSUBSCRIPT italic_p italic_e italic_r italic_t end_POSTSUBSCRIPT = roman_Σ start_POSTSUBSCRIPT italic_p italic_e italic_r italic_t end_POSTSUBSCRIPT / 2 italic_H = roman_Σ ( 1 + divide start_ARG roman_Δ roman_Σ end_ARG start_ARG roman_Σ end_ARG ) / 2 italic_H . (5)

If we assume the disc is marginally stable, we can obtain

MJ=42π33GQ1/2cs2H(1+ΔΣΣ).subscriptݑ€ݐ½42superscriptݜ‹33ݐºsuperscriptݑ„12subscriptsuperscriptݑ2ݑ ݐ»1ΔΣΣM_J=\frac4\sqrt2\pi^33G\fracQ^1/2c^2_sH\left(1+\frac\Delta% \Sigma\Sigma\right).italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = divide start_ARG 4 square-root start_ARG 2 end_ARG italic_π start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG 3 italic_G end_ARG divide start_ARG italic_Q start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT italic_H end_ARG start_ARG ( 1 + divide start_ARG roman_Δ roman_Σ end_ARG start_ARG roman_Σ end_ARG ) end_ARG . (6)

In Forgan & Rice (2011), we used the empirical result of Cossins et al. (2009) to estimate the fractional amplitude ΔΣ/ΣΔΣΣ\Delta\Sigma/\Sigmaroman_Δ roman_Σ / roman_Σ:

<ΔΣrmsΣ>=1βc.expectationΔsubscriptΣrmsΣ1subscriptݛ½ݑ=\frac1\sqrt\beta_c.< divide start_ARG roman_Δ roman_Σ start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT end_ARG start_ARG roman_Σ end_ARG >= divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG end_ARG . (7)

As we are now considering irradiated discs, it is more appropriate to use the empirical relationship determined by Rice et al. (2011):

<ΔΣrmsΣ>=4.47α.expectationΔsubscriptΣrmsΣ4.47ݛ¼=4.47\sqrt\alpha.< divide start_ARG roman_Δ roman_Σ start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT end_ARG start_ARG roman_Σ end_ARG >= 4.47 square-root start_ARG italic_α end_ARG . (8)

For the disc to be in thermal equilibrium, the radiative cooling, viscous heating and irradiation heating must balance. The dimensionless cooling time, βcsubscriptݛ½ݑ\beta_citalic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is modified thus:

βc=(τ+τ-1)Σcs2ΩσSB(T4-Tirr4)γ(γ-1),subscriptݛ½ݑݜsuperscriptݜ1Σsubscriptsuperscriptݑ2ݑ ΩsubscriptݜŽݑ†ݐµsuperscriptݑ‡4subscriptsuperscriptݑ‡4irrݛ¾ݛ¾1\beta_c=\frac(\tau+\tau^-1)\Sigma c^2_s\Omega\sigma_SB(T^4-T^4% _\rm irr)\gamma(\gamma-1),italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = divide start_ARG ( italic_τ + italic_τ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) roman_Σ italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT roman_Ω end_ARG start_ARG italic_σ start_POSTSUBSCRIPT italic_S italic_B end_POSTSUBSCRIPT ( italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_irr end_POSTSUBSCRIPT ) italic_γ ( italic_γ - 1 ) end_ARG , (9)

where Tirrsubscriptݑ‡irrT_\rm irritalic_T start_POSTSUBSCRIPT roman_irr end_POSTSUBSCRIPT represents the temperature of the local irradiation field.

2.2 The Fragmentation Criterion

To produce a generalised fragmentation criterion, we consider the timescale on which the local Jeans mass changes. We define:

ΓJ=MJM˙JΩ.subscriptΓݐ½subscriptݑ€ݐ½subscript˙ݑ€ݐ½Ω\Gamma_J=\fracM_J\dotM_J\Omega.roman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = divide start_ARG italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT end_ARG start_ARG over˙ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT end_ARG roman_Ω . (10)

If this quantity is small and negative, then the local Jeans mass decreases rapidly, and fragmentation becomes favourable. Equally, if this quantity is small and positive, the local Jeans mass increases rapidly, and fragmentation becomes unlikely.

The value of ΓJsubscriptΓݐ½\Gamma_Jroman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT estimates the number of local orbital periods that will elapse before fragmentation becomes likely (if at all). In a steady-state disc in perfect thermodynamic equilibrium, ΓJ→∞→subscriptΓݐ½\Gamma_J\rightarrow\inftyroman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT → ∞, and fragmentation will never occur. Discs that are not in equilibrium will assume non-infinite values of ΓJsubscriptΓݐ½\Gamma_Jroman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT, and fragmentation is either likely or unlikely.

We assume -5<ΓJ<05subscriptΓݐ½0-5<- 5
MJ=42π33GQ1/2cs3Ω(1+4.47α).subscriptݑ€ݐ½42superscriptݜ‹33ݐºsuperscriptݑ„12subscriptsuperscriptݑ3ݑ Ω14.47ݛ¼M_J=\frac4\sqrt2\pi^33G\fracQ^1/2c^3_s\Omega\left(1+4.47% \sqrt\alpha\right).italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = divide start_ARG 4 square-root start_ARG 2 end_ARG italic_π start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG 3 italic_G end_ARG divide start_ARG italic_Q start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG start_ARG roman_Ω ( 1 + 4.47 square-root start_ARG italic_α end_ARG ) end_ARG . (11)

We can calculate M˙Jsubscript˙ݑ€ݐ½\dotM_Jover˙ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT using the chain rule:

M˙J=∂MJ∂csc˙s+∂MJ∂ΩΩ˙+∂MJ∂αα˙+∂MJ∂QQ˙.subscript˙ݑ€ݐ½subscriptݑ€ݐ½subscriptݑݑ subscript˙ݑݑ subscriptݑ€ݐ½Ω˙Ωsubscriptݑ€ݐ½ݛ¼˙ݛ¼subscriptݑ€ݐ½ݑ„˙ݑ„\dotM_J=\frac\partial M_J\partial c_s\dotc_s+\frac\partial M_% J\partial\Omega\dot\Omega+\frac\partial M_J\partial\alpha\dot% \alpha+\frac\partial M_J\partial Q\dotQ.over˙ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = divide start_ARG ∂ italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG over˙ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT + divide start_ARG ∂ italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT end_ARG start_ARG ∂ roman_Ω end_ARG over˙ start_ARG roman_Ω end_ARG + divide start_ARG ∂ italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_α end_ARG over˙ start_ARG italic_α end_ARG + divide start_ARG ∂ italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_Q end_ARG over˙ start_ARG italic_Q end_ARG . (12)

We will assume Q˙=α˙=Ω˙=0˙ݑ„˙ݛ¼˙Ω0\dotQ=\dot\alpha=\dot\Omega=0over˙ start_ARG italic_Q end_ARG = over˙ start_ARG italic_α end_ARG = over˙ start_ARG roman_Ω end_ARG = 0, and hence

M˙J=MJ3c˙scssubscript˙ݑ€ݐ½subscriptݑ€ݐ½3subscript˙ݑݑ subscriptݑݑ \dotM_J=M_J3\frac\dotc_sc_sover˙ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT 3 divide start_ARG over˙ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG (13)

From equation (22) of Forgan & Rice (2011), we calculate

c˙scs=1/2(9αγ(γ-1)Ω4-1tcool),subscript˙ݑݑ subscriptݑݑ 129ݛ¼ݛ¾ݛ¾1Ω41subscriptݑ¡cool\frac\dotc_sc_s=1/2\left(\frac9\alpha\gamma(\gamma-1)\Omega4-% \frac1t_\rm cool\right),divide start_ARG over˙ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_ARG = 1 / 2 ( divide start_ARG 9 italic_α italic_γ ( italic_γ - 1 ) roman_Ω end_ARG start_ARG 4 end_ARG - divide start_ARG 1 end_ARG start_ARG italic_t start_POSTSUBSCRIPT roman_cool end_POSTSUBSCRIPT end_ARG ) , (14)

and hence derive:

ΓJ=(3/2(-1βc+9αγ(γ-1)4))-1.subscriptΓݐ½superscript321subscriptݛ½ݑ9ݛ¼ݛ¾ݛ¾141\Gamma_J=\left(3/2\left(-\frac1\beta_c+\frac9\alpha\gamma(\gamma-1)% 4\right)\right)^-1.roman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = ( 3 / 2 ( - divide start_ARG 1 end_ARG start_ARG italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG + divide start_ARG 9 italic_α italic_γ ( italic_γ - 1 ) end_ARG start_ARG 4 end_ARG ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT . (15)

As in Forgan & Rice (2011), we assume that the maximum value for αݛ¼\alphaitalic_α saturates at some value αsat=0.1subscriptݛ¼sat0.1\alpha_\rm sat=0.1italic_α start_POSTSUBSCRIPT roman_sat end_POSTSUBSCRIPT = 0.1, which is slightly higher than the canonical value of 0.06 (Gammie, 2001; Rice et al., 2005). We take this value as a conservative estimate given current uncertainties regarding convergence of 3D simulations of fragmentation (Meru & Bate, 2011; Lodato & Clarke, 2011; Rice et al., 2012). If the local value of α<αsatݛ¼subscriptݛ¼sat\alphaitalic_α
We should note that we do not consider infall in this analysis, although we do assume the discs reach a steady state where the disc accretion rate M˙˙ݑ€\dotMover˙ start_ARG italic_M end_ARG is the appropriate value to process material to maintain a constant disc mass (i.e. the local rate of change of surface density Σ˙=0˙Σ0\dot\Sigma=0over˙ start_ARG roman_Σ end_ARG = 0). From the above, we could suppose that a disc experiencing infall such that Σ˙>0˙Σ0\dot\Sigma>0over˙ start_ARG roman_Σ end_ARG >0 may give ΓJ<0subscriptΓݐ½0\Gamma_J<0roman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT <0, and as such irradiation could encourage fragmentation provided that Qݑ„Qitalic_Q does not also increase. We leave investigation of this possibility to future work.

It is unclear from this analysis whether irradiation should drive or stop fragmentation - consequently it is instructive to test this using semi-analytic disc models.

2.3 Simple Irradiated Discs with Local Angular Momentum Transport

As in Forgan & Rice (2011), we construct simple disc models in the same manner as Levin (2007) and Clarke (2009) to evaluate the dependence of the Jeans mass on disc parameters. We fix the disc’s accretion rate

M˙=3παcs2ΣΩ˙ݑ€3ݜ‹ݛ¼subscriptsuperscriptݑ2ݑ ΣΩ\dotM=\frac3\pi\alpha c^2_s\Sigma\Omegaover˙ start_ARG italic_M end_ARG = divide start_ARG 3 italic_π italic_α italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT roman_Σ end_ARG start_ARG roman_Ω end_ARG (16)

to be constant for all radii, and impose an outer disc radius. Assuming marginal instability (Q=2ݑ„2Q=2italic_Q = 2) and thermal equilibrium then fixes the disc surface density profile. If thermal equilibrium demands α>αsatݛ¼subscriptݛ¼sat\alpha>\alpha_\rm satitalic_α >italic_α start_POSTSUBSCRIPT roman_sat end_POSTSUBSCRIPT, then αݛ¼\alphaitalic_α is set to αsatsubscriptݛ¼sat\alpha_\rm satitalic_α start_POSTSUBSCRIPT roman_sat end_POSTSUBSCRIPT, and ΓJsubscriptΓݐ½\Gamma_Jroman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT can then become a finite, negative quantity. If, or when, -5<ΓJ<05subscriptΓݐ½0-5<- 5
We fix the star mass at 1M⊙1subscriptݑ€direct-product1M_\odot1 italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT, and consider four scenarios:

1.
No irradiation (Tirr=0Ksubscriptݑ‡ݑ–ݑŸݑŸݑŸ0ݐ¾T_irr=0Kitalic_T start_POSTSUBSCRIPT italic_i italic_r italic_r end_POSTSUBSCRIPT ( italic_r ) = 0 italic_K),

2.
Envelope irradiation, Tirr=10Ksubscriptݑ‡irrݑŸ10ݐ¾T_\rm irr=10Kitalic_T start_POSTSUBSCRIPT roman_irr end_POSTSUBSCRIPT ( italic_r ) = 10 italic_K at all radii,

3.
Envelope irradiation, Tirr=30Ksubscriptݑ‡irrݑŸ30ݐ¾T_\rm irr=30Kitalic_T start_POSTSUBSCRIPT roman_irr end_POSTSUBSCRIPT ( italic_r ) = 30 italic_K at all radii

4.
Irradiation from the central star.

For the third case, we use the radiation field expression given by Hayashi (1981):

Tirr=280K(M*1M⊙)(r1AU)-1/2.subscriptݑ‡ݑ–ݑŸݑŸݑŸ280ݐ¾subscriptݑ€1subscriptݑ€direct-productsuperscriptݑŸ1ݐ´ݑˆ12T_irr=280K\left(\fracM_{*}1M_\odot\right)\left(\fracr1AU\right)% ^-1/2.italic_T start_POSTSUBSCRIPT italic_i italic_r italic_r end_POSTSUBSCRIPT ( italic_r ) = 280 italic_K ( divide start_ARG italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT end_ARG start_ARG 1 italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT end_ARG ) ( divide start_ARG italic_r end_ARG start_ARG 1 italic_A italic_U end_ARG ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT . (17)

For αsat=0.1subscriptݛ¼sat0.1\alpha_\rm sat=0.1italic_α start_POSTSUBSCRIPT roman_sat end_POSTSUBSCRIPT = 0.1 and γ=5/3ݛ¾53\gamma=5/3italic_γ = 5 / 3, we can actually determine the maximum value of βcsubscriptݛ½ݑ\beta_citalic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT at which fragmentation occurs (i.e. where ΓJ=-5subscriptΓݐ½5\Gamma_J=-5roman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = - 5), which we find to be

βcrit<2.2.subscriptݛ½crit2.2\beta_\rm crit<2.2.italic_β start_POSTSUBSCRIPT roman_crit end_POSTSUBSCRIPT <2.2 . (18)

Note the conventional critical cooling time for this value of γݛ¾\gammaitalic_γ is βcrit=3subscriptݛ½crit3\beta_\rm crit=3italic_β start_POSTSUBSCRIPT roman_crit end_POSTSUBSCRIPT = 3. We have however assumed a value of αsatsubscriptݛ¼sat\alpha_\rm satitalic_α start_POSTSUBSCRIPT roman_sat end_POSTSUBSCRIPT around 60%percent6060\%60 % larger than the usual value, so our slightly smaller value of βcritsubscriptݛ½crit\beta_\rm crititalic_β start_POSTSUBSCRIPT roman_crit end_POSTSUBSCRIPT compares sensibly to the typically used cooling time criteria. Altering the critical value of ΓJsubscriptΓݐ½\Gamma_Jroman_Γ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT instead would also allow βcritsubscriptݛ½crit\beta_\rm crititalic_β start_POSTSUBSCRIPT roman_crit end_POSTSUBSCRIPT to be increased to a more standard value, but this would permit discs to fragment on timescales longer than might be considered physical. In any case, the resulting fragment masses are only very weakly sensitive to these considerations.

3.1 Disc Profiles

Figure 1 shows 2D contours of the disc-to-star mass ratio qݑžqitalic_q in all four scenarios, given the accretion rate and the disc outer radius. The differences between the non-irradiated case (top left in Figure 1) and the models discussed in Forgan & Rice (2011) are due to a different selection of Qݑ„Qitalic_Q (in this paper, we assume Q=2ݑ„2Q=2italic_Q = 2, and in the previous work we assumed Q=1ݑ„1Q=1italic_Q = 1), but the same broad features remain: a high disc-to-star mass ratio is required to maintain a self-gravitating disc with a modest accretion rate and reasonable outer disc radius. For example, in the non-irradiated disc, an accretion rate of 10-7M⊙yr-1superscript107subscriptݑ€direct-productsuperscriptyr110^-7M_\odot\mathrmyr^-110 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT roman_yr start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and outer radius of rout=30subscriptݑŸݑœݑ¢ݑ¡30r_out=30italic_r start_POSTSUBSCRIPT italic_o italic_u italic_t end_POSTSUBSCRIPT = 30 au demands a disc mass of 0.119 M⊙subscriptݑ€direct-productM_\odotitalic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT.

At high accretion rates, the lifetime of material in the disc becomes comparable to the orbital timescale. For this reason, we truncate the upper limit of the contours by demanding that

MdiscM˙>52πΩ,subscriptݑ€disc˙ݑ€52ݜ‹Ω\fracM_\rm disc\dotM>5\frac2\pi\Omega,divide start_ARG italic_M start_POSTSUBSCRIPT roman_disc end_POSTSUBSCRIPT end_ARG start_ARG over˙ start_ARG italic_M end_ARG end_ARG >5 divide start_ARG 2 italic_π end_ARG start_ARG roman_Ω end_ARG , (19)

which essentially requires the disc to exist for at least five orbital periods at the given radius. Any disc model which does not satisfy this is discarded. Not considered in the subsequent analysis. This explains the empty regions in the upper right portion of each plot.

Comparing the non-irradiated case to the other three cases, we can see that the high M˙˙ݑ€\dotMover˙ start_ARG italic_M end_ARG section of the parameter space (i.e., above M˙∼10-6M⊙yr-1similar-to˙ݑ€superscript106subscriptݑ€direct-productsuperscriptyr1\dotM\sim 10^-6M_\odot\mathrmyr^-1over˙ start_ARG italic_M end_ARG ∼ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT roman_yr start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT) remains similar. The effect of irradiation becomes more apparent at lower accretion rates, forcing the equilibrium disc structure to be more m

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