FRAGMENT PRODUCTION AND SURVIVAL IN IRRADIATED DISKS: A COMPREHENSIVE COOLING CRITERION

What does it mean for a disk to be "dominated" by irradiation? Qualitatively it means that the primary source of the disk's thermal energy is external irradiation rather than dissipation of accretion energy (which we compute as an artificial viscosity). In other words, the temperature to which the disk midplane is heated by external irradiation is higher than the equilibrium temperature that the disk could achieve by balancing accretional heating and cooling via thermal emission. As a result, irradiated disks will be roughly vertically isothermal. We shall show an example of this later in Figures 2 and 3.

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We choose this terminology over the more common, but less informative, distinction between passive (irradiated) and active (viscously heated), primarily because irradiated disks can have ongoing accretion—they are not entirely "passive."

Quantitatively, the boundary between an irradiated and viscously heated disk can be defined by balancing radiative cooling from the midplane with heating via dissipation of accretion energy. We quantify this dissipation in the standard manner, by employing an effective viscosity, ν = αc²_s/Ω, described by the parameter α ⩽ 1. The midplane temperature, T, generated by dissipation of energy due to accretion of a mass flux

$$\begin{equation} \dot{M} = 3\pi \Sigma \nu \end{equation} \tag{ 4 }$$

is given by

$$\begin{equation} \frac{1}{\tau _{R}}\sigma T^4 = \frac{3}{8\pi }\dot{M}\Omega ^2 = \frac{9}{8}\alpha c_s^2\Sigma \Omega \end{equation} \tag{ 5 }$$

in the optically thick case, relevant for Q = 1 disks. Inserting Equation (5) into Equation (3) yields

$$\begin{equation} t_{\rm cool} = \frac{1}{12\alpha }\frac{\gamma }{\gamma -1}\Omega ^{-1} \;. \end{equation} \tag{ 6 }$$

Equation (6) expresses the conditions for thermal equilibrium of a disk heated only by viscous dissipation in terms of the cooling time as defined in Equation (3). Irradiated disks have cooling times less than this critical value:

$$\begin{equation} t_{\rm cool} < \frac{1}{12\alpha }\frac{\gamma }{\gamma -1}\Omega ^{-1} \;, \end{equation} \tag{ 7 }$$

because they are heated to higher temperatures by irradiation than by dissipation of accretion energy. This critical disk temperature, in terms of density and α, is

$$\begin{equation} T > \left(\frac{9}{8}\frac{\alpha \Sigma }{\sigma }\frac{k}{\mu }{\tau _R}\Omega \right)^{1/3}. \end{equation} \tag{ 8 }$$

If stellar irradiation maintains the disk (or some region of it) above the temperature in Equation (8), it is irradiation-dominated. Typically the outer regions of disks are irradiation-dominated, while in their inner regions accretional heating is more important than irradiation. This radial dependence is due to the liberation of more energy as material in the disk is lowered deeper into the potential well of the central object.

To evaluate whether a disk in steady state satisfies Equation (8) at a particular disk location, one must set α to the value required to process the incoming material either from larger radii in the disk or the background environment (e.g., a protostellar core). Of course, if the α required to process the material is larger than that provided by any means of angular momentum transport, a steady state is not possible. We note that over time, a particular disk radius may transition from irradiation-dominated to viscously heated or vice versa as the disk evolves.

The boundary between viscously heated and irradiated disks is independent of the mechanism for angular momentum transport, so long as it can be quantified by an α. In this work we are concerned with transport due to GI. Unlike transport due to the magneto-rotational instability (MRI; Balbus & Hawley 1994), which is often modeled using a single value for α, GI-induced transport cannot be described by a fixed α. Instead, α is a function of disk parameters (in particular Q) and should be thought of as a response to the self-gravitating state of the disk (Lin & Pringle 1987). We return to this point in Section 3.2.

We now answer the first question posed above and in Figure 1: how do disks reach the boundary of linear instability?

In order to fragment, a disk must first reach Q = Q₀ ∼ 1, allowing linear collapse to proceed. Though dramatic events may sometimes drive disks into gravitationally unstable configurations on short timescales, typical disks likely evolve on long enough timescales that Q = 1 must be approached gradually if it is approached at all.

First, consider disks which are heated only through accretion energy. In the absence of any external heat source (e.g., irradiation), a disk can only be driven to fragment by rapid infall (Matzner & Levin 2005). In the case of a protostellar disk, for example, this infall comes from the protostellar cloud and its rate is set by conditions external to the disk.² We discuss in the next section how irradiation changes this picture, and in the Appendix we derive explicitly the evolution of Q with $d\!\dot{M}/dt$.

As long as infall rates are not too large, a Q ≳ 1 disk heated solely by viscous dissipation relaxes into an equilibrium state known as gravitoturbulence. Gravitoturbulence results from a feedback loop which may be understood as follows. When Q ∼ 1, if radiative cooling exceeds viscous heating, the disk will cool and get closer to Q = 1. As it gets closer to Q = 1, the onset of gravitational collapse leads to turbulence and accretion, which heats the disk and generates extra pressure support to counteract gravitational collapse. If the heating rate were to exceed the cooling rate such that Q ≫ 1, then the α generated by GI would decrease, the heating rate would decline, and the disk would again cool down. This feedback loop is gravitoturbulence.

In gravitoturbulent equilibrium, viscous heating and radiative cooling balance such that

$$\begin{equation} \beta = \frac{4}{9\gamma (\gamma -1)}\frac{1}{\alpha } \end{equation} \tag{ 9 }$$

(Gammie 2001). Here, β, defined in Equation (1), represents the cooling rate in terms of orbital periods. One may derive Equation (9) by equating the cooling rate per unit mass u/τ_c (Equation (2)) with the viscous heating rate per unit mass, (9/4)αc²_sΩ (cf. Equation (5), remembering to include two disk faces). We have used u = c²_s/(γ − 1) appropriate for an ideal gas.³

However, the equilibrium reflected in Equation (9) cannot be achieved for arbitrarily high mass infall rates. For GI-driven accretion, α is a function of disk properties (in particular of Q), dictated by the strength of self-gravity in the disk. Numerical simulations demonstrate that there exists a maximum transport rate provided by GI (Gammie 2001) and hence a maximum α < 1, which we call its saturation value α_sat.

The precise values of α_sat as a function of disk properties and the function α(Q) remain uncertain (though see Kratter et al. 2008; Vorobyov 2010). We emphasize that α is not a free parameter and neither is the temperature of the disk in equilibrium. Whether or not a disk can restabilize itself at ever higher $\dot{M}$ depends on both α(Q) and on the temperature dependence of the opacity. Unless κ∝T^>2, increasing $\dot{M}$ always drives Q down. Depending on the functional form of α(Q), a very steep temperature dependence may also cause Q to decline with increasing $\dot{M}$. The disks that we consider here typically have κ∝T^⩽2 (Semenov et al. 2003), but we explore this dependence fully in the Appendix. Thus in general, we expect rapid accretion to drive disks toward instability and in some cases fragmentation.

The distinction between a disk that can maintain gravitoturbulence and one that fragments is encompassed in the Viscous Criterion:

$$\begin{equation} \beta \le \frac{4}{9\gamma (\gamma -1)}\frac{1}{\alpha _{\rm sat}}. \end{equation} \tag{ 10 }$$

When Equation (10) is satisfied, cooling exceeds viscous heating and the disk is unable to reverse this imbalance by increasing its accretion rate and hence its heating rate. Thus material is being added into the disk faster than it can be drained to smaller radii, which causes Σ to increase proportionally faster than T^1/2, Q to decline, and fragmentation to proceed. We note that accretional energy must be deposited in the disk as material falls toward the star, independent of the physical processes involved in dissipation of that energy. Hence, the Viscous Criterion is robust unless dissipation of accretion energy is dramatically non-local.

Note that most simulations of gravitoturbulence involve isolated disks, i.e., not undergoing infall. These disks fragment because they are set up out of equilibrium (Lodato & Rice 2004, 2005). Cooling exceeds heating by fiat, and thus the disk moves toward Q = 1. In reality, the cooling rate is set by the thermal physics of the disk and is not an independent parameter.

Of course, as we addressed above, many disks have two heating sources, accretion energy and external irradiation. In fact, the Viscous Criterion is nearly identical to the boundary between these two regimes in Equation (7), modulo different numerical coefficients. The difference in coefficients derives from the fact that here we are concerned with the timescale for the disk to radiate away its internal energy, and therefore cool off. To distinguish between viscous and irradiated disks we instead consider whether a temperature perturbation in the midplane heats the disk or is radiated away (see the Appendix of Kratter et al. 2010b). A disk satisfying the Viscous Criterion can avoid fragmentation only if it is irradiation-dominated, because the external heat source alters the feedback loop—the temperature, and thus cooling rate—are no longer controlled by dissipation, but instead by the radiation field. We address how disks heated by irradiation are driven toward instability below.

What is required for an irradiated disk to be driven to Q = Q₀? Again, infalling material is required to drive the disk unstable. However, we suggest that for very low mass disks, the critical infall rate may be lower than previously thought.

Unlike their viscously heated counterparts, irradiated disks will not become gravitoturbulent as infall adds mass and decreases Q. Gravitoturbulence relies on a feedback loop between Q and disk temperature, but the temperature of an irradiation-dominated disk is independent of Q, modulo small perturbations. For simplicity, for the remainder of this section, we only consider disks which remain irradiation-dominated at all accretion rates.

To understand why irradiated disks cannot maintain gravitoturbulence, consider the following hypothetical disk, with Q = 1 around a 1.5 M_☉ star. Imagine that as the disk approaches Q = 1, the effective α generated by GI-driven accretion increases from 0.1 to α_sat = 1.0. For an order of magnitude increase in α, the equilibrium temperature for a disk heated through viscous dissipation rises from 5 to 20 K at 100 AU. However, the equilibrium temperature of the disk including external irradiation from a young A star increases from 47 to 50 K. Thus, the fractional contribution of GI to the total pressure support of the disk (∝T) is negligible. Now, we add more and more material to the disk. Even if the disk processes material at this maximum rate, and dissipates energy at this maximum rate, the disk cannot reheat itself via turbulence. External irradiation, which initially prevents the disk from collapsing, cannot adjust the disk temperature to avoid the Q = 1 threshold. Once Q crosses this threshold, fragmentation will proceed.

However, irradiation-dominated disks are not entirely unaffected by self-gravity as infalling material drives them toward Q = 1. Irradiated disks (or indeed disks with an isothermal equation of state, which behave similarly) respond to decreasing Q by generating gravitational torques in spiral arms (Boley et al. 2007; Kratter et al. 2010a). These torques can, in turn, drive rapid accretion through the disk.

The properties of spiral arm transport depend on the ratio of the disk mass (Adams et al. 1989), M_disk = πr²_outΣ, to the central mass, M_*. Here r_out is the outer radius of the disk, where most of the mass resides. For a Q = 1 disk, M_disk/M_* = H/r_out, where H = c_s/Ω is the disk scale height, so the disk-to-central object mass ratio is equivalent to the disk aspect ratio. Spiral arms that form in thin and thick disks have different properties.

Numerical simulations indicate that high-mass (thick) disks can process mass infall rates up to $\dot{M} \sim c_s^3/G$ (Kratter et al. 2010a; Offner et al. 2010), while at higher rates they fragment. Lower infall rates do not generate fragmentation because these disks become globally gravitationally unstable; the resulting spiral arms transport material efficiently, keeping Q > Q₀. In the language of Section 3.2, $\dot{M} \gtrsim c_s^3/G$ corresponds to driving the disk with an infall rate having α ≳ 1. Thus thick, irradiated disks, like viscously heated disks, must be forced with large infall rates in order to fragment.

However, if the disk is low in mass (thin), has inefficient transport by spiral arms, and is very near Q = 1, a modest infall rate could tip it over the fragmentation threshold faster than it can process even the modest amount of mass it receives. For this to occur, other mechanisms of angular momentum transport such as the MRI must not be able to process the infalling material. Whether low-mass (thin), irradiated disks can avoid effective angular momentum transport by spiral arms remains uncertain. Simulations suggest that strong global spiral arm transport may require M_disk/M_* ≳ 0.1 (Lodato & Rice 2004).

Since the actual criterion for global mode transport is unknown, it is difficult to provide a precise criterion: future numerical work can delineate the exact conditions for this regime, and indeed whether it exists at all. This work will need to be conducted with great care; N-body and smoothed particle hydrodynamics (SPH) simulations which show spontaneous instability at Q > 1 may suffer from Swing amplification of Poisson noise (E. D'Onghia et al. 2011, in preparation). Current state-of-the-art simulations (Boley et al. 2007; Cai et al. 2008) do not yet explore the full parameter space required to answer these questions. For earlier attempts to quantify this behavior, see Laughlin & Rozyczka (1996), Laughlin & Korchagin (1996), and Laughlin et al. (1997). For a numerical examination of cooling and fragmentation in irradiated disks, see Stamatellos & Whitworth (2009) and Stamatellos et al. (2011).

Can nature produce a Q ≈ 1 disk that remains irradiation-dominated, unaffected by the MRI, and low enough in mass to avoid global instability? We show in Section 6 that it is difficult, but not impossible, to achieve Q = 1 and maintain a sufficiently low-mass disk to suppress spiral arm formation.

Thus, we have answered the first of our two questions and shown how both classes of disks can reach the instability threshold. In summary, to fragment, viscously heated disks must be driven by mass infall at a rate corresponding to α > α_sat. Disks that are irradiation-dominated (and thus satisfy the Viscous Criterion) might be driven to fragment at lower accretion rates, particularly when such disks have low masses relative to their host stars (i.e., are thin). This possibility is interesting because such disks could generate fragments that remain near their initial masses rather than experiencing substantial growth. Numerical parameter studies of global mode formation in low-mass, irradiated disks will be required to evaluate this possibility. Other angular momentum transport mechanisms such as the MRI will prevent fragmentation from occurring for arbitrarily small infall rates.

Can fragments survive once the instability threshold is reached? We now consider the cooling criteria related to fragment survival: the Stalling Criterion and the Collisional Criterion.

Consider a disk that has arrived near enough to Q = 1 to produce perturbations that can seed collapse. In an isothermal disk, once a perturbation exists with Q < 1, collapse will occur on the free-fall timescale. For non-isothermal collapse we must examine the energy budget of the collapsing region as self-gravity tries to make it contract.

We neglect internal rotation in the following analysis as it is difficult to estimate its evolution in time analytically. We comment on the importance of rotation in Section 5. By neglecting rotation we provide optimistic limits for collapse and thus fragment survival.

When Q ∼ 1 and linear collapse has begun, self-gravity, pressure, and tidal forces from the central object exert comparable forces on the fragment, but self-gravity is currently winning. As collapse proceeds and the radius, x, of the fragment gets smaller, the ratio of the self-gravitational force to the tidal force increases as x⁻³, assuming that the enclosed mass in the fragment stays constant. However, the ratio of self-gravitational force to pressure force may increase or decrease. We define the virial ratio B in a clump with density ρ

$$\begin{equation} B = \frac{{\it GM}_{{\rm enc}}\rho /x^2}{P/x}, \end{equation} \tag{ 11 }$$

where M_enc = 4/3πρx³ and P is the internal pressure. We approximate the pressure gradient as P/x since the fragment is only one pressure scale height in extent.⁴ If B increases with time, then self-gravity will always dominate and free-fall collapse is ensured. If B decreases, pressure will eventually stall collapse.

For small enough β (fast cooling), B initially increases as collapse proceeds. In other words,

$$\begin{equation} -\frac{{d}}{{d} x}(B) > 0, \end{equation} \tag{ 12 }$$

where the negative sign comes from the fragment radius, x, decreasing as the object collapses. Fragments collapse in free fall when β is less than a critical value, which we now calculate.

Holding fragment mass, M_enc, fixed and substituting in for an ideal gas with pressure P = (γ − 1)ρu, Equation (12) yields the following inequality:

$$\begin{equation} -\frac{{d}}{{d} x}\left(\frac{1}{xu}\right) > 0. \end{equation} \tag{ 13 }$$

To proceed we need an expression for the spatial derivative of u

$$\begin{equation} \frac{du}{dx} = \left(\frac{P}{\rho ^2}\frac{d\rho }{dt} - \frac{u}{\tau _c} + \frac{9}{4}\alpha c_s^2\Omega \right) \frac{dt}{dx}, \end{equation} \tag{ 14 }$$

where the time derivative of ρ is given by

$$\begin{equation} \frac{d\rho }{dt}= -3 \frac{\rho }{x}\frac{dx}{dt}. \end{equation} \tag{ 15 }$$

The coefficient 3 in Equation (15) arises from the spatial derivative of the enclosed mass and equals the dimension of collapse.

The three terms in our expression for du/dx (Equation (14)) come from PdV work, cooling parameterized by the cooling time τ_c (cf. Equation (2)), and viscous heating. Balancing viscous heating against cooling gives rise to the Viscous Criterion. In order to isolate the effect of PdV work after fragmentation has occurred, we shall hereafter drop the last, viscous term, assuming that turbulent heating is inefficient within a fragment. This choice differentiates fragments from turbulent eddies.

For convenience we define the collapse timescale of the fragment:

$$\begin{equation} t_{ \rm coll} \equiv -\left(\frac{d{\rm {ln}}x}{{dt}}\right)^{-1}. \end{equation} \tag{ 16 }$$

Equation (13) may now be written in the following simple form:

$$\begin{equation} t_{\rm coll} \ge \tau _c (3\gamma - 4). \end{equation} \tag{ 17 }$$

Equation (17) confirms that the object cannot collapse on a timescale faster than its own cooling time. We recover the standard result from stellar astrophysics that if γ < 4/3, thermal pressure cannot prevent collapse. Also, an object cannot collapse on a timescale faster than its own free-fall time

$$\begin{equation} t_{\rm ff} = \frac{1}{\sqrt{G\rho }} = \frac{\sqrt{2\pi Q}}{\Omega }, \end{equation} \tag{ 18 }$$

where to derive the final expression, we have assumed that ρ = Σ/2H with H = c_s/Ω equal to the disk scale height. Hence, the fragment collapses on whichever timescale is longer: τ_c(3γ − 4) or t_ff.

Comparing these timescales, we find that free-fall collapse requires

$$\begin{equation} \beta \le \frac{\sqrt{2\pi Q} }{(3\gamma -4)}. \end{equation} \tag{ 19 }$$

As expected, pressure is negligible when the cooling time is of order an orbital time. For Q = 1 the limits on β for free-fall collapse are

$$\begin{eqnarray} &&\beta \lesssim 2.5,\quad \gamma = 5/3 \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} &&\beta \lesssim 13,\quad \gamma = 7/5. \end{eqnarray} \tag{ 21 }$$

These critical values of β will be compared to the actual radiative cooling time of fragments in different regimes in Section 6.

This calculation involves order unity coefficients which remain uncertain. Our primary goal is to establish the relevant physical parameters that must be evaluated to determine if fragments survive; in reality each value depends on the nonlinear evolution which follows linear instability. High-resolution hydrodynamic simulations will provide more insight than an analytic consideration of a range of fragment structures, densities, and cooling laws. Note that our use of β does not imply that the fragment cooling time must be equal to the background disk cooling time, although we expect the two to be quite similar just after the onset of fragmentation, which is the timescale addressed by this calculation.

Now we consider the opposite regime, in which cooling is slow, and the fragment undergoes Kelvin–Helmholtz contraction rather than free-fall collapse. We calculate the critical size, R_{H + p}, at which a pressure-supported fragment can withstand tidal shear. In the absence of pressure support, this radius would just be the Hill radius (or Roche lobe). The addition of pressure support is a small, though potentially significant, modification to this limit. Pressure partially negates the force of self-gravity, leaving the object more susceptible to disruption by tidal forces.

In binary stellar systems, modification of the Roche lobe due to pressure support is small because the scale height at the surface of the star is much smaller than the size of the star itself. For fragments, this is not the case—the pressure scale length of the gas is comparable to the size of the fragment and pressure should be considered. Thus we are justified in our approximation of the pressure gradient here as P/x. We calculate R_{H + p} for a compressible gas along the line of sight to the star, the direction in which tidal gravity is most disruptive.

Consider a fragment that has reached the Kelvin–Helmholtz contraction regime. The system is evolving on a cooling time, which we assume is substantially longer than a dynamical time. This implies that self-gravity, pressure, and tidal forces are roughly in balance:

$$\begin{equation} \frac{{\it GM}_{ \rm enc} \rho }{x^2} - \frac{P}{x} - 3\Omega ^2 x \rho = 0. \end{equation} \tag{ 22 }$$

Because the fragment is contracting slowly, it remains in approximate equilibrium

$$\begin{equation} \frac{{d}}{{d} x}\left(\frac{{\it GM}_{\rm enc} \rho }{x^2} - \frac{P}{x} - 3\Omega ^2 x \rho \right) = 0, \end{equation} \tag{ 23 }$$

and x(t) is an undetermined function describing contraction. For simplicity, we rewrite this expression in terms of the dimensionless variable B

$$\begin{equation} \frac{{d}}{{d} x}\left[1 - B^{-1} - \left(\frac{x}{R_H}\right)^3\right] = 0, \end{equation} \tag{ 24 }$$

where R_H = a(M_enc/3M_*)^1/3 is the Hill Radius.

We plug Equations (14), (15), and (22) into Equation (24) to calculate the contraction time as a function of τ_c and fragment size. Equation (24) reduces to

$$\begin{equation} \left(1-\left(\frac{x}{R_H}\right)^3\right)\left(\frac{t_{\rm coll}}{\tau _c}-3\gamma +4\right)+3\left(\frac{x}{R_H}\right)^3 =0. \end{equation} \tag{ 25 }$$

Substituting in for different values of γ we find

$$\begin{eqnarray} t_{\rm {coll}} = \left\lbrace \begin{array}{@{}cl}\displaystyle \tau _c \frac{4y-1}{y-1} \;\;\;&, \gamma = 5/3 \\\ms \displaystyle \tau _c \frac{16y-1}{5y-5} &, \gamma = 7/5, \end{array}\right. \end{eqnarray} \tag{ 26 }$$

where

$$\begin{equation} y=\left(\frac{x}{R_H}\right)^3. \end{equation} \tag{ 27 }$$

We can find the critical value of y for which the clump will keep contracting by finding where t_coll, given in Equation (26), changes sign: a negative contraction time signals expansion. This provides an estimate of the maximum size fragment that can be pressure-supported but not torn apart by tides:

$$\begin{eqnarray} &&{x} \le R_{\rm H+p}= \left(\frac{1}{4}\right)^{1/3} R_H, \gamma = 5/3 \end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray} &&{x} \le R_{\rm H+p}= \left(\frac{1}{16}\right)^{1/3} R_H, \gamma = 7/5. \end{eqnarray} \tag{ 29 }$$

To interpret the severity of the hurdle embodied in Equations (28) and (29), we must relate R_{H + p} to the initial size and mass of the fragment. A debate regarding the appropriate choice of these numbers is ongoing in the literature (see, e.g., Boley et al. 2010). The choices we make here are not the only sensible possibilities. We take the diameter of the collapsing region, D_coll, to be half of the Toomre wavelength, λ_T, at Q = Q₀ ≈ 1, envisioning that only half of the wavelength participates in forming an overdensity. Since λ_T = 2πH, D_coll = πH. We expect this length scale to be at a minimum 2H for collapse to be roughly symmetric in the vertical and planar directions.⁵ The radius of the collapsing region r_coll = D_coll/2 = λ_T/4 = (π/2)H, and the mass enclosed in the cylinder circumscribed by this radius is

$$\begin{equation} M_{\rm frag} = \Sigma _{Q=1} \pi (\lambda _T/4)^2. \end{equation} \tag{ 30 }$$

The Hill radius for a fragment with this mass is

$$\begin{equation} R_{H,\lambda _T} = \left(\frac{\pi ^2}{12Q_0}\right)^{1/3} H \end{equation} \tag{ 31 }$$

so that

$$\begin{equation} \frac{R_{H,\lambda _T}}{\lambda _T/4} = \left(\frac{2}{3\pi Q_0}\right)^{1/3} \approx 0.6. \end{equation} \tag{ 32 }$$

Thus, λ_T/4 cannot be an estimate of the initial radial extent of the fragment for any extended length of time. Rather, λ_T/4 likely represents the initial feeding zone for the fragment, not the fragment itself (Rafikov 2005). A collapsing fragment will only appear as a bound object when x ≲ R_H.

This interpretation is not an artifact of our choice of r_coll. In a Q = 1 disk, the only length scale on which one can create an object of mass Σπx² that lives within its own Hill radius is smaller than the disk scale height:

$$\begin{equation} R_{H, \rm {sc}} = \frac{H}{3Q}. \end{equation} \tag{ 33 }$$

Creating such a fragment is inconsistent as it implies that the vertical and radial extents of the fragment differ by a factor of three. Apparently, one cannot view fragments as chunks of the disk of radius x extracted as with a cookie cutter from a sheet of dough. Perturbations must grow from the Q = 1 state and contract.

Can such a contracting fragment achieve the radius required by Equations (28) or (29) before pressure stalls collapse? In general, answering this question requires following the nonlinear evolution of the fragment, incorporating changes in the cooling time throughout this process. Full consideration of this evolution is beyond the scope of the current work. However, we obtain some guidance by considering the collapse of a fragment in the adiabatic (i.e., infinitely slow cooling) limit.

Imagine that at the time of its initial linear collapse a fragment has virial ratio B = B₀ > 1 but −dB/dx < 0 so that pressure is growing faster than self-gravity. Assuming that rotational support does not become important (see Section 5), collapse will stall when B ≈ 1. The fragment will survive if tidal forces are not important at this stalling radius (i.e., if the stalling radius is smaller than R_{H + p}). For an adiabatic gas, P∝ρ^γ, so

$$\begin{equation} B \propto x^{(3\gamma -4)}, \end{equation} \tag{ 34 }$$

where we have assumed a constant M_enc so that ρ∝x⁻³. Hence, the fragment stalls when its radius has been reduced by a factor of approximately B⁻¹₀ for γ = 5/3 or B⁻⁵₀ for γ = 7/5.

For r_coll = πH/2 and Q₀ = 1, B₀ = π/2. The fragment will initially start collapsing. In a γ = 5/3 gas, the fragment only collapses to ∼60% of its original size in the adiabatic limit (cooling would allow additional collapse). However, for γ = 7/5, the collapse factor is nearly 10. In fact, to contract from λ_T/4 ∼ 1.7 R_H to 0.4 R_H (cf. Equation (29)) requires only a ∼34% enhancement of gravity over pressure (B₀ ∼ 1.34) in the limit of infinite cooling time. This large factor suggests that even if the cooling time of the fragment becomes long, if its equation of state is soft, it can survive. This further suggests that fragment survival is trivial in disks that have both soft equations of state (including γ = 7/5, appropriate for molecular gas) and strong external irradiation.

That the γ = 5/3 case is less promising for fragmentation is neither surprising nor as significant. We have only considered the adiabatic limit. Nevertheless we infer that fragmentation is much more likely to be inhibited by the Stalling Criterion for stiffer equations of state. Note that the slow cooling limit differs from the adiabatic case in that the former is not dissipationless. A truly adiabatic fragment could rebound after contraction. We use this limit merely to show that slowly cooling fragments can contract below the stalling radius for soft equations of state. Again, Equation (34) reveals that fragmentation is difficult to inhibit in radiation pressure-dominated disks with γ = 4/3. This result is consistent with recent simulations by Jiang & Goodman (2011).

Shlosman & Begelman (1989) argue that slowly cooling fragments remain susceptible to disruption by neighboring fragments if the disk breaks up into objects separated by an order of their own size, as expected from linear theory. Since the time to shear across a Hill radius at the Keplerian shearing velocity is of order an orbital period, fragments must contract on roughly the orbital timescale in order to avoid collisions. For a Q = 1 disk, the Keplerian shear energy at a separation of the Hill radius is of order the binding energy of a fragment, implying that encounters are likely disruptive. More generally, Shlosman & Begelman (1989) suggest

$$\begin{equation} \beta \lesssim \frac{\Sigma _{\rm Q=1}}{\Sigma }\approx 1, \end{equation} \tag{ 35 }$$

where Σ_{Q = 1} is the critical surface density for fragmentation. This relation derives from the proportionality between Σ and the wavelengths of growing modes in an unstable disk. Higher values of Σ provide a wider range of growing modes.

This argument ignores an important effect, which is that for soft equations of state (e.g., γ = 7/5), fragments with β > 1 can still collapse on an orbital timescale. Moreover, for fragments stalled at R_{H + p}, the binding energy should be larger than the collisional energy by a factor of two, so fragments may avoid collisional disruption (and may not collide at all) for soft equations of state.

Global simulations suggest that in systems with disk-to-central mass ratios ≳ 10%, fragments form in relative isolation within spiral arms (Lodato & Rice 2005; Boley et al. 2007; Kratter et al. 2010a), reducing the overall frequency of collisions. Finally, Goodman & Tan (2004) and Levin (2007) have argued that collisions might cause growth rather than disruption. These collisions have been observed in some global models (Boley et al. 2010; Stamatellos & Whitworth 2009).

The most likely candidate for disruption that we have neglected is rotation. Gauging the initial angular momentum of a fragment and predicting its evolution is challenging. If angular momentum is conserved in a collapsing fragment, the object will eventually become rotationally supported. A. Dekel et al. (2011, in preparation) report that fragments in isothermal disks collapse by a factor of 32/π ∼ 10 in radius before becoming rotationally supported. Boley et al. (2010) found that in SPH simulations with radiative cooling, the rotational support was only 20% of the gravitational binding energy.

In reality, subdisks eventually form around fragments, transporting away some fraction of the internal angular momentum. Since the disks are small, the viscous time is relatively short, about Ω⁻¹_disk/α_disk, allowing them to drain faster than the parent disk. Recent numerical work also suggests that gravitational torques can likely prevent the object from rotating faster than 50% of break up (Lin et al. 2011). It is also possible that in the process of angular momentum transport, some mass is carried away. We make no attempt to account for this.

If the above estimates are correct, then rotational stalling likely occurs at radii smaller than R_{H + p}, implying that rotational support does not inhibit fragmentation. However, more numerical work is needed to confirm this conclusion.

Even if a fragment can survive at its current location, migration may bring it closer to the host star and shrink R_{H + p} below the size of the fragment. Because migration-induced disruption occurs on timescales longer than a few orbital times, even initially free-falling fragments might in principle suffer from late disruption due to this process, depending on the extent of the migration (Vorobyov & Basu 2005; Boley et al. 2010).

Close interactions with spiral arms generate a tidal force roughly equivalent to that from the central star, so collapsing in free fall or shrinking to R_{H + p} should protect fragments from this means of disruption. In the vicinity of an arm of width λ_arm, the ratio of tidal gravity from the arm to tidal gravity from the star is approximately (Σ_armλ²_arm/M_*)(r/λ_arm)³ ∼ (H_arm/λ_arm)/(πQ_arm).

We consider the case of a fragmenting disk surrounding an A star. This choice is motivated by both theory and observations which suggest that A stars may have companions consistent with formation by GI (Kratter et al. 2008, 2010b; Hinkley et al. 2010). We calculate the equilibrium disk temperature for a Q = Q₀ = 1 disk, including both viscous heating and irradiation from the star.

We begin by setting α = α_sat, its saturation value for the gravitoturbulent regime, and we identify the irradiation-dominated regions within the disk given this choice. Because α_sat is not well determined, we consider a range of possible values. Because gravitoturbulence cannot support α > α_sat, disk regions which are irradiation-dominated for α = α_sat will be irradiation-dominated in all Q ∼ 1 disks. In contrast, regions of the disk which are viscously heated in this limit are not necessarily viscously heated at lower mass infall rates, $\dot{M}_{\rm inf}$, onto the disk, so after considering this limit, we turn to the lower values of α which can be generated by low $\dot{M}_{\rm inf}$.

We use the stellar irradiation model of Chiang & Goldreich (1997), which determines the incident flux on to the disk as a function of radius.⁶ The disk midplane temperature is found by solving the following equation:

$$\begin{equation} \sigma T^4 = \frac{3}{8}\tau _R F_\nu + \sigma T_{\rm irrad}^4, \end{equation} \tag{ 36 }$$

where the energy flux due to viscous dissipation in the midplane is

$$\begin{equation} F_\nu = \frac{3}{8\pi }{\dot{M}\Omega ^2}, \end{equation} \tag{ 37 }$$

the mass accretion rate through the disk when Q = 1 is $\dot{M}\approx 3 \alpha c_s^3/G$, and

$$\begin{eqnarray} T_{\rm irrad}& = &\left[\frac{1}{(7\pi \sigma _B)^2}\frac{k}{\mu }\frac{1}{{\it GM}_*} \right]^{1/7}L_*^{2/7}r^{-3/7}. \end{eqnarray} \tag{ 38 }$$

In the following plots, we use

$$\begin{eqnarray} T_{\rm irrad} &=& 40 \,{\rm K} \left(\frac{r}{50\,\rm {AU}}\right)^{-3/7} \left(\frac{L_*}{5\, L_\odot }\right)^{2/7}\left(\frac{M_*}{1.5\,M_{\odot }}\right)^{1/7}\!\!{,}\qquad \end{eqnarray} \tag{ 39 }$$

where μ = 2.33m_H and m_H is the mass of a hydrogen atom. From the tables accompanying Landin et al. (2009), a pre-main-sequence A star has intrinsic luminosity L_* ≈ 2.5 L_☉ for the ages of 1–3 Myr, with L_☉ equal to the current luminosity of the Sun. Infall is much more likely to drive disks unstable at earlier times, and at (3–8) × 10⁵ years, the intrinsic luminosity of a 1.5 M_☉ star is approximately 7 L_☉. Accretion luminosity will exceed the intrinsic stellar luminosity for accretion rates onto the star ≳ 10⁻⁷ M_☉ yr⁻¹. The stellar accretion rate may or may not match the mass infall rate $\dot{M}_{\rm inf}$. We choose L_* = 5 L_☉ as an intermediate value. We use the cooling time defined in Equation (3).

To derive the most optimistic limits on fragmentation at small masses, we imagine that dust grains have grown to ∼300 μm so that the opacity κ ≈ 0.24 cm² g⁻¹ is constant with temperature (see Kratter et al. 2010b for a discussion of this choice and why it provides optimistic limits at small fragment masses). Given this disk model, we find the characteristic fragment mass Σπr²_coll for the disk where fragments can free-fall collapse. Again to generate the smallest possible fragment masses, we use r_coll = πH/2 (cf. Section 4.3; Boley et al. 2010).

Figure 2 shows the temperature profiles, critical radii where the cooling criteria are satisfied, and initial fragment masses for a disk with α_sat = 0.05. We show the corresponding temperatures, radii, and fragment mass scales for α_sat = 0.005 and 0.1 in Figure 3. For α_sat > 0.05, which is expected for GI-driven turbulence, the location at which the disk becomes irradiation-dominated is effectively the same as the fragmentation boundary typically calculated from the Viscous Criterion.

For lower values of α_sat fragmentation is pushed inward, and there are disk locations that remain irradiation-dominated, but where cooling prohibits free-fall collapse. Fragmentation is more uncertain in these regimes, but plausible if α_sat ever takes such low values.

Recall that the critical value α_sat depends on the transport mechanism, so in irradiation-dominated regions where global modes control transport, the saturation value for angular momentum transport will be different from the value set by gravitoturbulence and likely depends on disk mass (Laughlin et al. 1997). Low-mass, irradiation-dominated disks with low infall rates might be able reach Q₀ as discussed in Section 3.3. In this case, fragmentation could be pushed significantly closer to the star.

To determine whether this low disk mass, low accretion rate regime might be relevant, we calculate the equilibrium temperature for a Q = 1 disk given a constant accretion rate (not fixed α_sat), and find the irradiation-dominated regions of the disk. In Figure 4 we show the temperatures, disk masses, fragment masses, and corresponding values of α for three different accretion rates. We must compare these with α_eff generated by the MRI, as noted below.

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As discussed in Section 3.3, Q = Q₀ is only likely to be achieved for irradiated disks with M_d/M_* = H/r ≲ 0.1, evaluated at r = r_out, where this critical value of 0.1 remains substantially uncertain. Using the Chiang & Goldreich (1997) irradiation temperature model

$$\begin{eqnarray} \frac{H}{r} &\approx & \left[\frac{1}{(7\pi \sigma _B)^2}\left(\frac{k}{\mu }\right)^8\right]^{1/14} \left(\frac{1}{GM_*}\right)^{4/7}L_*^{1/7} r^{2/7} \nonumber \\ &\approx & 0.07 \left(\frac{M_*}{M_\odot }\right)^{-4/7} \left(\frac{L_*}{5\,L_\odot }\right)^{1/7} \left(\frac{r_{\rm out}}{50\;{\rm AU}}\right)^{2/7}, \end{eqnarray} \tag{ 40 }$$

where we have assumed that the disk is optically thick at the outer edge when Q ∼ 1. From the pre-main-sequence evolutionary tracks of Landin et al. (2009), stars with M_* < 2 M_☉ have intrinsic luminosities L_* ≈ 3.5 L_☉(M_*/M_☉)^1.6 at an age of (3–8) ×  10⁵ years and L_* ≈ 1.4 L_☉(M_*/M_☉)^1.5 at 1–3 Myr. These scalings suggest that at a given age, the value of H/r decreases somewhat with increasing stellar mass, implying that spiral arm-driven transport may be more effective for disks around G stars than for those around A stars, particularly when low accretion rates do not generate significant stellar accretion luminosity.

Figure 4 illustrates that disks can plausibly be low enough in mass to avoid global instabilities, but require more angular momentum transport than provided by the MRI. The quantity α_eff is averaged over the entire disk column—in dead zones experiencing only surface accretion, the α_eff which can be supported by the MRI can be quite low (Gammie 1996). In this case, the disk could be driven unstable while accreting at a lower rate than required in the viscous regime. For such low rates to drive fragmentation, the disk must already be near Q = 1 such that low accretion rates need not significantly alter the disk mass.

Since the exact criterion for global mode transport is unknown (see Section 3.3), it is difficult to evaluate the importance of this regime. We suggest that future simulations search for the maximum mass at which Q ∼ 1, irradiated or isothermal disks do not show spiral arm-driven transport.

We have shown regimes in which planetary mass fragments might form and survive. We emphasize that newly born fragments must still be prevented from accreting surrounding disk material if they are to remain low in mass (Kratter et al. 2010b). The isolation mass for such fragments remains in the brown dwarf to stellar mass regime.

Contrary to the protostellar case, the question of interest in galactic scale disks is "can fragmentation be prevented?" rather than "can fragmentation be relevant for planets?" Relaxing the cooling criterion in the strongly irradiated portions of disks is of little import as these regions already satisfy the more restrictive Viscous Criterion for large α_sat, even at a fraction of a parsec (Goodman 2003; Levin 2007). If galactic disks are irradiation-dominated at radii as small 0.01–0.1 pc, fragments still collapse in free fall for modest black hole masses (10⁶ M_☉). The inner 0.01–0.1 pc of hotter, more massive AGN disks (M_BH = 10⁷–10⁸) are likely dominated by radiation pressure at these radii, shifting to γ = 4/3, where free-fall collapse is ensured. Recent numerical simulations by Jiang & Goodman (2011) are consistent with this analysis: they find fragmentation at long cooling times when radiation pressure dominates.

This work should be a useful guide for future numerical simulations intended to study fragmentation in irradiation-dominated disks. Two major uncertainties limit our conclusions for protostellar disks.

• 1.  
  What is the value of α_sat in a viscously heated Q ∼ 1 disk?
• 2.  
  What is the maximum GI-driven transport rate as a function of H/r in an irradiated Q ∼ 1 disk?

These two uncertainties are effectively the same question, but since the physics of transport in these regions differs, they must be considered independently. Simulations will also be required to confirm our conclusion that regardless of whether they collapse in free fall, fragments can likely survive if γ = 7/5 because they collapse to smaller than R_{H + p} on an order of a free-fall time. Hence, disks with soft equations of state must simply reach Q = Q₀ to fragment. Confirmation of this conclusion will require a full treatment of radiative transfer and disk opacity. As noted earlier, the impact of fragment rotation must also be explored. Current state of the art simulations of disks with radiative transfer do not yet cover the range of parameter space required to answer the above questions (Boss 2007; Cai et al. 2008, 2010; Boley et al. 2010; note that the latter two authors have questioned the conclusions of the former based on differing treatments of the photosphere).

Answers from such simulations will enable us to distinguish between the following outcomes in each regime.

• 1.  
  Viscously heated regions. If α_sat ≳ 0.1, then standard applications of the cooling time criterion give accurate results. If α_sat ≲ 0.1, then the standard Viscous Criterion will determine the boundary for fragmentation, but the critical cooling time will be longer.
• 2.  
  Irradiation-dominated regions. If both the maximum GI-driven transport rate and α_eff due to MRI are too low to process the infall rate onto a protostellar disk, fragmentation may be possible for thin disks at infall rates lower than ∼c³_s/G. If this occurs, low-mass fragments which do not experience substantial growth might be produced. Otherwise, high accretion rates are required to drive these disks unstable.