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tokens: employ the same beta cooling prescription with , and determine the fragmentation boundary, , as a function of the adiabatic index γ . Table 1 shows agreement between and . The match is remarkable, especially with the global 3D simulations of Rice et al. ( 2005 ), since Equation ( 54 ) is derived for 2D disks in the local limit. Table 1 Characteristic Cooling Times as a Function of γ 4.1.2. <inline-formula> <tex-math> </tex-math> <inline-graphic xlink:href="apj523559ieqn106.gif"></inline-graphic> </inline-formula> Standard beta cooling with corresponds to “thermal relaxation”: the temperature is restored to its initial value over the cooling time (Lin Youdin 2015 ; Mohandas Pessah 2015 ). In this case no additional heat source need be invoked to define an inviscid steady state. From Equation ( 47 ), the instability condition is . This is the same as the classic Toomre condition for an isothermal disk (which may be obtained from Equation ( 46 ) by taking ). In this respect, a fully irradiated disk, in which the equilibrium temperature is set externally, behaves isothermally regardless of the cooling time (Gammie 2001 ; Johnson Gammie 2003 ). 4.2. Viscous Disk We now consider a viscous disk with parameters in our adopted viscosity law, Equation ( 15 ). In the 2D case this implies is constant. We check in Appendix B that our dispersion relation reduces to previous results for viscous GI in the isothermal limit (by taking ). It is useful to consider several limiting cases. To see the effect of cooling and irradiation, we simplify the dispersion relation, Equation ( 45 ), by assuming . Then for we findwhich coincides with Gammie's Equation ( 18 ) for vanishing wavenumber. For we findFor a rough measure of the maximum growth rate can be obtained by equating Equations ( 55 ) and ( 56 ). 5 5 If θ is not small and/or Q is large then one may just use Equation ( 55 ) to maximize S over K , see the curve in the bottom panel of Figure 2 . This exercise yields To compute growth rates numerically, we consider a model with , and given by thermal equilibrium (Equation ( 24 )). Furthermore, we relate the strength of self-gravity and viscosity byto mimic a gravito-turbulent basic state, where one might expect the dimensionless stress (Lin Pringle 1987 ). Figure 2 shows growth rates as a function of the wavenumber obtained from the dispersion relation Equation ( 45 ). The limiting behavior for small/large K are well-captured by Equations ( 55 ) and ( 56 ). Comparing the two panels shows that increasing the irradiation level ( θ ) suppresses small-scale perturbations. Figure 2. Growth rates for the 2D viscous problem as a function of the radial wavenumber, k , for a range of cooling times, β . The dashed and dotted lines correspond to asymptotic behaviors for small and large k , respectively, computed from Equations ( 55 ) and ( 56 ). Top: without a floor temperature (); bottom: with a floor temperature set to of the equilibrium temperature. Figure 3 shows the