Assumptions

Axisymmetric: As yet, there is no firm observational evidence for a circumbinary disk around a SMBH binary. Though certain aspects of the formation of these disks have been studied, a detailed understanding of their formation is currently beyond current numerical abilities. It is thus generally supposed that when gas is present, a gaseous disk will form based on roughly the same physics that would yield a standard accretion disk in the case of a single SMBH. In a binary, however, torques are exerted on the inner regions of the disk by the binary. [20], [36], [21], [22], and various other authors, have shown that when such a disk is co-rotating with the binary these torques inject angular momentum into the disk at resonant radii and effectively “push” material outwards. This process evacuates a region surrounding the binary whose size depends on the mass ratio of the binary. In the case of equal mass mergers [21] used simulations to model these circumbinary disks and showed that the inner radius of the disk,rinner, is about twice the binary separation. This was also recently

shown analytically by [22]. When merger timescale of the binary becomes shorter than the viscous timescale of the disk, the disk effectively decouples from the binary, and diffuses inwards on a viscous timescale.

Following decoupling, the disk will evolve viscously in what will effectively be an axisymmetric gravitational potential. Then, by the time of merger, viscosity will have smoothed the disk to a state which is, to reasonable approximation, axially symmetric.

rinner= 100rS: Though we assumerinner= 100rSin our simulations, as discussed in Chapter 2,

the analytic solution is self similar and scales, along with all of the relevant positions and times in the flow, with wherever the peak of the density distribution is located. Thus, should the true inner radius of the disk be farther in or out, one need only scale our results be the appropriate factor such that the two inner radii match.

This point is made abundantly clear in Sec. 3.4.2 where our simulations using these dimen- sionless variables compare favorably, after scaling, with the 3D relativistic magneto-hydrodynamic simulations run by [23] withrinner= 10rS.

Our choice ofrinner= 100rSis appropriate for an equal-mass ratio merger [22, 36, 78]. However,

as the mass ratio q _≤ 1 decreases, rinner will also decrease, until gas can eventually accrete onto

the SMBHs [22]. For example, [22] find that, for 108_M

⊙ black hole, a gap will form in the disk for

mass ratios greater than q = 3_×10−3_{. In this case the binary decouples from the disk at binary}

separation rbin ≈ 15. They consider several mass ratios, and find that the radius of decoupling

changes smoothly withq: decoupling occurs at rbin≈24 forq= 10−2 andrbin≈56 for q= 10−1.

Even in these extreme cases, the response of the disk exterior to the binary will undergo a response to what we describe here.

Note that as the mass ratio becomes more extremal, the mass decrement occurring in less than a dynamical time of the inner disk decreases for two reasons: The first is that as rinner decreases,

the dynamical time of the disk decreases (_∝r_inner3/2 ), so that only a smaller fraction of the mass loss occurs in less than a dynamical time. Moreover, as the mass ratio becomes more extremal the total mass-energy radiated during inspiral, merger and ring down also decreases. In fact, in the limit of q_�1 both of these points are accentuated, because, in contrast to comparable-mass mergers, only a negligible amount of the mass-energy_∝q2 _{is radiated during the final plunge, merger, and ring}

down.

Newtonian: We do not consider relativistic effects in our simulations. According to [22], as long as q≥10−2 _{this leaves r}

inner ≥12rS. Note that although these values are calculated for a SMBH

primary mass of 108M⊙, [36] showed thatrinnerhas only a weak dependence on the primary’s mass.

Thus, general relativistic effects are negligible.

Thin Disk: In most circumstances, the disk is predicted [36] to be cool enough to be described using the well-developed theory of geometrically thin disks, where the scale height h, is less than the radial distancer.

We consider a disk in the central potential of a mass M0, and model the disk as geometrically

Because the dependence onN is relatively weak, to simplify the analysis in this chapter we have chosen N = 1 for all of our simulations. This makes the effects of the varied parameters clearer. Moreover, it is then also the case that

h r = ˜ h ˜ r =H . (3.2)

One-Dimensional: Because we are assuming an axially symmetric disk (see first assumption above), which is geometrically thin (previous assumption) the problem is reduced to one dimension. We ignore turbulence and MHD effects. In 1D, the waves and shocks in the disk necessarily propagate purely radially, with no vertical variation or refraction. This is unlikely to be accurate over many dynamical times, but as is corroborated in Sec. 3.4.2, it should be reasonable on the short timescales of interest to us.

Non-Self-Gravitating: We assume that the mass of the inner regions of the disk is significantly less than the mass of the SMBHs. This assumption is consistent with the results of [22] which find disk masses with masses around 103_M

⊙.

Impulsive: The disk responds adiabatically to mass-energy loss of the central binary over timescales longer than the dynamical time, slowly expanding but remaining circular to accommodate the change (see Chapter 2 for a more in-depth discussion of this). However, the mass-energy lost on timescales shorter than the dynamical time is perceived by the disk to be an impulsive change to the central potential and leads to the effects discussed in this chapter. Thus, we changed the central mass instan- taneously at the beginning of every simulation with initial conditions appropriate for a steady-state disk orbiting the original mass.

Adiabatic: For simplicity, our simulations do not account for radiative losses, and thus strictly apply only to disks whose cooling time is longer than the timescales considered.