4.4 Non-axisymmetric nuclei

The estimates made above were based on spherical models of nuclei. The total number of stars in a full loss cone can be much larger if the nucleus is flattened and axisymmetric [124 ], when only one component of the angular momentum is conserved. In very flattened nuclei (with ellipticities $e ~ 0.5$ ), single emptying of an initially full loss cone can in some cases be sufficient to drive the binary to coalescence [228

]. However loss cone dynamics can be qualitatively different in non-axisymmetric (triaxial or bar-like) potentials, since a much greater number of stars may be on “centrophilic” - box or chaotic - orbits which take them arbitrarily near to the SBH(s) [159 , 67 , 196 , 212 , 228 ]. Stars on centrophilic orbits of energy $E$ experience pericenter passages with $r < d peri$ at a rate $~ A(E)d$ [143 ]. If the fraction of stars on such orbits is appreciable, the supply of stars into the binary’s loss cone will remain essentially constant, even in the absence of collisional loss-cone refilling. Such models need to be taken seriously given recent demonstrations [169 , 90 , 170

] that galaxies can remain stably triaxial even when composed largely of centrophilic orbits. Furthermore imaging of galaxy centers on parsec scales reveals a wealth of features in the stellar distribution that are not consistent with axisymmetry, including bars, nuclear spirals, and other misaligned features [221 , 164 , 40 ].

The total rate at which stars pass within a distance $Ra$ of the massive binary is

where $Mc(E)dE$ is the mass on centrophilic orbits in the energy range $E$ to $E + dE$ . In a nucleus with $r ~ r-2$ , the implied feeding rate into a radius $rinfl$ is roughly

--s3 M*~~ fc-- (38) G ( )3 ~~ 2500M yr- 1f- ---s----- , (39) o. c 200km s3

where $-- fc$ is the fraction of stars on centrophilic orbits. If this rate were maintained, the binary would interact with its own mass in stars in a time of only $5 ~ 10 yr$ , similar to the decay time estimated above (Equation 23

) for a binary in a fixed background. In fact, the feeding rate would decline with time as the centrophilic orbits were depleted. Solving the coupled set of equations for $a(t)$ and $Mc(t)$ , one finds that at late times, the binary separation in a $r oc r-2$ nucleus varies as [170 ]

Comparison with Table 2 shows that this is the same time dependence as for the “full loss cone” regime of spherical nuclei. Placing just a few percent of a galaxy’s mass on centrophilic orbits is sufficient to overcome the final parsec problem and induce coalesence, if the stellar density profile is steep and if the chaotic orbits are present at all energies. This example is highly idealized, but shows that departures from axial symmetry in galactic nuclei can greatly affect the rate of decay of a binary SBH.