How does gas in accretion disks of supermassive black holes create orbital torques on smaller black holes within the disks causing them to migrate?

Question

This answer to What enhances the capture and merge rates of pairs of small black holes orbiting around supermassive black holes? links to Migration Traps in Disks Around Supermassive Black Holes which includes the following sentence in the introduction:

Migration toward trapping orbits may be such a mechanism. Objects orbiting within differentially rotating disks exchange angular momentum with the gas around them as they orbit, which results in a torque, typically causing the objects to migrate.

Question: How exactly do density gradients in accretion disks of supermassive black holes create torques on smaller black holes orbiting within the disks? Is it through simple gravitational forces, GR effects (maybe some "spin-orbit coupling" thing?), aerodynamic drag, accretion by the small black hole in a velocity gradient, something else, or some combination of these?

One sentence in the linked answer is

These "migration traps" (Bellovary et al. 2016) are caused by torques exerted on the orbiting objects (in this case black holes) caused by gas accretion.

and I think it means that the orbital torque (torque on the small black hole about the SBH's center) comes from gas accretion by the small black hole. If this is the case, is it possible to elaborate a bit?

Answer 1

Not really my area, but this question is probably related to the planetary migration in circumstellar disks. In this case, the migration is caused by gravitational interactions between the planet and the gas in the disk. There are two explanations for this effect

• Impulse approximation: consider a parcel of gas in the corrotating frame. If the gas is close enough to the orbit of the planet in an internal orbit, it will eventually reach the planet and its gravitational pull will move it to an external orbit. From the change of velocity in the gas, you can calculate the torque exerted to the planet. This approach is relatively simple in terms of math, but uses a lot of assumptions which are not true. However, it can be used to get an order of magnitude estimation.

• Resonant torques: Decompose the torque exerted to the planet in the sum of individual torques at resonant locations in the disk. Much more complicated, but better results.

You can find the complete deduction in Astrophysics of Planet Formation by Armitage, or the compact version in this paper.

There are surely differences between planets and black holes (GR corrections, larger accretion rate, etc.), but I'd assume the physical principles are basically the same

Answer 2

A gravitating object in a disk creates a wake on the inner edge of its orbit as well on the outer edge of its orbit. The inner wake is the leading wake, while the outer wake is trailing. The torques created by these two wakes would cancel, if the disk had no shear. Yet disks in astrophysical contexts are subject to Keplerian shear - and thus the torques excerted by the two wakes are not exactly symmetrical so that momentum transfer happens from disk to object and vice versa which can lead to migration. This is a phenomenon often observed in protoplanetary disks and as such often discussed and analysed in that context:

Looking at in detail, there's at least three different types of migration for an object embedded in a disk, type 1 is the linear case for an object of small mass, type 2 and 3 deal with a heavy object and mass flow through the gap. These cases are discussed in some detail in this paper by Papaloizou et al and many others like also the Armitage papers as quoted by Pablo Lemos in the other answer.