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As almost every source that I'd found explains, it was a challenge for scientists to understand the mechanism by which the total angular momentum in the accretion disc was conserved, while taking the latter (total conservation of angular momentum) in an accretion disc as a matter of fact. However, clearly, if such a large amount of friction as that it can generate enough light to outshine a thousand of Milky Ways (and a lot more X-ray radiation and heat) is present, it will radiate away enough of the accretion disk's energy for it to slow down and drop orbits, which is essentially losing angular momentum to radiation? Or is there any other way we know the total angular momentum of the accretion disc has to be conserved?

As you can probably see, I am not very advanced in this area of physics, so a simple as possible answer (quickly deducable or intuitive equations - or anything up to A-level (maybe basic university) physics - count as being simple) would be largely appreciated.

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This is a curious question, since stars, black holes, white dwarfs, quasars etc do accrete matter and do find a way to transfer orbital angular momentum outwards whilst transferring mass inwards and that process does occur because of viscosity, but the angular momentum is not transported by radiation as you suggest.

The energy radiated comes about because of the inward mass flow, not vice-versa. The amount of angular momentum lost in the form of light is totally negligible. To carry angular momentum the light would need to be circularly polarised. Circular polarisation of light from sources such as quasars is quite limited (e.g. see Hutsmekers 2010) and even if the light were 100% polarised it could only carry $\hbar$ of angular momentum per photon. Thus, the angular momentum per energy unit would be $\omega^{-1}$, where $\omega$ is the frequency of the light, compared to the angular momentum per energy unit of a mass in a circular orbit of $2\Omega^{-1}$ (e.g. see here), where $\Omega$ is the orbital frequency. Since $\omega \gg \Omega$, then the angular momentum of the light is irrelevant - a body in orbit that loses energy just by emitting light loses a negligible amount of angular momentum and therefore cannot move inwards by this mechanism alone. To put it another way, a hot body in orbit that radiates away its thermal energy just turns into a cold body in the same orbit.

To fall into a lower orbit then a torque needs to be exerted because the specific angular momentum of an object in orbit is $\propto r^{1/2}$. Angular momentum is transferred outwards by torques associated with viscosity - at the microscopic level this is still poorly understood. Material in closer orbits moves faster than material further out. Turbulence or magneto-rotational instabilities may be capable of supplying the required viscosity. Angular momentum can also be lost from the system as a whole via winds from the disc surface or rotating jets propelled along the rotation axis; but not by electromagnetic radiation.

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    $\begingroup$ Quick pedantic note that not all angular momentum of light comes from its polarization - it can have orbital angular momentum as well. This is also irrelevant, as the emission is incoherent, and there's no chance to create a meaningful corkscrew wavefront around the rotation axis in the far field. $\endgroup$ – Emilio Pisanty May 9 '17 at 19:44
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    $\begingroup$ That said, thermal radiation along the orbital plane is Doppler shifted differently when emitted forward vs backward, by an amount proportional to $\Omega$. Dos this not introduce the possibility of nonzero light OAM contributions? $\endgroup$ – Emilio Pisanty May 9 '17 at 20:17
  • $\begingroup$ @EmilioPisanty There indeed is this kind of effect but it is a relativistic correction which is negligible unless the fluxes are very large. It appears for instance in the Page-Thorne paper eq. (27), compare with (2.3) in Shakura-Sunyaev. $\endgroup$ – Void May 9 '17 at 23:18
  • $\begingroup$ @EmilioPisanty I was going to post something along the lines of it is probably a small, but non-negligible issue for material at the ISCO and not at larger radii, but maybe I'll look at Void's sources first. $\endgroup$ – Rob Jeffries May 9 '17 at 23:57
  • $\begingroup$ @Rob sure. It would also be interesting to know what form that angular momentum takes in the far field of the radiation, but maybe that's worth its own question. $\endgroup$ – Emilio Pisanty May 10 '17 at 0:36

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