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Please excuse an amateur question. While trying to think of anything but what was happening during a dental procedure my mind turned to a model of a star close to a rotating black hole and the effects on the drawn in matter.

While it is obvious such matter would be excited to high temperature could the combination of rotation and excitation be enough to induce a sustained fusion reaction?

If so would this produce enough energy to maintain a fusion 'ring' at the event horizon - essentially a doughnut star?

Would there be enough of a reaction to start producing lighter elements?

Pure curiosity generated by an attempt to distract myself

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    $\begingroup$ Good question anyway, but ++1 for astronomy as a distraction at the dentist! $\endgroup$ – Chappo Apr 19 '18 at 8:16
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    $\begingroup$ I want to say yes, fusion happens in the accretion disk due to the very high orbital speeds and crushing of matter falling into what's comparatively a very tiny astrological object, at least, around a stellar mass black hole. Any fusion energy released is considerably lower than the potential energy of any in-falling matter, so even if fusion happens readily it would only contribute to a small percentage of the gamma rays that escape from the disk. "Donut star", I think we should stick with accretion disk, as it's more violent than a star. As I'm not really sure, I'm just going to comment. $\endgroup$ – userLTK Apr 20 '18 at 2:13
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The accretion of material onto (into) black holes (and neutron stars) provides environments that are both very hot and (relatively) dense. Under these circumstances it is possible for nuclear fusion to occur, the question is whether this is significant, both energetically or as a means of producing new chemical elements (nucleosynthesis).

The answer to the first of these questions is relatively straightforward. As material falls towards the black hole, its angular momentum forces it to form an accretion disk. Viscous processes heat the disk and provide torques, cause the material to lose energy and angular momentum and eventually allow it to fall into the black hole. Much of the gravitational potential energy (GPE) gained as the material falls towards the black hole does end up heating the material.

The innermost stable circular orbit of a black hole is at 3 Schwarzschild radii $=6GM/c^2$, where $M$ is the black hole mass. The GPE released for material of mass $m$ falling to this radius is $\sim GMmc^2/6GM = mc^2/6$. i.e. fully one sixth of the rest mass energy of the material could be released as heat.

Compare this with nuclear fusion. The fusion of hydrogen into helium only releases 0.7% of the rest mass as energy that can heat the accretion disk.

So from the energetic point of view, fusion reactions are negligible, unless they can occur much further out in the disk

The question about nucleosynthesis yields is more complex. The more massive a black hole and the higher the accretion rate, then in general the higher the disk temperature and density and the higher the fusion rate. But it also depends on the details of the cooling processes that are possible and how much material is advected into the black hole. Hu & Peng (2008) present some models of accretion onto a 10 solar-mass black hole and suggest that it may be possible to produce certain rare isotopes by this mechanism. Stellar-sized black holes probably need very substantially super-Eddington accretion rates to achieve the necessary temperatures to sustain nuclear fusion (i.e. much greater accretion rates than are possible by radiation-pressure opposed spherical accretion flows), according to Frankel (2016). Such rates are likely only in the cases where black holes disrupt a binary companion, rather than through a steady accretion flow.

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    $\begingroup$ I took note of "fully one sixth of the rest mass energy of the material could be released as heat". $\endgroup$ – John Duffield Apr 20 '18 at 15:17
  • $\begingroup$ @JohnDuffield Maybe I should have said "up to 1/6", since some can obviously be advected into the black hole. $\endgroup$ – Rob Jeffries Apr 20 '18 at 16:31
  • $\begingroup$ maybe you should have said up to 1/1! $\endgroup$ – John Duffield Apr 21 '18 at 13:58
  • $\begingroup$ @JonDuffield The most efficiently that rest mass energy can be converted into heat/radiation for a non-rotating black hole with an accretion disc is actually about 6%. It can rise to a maximum of 42% for a mximally spinning black hole. $\endgroup$ – Rob Jeffries Apr 21 '18 at 15:03
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Heat in accretion disk happens due to friction and friction happens only when there is relative motion. So in that accretion disk a lot of particles are moving relative to each other with high velocities, so fusion should not happen, because for that particle should come together. Even in star (like our sun), mass of star is not enough to produce fusion, and it needs the help of quantum tunneling, so we cannot say that pressure is available within that accretion disk so as to overcome repulsion of nuclear force.

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    $\begingroup$ It may be more helpful to talk about density and temperature instead of pressure. The temperature determines how many of the particles have enough energy for fusion and the density affects the overall rate. It is possible to have fusion at pressures that are much lower than in stellar cores, e.g. in man-made fusion reactors. $\endgroup$ – Hannes Apr 20 '18 at 10:29

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