Accretion for Gamma > 5/3 | Black Hole Solutions

[Vrbic]

Hello,
I would like to know, what happen with accretion (cloud) when drop toward a black hole. In Bondi case and terminology. If I understand:
1) We describe the cloud falling to star (black hole) by EoS $P(\rho)=K\rho^\Gamma$.
2) This accretion is steady it means $\dot{M}=4\pi r^2u\rho=const.$.
3) Accretion to black hole is possible just for supersonic speed => solution of Bondi equation has to come through critical sound speed point.
4) For $\Gamma>\frac{5}{3}$ doesn't exist critical point and accretion is not supersonic => no steady accretion on black hole for cloud with $\Gamma>\frac{5}{3}$.
Are these 4 claims right?
Bondi solution is dealing just with $\Gamma<\frac{5}{3}$. It is quit hard constrain.
My question are:
a) What is the matter with cloud with $\Gamma>\frac{5}{3}$ during falling to black hole? Reaches the event horizon?
b) How do we must change in assumptions for dealing with cloud with $\Gamma>\frac{5}{3}$?
I was checking in literature for this part of problem, but I didn't find anything important. Please let me know, comments or links.
Thank you.

 

[Ambitwistor]

Hello,

Thank you for your question. Accretion is a complex process that involves the movement of matter towards a central object, such as a black hole. In the case of Bondi accretion, the matter is accreting onto a non-rotating black hole.

To answer your first question, when a cloud of matter falls towards a black hole, it will eventually reach the event horizon. This is because the gravitational pull of the black hole becomes stronger as the cloud gets closer, and at the event horizon, the escape velocity becomes equal to the speed of light. This means that the matter will fall into the black hole and cannot escape.

Your second and third claims are correct. The accretion rate, \dot{M}, is constant in the Bondi accretion model, and for accretion onto a black hole, the matter must be traveling at supersonic speeds. This is because the critical point, where the speed of sound becomes equal to the escape velocity, is necessary for the Bondi solution to hold.

For your fourth claim, it is true that for \Gamma>\frac{5}{3}, there is no critical point and the accretion will not be supersonic. This means that the Bondi solution cannot be applied in this case. Instead, other models, such as the Hoyle-Lyttleton accretion model, must be used.

To answer your additional questions, for a cloud with \Gamma>\frac{5}{3}, the matter will still fall towards the black hole, but it will not reach the event horizon. It will instead form an accretion disk around the black hole. In this case, the assumptions for the Bondi model must be modified to account for the different behavior of the matter.

I hope this helps answer your questions. For more information on accretion onto black holes, I suggest looking into papers on the Hoyle-Lyttleton model or the more general Bondi-Hoyle-Lyttleton model. Thank you for your interest in this topic.