Spherical Accretion Mass inflow Rate

[PhotonKing]

Hello everyone I am a new user around here. I have been attempting to understand spherical accretion and create a simple model of that process but I am running into a problem. This is getting at the heart of the issue. So far I have come up with two concentric spheres, the outer sphere is the shell of gas the inner sphere is a black hole. I have assumed that the shell sphere will shrink in radius at a rate determined by gravity and it's distance from the center of the black hole. So the accretion shell is not self gravitating. My trouble is figuring how much mass is actually accreted onto the black hole, ie the accretion rate. Any help or hints would be extremely helpful. I have given a link to my work so far in a google document. Let me know what you all think. I am doing this to better understand Bondi accretion. The radius of the black hole depends only on mass, so if I know the mass flow rate I should be able to solve these equations for, Volume of the shell as a function of time, Density of the shell as a function of time, Recession velocity of the shell's collapse as a function of time, perhaps even pressure and temperature. Any help would be hot. Here's a link to my google doc, this isn't a homework question it's a question about research I'm involved in. I could simply take the result from the Bondi paper but i'd like a better understanding of how they got to that result. I have been stumped at finding a solution to this for a while now.

The result of the accretion rate given in the paper is:

dM/dt=4πr²vρ

Link to the paper: https://drive.google.com/drive/folders/0B-bI1gdIXqgrSnZsdHlZb1BUdTg

My work:

https://docs.google.com/document/d/1i-A8WVXsmkPKDAVt77Xu_Xpg2pu1dI07nzlmqB4FsEw/edit?usp=sharing

Thanks

 

[AndyW]

for sharing your work so far on this topic! It seems like you have a good understanding of the basics of spherical accretion and have made some progress in creating a simple model. However, as you mentioned, the challenge lies in determining the actual accretion rate onto the black hole.

To calculate the accretion rate, you will need to consider the mass flow rate, which is the amount of mass passing through a given area in a given time. This can be calculated using the density, velocity, and surface area of the accretion shell. The equation you provided from the paper, dM/dt=4πr2vρ, is a good starting point. It relates the accretion rate (dM/dt) to the density (ρ), velocity (v), and surface area (4πr2) of the accretion shell.

You may also want to consider the Bondi radius, which is the distance from the black hole at which the accretion flow becomes supersonic. This can help you determine the size of your accretion shell and the velocity of the gas particles at different distances from the black hole.

In addition, you may want to take into account the effects of pressure and temperature on the accretion process. These can affect the density and velocity of the gas particles, which in turn will impact the accretion rate.

I would recommend further researching the equations and theories behind Bondi accretion, as well as consulting with other scientists in the field. It may also be helpful to run simulations or experiments to test your model and make adjustments as needed.

Overall, your approach to better understand Bondi accretion through your own research is commendable. Keep up the good work and don't hesitate to reach out for help or clarification when needed. Good luck!