Non-Dimensional Accretion Rate λ: Overview and Questions

[Vrbic]

Hello, I have question about "so called" non-dimensional accretion rate, mostly known as λ. In Shapiro,Teukolsky 14.3. is made an overview but, λ is just defined, but I don't know why? Why do we need it? What is good for?

 

[Vrbic]

I have another problem with derivation of this accretion. Somebody who did it by his self and understand all?

 

[PeterDonis]

Vrbic, can you give a link to a reference for what you are talking about?

 

[Vrbic]

From Press, Teukolsky - Black holes, white dwarfs and neutron stars. Chapter 14.3. Hydrodynamic spherical accretion. I would like to recalculate all and produce same graph ( u/a vs. r) as is there. Theoretically, I understand all they did, but I have problem with the graph. From which eq. and how exactly solve it.
Thank you for reply.

 

[PeterDonis]

Unfortunately I don't have that book. Someone else here might, or you could try to post the particular equations you are concerned about.

 

[Vrbic]

Unfortunately I don't have that book. Someone else here might, or you could try to post the particular equations you are concerned about.

Here is picture few relevant pages. https://www.dropbox.com/s/3scz1yphi91skdu/Teukolsky 14.3..png?dl=0 I am talking about figure 14.1. at the end.

 

[Calion]

Bondi's original paper (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1952MNRAS.112..195B&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf ) investigates steady state solutions for spherically symmetric accretion (e.g. ρ(r), v(r) not changing with time). It turns out that there is not a unique solution to this problem, and you are left with this free parameter λ. However, there is a maximum value of λ=λ_c, above which no solution is possible. λ=0 gives the lowest accretion rate (all the gas is at rest, so 0) and λ_c gives the maximum rate. Values near λ_c are expected for the flow, since nothing is stopping the gas from falling in. The values in the table in your book are for λ_c, and are near unity.

 

[Vrbic]

Bondi's original paper (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1952MNRAS.112..195B&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf ) investigates steady state solutions for spherically symmetric accretion (e.g. ρ(r), v(r) not changing with time). It turns out that there is not a unique solution to this problem, and you are left with this free parameter λ. However, there is a maximum value of λ=λ_c, above which no solution is possible. λ=0 gives the lowest accretion rate (all the gas is at rest, so 0) and λ_c gives the maximum rate. Values near λ_c are expected for the flow, since nothing is stopping the gas from falling in. The values in the table in your book are for λ_c, and are near unity.

Thank you very much for nice explanations. If I may I have another but technical query. I've tried to solve this problem numerically by Wolfram Mathematica. I led the book of Shapiro, Teukolsky (upper link) and used NSolve function with initial condition in critical point (sound barrier point - rs): a(rs)=u(rs) and u(rs)= us, where us is value of u(r) in sonic point coming from condition of smoothness of u'(r) in sonic point. But it diverges in initial point. Where should be problem? In initial conditions or in method used by Mathematica? If I tried some initial conditions in infinity (u(1000rs)=1/1000-initial velocity is going to zero) it solved something but a(r) - sonic speed in the matter was for example negative or other discrepancy. Do you have any comments or something like that?