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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?
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.PeterDonis said:Unfortunately I don't have that book. Someone else here might, or you could try to post the particular equations you are concerned about.
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?Calion said: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.
Non-dimensional accretion rate λ is a measure of the rate at which matter is being accreted onto an object, relative to its size and gravitational potential. It is often used in astrophysics to study the accretion of matter onto stars, black holes, and other celestial bodies.
Non-dimensional accretion rate λ is calculated by dividing the actual accretion rate by the theoretical maximum accretion rate. The theoretical maximum accretion rate is determined by the object's size and gravitational potential, and is based on the assumption that all the matter in the surrounding environment can be accreted onto the object.
Non-dimensional accretion rate λ allows scientists to compare the accretion rates of different objects, regardless of their size or gravitational potential. This helps us understand the processes and mechanisms involved in accretion, and can provide insights into the formation and evolution of celestial bodies.
Non-dimensional accretion rate λ is affected by various factors, including the properties of the accreting object (such as its size and gravitational potential), the properties of the accreting matter (such as its density and velocity), and the physical conditions of the surrounding environment (such as the gas density and temperature).
Non-dimensional accretion rate λ is used in a wide range of research studies in astrophysics, including studies of star and planet formation, black hole accretion, and the dynamics of galactic structures. It can also be used as a tool to test and refine theoretical models of accretion processes.