Spherically symmetric accretion

Let me again point you the the textbook by Vladimir Karas; Section 4.1. The reader is alsoencouraged to go through the introduction of Section 4. My hand-notes are also available online, although I think they do not add anything to the textbook.

Let me emphasise a few points:

• equation (4.6) is also known as Parker's equation and is used to describe a basic model of stellar winds (see also lecture Astrophysics II)
• logical arguments are used to determine qualitative structure of a subset of solutions of Parker's equation; exact solutions are dependent on boundary conditions and the particular form of the equation of state and can be obtained only by means of numerical integration (except for some singular cases)
• the exact coordinates of the sonic point, $r_{\mathsf{s}}$, can be obtained only through integration of the Parker's equation. Instead, characteristic accretion radius, $r_{\mathsf{acc}}\approx GM∕c_{\mathsf{s}}^2\left(\infty \right)$, can be obtained directly from parameters of the system and it is commonly used as an estimate of the radius of region in which gravity of the central object dominates motion of the fluid and from which the fluid is typically accreted onto it in real systems (which typically do not obey the asymptotic boundary conditions considered in simple, spherically symmetric model)