A regular sonic transition selects spherical accretion

Work in the rest frame of a Newtonian point-mass accretor. Assume steady spherical flow, negligible gas self-gravity, no dynamically important angular momentum or magnetic field, and a static uniform reservoir at large radius. Let u be the positive inward speed, so the radial vector velocity is minus u. In the isothermal model pressure equals density times a constant sound speed squared. Keeping this temperature constant requires heat exchange; it is not the same thermodynamic closure as an adiabatically compressed gas.

Define the Bondi radius as rB=GM/cₛ². Some authors put a factor of two in a capture-radius definition, so the convention must accompany any quoted size. Balancing gravity with thermal motions gives this characteristic scale, while continuity suggests a rate proportional to density times (GM)² divided by sound speed cubed. A moving accretor encountering a supersonic wind is a different boundary problem. Substituting a relative speed into a dimensional expression does not turn that problem into the exact static-reservoir Bondi solution.

A transonic branch has u=cₛ at some finite radius. A smooth derivative there requires the right-hand side of the wind equation to vanish too, selecting rₛ=GM/(2cₛ²)=rB/2. This is a regularity condition, not an extra force. Subsonic breeze-like branches and other mathematical solutions exist for different inner boundary choices. The accretion solution considered here connects a static outer reservoir to an inward supersonic branch that the chosen central sink can absorb.

The isothermal Bernoulli relation contains cₛ² ln(ρ/ρ∞), with a dimensionless logarithm. At infinity the inward speed and potential vanish. Evaluating the same constant at the sonic point yields ρₛ/ρ∞=exp(3/2). Substituting that density, speed and radius into continuity fixes the rate. A polytropic or radiatively heated flow has a changing sound speed and generally a different coefficient and critical structure. In particular one cannot carry this isothermal density enhancement unchanged into every adiabatic model.

The sound-speed dependence makes a Bondi estimate sensitive to how the distant gas temperature is measured. Unresolved density gradients, multiphase gas, feedback heating, turbulent support and residual rotation can all change the supplied rate. A small angular momentum can become important after contraction and produce a disk inside the capture scale. A black-hole horizon belongs to relativistic inner geometry, whereas rB reflects outer gas conditions. Multiplying the model capture rate by a guessed constant efficiency does not by itself establish a measured bolometric luminosity.

The function lab checks just the mass dependence. Hold the reservoir density and sound speed fixed, and identify x with central mass divided by a reference mass. Reset and set both controls to a=1,x=1, predicting y=1. Reset and set a=1,x=2, predicting y=4. This matches the quadratic mass scaling of the conditional capture rate. It does not integrate a flow, locate a sonic point, include feedback, or determine what fraction of captured material reaches an inner object.