Isothermal Bondi equation

For inward speed

u > 0

, steady spherical mass conservation and radial momentum balance give

(u - \frac{c _{s}^{2}}{u}) \frac{d u}{d r} = \frac{2 c _{s}^{2}}{r} - \frac{GM}{r ^{2}} .

Bondi sonic point ( $r_{s}$ )

The isothermal Bondi sonic point is

r_{s} = GM / (2 c_{s}^{2})

, where the regular transonic solution has

u = c_{s}

d u / d r

and the remaining right-hand side of the Isothermal Bondi equation vanish, allowing a finite derivative through the sonic transition.

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