= Quadratic accretion in a central Keplerian potential
{title2=$\dot M=\lambda M^2,\quad M(t)=M_0/(1-\lambda M_0t)$}
In a fixed central <Newtonian gravitational potential>, pressureless <free fall> has speed $u=aR^{-1/2}$, and randomly moving stars have typical gas-relative speed $V=bR^{-1/2}$. A steady, undepleted spherical <mass flux> $A$ gives $\rho=A R^{-3/2}/(4\pi a)$. The <Bondi--Hoyle--Lyttleton accretion rate> therefore becomes $\dot M=\lambda M^2$ with $\lambda=G^2A/(ab^3)$, independent of radius. Integrating $d(1/M)/dt=-\lambda$ gives the displayed solution before $\lambda M_0t=1$. Its finite-time divergence signals failure of the fixed potential, undepleted gas supply and negligible feedback assumptions; it is not a physically unlimited mass supply.
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