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Write the radial velocity as $u(r)$, allowing $u<0$ for accretion flow, and let $c^2=\gamma p/\rho$ be the squared adiabatic sound speed. The steady continuity equation, radial Euler equations for an inviscid fluid and entropy advection equation giveOn a nonzero smooth flow branch, the polytropic equation of state is $p=K{\rho }^{\gamma }$ with constant $K$. Thus the two useful first integrals for spherically symmetric adiabatic flow areHere $\mathcal{J}$ is the signed mass flux and $\mathcal{B}$ is the Bernoulli function. The specific enthalpy is $c^2/(\gamma -1)$; the prescribed Newtonian gravitational potential has no contribution from the gas's own gravity. Eliminating the density derivative using mass conservation gives the differential formThis sonic point equation displays both the singular coefficient and the numerator that must vanish for a smooth transonic branch.

For spherically symmetric adiabatic flow in a power-law gravitational potential, let $X=(r_{\ast }/u_{\ast })u_{\ast }^{\prime }$ and $\delta =\gamma -1$. Differentiation at a sonic point gives $(2+\delta )X^2+4\delta X+4\delta -2\beta =0$. Its discriminant is $8[2\beta -(4-\beta )\delta ]$; a positive value gives the two local crossing directions.