= Solution
Write the inward radial <velocity> as $u_R=-v(R)$, with $v>0$. Steady spherical <mass conservation> and the radial <Euler equations for an inviscid fluid> give
$$
\boxed{\dot M=4\pi R^2\rho v=\text{constant}>0,\qquad vv'=-\frac1\rho p'-\Phi'.}
$$
For a <barotropic fluid>, define the <sound speed> $c_s^2=dp/d\rho>0$ and the <barotropic enthalpy function> by $dh=dp/\rho$. Choosing $\Phi(\infty)=0$, integration gives the <Bernoulli equation>
$$
\frac{v^2}{2}+h(\rho)+\Phi=h(\rho_\infty),\qquad p(\rho_\infty)=p_0.
$$
The continuity equation gives $\rho'/\rho=-2/R-v'/v$. Eliminating <mass density> from the momentum equation produces the <barotropic spherical accretion equation>
$$
\boxed{\left(v-\frac{c_s^2}{v}\right)v'=\frac{2c_s^2}{R}-\Phi'.}
$$
The <sonic point> is where the inward <Mach number> $v/c_s$ equals one. A smooth solution cannot cross the vanishing coefficient on the left with a nonzero right-hand side, so a regular sonic radius must obey
$$
v_s=c_{s,s},\qquad c_{s,s}^2=\frac{R_s\Phi'(R_s)}2.
$$
Conversely a continuous flow that is subsonic at infinity and supersonic inward must cross unit <Mach number>; smooth passage selects a simultaneous zero of the two sides, hence a regular <sonic point>. The associated critical accretion branch, rather than an arbitrary subsonic solution with smaller flux, is selected by these outer and inner requirements.
For the <polytropic equation of state>, set $\Gamma=1+1/m$ to distinguish its exponent from any separately specified perturbation exponent. Assume $K>0$ and first take the usual positive index $m>0$. Then
$$
c_s^2=\Gamma K\rho^{1/m},\qquad h=mc_s^2,\qquad \rho_\infty=(p_0/K)^{m/(m+1)},\qquad c_\infty^2=\Gamma p_0/\rho_\infty.
$$
For the attractive mixed potential, assume $\lambda,\mu>0$. The sonic condition and <Bernoulli equation> give
$$
c_{s,s}^2=\frac\lambda{R_s^2}+\frac\mu{2R_s},\qquad (m+\tfrac12)c_{s,s}^2-\frac\lambda{R_s^2}-\frac\mu{R_s}=mc_\infty^2.
$$
Hence <polytropic accretion in a mixed inverse-power potential> reduces to
$$
mc_\infty^2R_s^2-\frac{2m-3}{4}\mu R_s-\frac{2m-1}{2}\lambda=0.
$$
For $0<m\leq1/2$ all nonzero coefficients in the left side are nonnegative for $R_s>0$, with at least one strictly positive: there is no finite positive root. For $m>1/2$ the constant term is negative and the leading coefficient positive, giving exactly one positive root. Define $B=(2m-3)\mu/4$ and $C=(2m-1)\lambda/2$. Then
$$
\boxed{m>\tfrac12\quad(1<\Gamma<3),\qquad R_s=\frac{B+\sqrt{B^2+4mc_\infty^2C}}{2mc_\infty^2}.}
$$
This is the finite-radius transonic range for the two positive attractive terms.
The critical point has real distinct crossing slopes. To check this rather than just solve the sonic equalities, put $U=\lambda/R_s^2$, $V=\mu/(2R_s)$ and $x=R_sv_s'/v_s$. Differentiating the <barotropic spherical accretion equation> at its simultaneous zero gives
$$
(2m+1)x^2+4x+4+2m-m\frac{6U+4V}{U+V}=0.
$$
Its <discriminant> is $8m[2(2m-1)U+(2m-3)V]/(U+V)$. The sonic <Bernoulli equation> says $(2m-1)U+(2m-3)V=2mc_\infty^2$, so this <discriminant> is strictly positive for $m>1/2$. The accretion crossing is the slope for which $d\log(v/c_s)/d\log R<0$, namely the smaller root. The two slopes have opposite signs of this Mach-number derivative. This is the regular transonic saddle, not a singular vertical tangent.
The corresponding <mass accretion rate> follows without an additional arbitrary <mass density> normalization:
$$
\rho_s=\rho_\infty\left(\frac{c_{s,s}^2}{c_\infty^2}\right)^m,\qquad \boxed{\dot M=4\pi R_s^2\rho_\infty c_{s,s}\left(\frac{c_{s,s}^2}{c_\infty^2}\right)^m.}
$$
One may equivalently replace $\rho_s$ by $[c_{s,s}^2/(\Gamma K)]^m$. With a nonzero $\lambda$ term, inner free fall has $v\sim\sqrt{2\lambda}/R$ and continuity gives $\rho\propto R^{-1}$. Thus $v^2/c_s^2\propto R^{-2+1/m}$ increases without bound precisely in the positive-index range $m>1/2$, consistent with the required inner supersonic branch.
The assumptions on the parameters matter. If $\lambda=0$ and $\mu>0$, the finite sonic radius instead requires $m>3/2$, the usual <Bondi accretion> range; $m=3/2$ is the zero-radius limiting critical case. If $\mu=0$ and $\lambda>0$, the condition remains $m>1/2$. With arbitrary signs of $\lambda,\mu$, one must use the quadratic and retain only positive roots with $c_{s,s}^2>0$ and real admissible crossing slopes; attraction was not explicitly given as a sign condition in the source.
If positive polytropic index was not intended as an implicit physical convention, there is a further mathematical range. For $K>0$, positive compressibility also permits $m<-1$, giving $0<\Gamma<1$. Both potential terms attractive then give a positive sonic root as well:
$$
R_s=\frac{B-\sqrt{B^2+4mc_\infty^2C}}{2mc_\infty^2}>0.
$$
The same <mass accretion rate> formula applies; its crossing-slope <discriminant> is positive. The range $-1<m<0$ has $c_s^2<0$ and no physical acoustic sonic point, $m=-1$ has constant <pressure> and zero <sound speed>, and $m=0$ does not define the printed equation of state. Thus the usual answer $m>1/2$ assumes $m>0$ and positive attractive coefficients; these conventions should not be confused with additional printed restrictions.
Back to article page
