= Solution
Take $u<0$ for inward radial <velocity>, and assume a constant <specific-heat ratio> $\gamma>1$. The <isentropic flow> has $p=K\rho^\gamma$ and <adiabatic sound speed> $c^2=\gamma K\rho^{\gamma-1}$. Steady spherical <mass conservation> and the radial <Euler equation for fluid motion> give
$$
4\pi r^2\rho u=-\dot M,\qquad
\frac{\rho'}\rho=-\frac2r-\frac{u'}u,\qquad
uu'=-c^2\frac{\rho'}\rho-\frac{GM}{r^2}.
$$
Eliminating the <mass density> <derivative> gives $(u-c^2/u)u'=2c^2/r-GM/r^2$, or
$$
\boxed{\frac{u'}u=\frac1r\frac{GM/r-2c^2}{c^2-u^2}.}
$$
Differentiating $c^2\propto\rho^{\gamma-1}$ gives the corresponding <sound speed> equation:
$$
\boxed{\frac{c'}c=-\frac{\gamma-1}{2}\left(\frac2r+\frac{u'}u\right)
=-\frac{\gamma-1}{2r}\frac{GM/r-2u^2}{c^2-u^2}.}
$$
At a smooth <sonic point>, $u_s^2=c_s^2$. A finite <derivative> requires the numerator of the flow equation to vanish too. Thus
$$
\boxed{u_s^2=c_s^2=\frac{GM}{2r_s}.}
$$
This regularity requirement concerns a smooth <transonic branch>, rather than an arbitrary singular solution reaching unit <Mach number>.
Integrating the momentum equation using the <specific enthalpy> $c^2/(\gamma-1)$ yields the <Bernoulli equation>
$$
\frac{u^2}{2}+\frac{c^2}{\gamma-1}-\frac{GM}{r}
=\frac{c_\infty^2}{\gamma-1},\qquad
c_\infty^2=\gamma K\rho_\infty^{\gamma-1}.
$$
For a finite positive <mass accretion rate>, its large-radius solution has
$$
u=-\frac{\dot M}{4\pi\rho_\infty r^2}[1+O(r^{-1})],\qquad
c^2=c_\infty^2+\frac{(\gamma-1)GM}{r}+O(r^{-4}),
$$
$$
\rho=\rho_\infty\left[1+\frac{GM}{c_\infty^2r}+O(r^{-2})\right].
$$
These expressions satisfy both the flow equations asymptotically and show \b[$u\to0$ and $\rho\to\rho_\infty$]. In particular $u'/u\sim-2/r$ agrees with the first boxed equation. Combining the <Bernoulli equation> with sonic regularity additionally gives $c_s^2=2c_\infty^2/(5-3\gamma)$, so a finite positive <sonic point> occurs in the range identified below.
For the small-radius inward <free fall> branch, neglecting enthalpy against $GM/r$ in the <Bernoulli equation> predicts $u^2\sim2GM/r$. The <continuity equation> then fixes $\rho\propto r^{-3/2}$, and consequently $c^2\propto r^{-3(\gamma-1)/2}$. The neglected-to-leading ratio is
$$
\frac{c^2}{u^2}\propto r^{(5-3\gamma)/2}.
$$
It tends to zero precisely when
$$
\boxed{\gamma<\gamma_{\mathrm{crit}}=\frac53.}
$$
This makes the assumed <supersonic flow> and leading <free fall> self-consistent. At equality, the ratio is of constant order rather than vanishing; above it, the proposed asymptotic balance fails. The isothermal limit $\gamma=1$ also permits free fall, with logarithmic rather than power-law enthalpy.
For the stellar funnel, write the magnetic scalar potential as $\psi=\mu\cos\theta/r^2$, where $R=r\sin\theta$ and $z=r\cos\theta$. Its <magnetic dipole field> has $B_r=-2\mu\cos\theta/r^3$ and $B_\theta=-\mu\sin\theta/r^3$, so a <magnetic-field-line equation> is
$$
\frac{dr}{r\,d\theta}=\frac{B_r}{B_\theta}=2\cot\theta,
\qquad r=\ell\sin^2\theta.
$$
The edge of the polar <flux tube> meets the stellar surface at $\theta_*\simeq a/R_*$. Thus, near the pole, $\theta^2\simeq\theta_*^2r/R_*$, and its transverse radius is $R\simeq r\theta$. This proves the <dipolar flux-tube area> law
$$
\boxed{A(r)\simeq\pi r^2\theta^2=\pi a^2\left(\frac r{R_*}\right)^3.}
$$
Equivalently <magnetic flux> conservation gives $A|\mathbf B|=\mathrm{constant}$ and $|\mathbf B|\propto r^{-3}$. The stipulated small cross section keeps the tube close to the polar axis, where its direction is approximately radial; the approximation is not valid all the way to arbitrarily large $r$.
Outside the star the prescribed field is a gradient, hence $\nabla\times\mathbf B=0$ and $\mathbf j=0$. Its <Lorentz force density> is therefore \b[$\mathbf j\times\mathbf B=0$]. The magnetic-<pressure> and magnetic-tension parts cancel in this exact current-free dipole. Field-aligned <velocity> also has $\mathbf u\times\mathbf B=0$, consistent with a steady prescribed field. The following calculation uses the narrow polar funnel approximation.
Now steady <mass conservation> reads $\rho u A=\mathrm{constant}$. With $A\propto r^3$, replace the spherical factor 2 by 3:
$$
\left(u-\frac{c^2}{u}\right)u'=\frac{3c^2}{r}-\frac{GM}{r^2}.
$$
The <Bernoulli equation> is unchanged because the magnetic field supplies no work or force. On a gravity-dominated <free fall> branch, $u^2\sim2GM/r$, but $\rho\propto r^{-5/2}$ and $c^2\propto r^{-5(\gamma-1)/2}$. Hence
$$
\frac{c^2}{u^2}\propto r^{(7-5\gamma)/2},\qquad
\boxed{\gamma_{\mathrm{crit}}=\frac75\quad\text{for the dipolar funnel}.}
$$
The same bound follows from regular <transonic accretion in a power-law tube>: $u_s^2=c_s^2=GM/(3r_s)$ and the reservoir-matched <Bernoulli equation> gives $c_s^2=2c_\infty^2/(7-5\gamma)$. The stellar surface cuts off the mathematical small-$r$ limit; the criterion refers to a funnel that becomes gravity dominated and crosses its <sonic point> before reaching that surface.
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