= Transonic accretion in a dipolar flux tube
The <dipolar flux-tube area> has $A(r)=A_\star(r/R_\star)^3$. Applying <transonic accretion in a power-law tube> with $n=3$ and a <polytropic equation of state> gives
$$
c_s^2=\frac{2c_0^2}{7-5\gamma},
\qquad
r_s=\frac{GM(7-5\gamma)}{6c_0^2},
\qquad
\rho_s=\rho_0\left(\frac2{7-5\gamma}\right)^{1/(\gamma-1)}.
$$
Here $c_0$ and $\rho_0$ are the reservoir <sound speed> and <mass density>, and $c_s$ is the <sound speed> at the <sonic point>. The <mass accretion rate> is
$$
\dot M=A_\star\left(\frac{r_s}{R_\star}\right)^3\rho_s c_s.
$$
For two equal polar caps of surface angular radius $\theta_\star\ll1$, $A_\star\simeq2\pi R_\star^2\theta_\star^2$. One loaded cap gives half the total rate. The finite-radius <transonic branch> exists for $1<\gamma<7/5$, must cross outside the star ($r_s>R_\star$), and requires $\theta_\star^2r_s/R_\star\ll1$ for the sonic region to remain a narrow <flux tube>.
Back to article page
