= Cold radial magnetohydrodynamic wind integral
{title2=$f(x,y)=\mathrm{constant}$}
For a cold equatorial <axisymmetric magnetohydrodynamic wind> with $B_p=CR^{-2}$ in the gravity of a <point mass> $M$, smooth crossing of the <Alfvén surface> gives $\rho_a=\mu_0k^2$ and $\ell=\omega R_a^2$. Put $x=R/R_a$, $y=\rho/\rho_a$, $\alpha=\omega^2R_a^3/(GM)$ and $\beta=C^2/(\mu_0\rho_a GM R_a^3)$. The corotating <energy>, scaled by $GM/R_a$, is
$$
f=\frac\alpha2\left[\frac{(x-x^{-1})^2}{(y-1)^2}-x^2\right]+\frac\beta{2x^4y^2}-\frac1x.
$$
The azimuthal <velocity> relative to field-line rotation is $\omega R_a(x-x^{-1})/(y-1)$ and the poloidal speed is $C/(\sqrt{\mu_0\rho_a}R_a^2x^2y)$. The apparent singularity at $x=y=1$ is interpreted through a smooth limiting slope. A nondegenerate magnetosonic crossing on a constant-$f$ curve requires $f_x=f_y=0$ and a real admissible tangent slope.
Back to article page
