= Solution
In <magnetically channelled accretion>, an outer <accretion disk> supplies nearly Keplerian material. Near the <magnetospheric truncation radius>, magnetic stresses disrupt the disk and guide matter along field lines onto polar <accretion columns>. Shocks and settling near the surface release gravitational binding <energy>, and an oblique magnetic geometry makes the rotating emission pattern pulsate.
The <corotation radius> satisfies $\Omega_K(r_{\rm co})=\Omega_*=2\pi/P$, hence
$$
\boxed{r_{\rm co}=\left(\frac{GMP^2}{4\pi^2}\right)^{1/3}=2.07\times10^6\ \mathrm m}.
$$
The <accretion Alfvén radius> estimates where magnetic stress becomes comparable with the inflow's dynamical stress; it is the magnetic truncation scale in this application, not the Alfvén critical point of an outward stellar wind. The supplied seventh-root scaling follows from a dipole's rapidly decreasing <magnetic pressure> and an accretion-pressure estimate. Its coefficient depends on disk geometry. Ordinary inward surface <accretion> requires the magnetosphere to remain outside the star and usually inside or near corotation, $R\lesssim r_{\rm Alf}\lesssim r_{\rm co}$. Beyond corotation, a rapidly moving magnetosphere can transfer <angular momentum> outward and inhibit inflow through the <propeller effect>.
\Image[/past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015-magnetic-accretion-geometry.png]
{title=Magnetic channeling from a truncated accretion disk to neutron-star polar caps, with the corotation condition shown separately}
Back to article page
