An Introduction to Modern Astrophysics · Problem 18.18

The Alfvén radius marks where a magnetized stellar wind's magnetic pressure balances the ram pressure of infalling material—it's the boundary where the star's magnetic field stops the accretion flow. To derive it, you'll equip the magnetic pressure ($B^2/8\pi$) against the dynamic pressure of the accreting gas, using the dipole field strength at that radius and the mass accretion rate to set up the balance equation. For the spin-up rate, recognize that angular momentum transfer from the accreting material spinning up the pulsar depends on the torque exerted at the Alfvén radius, where material couples to the magnetic field and begins co-rotating with the star.

The Alfvén radius marks where a magnetized stellar wind's magnetic pressure balances the ram pressure of infalling material—it's the boundary where the star's magnetic field stops the accretion flow. To derive it, you'll equip the magnetic pressure ($B^2/8\pi$) against the dynamic pressure of the accreting gas, using the dipole field strength at that radius and the mass accretion rate to set up the balance equation. For the spin-up rate, recognize that angular momentum transfer from the accreting material spinning up the pulsar depends on the torque exerted at the Alfvén radius, where material couples to the magnetic field and begins co-rotating with the star.