Magnetic Braking

In this entry we describe a mechanism that can cause stars to spin down. Let us consider a star with radius $R$ that loses mass at a rate ${\dot {M}}$. The star also has some magnetic field, which exerts a torque on the wind, so that the wind keeps rotating at the spin frequency of the star, even when it is at much larger radii. This means that the angular momentum of the wind increases, at the expense of angular momentum of the star. In other words, this mechanism causes the star to spin down. The maximum radius where the magnetic field can keep the wind in co - rotation is called the Alfven radius, and it can be obtained by comparing the magnetic pressure and the ram pressure of the wind.

In vacuum, the magnetic field at a distance $r$ from the star would decline like a dipole field $b\propto r^{-3}$. However, since the stellar wind is highly ionised, then the magnetic field declines like the mass density, which in our case is the same as a monopole field $b\propto r^{-2}$. If $B$ is the magnetic field on the surface of the star, then the magnetic field at a larger radius is

$b\approx B{\frac {R^{2}}{r^{2}}}$

The magnetic pressure is

$P_{m}\approx b^{2}\approx B^{2}{\frac {R^{4}}{r^{2}}}$

The ram pressure is given by

$P_{r}\approx {\frac {{\dot {M}}v}{r^{2}}}$

where $v$ is the wind velocity. Comparing the two pressures yields

$P_{m}\approx P_{r}\Rightarrow r\approx R{\sqrt {\frac {B^{2}R^{2}}{{\dot {M}}v}}}$

For our sun, the Alfven radius is roughly 15 solar radii.

In order for the spin to decline considerably, the necessary mass loss is

${\frac {\Delta M}{M}}\approx {\frac {R^{2}}{r^{2}}}$

For the sun, the fractional mass loss is about 0.5%, and according to some models the sun might have lost this much mass over its lifetime.