Magnetic braking (astronomy)

The currently accepted theory of the solar system's evolution states that the Solar System originates from a contracting gas cloud. As the cloud contracts, the angular momentum ${\displaystyle L}$ must be conserved. Any small net rotation of the cloud will cause the spin to increase as the cloud collapses, forcing the material into a rotating disk. At the dense center of this disk a protostar forms, which gains heat from the gravitational energy of the collapse. As the collapse continues, the rotation rate can increase to the point where the accreting protostar can break up due to centrifugal force at the equator.

Thus the rotation rate must be braked during the first 100,000 years of the star's life to avoid this scenario. One possible explanation for the braking is the interaction of the protostar's magnetic field with the stellar wind. In the case of our own Sun, when the planets' angular momenta are compared to the Sun's own, the Sun has less than 1% of its supposed angular momentum. In other words, the Sun has slowed down its spin while the planets have not.

Ionized material captured by the magnetic field lines will rotate with the Sun as if it were a solid body. As material escapes from the Sun due to the solar wind, the highly ionized material will be captured by the field lines and rotate with the same angular velocity as the Sun, even though it is carried far away from the Sun's surface, until it eventually escapes. This effect of carrying mass far from the centre of the Sun and throwing it away slows down the spin of the Sun.[1][2] The same effect is used in slowing the spin of a rotating satellite; here two wires spool out weights to a distance slowing the satellites spin, then the wires are cut, letting the weights escape into space and permanently robbing the spacecraft of its angular momentum.

As ionized material follows the Sun's magnetic field lines, due to the effect of the field lines being frozen in the plasma, the charged particles feel a force ${\displaystyle \mathbf {F} }$ of the magnitude:

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$

where ${\displaystyle q}$ is the charge, ${\displaystyle \mathbf {v} }$ is the velocity and ${\displaystyle \mathbf {B} }$ is the magnetic field vector. This bending action forces the particles to "corkscrew" around the magnetic field lines while held in place by a "magnetic pressure"${\displaystyle P_{B}}$, or "energy density", while rotating together with the Sun as a solid body:

${\displaystyle P_{B}={\frac {B^{2}}{2\mu _{0}}}}$

Since magnetic field strength decreases with the cube of the distance there will be a place where the kinetic gas pressure ${\displaystyle P_{g}}$ of the ionized gas is great enough to break away from the field lines:

${\displaystyle P_{g}=nmv^{2}}$

where n is the number of particles, m is the mass of the individual particle and v is the radial velocity away from the Sun, or the speed of the solar wind.

Due to the high conductivity of the stellar wind, the magnetic field outside the sun declines with radius like the mass density of the wind, i.e. decline as an inverse square law.[3] The magnetic field is therefore given by

${\displaystyle B(r)=B_{s}{\frac {R^{2}}{r^{2}}}}$

where ${\displaystyle B_{s}}$ is the magnetic field on the surface of the sun and ${\displaystyle R}$ is its radius. The critical distance where the material will break away from the field lines can then be calculated as the distance where the kinetic pressure and the magnetic pressure are equal, i.e.

${\displaystyle P_{B}=P_{g}\Rightarrow }$

${\displaystyle {\frac {B(r_{c})^{2}}{2\mu _{0}}}=nmv^{2}\Rightarrow }$

${\displaystyle {\frac {B_{s}^{2}R^{4}}{2\mu _{0}r_{c}^{4}}}=nmv^{2}}$

If the solar mass loss is omni-directional then the mass loss ${\displaystyle nm={\frac {dM/dt}{4\pi r^{2}v}}}$; plugging this into the above equation and isolating the critical radius it follows that

${\displaystyle r_{c}=R\left({\frac {2\pi B_{s}^{2}R^{2}}{\mu _{0}v{\dot {M}}}}\right)^{1 \over 2}}$

• Stellar magnetic field
• Stellar rotation