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I've been asked to estimate how much mass a neutron star in a low mass X-ray binary would have to accrete in order to be spun up to ~300 Hz. I found the conservation formula

$\frac{d(I\omega)}{dt} = \dot{M}\sqrt{GMR}$,

in some lecture notes, which I'll probably use with moment of inertia $I=\frac{2}{5}MR^2$ for a solid sphere. However, I need help understanding where this formula comes from.

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Assume that the material, just before it is accreted, is in a circular orbit one centimeter above the surface of neutron star.

What is the accreting material's angular momentum per unit mass? That is equal to its radius R (ignoring the extra cm) multiplied by its velocity.

What is the velocity? Just use Newtonian mechanics with centripetal force = gravity

v^2 / R = MG/(R^2)

Stick it all together, and the change in angular momentum is equal to the amount of material that accretes. If you take the time derivative of both sides of the equation you get the formula you listed, but you are going to integrate over time, so you only care about the total angular momentum and total accreted mass.

Do a cross-check and adjust if the additional mass is a large fraction of the initial mass.

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  • $\begingroup$ Shouldn't it be v^2 / R in your formula? $\endgroup$ – lukel Nov 19 '18 at 22:39

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