Moment of Inertia in the Slowly-Rotating Approximation

O2scl

The differential equations for slow rigid rotation are solved by tov_solve if o2scl::tov_solve::ang_vel is set to true.

In the case of slow rigid rotation with angular velocity \(\Omega\), the moment of inertia is

\[I = \frac{8 \pi}{3} \int_0^R dr~r^4\left(\varepsilon+P\right) \left(\frac{\bar{\omega}}{\Omega}\right) e^{\Lambda-\Phi} = \frac{8 \pi}{3} \int_0^R dr~r^4\left(\varepsilon+P\right) \left(\frac{\bar{\omega}}{\Omega}\right) \left(1-\frac{2 G m}{r}\right)^{-1/2} e^{-\Phi}\]

where \(\omega(r)\) is the rotation rate of the inertial frame, \(\Omega\) is the angular velocity in the fluid frame, and \(\bar{\omega}(r) \equiv \Omega - \omega(r)\) is the angular velocity of a fluid element at infinity. The function \(\bar{\omega}(r)\) is the solution of

\[\frac{d}{dr} \left( r^4 j \frac{d \bar{\omega}}{dr}\right) + 4 r^3 \frac{d j}{dr} \bar{\omega} = 0\]

where the function \(j(r)\) is defined by

\[j = e^{-\Lambda-\Phi} = \left( 1-\frac{2 G m}{r} \right)^{1/2} e^{-\Phi} \, .\]

Note that \(j(r=R) = 1\). The boundary conditions for \(\bar{\omega}\) are \(d \bar{\omega}/dr = 0\) at \(r=0\) and

\[\bar{\omega}(R) = \Omega - \left(\frac{R}{3}\right) \left(\frac{d \bar{\omega}}{dr}\right)_{r=R} \, .\]

One can use the TOV equation to rewrite the moment of inertia as

\[I= \left(\frac{d \bar{\omega}}{dr}\right)_{r=R} \frac{R^4}{6 G \Omega} \, .\]

The star’s angular momentum is just \(J = I \Omega\).