91Ó°ÊÓ

If a system experiences no external torques from the environment, the total angular momentum is conserved.

$\mathrm{Δ}\stackrel{→}{L}→t=0$

Apply the law of conservation of angular momentum to a system whose moment of inertia changes gives:

$I_j{\mathrm{Ó\neg }}_j=l_f{\mathrm{Ó\neg }}_f=\text{constant}$

Given that, initial radius of the original star is $R_j=7.0×{10}^5\mathrm{km}$, and the final radius of the neutron star is $R_f=16\mathrm{km}$. The angular speed of the original star is

The star is modelled as a uniform, solid, rigid sphere both before and after the collapse.

Let the initial state of the system before the collapse and the final sate be after the collapse.

Consider there are no astronomical bodies are in the vicinity of the star, no forces or torques are exerted on the star. Thus, apply conservation of momentum principle from equation to the star between the initial and final states:

$$\begin{gathered} I_j{\mathrm{Ó\neg }}_i=l_f{\mathrm{Ó\neg }}_f \\ \frac{2}{5}M{R}_i^2{\mathrm{Ó\neg }}_i=\frac{2}{5}M{R}_f^2{\mathrm{Ó\neg }}_f \\ {R}_i^2{\mathrm{Ó\neg }}_i=R_f^2{\mathrm{Ó\neg }}_f \end{gathered}$$

Solve for ${\mathrm{Ó\neg }}_f$as follows:

$$\begin{gathered} {\mathrm{Ó\neg }}_f={\mathrm{Ó\neg }}_i{\left(\frac{R_i}{R_f}\right)}^2 \\ =2.424×{10}^{-6}\left(\frac{7.0×{10}^5}{16}\right) \\ =4.6×{10}^3\mathrm{rad}/\mathrm{s} \end{gathered}$$

Hence, the final angular speed is $4.6×{10}^3\mathrm{rad}/\mathrm{s}$.