To calculate the new rotational period of the star, we can apply the conservation of angular momentum principle. Angular momentum is conserved because there is no external torque acting on the system. The formula for angular momentum is:

$L = I \cdot \omega$

Where:

• $I$ is the moment of inertia of a sphere, $I = \frac{2}{5} M R^2$,
• $\omega$ is the angular velocity, $\omega = \frac{2\pi}{T}$, with $T$ as the rotational period.

Since angular momentum is conserved: $I_{\text{initial}} \cdot \omega_{\text{initial}} = I_{\text{final}} \cdot \omega_{\text{final}}$

Expanding this: $\left(\frac{2}{5} M R_{\text{initial}}^2\right) \cdot \frac{2\pi}{T_{\text{initial}}} = \left(\frac{2}{5} M R_{\text{final}}^2\right) \cdot \frac{2\pi}{T_{\text{final}}}$

The mass $M$ and the constants $\frac{2}{5}$ and $2\pi$ cancel out, leaving: $\frac{R_{\text{initial}}^2}{T_{\text{initial}}} = \frac{R_{\text{final}}^2}{T_{\text{final}}}$

Rearranging for $T_{\text{final}}$: $T_{\text{final}} = T_{\text{initial}} \cdot \frac{R_{\text{final}}^2}{R_{\text{initial}}^2}$

Substituting the known values:

• $T_{\text{initial}} = 23 \, \text{days} = 23 \times 24 \times 3600 \, \text{s} = 1,987,200 \, \text{s}$,
• $R_{\text{initial}} = 7.0 \times 10^5 \, \text{km} = 7.0 \times 10^8 \, \text{m}$,
• $R_{\text{final}} = 6.5 \times 10^3 \, \text{km} = 6.5 \times 10^6 \, \text{m}$.

$T_{\text{final}} = 1,987,200 \cdot \left(\frac{6.5 \times 10^6}{7.0 \times 10^8}\right)^2$

$T_{\text{final}} = 1,987,200 \cdot \left(\frac{6.5}{700}\right)^2$

$T_{\text{final}} = 1,987,200 \cdot \left(\frac{65}{7000}\right)^2$

$T_{\text{final}} = 1,987,200 \cdot \left(\frac{65}{7000}\right)^2 = 1,987,200 \cdot 0.000086$

$T_{\text{final}} \approx 170.3 \, \text{s}$

Final Answer:

The new rotational period would be approximately 170.3 seconds.

Would you like a deeper explanation or have additional questions?

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Related Questions:

1. How is angular momentum conserved in astrophysical phenomena?
2. What assumptions are made when modeling stars as spheres of uniform density?
3. How does the conservation of angular momentum apply to other collapsing objects like neutron stars?
4. What would happen if the star ejected mass during collapse?
5. How do scientists measure the rotational period of stars?

Tip: Always check units when solving astrophysical problems to ensure accuracy!