PHYS6562 F5 Daily Quaetion

For discrete system, the entropy $S_{discrete} = -k_B \sum p_i\log p_i$ $S_{discrete} = \frac{3}{2}k_B \log 2$
Entropy for being hit by a comet with $$\rho_c(\theta,\phi) = \frac{1}{4\pi}$$: $S_c = -k_B \iint \rho_s \log (\rho_s) d\Omega = k_B \log (4\pi)$ Entropy for being hit by an asteroid with $$\rho_a(\theta,\phi) = \frac{cos(\theta)}{\pi^2}$$: $S_a = -k_B \iint \rho_a \log (\rho_a) d\Omega = -k_B \int \left( \frac{2\cos^2\theta}{\pi}\log(\cos\theta) - \frac{4\cos^2\theta}{\pi}\log\pi \right)d\theta$ $S_a = -k_B \left(\frac{1}{2}(1-\log 4) - 2\log(\pi) \right) = k_B \log \left(\frac{2\pi^2}{\sqrt{e}} \right)$
The entropy difference $\Delta S = S_c - S_a = k_B\log\left(\frac{2\sqrt{e}}{\pi} \right)$