For $D_{\rm im}\geq X_cD$, the fraction removed from the largest remnant is
$$
1-f_{\rm lr}=1-\frac12
\left(\frac{X_cD}{D_{\rm im}}\right)^{3.72},
$$
because $Q/Q_D^*=(D_{\rm im}/X_cD)^3$. Again using $R_{\rm col}\simeq AD^2D_{\rm im}^{-\alpha}$ near the dominant lower limit,
$$
\frac{\dot m_{\rm cat}}m
\simeq AD^2\int_{X_cD}^{\infty}
\left[1-\frac12\left(\frac{X_cD}{D_{\rm im}}\right)^{3.72}\right]
D_{\rm im}^{-\alpha}\,dD_{\rm im}
$$
$$
=AX_c^{1-\alpha}D^{3-\alpha}
\left(\frac1{\alpha-1}-\frac1{2(\alpha+2.72)}\right).
$$
Consequently the <catastrophic collision mass-loss timescale> is
$$
\boxed{\frac m{\dot m_{\rm cat}}
\simeq\frac{2(\alpha+2.72)}{\alpha+6.44}\frac1{R_{\rm cc}(D)}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 1 / vi / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-316/1/vi/solution