রোশ লিমিট

$$ \begin{aligned} F_{\text{tidal}} &\simeq \frac{2 G M_p R_m}{r^3} \\[4pt] \frac{G M_m}{R_m^{2}} &= \frac{2 G M_p R_m}{r^3} \\[4pt] \Rightarrow\ \frac{M_m}{R_m^{2}} &= \frac{2 M_p R_m}{r^3} \\[4pt] M_m &= \frac{4}{3}\pi R_m^{3}\rho_m,\qquad M_p=\frac{4}{3}\pi R_p^{3}\rho_p \\[4pt] \Rightarrow\ \frac{\frac{4}{3}\pi R_m^{3}\rho_m}{R_m^{2}} &= \frac{2\left(\frac{4}{3}\pi R_p^{3}\rho_p\right)R_m}{r^{3}} \\[4pt] \Rightarrow\ \frac{4}{3}\pi R_m \rho_m &= \frac{8}{3}\pi \frac{R_p^{3}\rho_p R_m}{r^{3}} \\[4pt] \Rightarrow\ \rho_m &= 2\,\rho_p\,\frac{R_p^{3}}{r^{3}} \\[4pt] \Rightarrow\ r^{3} &= 2\,\frac{\rho_p}{\rho_m}\,R_p^{3} \\[4pt] \therefore\quad r_{\text{crit}} &= 2^{1/3}\!\left(\frac{\rho_p}{\rho_m}\right)^{1/3} R_p \\[8pt] r_{\text{crit}} &\approx 2.456\left(\frac{\rho_p}{\rho_m}\right)^{1/3} R_p \quad (\text{Roche}) \end{aligned} $$