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--- bn:un:tidal-force [2025/08/11 04:24] – [4. জায়ান্ট গ্রহদের রিং] asad
+++ bn:un:tidal-force [2025/08/11 04:25] (current) – [4. জায়ান্ট গ্রহদের রিং] asad
@@ Line 55: / Line 55: @@
 {{https://upload.wikimedia.org/wikipedia/commons/e/e5/Galilean_moon_Laplace_resonance_animation_2.gif?nolink}}
-===== - জায়ান্ট গ্রহদের রিং =====
-$$
-\begin{aligned}
-F_{\text{tidal}} &\simeq \frac{2 G M_p R_m}{r^3} \\[4pt]
-\frac{G M_m}{R_m^{2}} &= \frac{2 G M_p R_m}{r^3} \\[4pt]
-\Rightarrow\ \frac{M_m}{R_m^{2}} &= \frac{2 M_p R_m}{r^3} \\[4pt]
-M_m &= \frac{4}{3}\pi R_m^{3}\rho_m,\qquad M_p=\frac{4}{3}\pi R_p^{3}\rho_p \\[4pt]
-\Rightarrow\ \frac{\frac{4}{3}\pi R_m^{3}\rho_m}{R_m^{2}} &= \frac{2\left(\frac{4}{3}\pi R_p^{3}\rho_p\right)R_m}{r^{3}} \\[4pt]
-\Rightarrow\ \frac{4}{3}\pi R_m \rho_m &= \frac{8}{3}\pi \frac{R_p^{3}\rho_p R_m}{r^{3}} \\[4pt]
-\Rightarrow\ \rho_m &= 2\,\rho_p\,\frac{R_p^{3}}{r^{3}} \\[4pt]
-\Rightarrow\ r^{3} &= 2\,\frac{\rho_p}{\rho_m}\,R_p^{3} \\[4pt]
-\therefore\quad r_{\text{crit}} &= 2^{1/3}\!\left(\frac{\rho_p}{\rho_m}\right)^{1/3} R_p \\[8pt]
-r_{\text{crit}} &\approx 2.456\left(\frac{\rho_p}{\rho_m}\right)^{1/3} R_p \quad (\text{Roche})
-\end{aligned}
-$$