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  4. The two planets with radii R1,R2 have densities ρ1,ρ2 and atmospheric pressures P1 and P2 respectively. Therefore the ratio of masses of their atmospheres, neglecting variation of g within the limits of atmosphere, is

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Q. The two planets with radii $R_1,R_2$ have densities $\rho_1,\rho_2$ and atmospheric pressures $P_1\,and\,P_2$ respectively. Therefore the ratio of masses of their atmospheres, neglecting variation of $g$ within the limits of atmosphere, is

Gravitation

Solution:

Acceleration due to gravity,
$
\begin{array}{l}
g =\frac{ GM }{ R ^{2}}=\frac{4}{3} \pi \rho GR \\
\therefore \frac{ g _{1}}{ g _{2}}=\frac{\rho_{1} R _{1}}{\rho_{2} R _{2}}
\end{array}
$
Atmospheric pressure can be given by $p =\frac{ W }{ S }$
where, $W =$ weight of atmosphere,
$S =$ Surface area of the planet
$
\therefore \frac{ m _{1}}{ m _{2}}=\frac{\rho_{1} S _{1} g _{2}}{\rho_{2} S _{2} g _{1}}=\frac{ p _{1} \cdot\left(4 \pi R _{1}^{2}\right)}{ p _{2}\left(4 \pi R _{2}^{2}\right)} \cdot \frac{\rho_{2} R _{1}}{\rho_{1} R _{1}}=\frac{ p _{1} R _{1} \rho_{2}}{ p _{2} R _{2} \rho_{1}}
$