The planets with radii R₁ and R₂ have densities ${\rho }_1$ and${\rho }_2$ , respectively. Their atmospheric pressures are p₁ and p₂ , respectively. Therefore, the ratio of masses of their atmospheres, neglecting variation of g within the limits of atmosphere is                                                      [JIPMER 2017]

Acceleration due to gravity,

$g=\frac{GM}{R^2}=\frac{4}{3}\pi \rho GR \left(\because \rho =\frac{M}{V}\text{ and }V=\frac{4}{3}\pi R^3\right)$

$\therefore \frac{g_1}{g_2}=\frac{{\rho }_1{R}_1}{{\rho }_2{R}_2} ..............\left(\mathrm{i}\right)$

Atmospheric pressure can be given by  $p=\frac{w}{S} .......\left(\mathrm{ii}\right)$

where, w = weight of atmosphere

and S = surface area of the planet.

From w = mg, we get

$\frac{m_1}{m_2}=\frac{w_1{g}_2}{w_2{g}_1}$

$\therefore \frac{m_1}{m_2}=\frac{p_1{S}_1{g}_2}{p_2{S}_2{g}_1} [From Eq. (ii\left)\right]$

$=\frac{p_1·\left(4\pi R_1^2\right)}{p_2\left(4\pi R_2^2\right)}·\frac{{\rho }_2{R}_2}{{\rho }_1{R}_1} [From Eq. (i\left)\right]$

$=\frac{p_1{R}_1{\rho }_2}{p_2{R}_2{\rho }_1}$