Question

The two planets with radii $R_1$ and $R_2$ have densities ${\rho }_1$ and ${\rho }_2$, and atmospheric pressures $P_1$ and $P_2$ respectively. Find the ratio of masses of their atmosphere. [neglecting the variation of $g$ and $\rho$ within the limits of atmosphere and height of atmosphere $h<<R$]

• A. $\frac{P_1{R}_2{\rho }_1}{P_2{R}_1{\rho }_2}$
• B. $\frac{P_1{R}_2{\rho }_2}{P_2{R}_1{\rho }_1}$
• C. $\frac{P_1{R}_1{\rho }_1}{P_2{R}_2{\rho }_2}$
• D. $\frac{P_1{R}_1{\rho }_2}{P_2{R}_2{\rho }_1}$

Answer 1

The correct option is D $\frac{P_1{R}_1{\rho }_2}{P_2{R}_2{\rho }_1}$

Consider a planet where,

$m=$Mass of atmosphere,

$h=$Height of atmosphere,

$P=$Pressure at the surface of the planet,

${\rho }_{atm}=$density of atmosphere.


$R=$Radius of planet and

$g=$Accelaration due to gravity.

Now, atmospheric pressure on the planet,

$P={\rho }_{atm}gh$

Where,

${\rho }_{atm}=\frac{M_{atm}}{V_{atm}}=\frac{m}{\frac{4}{3}\pi [{(R+h)}^3-R^3]}$


$\Rightarrow P=\frac{mgh}{\frac{4}{3}\pi [{(R+h)}^3-R^3]}$

$\Rightarrow P=\frac{mgh}{\frac{4}{3}\pi [R^3{(1+\frac{h}{R})}^3-R^3]}$

As, $h<<<R$, So

$[{(1+\frac{h}{R})}^3\approx 1+\frac{3h}{R}]$

$\Rightarrow P=\frac{mgh}{\frac{4}{3}\pi [R^3(1+\frac{3h}{R})-R^3]}$

$\Rightarrow P=\frac{mgh}{\frac{4}{3}\pi R^2\times 3h}$

$\Rightarrow P=\frac{mg}{4\pi R^2}$

As, we know that, $g=\frac{GM}{R^2}$ where, $M$ is mass of planet.

$\Rightarrow P=\frac{m}{4\pi R^2}\frac{GM}{R^2}$

$\Rightarrow P=\frac{Gm}{4\pi R^4}M$

Substituting the value of mass of planet $M$,

$\Rightarrow P=\frac{Gm}{4\pi R^4}\times \frac{4}{3}\pi R^3\rho$,

where, $\rho$ is the density of planet.

$\Rightarrow P=\frac{Gm\rho }{3R}$

$\Rightarrow m=\frac{3}{G}\frac{PR}{\rho }$

Since, $\frac{3}{G}$ is constant, so

$\Rightarrow m\propto \frac{PR}{\rho }$

If $m_1$ and $m_2$ are the masses of atmospheres for the given planets in the problem, then

$\therefore \frac{m_1}{m_2}=\frac{P_1{R}_1{\rho }_2}{P_2{R}_2{\rho }_1}$

Hence, option (d) is the correct answer.

$\text{Key Concept:}$ Acceleration due to gravity on the surface of a planet is given by $$g=\frac{GM}{R^2}$$$\text{Why this question:}$ To make students understand that how the atmospheric mass depends on the different parameters of the planet and how to deduce the relation between atmospheric pressure and gravitational field parameters.