Question

A planet of mass $M$ and radius $r$ is surrounded by an atmosphere of constant density consisting of a gas of molar mas $\mu$. Determine the temperature $T$ of the atmosphere on the surface of the planet, if the height of the atmosphere is $h$ and $R$ is the gas constant ($h<<r$)

• A. $\frac{R{r}^2}{\mu Mh}$
• B. $\frac{\mu GMh}{R{r}^2}$
• C. $\frac{\mu h{r}^2}{GMR}$
• D. $\frac{GM}{\mu hR{r}^2}$

Answer 1

The correct option is B $\frac{\mu GMh}{R{r}^2}$

Pressure of atmosphere varies as, $P=P_o{e}^{-\frac{mgz}{kT}}$.

Scale height of atmosphere is defined as height at which pressure falls to $\frac{1}{e}$ of its value at sea level.

Hence, $h=\frac{kT}{mg}$

$T=\frac{mgh}{kT}$

Since, $g=\frac{GM}{r^2}$

$T=\frac{mGMh}{k{r}^2}$

Since, $k=\frac{R}{N_A}$

$T=\frac{\mu GMh}{R{r}^2}$