Atmosphere of a planet contains only an ideal gas of molar mass `M_(0)`. The temperature of the atmosphere varies with height such that the density of atmosphere remains same throughout. The planet is a uniform sphere of mass M and radius ‘a’. Thickness of atmosphere is small compared to ‘a’ so that acceleration due to gravity can be assumed to be uniform throughout the atmosphere.
(a) Find the temperature difference between the surface of the planet and a point at height H in its atmosphere
(b) If the rms speed of gas molecules near the surface of the planet is half the escape speed, calculate the temperature of the atmosphere at a height H above the surface. Assume `(C_(p))/(C_(V)) = gamma` for the atmospheric gas

Answer

(a) `DeltaT=(GMM_(0)H)/(Ra^(0)) " " (b) T_(H)=(GMM_(0))/(R_(a))[(1)/(2 gamma)-(H)/(a)]`