Question

The thermosphere of a Jovian planet can be very hot; for example, the temperature of Jupiter is more than 400 K at a height of about 200 km above the cloud tops.

Consider a planet around a Sun-like star, with a mass 49 times that of Earth and a radius 9.9 times that of Earth. What is the maximum temperature this planet can be before it starts losing hydrogen to Jeans escape over a few billion years? Report your answer in Kelvin.

Answer 1

Jeans escape occurs when the tail end of the maxwell's velocity distribution (consisting of a small number of molecules) exceeds the escape velocity of the planet at the exobase height. Here the collisions are very less and the thermal energy that results in the thermal velocity of a molecule according to maxwell's distribution is sometimes enough to propel some number of molecules at speeds greater than the escape velocity resulted in the loss of atmosphere. A qualitative diagram is shown below and then the Jeans Rule of Thumb is used to express the temperature at which a planet can retain its hydrogen for a few billion years.