A planet has been discovered around a Sun-like star with an estimated mass of 63

Question

A planet has been discovered around a Sun-like star with an estimated mass of 63.7 times the Earth's mass and an estimated radius of 12.9 times the Earth's radius. It has been hypothesized that the planet has abundant hydrogen in its atmosphere. In order for the planet to maintain a hydrogen atmosphere for the next few billion years, the planet's temperature must be less than some maximum temperature. Recall that the mass of Earth is 5.97 × 1024 kg and the radius of Earth is 6.37 × 106 m. The universal gravitational constant is 6.67 × 10-11 N·m2/kg2. The mass of a hydrogen atom is 1.67 × 10-27 kg. Boltzmann's constant is defined as 1.38 × 10-23 J/K. What is the maximum temperature of the atmosphere? ________k

Explanation / Answer

The maximum temperature of the atmosphere which will be given as :

using an equation, we have

U = (3/2) kB T

G mh me / re = (3/2) kB T

where, G = universal gravitational constant = 6.67 x 10-11 Nm2/kg2

mh = mass of hydrogen atom = 1.67 x 10-27 kg

me = mass of earth = 5.97 x 1024 kg

re = radius of earth = 6.37 x 106 m

kB = boltzmann constant = 1.38 x 10-23 J/K

then, we get

(6.67 x 10-11 Nm2/kg2) (1.67 x 10-27 kg) (63.7 x 5.97 x 1024 kg) / (12.9 x 6.37 x 106 m) = (3/2) (1.38 x 10-23 J/K) T

(4236 x 10-14 Nm2) / (82.1 x 106 m) = (2.07 x 10-23 J/K) T

(51.6 x 10-8 J) = (2.07 x 10-23 J/K) T

T = (51.6 x 10-8 J) / (2.07 x 10-23 J/K)

T = 24.9 x 1015 K

T = 2.5 x 1016 K

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