Suppose that there is a planet in the solar system with a semi-major axis of a, of 77,2 A.U.that is an icy?
Question:
Answer:
The sun emits energy at a rate of P = 4x10^26 W. The amount of power density that reaches the icy planey is then this
Pd = P/(4*pi*R^2)
where R is the distance from the sun to the planet. This is the total flux of energy emitted by the sun spread over the surface area of a sphere of radius R. The albedo (A) of a planet indicates how much of the incident energy is reflected back into space. The absorbed power density is then
Pa = (1 - A)*Pd = (1 - A)*P/(4*pi*R^2).
The absorbed energy heats the planet and also leads to radiation. The radiated power follows the Stefan-Boltzmann law
Pr = e*s*T^4
where e is the emissivity, s is Stefan's constant, and T is the temperature of the radiating body. Stefan's constant is s = 5.67x10^-8 W/(m^2*K^4), and for a blackbody, the emissivity is one.
At equilibrium, the absorbed power and the radiated power are equal
(1 - A)*P/(4*pi*R^2) = s*T^4
which can be solved for T, the temperature of the planet.
The temperature of the planet will impart kinetic energy to gas molecules in the air. The speed of the molecules will be distributed with an rms spedd os
Vrms = sqrt(3*k*T/m) = sqrt(3*R*T/M)
where k is Boltzmann's constant, R is the universal gas constant, m is the mass of the molecule, and M is the molar mass.
For a planet the escape velocity for an object is given by
Vesc = sqrt(2*G*M/r)
where G is the universal gravitation constant, M is the mass of the planet, and r is the planet's radius.
If the speed of the gas molecules is greater than the escape velocity of the planet, then that planet will not be able to retain that gas at the given temperature.
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