Satelite and heat from sun and earth.

[prehisto]

Homework Statement

Assume that a satellite can be modeled as a thermally leading sphere with radius 0.5m and its energy consumption is 200W, and its surface can be described with α=0.3 and ε=0.9.
Determine the equilibrium temperature of satellite in these cases:
a) Satellite is between Earth and sun;
b) Earth is between satellite and sun;
c) Satellite is far away from stars or planets.
Assume that Suns radiation is 1367 Watt/m^2, from Earth reflected 410 Watt/m^2 and that Earth's thermal radiation can be modeled as black body radiation with temperature T=255 K

Homework Equations

The Attempt at a Solution

I know this law R_T=δ*b(T^4-T_∞^4) and the equilibrium state is when radiation=R_T,but that's only the case when one side of the body is isolated. But here i have radiation coming from 2 directions. So can someone please help me, with some ideas ?

 

[anathema]

First, we need to calculate the total incoming radiation on the satellite from both the sun and the Earth. For part a), the satellite is between the Earth and the sun, so it will receive radiation from both sources. The total incoming radiation can be calculated as follows:

R_in = R_sun + R_earth

Where R_sun is the radiation from the sun and R_earth is the radiation reflected by the Earth. We can use the given values to calculate R_sun and R_earth:

R_sun = 1367 Watt/m^2
R_earth = 410 Watt/m^2

Therefore, R_in = 1367 + 410 = 1777 Watt/m^2.

Now, we can use the Stefan-Boltzmann law to calculate the equilibrium temperature of the satellite. The law states that the rate of thermal radiation (R_T) from a black body is equal to the emissivity (ε) times the Stefan-Boltzmann constant (σ) times the temperature raised to the fourth power (T^4):

R_T = ε * σ * T^4

R_T can also be expressed as the difference between the incoming radiation and the outgoing radiation (R_in - R_out). In this case, the outgoing radiation is equal to the thermal radiation emitted by the satellite:

R_out = ε * σ * T_sat^4

Setting these two equations equal to each other and solving for T_sat gives us:

T_sat = (R_in / (ε * σ))^(1/4)

Substituting in the given values:

T_sat = (1777 / (0.9 * 5.67 * 10^-8))^(1/4) = 354.8 K

Therefore, the equilibrium temperature of the satellite between the Earth and the sun is approximately 354.8 K.

For part b), the satellite is on the opposite side of the Earth from the sun, so it will only receive radiation from the Earth. The total incoming radiation can be calculated as follows:

R_in = R_earth = 410 Watt/m^2

Using the same equation as above, we can calculate the equilibrium temperature of the satellite:

T_sat = (R_in / (ε * σ))^(1/4) = (410 / (0.9 * 5.67 * 10^-8))^(1/4) = 252.5 K

Therefore, the equilibrium temperature of the