Temperature on earth if radiation off earth equals radiation recieved from the sun?

What would the temperature on earth be if we assume that its temperature makes the outgoing radiation equal to the radiation vi receive from the sun? The earth is not a perfect black-body and some of the the outgoing radiation will be reflected by the atmosphere.



The power density of the sun's radiation on the surface of the earth is approximately 1.4 kW/m^2 when the radiation is perpendicular to the ground.



And in addition, what would the earth temperature be if we had sunlight 24 hours a day everywhere on earth?



Pointers or complete solutions would be greatly appreciated.Temperature on earth if radiation off earth equals radiation recieved from the sun?
The idea is the following: the total power emission of a black body is P = s T^4 where s is the stefan boltzmann constant. Thus, the sun, which we can imagine to be at the center of the earth's circular orbit radiates a power s T^4. now if we assume the radiation is isotropic (independent of direction) how much power is incident on the earth? consider the energy flux (power/unit area) emitted from the sun, ie imagine the power radiating like a big sphere from the surface of the sun- for power to be conserved the power density must decrease like 1/r^2 where r is the distance from the sun. so the power density at a distance R from the sun is



p = P/A = s T^4 / (4 pi R^2)



this power density p is incident on the earth's cross section, and so the total power absorbed by the earth is



p * pi re^2



where re is the radius of the earth. the earth will be at thermal equilibrium when it is also emitting this much power



ie when



Pe = s Te^4 = s T^4 (pi re^2) / (4pi R^2)



where Te is the temperature of the sun. So just solve for Te!