Question

Find a relationship between the equilibrium temperature of a spherical object and the distance from the sun in terms of the Sun's emission temperature

Answer 1

example of this is thermal equilibrium of water filled with ice, wherein the 'system' is the warm room, glass, ice and water. The temperature of the ice will increase thereby decreasing the temperature of water. So, evidently the amount of heat lost by the hotter object will be given to the colder object.

Qhot to Qcold (Law of Conservation of Energy)
It may seem that Qlost = Qgained, but this is not completely correct.
(Or use the Law of Conservation of Energy: ∆E = 0(closed system) thereby we can deduce
the thermal equilibrium formula

Q = m x cp x ∆T

where Q = Heat Flow (Heat lost or Heat gained)
m = Mass of the substance
cp = Specific heat capacity
∆T = (Tf - Ti) = Difference in temperature

Although much hotter on the inside, we can closely approximate the surface of the sun, from which its emission occurs, as a black body at a temperature of about 5800 K. The Stefan-Boltzmann equation then gives the energy flux emitted at the sun’s surface.

S_S = (5.67 × 10^–8 W·m^–2·K^–4)(5800 K)⁴ = 63 × 10⁶W·m^–2

The surface area of a sphere with a radius r is 4πr². If r_S is the radius of the Sun, the total energy it emits is S_S4πr_s². As the radiation is emitted from this spherical surface, it is spread over larger and larger spherical surfaces, so the energy per square meter decreases