Time to reach thermal equilibrium with radiation

[hellfire]

I would like to do some calculations for the evolution of the temperature of the universe with a homogeneous distribution of light sources in some simple models. For example, starting with the simplest one, consider a universe with an origin of time and with a static space. The integral of the bolometric flux received from different shells from r = 0 up to r = c T (T the age of the universe) is finite and there is no Olbers' paradox. However, such a model "tends" to a paradox as T -> infinity with a diverging flux integral. Consider an object which is created at the same time than the universe (t = 0). How can be calculated the time (or time scale) for this body to reach nearly thermal equilibrium with the radiation emitted by the light sources (and on what does this depend)? Let's call this time T_eq. Consider now a body created (or "inserted in this universe") at t > T_eq. How long does it take for this body to reach thermal equilibrium?

 

[turbo]

Let's start with Charles-Edouard Guillaime and this translation of his article `La Temp'erature de L'Espace', La Nature (1896)

Captain Abney has recently determined the ratio of the light from the starry sky to that of the full Moon. It turns out to be 1/44, after reductions for the obliqueness of the rays relative to the surface, and for atmospheric absorption. Doubling this for both hemispheres, and adopting 1/600,000 as the ratio of the light intensity of the Moon to that of the Sun (a rough average of the measurements by Wollaston, Douguer and Z\"ollner), we find that the Sun showers us with 15,200,000 time more vibratory energy than all the stars combined. The increase in temperature of an isolated body in space subject only to the action of the stars will be equal to the quotient of the increase of temperature due to the Sun on the Earth's orbit divided by the fourth root of 15,200,000, or about 60. Moreover, this number should be regarded as a minimum, as the measurements of Captain Abney taken in South Kensington may have been distorted by some foreign sources of light. We conclude that the radiation of the stars alone would maintain the test particle we suppose might have been placed at different points in the sky at a temperature of 338/60 = 5.6 abs. = $-207^\circ$.4 centigrade. We must not conclude that the radiation of the stars raises the temperature of the celestial bodies to 5 or 6 degrees. If the star in question already has a temperature that is very different from absolute zero, its loss of heat is much greater. We will find the increase of temperature due to the radiation of the stars by calculating the loss using Stefan's law. In this way, we find that for the Earth, the temperature increase due to the radiation of the stars is less than one hundred-thousandth of a degree. Furthermore, this figure should be regarded as an upper limit on the effect we seek to evaluate.

 

[hellfire]

Thank you turbo-1, but at first I am looking for a formula for the time it takes to reach thermal equilibrium in a radiation field.