Planck's Equation relates the rate of emission of radiation at a particular wavelength to the temperature and the wavelength. From this you can derive Stefan-Boltzmann's Law and Wien's Law. [1,2]

The Stefan-Boltzmann Law relates the total rate of radiation of a black body (all wavelengths combined) to the fourth power of the absolute temperature. Wien's Law relates the wavelength of the point of maximum intensity in the continuous spectrum of a black body inversely proportional to the body's absolute temperature.

Lambert and Beer proposed that the amount of monochromatic radiant energy absorbed or transmitted by a media is an exponential function of the concentration of the absorbing substance present in the path of the radiant energy. [3]

Another concept key to understanding the effective radiative temperature on the planet Mars deals with the reflectivity of the planet's surface and atmosphere. Reflectivity is the fraction of incident light reflected (an effective reversal in the photon velocity vector) from a surface. Normal reflectivity is that property whereby light is caused to be reflected 180 degrees from a beam incident normally (perpendicular) to the reflecting surface.

Another concept to consider is called the Bond albedo. The Bond albedo is the ratio of total incident light to total reflected light. It is a measure of global reflection and absorption of sunlight and is often numerically smaller than the normal reflectivity. [1]

The albedo of a planet should, to a first approximation, be able to tell us the effective temperature for a planet by using the Stefan-Boltzmann Law which expresses the radiation emitted by a "perfect radiator" at a specific temperature. Unfortunately, the surface of any planet is not uniform and thus its surface temperature should vary in a non-uniform manner. Astronomers define the effective temperature as that temperature which satisfies the energy balance of the planet when the absorbed sunlight is equated to the emitted thermal power.

In utilizing the albedo-to-temperature relationship, the area of the planet is a necessary parameter. The area of a flat surface is pi time the radius squared buth that of a surface of a sphere is four times the flat surface area. We may simplify the situation by assuming that the planet radiates energy from the entire surface but only absorbs the energy on one face of the planet. [1,2]

Another type of albedo is the geometric albedo. The geometric albedo is the ratio of the global brightness, viewed in the direction of the sun, to that of an hypothetical surface which is white, diffusely reflecting, and possesses the same surface area at the same distance from the sun. [1]

Numerically, the geometric albedo is similar to the standard reflectivity, and because of how it is defined, it is possible to have a geometric albedo greater than one, because the body reflects more light than the reference white body sphere. In fact, sometimes a ratio of Bond albedo and geometric albedo is used, and this is also known as the phase integral. [1]

Other factors to consider revolve around the orbital dynamics of the planet. Kepler's Laws of Planetary Motion can be used in this instance. Kepler's First Law states that all planets follow ellipses about the sun, with the sun at one of the foci of the ellipse. Kepler's Second Law states that the line joining the centers of mass of the sun and a planet covers an area that increases at a constant rate as the planet moves in its orbit (i.e. the radius vector sweeps over equal areas in equal periods of time). Kepler's Third Law states that the square of the sidereal period divided by the cube of the planet's mean distance from the sun forms a ratio that is the same for all of the planets (i.e. a constant). [1,2]

Mars, like Earth, re-radiates some of the solar energy it receives as thermal radiation back into space. The thermal radiation from Mars arises mostly from its surface since its atmosphere is so much finer than Earth's. [4]

The Martian atmosphere emits weakly under clear sky conditions. When dust storms are present, the atmosphere absorbs sunlight directly and also radiates more effectively. The effect of increasing dust in the atmosphere is to reduce diurnal temperature variation at the surface. However, the dust also enhances atmospheric temperature variation. [4]

Other factors effecting thermal radiation from the surface of Mars include the dissipation of mechanical energy of winds, of wind waves, atmospheric tides and the energy transferred by the precipitation of carbon dioxide (and water, if present). Due to the time limitations, not all of these factors are likely to be incorporated into the model to be developed, which will simply be used as a set of input parameters for the life simulation code to be developed.

References are linked here.