Flux

The "brightness" of a star is actually measured in terms of the radiant flux $$F$$ received from the star. The radiant flux is the total amount of light energy of all wavelengths that crosses a unit area oriented perpendicular to the direction of the light's travel per unit time.

Units of Flux
$$F = \frac {[energy]}{[time][distance]^2}$$

Relationship to Luminosity
Imagine a star of luminosity $$L$$ surrounded by a huge spherical shell of $$r$$. Then, assuming that no light is absorbed during its journey out to the shell, the radiant flux $$F$$, measured at a distance $$r$$ follows the equation detailed below

$$F = \frac {L}{4 \pi r^2}$$

Reliance on Surface Area
The flux changes depending on how much surface area the luminosity is being directed towards.

When the luminosity is being directed into half the celestial sphere, the flux can be described with:

$$F = \frac {L}{2 \pi r^2}$$

Luminosity of an Object Receiving Flux
When an object receives flux and reflects it, it will inherently have some luminosity. Assuming that an object receives $$F _r$$ and reflects with some factor $$\alpha$$ (0 is total absorbing, 1 is total reflecting) and the object has some cross sectional area $$A$$, then it's luminosity can be expressed as:

$$L = \alpha A F _r$$

Blackbody Surface Flux
The amount of surface flux emitted by a blackbody can be characterized with the following equation:

$$F _{blackbody} = \sigma T^4$$

Flux Retained Through a Medium
Whenever flux goes through some medium, there is some of it that will inevitably be absorbed. The equation that governs how much flux will leave the medium is described below:

$$F = F_o e ^ {-\tau}$$