Here are some terms pertaining to telescope observations: aperture area (), solid angle on sky (), exposure time (), collects energy (), over waveband (), but .
is the specific intensity per unit frequency.
Flux density is power per unit frequency passing through a differential area whose normal is . Thus, flux density is:
Proof that Specific Intensity is conserved along a ray
The power received by the telescope is:
where is the intensity as a function of right-ascension () and declination (). Say that is the surface luminosity of a patch of sky (that is, the emitted intensity). Then power emitted by patch of sky is:
Recognizing that : This derivation assumes that we are in a vacuum and that the frequencies of photons are constant. If frequencies change, then though specific intensity is not conserved, is. Also, for redshift , so intensity decreases with redshift. Finally: is conserved along a ray, where is the index of refraction.
The Black Body
A blackbody is the simplest source: it absorbs and reemits radiation with
100% efficiency. The frequency content of blackbody radiation is given by
the Planck Function:
The # density of photons having frequency between and has
to equal the # density of phase-space cells in that region, multiplied by
the occupation # per cell. Thus:
so we have it. In the limit that :
Note that this tail peaks at . Also,