# Radiative Processes in Astrophysics

## Lecture 1

### Units

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*:

### Derivation

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:

However,

so we have it. In the limit that :

If :

Note that this tail peaks at . Also,