Difference between revisions of "Radiative Processes in Astrophysics"

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</math>
 
</math>
  
=== Derivation ===
+
'''Derivation:'''
 
 
 
The # density of photons having frequency between <math>\nu</math> and <math>\nu+d\nu</math> has
 
The # density of photons having frequency between <math>\nu</math> and <math>\nu+d\nu</math> has
 
to equal the # density of phase-space cells in that region, multiplied by
 
to equal the # density of phase-space cells in that region, multiplied by

Revision as of 20:42, 9 February 2010

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,