Difference between revisions of "Radiation Lecture 01"

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(Created page with '<latex> \documentclass[11pt]{article} \def\inv#1{\frac{1}{#1}} \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \section*{Units} Here are some te…')
 
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\begin{document}
 
\begin{document}
  
\section*{Units}
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\subsection*{Units}
  
 
Here are some terms pertaining to telescope observations:\par
 
Here are some terms pertaining to telescope observations:\par
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$$\boxed{F_\nu\equiv\int I_\nu\cos\theta d\Omega}$$
 
$$\boxed{F_\nu\equiv\int I_\nu\cos\theta d\Omega}$$
  
\section*{ Proof that Specific Intensity is conserved along a ray }
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\subsection*{ Proof that Specific Intensity is conserved along a ray }
  
 
The power received by the telescope is:
 
The power received by the telescope is:
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is conserved along a ray, where $\eta$ is the index of refraction.
 
is conserved along a ray, where $\eta$ is the index of refraction.
  
\section*{ The Blackbody }
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\subsection*{ The Blackbody }
  
 
A blackbody is the simplest source: it absorbs and reemits radiation with
 
A blackbody is the simplest source: it absorbs and reemits radiation with
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Note that this tail peaks at $\sim {3kT\over h}$. Also,
 
Note that this tail peaks at $\sim {3kT\over h}$. Also,
 
$$\nu B_\nu=\lambda B_\lambda$$
 
$$\nu B_\nu=\lambda B_\lambda$$
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Revision as of 02:33, 14 February 2010

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 Blackbody

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 Planck Function for Black Body Radiation)

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 :

Wein tail

If :

Rayleigh-Jeans tail Note that this tail peaks at . Also,