---
aliases:
- /2008/03/relation-between-power-density-and
categories:
- astrophysics
date: 2008-03-28 18:29
layout: post
slug: relation-between-power-density-and
title: Relation between Power density and temperature in an antenna

---

<p>
 Considering an antenna placed inside a blackbody enclosure at temperature T, the power received per unit bandwidth is:
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 $latex \omega = kT$
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 <br/>
 where k is Boltzmann constant.
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 <br/>
 This relationship derives from considering a constant brightness $latex B$ in all directions, therefore Rayleigh Jeans law tells:
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 <br/>
 $latex B = \dfrac{2kT}{\lambda^2}$
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 <br/>
 Power per unit bandwidth is obtained by integrating brightness over antenna beam
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 <br/>
 $latex \omega = \frac{1}{2} A_e \int \int B \left( \theta , \phi \right) P_n \left( \theta , \phi \right) d \Omega  $
 <br/>
 <br/>
 therefore
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 <br/>
 $latex \omega = \dfrac{kT}{\lambda^2}A_e\Omega_A $
 <br/>
 <br/>
 where:
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</p>
<ul>
 <br/>
 <li>
  $latex A_e$ is antenna effective aperture
 </li>
 <br/>
 <li>
  $latex \Omega_A$ is antenna beam area
 </li>
 <br/>
</ul>
<br/>
$latex \lambda^2 = A_e\Omega_A $ another post should talk about this
<br/>
<br/>
finally:
<br/>
<br/>
$latex \omega = kT $
<br/>
<br/>
which is the same noise power of a resistor.
<br/>
<br/>
source : Kraus Radio Astronomy pag 107