Relation between Power density and temperature in an antenna

Considering an antenna placed inside a blackbody enclosure at temperature T, the power received per unit bandwidth is:
$latex\omega =kT$

where k is Boltzmann constant.

This relationship derives from considering a constant brightness $latexB$ in all directions, therefore Rayleigh Jeans law tells:

$latexB={\displaystyle \frac{2kT}{{\lambda }^2}}$

Power per unit bandwidth is obtained by integrating brightness over antenna beam

$latex = A_e B ( , ) P_n ( , ) d $

therefore

$latex = A_e_A $

where:

• $latex{A}_e$ is antenna effective aperture


• $latex{\mathrm{\Omega }}_A$ is antenna beam area

$latex ^2 = A_e_A $ another post should talk about this

finally:

$latex = kT $

which is the same noise power of a resistor.

source : Kraus Radio Astronomy pag 107