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 \(latex B\) in all directions, therefore Rayleigh Jeans law tells:
\(latex B = \dfrac{2kT}{\lambda^2}\)
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
- \(latex \Omega_A\) 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