
\magnification=\magstep1

$$
\eqalign{
{\rm power\ \ P} \quad &= \quad {\rm { { energy } \over { time }}} \cr
                      &\phantom{=} \cr
                      &= \quad {\rm { { J } \over { s }}} 
}
$$



$$
\eqalign{
{\rm flux\ \ F} \quad &= \quad {\rm { { energy } \over { area \ time }}} \cr
                      &\phantom{=} \cr
                      &= \quad {\rm { { J } \over { m^2 \thinspace s }}} 
}
$$


$$
\eqalign{
{\rm flux\ density\ }S_{\nu} \quad &= \quad {\rm { { energy } \over { area \ time \ frequency}}} \cr
                      &\phantom{=} \cr
                      &= \quad {\rm { { J } \over { m^2 \thinspace s \thinspace Hz}}} 
}
$$

$$
{\rm 1\ Jansky} \quad = \quad 10^{-26} \ {\rm { { J } \over { m^2 \thinspace s \thinspace Hz}}} 
$$

$$
\eqalign{
{\rm Intensity\ }I_{\nu}\ \quad &= \quad {\rm { { energy } \over { area \ time \ frequency\ solid{\cdot}angle }}} \cr
                      &\phantom{=} \cr
                      &= \quad {\rm { { J } \over { m^2 \thinspace s \thinspace Hz \thinspace sr}}} 
}
$$

$$
{\rm solid\ angle\ } \Omega \quad = \quad \pi \theta^2 \quad \quad {\rm sr} 
$$

$$
{\rm measured\ power\ P } \quad = \quad \int I_{\nu} \ d\Omega \ d\nu \ dA
$$

$$
L_{\rm Hz} \quad \sim \quad 33 \ {\rm {{J}\over{s\ Hz}}}
$$

$$
I_{\nu} \quad = \quad { { {L_{\rm Hz}} \over { 4 \pi r^2 \ \Omega} } }
$$


$$
{\rm flux\ density\ \ F_{\nu}} \quad = \quad X {\nu}^2
$$


$$
{\rm flux\ density\ \ F_{\nu}} \quad = \quad \left( {{2 k T}\over{c^2}} \right)  {\nu}^2
$$

$$
{\rm intensity\ \ I_{\nu}} \quad = \quad \left( {{2 k T}\over{c^2}} \right)  {\nu}^2 \quad \quad \quad {\rm ?}
$$

$$
T \quad = \quad \left( {{c^2} \over {2 k {\nu}^2}} \right) \ I_{\nu} \quad \quad \quad {\rm if\ a\  blackbody}
$$

$$
{\rm I_{\nu}} \quad = \quad {{P} \over { A \ \Delta \nu \ \Omega} } 
$$

$$
{\rm brightness\ temperature\ T_{B}}  \quad \equiv  \quad 
       \left( {{c^2} \over {2 k {\nu}^2}} \right) \ I_{\nu}
$$

$$
{\rm beam\ width\ } \theta_{b} \quad \sim \quad {{\lambda}\over{D}}
$$

$$
{\rm beam\ solid\ angle\ } \Omega_{b} \quad \sim \quad
                     \pi \left( { {\theta_{b}}\over{2}} \right)^2
                      \quad \sim \quad {{\pi}\over{4}} \left( {{\lambda}\over{D}} \right)^2
$$

$$
\Omega_{b} \quad \sim \quad { {{\pi}\over{4}} } 
                            \left( {{\lambda}\over{D}} \right)^2 
                    \quad \sim \quad
                            { {\pi \thinspace c^2} \over {4 \thinspace D^2 \thinspace \nu^2} } 
$$

$$
{\rm diluted\ } I_{\nu}^{*} \quad = \quad {{P} \over { A \ \Delta \nu \ \Omega_{b}} }  \quad = \quad I_{\nu} \ \left( {{\Omega_{\rm source}} \over {\Omega_b}} \right) 
$$

$$
{\rm diluted\ T_{B}^{*}}  \quad =  \quad 
       \left( {{c^2} \over {2 k {\nu}^2}} \right) \ I_{\nu}^{*} 
       \quad = \quad {\rm T_B} \left( {{\Omega_{\rm source}} \over{\Omega_b}} \right) 
$$

$$
{\rm usually\ } \quad \quad \quad {\rm T_{B}^{*} } \quad < \quad 
                                  {\rm T_{B} }
$$

$$
{\rm antenna\ temperature\  T_{a}' } \quad = \quad {\rm T_{B}} 
                       \left( { {\Omega_{\rm source} } \over {\Omega_b}} \right)
$$





\bye
