power P = \frac{energy}{time}

= \frac{J}{s}

flux F = \frac{energy}{area time}

= \frac{J}{m^{2} s}

flux density S_{ν} = \frac{energy}{area time frequency}

= \frac{J}{m^{2} s Hz}

1 Jansky = 10^{−26} \frac{J}{m^{2} s Hz}

Intensity I_{ν} = \frac{energy}{area time frequency solid·angle}

= \frac{J}{m^{2} s Hz sr}

solid angle Ω = πθ^{2} sr measured power P = ∫ I_{ν} dΩ dν dA

L_{Hz} ∼ 33 \frac{J}{s Hz}

I_{ν} = \frac{L_{Hz}}{4πr^{2} Ω}

flux density F_{ν} = Xν^{2} flux density F_{ν} = (\frac{2kT}{c^{2}} ) ν^{2}

intensity I_{ν} = (\frac{2kT}{c^{2}} ) ν^{2} ?

T = (\frac{c^{2}}{2kν^{2}} ) I_{ν} if a blackbody

I_{ν} = \frac{P}{A Δν Ω}

brightness temperature T_{B} ≡ (\frac{c^{2}}{2kν^{2}} ) I_{ν}

beam width θ_{b} ∼ \frac{λ}{D}

beam solid angle Ω_{b} ∼ π (\frac{θ_{b}}{2} )^{2} ∼ \frac{π}{4} (\frac{λ}{D} )^{2}

Ω_{b} ∼ \frac{π}{4} (\frac{λ}{D} )^{2} ∼ \frac{π c^{2}}{4 D^{2} ν^{2}}

diluted I_{ν}^{∗} = \frac{P}{A Δν Ω_{b}} = I_{ν} (\frac{Ω_{source}}{Ω_{b}} )

diluted T_{B}^{∗} = (\frac{c^{2}}{2kν^{2}} ) I_{ν}^{∗} = T_{B} (\frac{Ω_{source}}{Ω_{b}} )

usually T_{B}^{∗} < T_{B}

antenna temperature T′_{a} = T_{B} (\frac{Ω_{source}}{Ω_{b}} )