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Temperatures

The power $ P_a$ received by an antenna in a bandwidth $ \delta\nu$ is conveniently expressed in term of antenna temperature, $ T_a$ , which is the temperature for which an equivalent resistor would give the same power $ P_a$ , following the Nyquist noise formula:

$\displaystyle P_a = k T_a \delta\nu$ (2)

As the quantity $ T_a$ is affected by the atmospheric absorption $ e^{-\tau_{atm}}$ and the antenna foward coupling factor $ \eta_f$ , we define the quantity $ T_a^*$ through:

$\displaystyle T_a^* = \frac{(1+g_{im})}{\eta_f}e^{\tau_{atm}}T_a$ (3)

The $ (1+g_{im})$ , where $ g_{im}$ is the image band rejection factor, accounts for a single sideband signal.

Brightness temperature, $ T_b$ , is the Rayleigh-Jeans temperature $ T_b$ of an equivalent blackbody which would give the same power per unit area per unit frequency and per unit solid angle $ I_{\nu}$ as the celestial source:

$\displaystyle T_b = \frac{c^2}{2 k\nu^2} I_{\nu}$ (4)

For a resolved source, the antenna temperature is equal to the sky brightness temperature. For an unresolved source, the coupling between antenna temperature $ T_a^*$ and source flux density $ S_{\nu}$ is:

$\displaystyle T_a^*=\frac{\eta_a A}{2k}S_{\nu} =\frac{1}{J} S_{\nu}$ (5)

where $ J = \frac{2k}{\eta_a A}$ is the antenna efficiency.

The noise temperature is the sum of the various noise contributions:

$\displaystyle T_{noise} = T_{bg} + T_{sky} + T_{spill} + T_{rec}$ (6)

where $ T_{bg}$ is the cosmic background, $ T_{sky}$ is the sky noise, $ T_{spill}$ is the ground noise pickup and $ T_{rec}$ is the receiver noise temperature. The forward efficiency $ \eta_f$ is a property of the antenna that indicates how much coupling there is in the forward hemisphere with respect to the full $ 4\pi$ sphere. With an atmosphere at a physical tempeture $ T_{atm}$ and with an opacity $ \tau_{atm}$ at the observed frequency, the sky noise and ground pickup temperature are expressed as:
$\displaystyle T_{sky}$ $\displaystyle =$ $\displaystyle \eta_f (1-e^{-\tau_{atm}})T_{atm}$ (7)
$\displaystyle T_{spill}$ $\displaystyle =$ $\displaystyle (1-\eta_f)T_{ground}$ (8)

$ T_{sys}$ is the system temperature, the noise equivalent temperature $ T_{noise}$ of the receiving chain refered to a perfect antenna located outside the atmosphere and for a single sideband signal:

$\displaystyle T_{sys} = \frac{1+g_{im}}{\eta_f}e^{\tau_{atm}} T_{noise}$ (9)


next up previous contents index
Next: Radiometric equation Up: Noise Previous: Noise   Contents   Index
Gildas manager 2022-01-17