= Microwave radiometer calibration example
{title2=$S=g(T+T_r)$}
Consider an original idealized narrow-band model for one matched spatial mode and one <polarization of waves>. In the classical <Rayleigh-Jeans law> regime, an available thermal <power> can be written $P=k_BT\Delta\nu$, where $\Delta\nu$ is the effective <bandwidth> and $T$ the equivalent input <noise temperature>. Assume a linear detector with stable positive gain and absorb its <bandwidth>, <Boltzmann constant> and response factor into $g$. Then the output is $S=g(T+T_r)$, with input-referred <receiver noise temperature> $T_r$. These are model assumptions, not a reconstruction of the historical electronics.
Two references with known <noise temperatures> $T_h\ne T_c$ give
$$
S_h=g(T_h+T_r),\qquad S_c=g(T_c+T_r),
$$
so the <two-load radiometer calibration> yields
$$
g=\frac{S_h-S_c}{T_h-T_c},\qquad
T_r=\frac{T_hS_c-T_cS_h}{S_h-S_c},\qquad
T_s=T_c+(T_h-T_c)\frac{S_s-S_c}{S_h-S_c}.
$$
Subtracting the reference equations eliminates the unknown receiver term; substituting back determines it. An arbitrary stable output offset can likewise be eliminated from the interpolation for $T_s$, although separating that offset from $T_r$ then requires additional information.
For deliberately chosen inputs, set $T_h=300\,\mathrm K$, $T_c=4.2\,\mathrm K$, $T_r=40\,\mathrm K$ and $g=2\,\mathrm{mV/K}$. These are equivalent <noise temperatures>; physical load thermometry must be converted consistently at the observing <frequency>. The reference outputs are $S_h=680\,\mathrm{mV}$ and $S_c=88.4\,\mathrm{mV}$. A sky-input output $S_s=90\,\mathrm{mV}$ gives $T_s=5.0\,\mathrm K$. If independent estimates assign $1.4\,\mathrm K$ to atmosphere, $0.5\,\mathrm K$ to antenna loss and $0.1\,\mathrm K$ to ground pickup in this lumped additive model, the residual is $3.0\,\mathrm K$.
\Image[/cmb-radiometer-calibration-and-planck-correction.png]
{title=Original calibration arithmetic and the distinction between an antenna temperature and a thermodynamic temperature}
{description=Chosen reference temperatures, receiver response and additive contributions; none are fitted historical measurements. The final panel calculates the <Rayleigh-Jeans law> equivalent <antenna temperature> of a stipulated $3\,\mathrm K$ <blackbody>. A change in <frequency> changes that equivalent temperature even though the underlying thermodynamic temperature is fixed.}
The subtraction exposes what a noise budget must achieve. A $1\,\mathrm K$ error in the assigned atmospheric term becomes a $1\,\mathrm K$ error in the residual, even with exact detector calibration. Multiplicative attenuation requires a different correction: if an isothermal atmosphere has transmission $t$, its ideal contribution is $tT_{\rm cosmic}+(1-t)T_{\rm atm}$ in the same classical equivalent-temperature convention. Recovering $T_{\rm cosmic}$ then requires both subtracting atmospheric emission and dividing by $t$. The additive toy budget applies when its terms already refer to the same receiver input.
There is also a conditioning issue. If $S_h-S_c$ is small, a fixed output error strongly changes the inferred <temperature>. Widely separated reference temperatures improve the scale determination only if the receiver remains linear over that range; measuring the reference repeatedly tests stability rather than proving linearity. The experiment therefore motivates a model, a calibration and a set of independent checks, each with its own assumptions.
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