Single Dish Math

Single Dish Math#

Here we briefly review some of the equations governing Single Dish math.

The Model#

The power being recorded by a backend can be described by

(1)#\[P=G\left(T_{\mathrm{A}}+T_{\mathrm{sys}}\right)\]

with \(G\) the telescope gain, \(T_{\mathrm{A}}\) the antenna temperature and \(T_{\mathrm{sys}}\) the system temperature.

During observations, a noise diode or a hot load can be used to determine the telescope gain or system temperature. When the noise diode is firing, the power becomes

(2)#\[P^{\mathrm{cal}}=G\left(T_{\mathrm{A}}+T_{\mathrm{sys}}+T_{\mathrm{cal}}\right)\]

with \(T_{\mathrm{cal}}\) the equivalent temperature of the noise diode or hot load, a quantity that is known beforehand or that must be derived from calibration observations.

System Temperature#

For observations with a noise diode or hot load the system temperature can be computed using

(3)#\[T_{\mathrm{sys}}=T_{\rm{cal}}\left[\frac{P_{\rm{ref}}}{P_{\rm{ref}}^{\rm{cal}}-P_{\rm{ref}}}\right]+\frac{T_{\rm{cal}}}{2}\]

where \(P_{\rm{ref}}\) is the power measured towards a reference position, a region without signal. The factor \(T_{\rm{cal}}/2\) accounts for the noise diode being fired half of the time — this factor is ommited if using a hot load. To minimize the “noise” in the computation, dysh takes the average of the numerator and the denominator over the inner 80% of the channels. So, in practice the system temperature is computed as

(4)#\[T_{\mathrm{sys}}=T_{\rm{cal}}\left[\frac{\langle P_{\rm{ref}}\rangle}{\langle P_{\rm{ref}}^{\rm{cal}}-P_{\rm{ref}}\rangle}\right]+\frac{T_{\rm{cal}}}{2}\]

where the \(\langle\rangle\) operator denotes an average. Thus, the system temperatures computed are scalars. In dysh the function responsible for this calculation is dysh.spectra.core.mean_tsys(). For more details on how to compute \(T_{\mathrm{cal}}\) for a hot load see GBT memo #302.

Antenna Temperature#

The antenna temperature is computed using

(5)#\[T_{\rm{A}}=T_{\rm{sys}}\frac{P-P_{\rm{ref}}}{P_{\rm{ref}}}\]

with \(P\) the power of the signal (e.g., towards the target in a position switched observation).

Radiometer Equation#

The radiometer equation equates the noise to the system temperature, integration time and bandwidth

(6)#\[\sigma(T)=\frac{T_{\rm{sys}}}{\sqrt{\Delta t \Delta\nu}}\]

with \(\Delta t\) the integration time and \(\Delta\nu\) the bandwidth. This is the standard deviation of the signal if it was purely thermal noise.

Weights#

By default, dysh uses the inverse variance of the thermal noise as weights

(7)#\[w=\frac{\Delta t \Delta\nu}{T_{\rm{sys}}^{2}}\]

Brightness Scales#

The definitions of the brightness scales used by dysh are in GBT memo #302.

Miscellaneous#

Smoothing the Reference#

During calibration it is possible to smooth the reference using the smoothref argument (this is an argument to the calibration routines, e.g., getps()). For purely thermal noise this would reduce the noise in the calibrated spectrum by a factor

(8)#\[\sqrt{\frac{N+1}{2N}}\]

where \(N\) is the width, in channels, of the smoothing kernel. So, if using smoothref=3 the noise should be reduced by \(\sqrt{4/6}\), or 0.82.

Exposure Time#

Effective exposure time after calibrating using a noisy reference power

(9)#\[\Delta t=\frac{t_{\rm{sig}}t_{\rm{ref}}}{t_{\rm{sig}}+t_{\rm{ref}}}\]

but this is assuming the reference was not smoothed, as from the previous section. The correct equation for a smoothed reference (with \(N\) the number of channels to smooth with) would be

(10)#\[\Delta t= t_{\rm{sig}} \frac{N.t_{\rm{ref}}}{t_{\rm{sig}}+N.t_{\rm{ref}}}.\]

Ruze Equation#

The Ruze equation relates the gain of an antenna to the root mean square (\(\delta\)) of the antenna’s random surface errors.

(11)#\[G = G_0 \exp{ (-(4\pi\delta / \lambda)^2) }\]
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