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\begin{document}

\title{IRAM Memo 2025-1\\[3\bigskipamount]
  NOEMA sensitivity estimator}%
\author{J. Pety$^{1,2}$, J. Boissier$^{1}$, S. Bardeau$^{1}$, E. Reynier$^{1}$\\[0.5cm]
  1. IRAM (Grenoble) \\
  2. Observatoire de Paris}%
\date{Feb, 6$^{th}$ 2025\\
  Version 1.0}

\maketitle

\begin{abstract}
  This memo describes the equations used in the NOEMA sensitivity estimator
  available on the online sensitivity estimator
  (\texttt{https://oms.iram.fr/tse/}) and used for proposal submission in
  the Proposal Management System.

  This memo is a full rewriting of the older memos dedicated to the subject
  to consolidate the equations and to take into account the new dual band
  and frequency cycling modes. The different versions of the memo 2015-2 are
  only kept in order to know how the sensitivity estimates were done in
  past proposal sessions.
\end{abstract}

\newpage{}

\tableofcontents{}

\newpage{}

\section{NOEMA instrumental modes}

The most usual instrumental mode for a radio-interferometer is to observe
with a single band of receiver a single LO frequency. This mode will be
called ``standard'' mode hereafter. NOEMA introduces two other
possibilities:
\begin{description}
\item[In dual band mode,] two frequencies (one at 3 and one at
  1\,mm) are observed simultaneously. The beam has thus to be split into
  two beams with the help of a dichroic. This adds some instrumental noise
  that we will encode as a higher value of the receiver noise.

  As the dichroic can be removed from the optical path when doing single
  band observations, this receiver noise increases will only happen when
  the sensitivity estimation is done in dual band mode.
\item[In frequency cycling mode,] the tuning frequency is regularly cycled
  between \nfreqphase{} predefined values inside the same receiver RF
  band. This implies that the on-source observing time must be split
  between the different tuning of the frequency cycling. To do this, the
  user will have to give the percentage of the time required per
  tuning. The sum of the percentage will have to be equal to 100\%. By
  default, PMS will divide equally the on-source time between the tunings,
  but the user will have the possibility to modify this time repartition.

  After the setup phase, each cycle observed at a given frequency must be
  surrounded by gain calibration observations at the same frequency.  This
  means that the observing efficiency decreases with respect to the
  standard instrumental mode: in practice this is like doubling the number
  of calibrators, since each calibrator will have to be observed at the 2
  frequencies (the frequencies of the previous and of the next cycle,
  whatever the number of cycles).
\end{description}
These two additional possibilities can in principle be combined during
the same observation. 

This memo describes how the sensitivity estimator tool used in the Proposal
Management System (\url{https://oms.iram.fr/oms/?pms}) encodes the
sensitivity estimation for these three instrumental
modes. Sections~\ref{sec:sensitivity:point-source},
\ref{sec:sensitivity:extended}, and~\ref{sec:system-temperature} first
reminds the basic equations used to compute the sensitivity of a
radio-interferometer, namely, the interferometric point source and
extended sensitivity, as well as the system temperature that characterizes
the noise added by the receiver, the telescope optics, and the
atmosphere. Section~\ref{sec:in-practice} details the actual computation as
a function of the observing mode.

\section{The interferometric point source sensitivity}
\label{sec:sensitivity:point-source}

The interferometric point source sensitivity is defined as the rms noise
level that should be observed when an actual point source is observed by
an interferometer.

\subsection{Power and sensitivity measured at the correlator output for one
  baseline}

After the atmopheric calibration that converts the measurement scale from
the correlator output (in counts) to the \Tas{} scale, the output of the
correlator for one correlation is a power equivalent temperature (in the
Rayleigh-Jeans domain), which is sampled at a rate of $2\dnu$, where \dnu{}
is the frequency bandwidth over which the power is measured. As explained
in the Sect.~\ref{sec:system-temperature}, the standard deviation of each
power measurement is given by the system temperature power (\Tsys). During
the integration time (\tint), $2\dnu\,\tint$ independent samples of the
signal power are measured to ensure the Nyquist sampling of the signal in
the bandwidth \dnu. The signal power is averaged over these independent
samples. The uncertainty on the averaged signal power, named sensitivity
(\noise[K]), is thus standard deviation of the average or
\begin{equation}
  \noise[K] = \frac{\Tsys}{\sqrt{2\,\dnu\,\tint}}.
  \label{eq:noise:one:baseline:1}
\end{equation}
One subtlety is that the system temperature characterize one baseline
between, eg., antennas $i$ and $j$. Instead
of just \Tsys{}, we should write in all generality $\Tsys^{ij}$ with
\begin{equation}
  \Tsys^{ij} = \sqrt{\Tsys^{i}\,\Tsys^{j}},
\end{equation}
where $\Tsys^{i}$ and $\Tsys^{j}$ are the single dish system temperature of
antenna $i$ and $j$. For simplicity, we will keep the notation \Tsys{}
hereafter.

\subsection{Quantization efficiency}

The Analog-to-Digital-Converter (ADC) will quantize the analog signal on a
finite number of bits. This can be seen as an additional source of noise
that is modeled as a spectrometer efficiency $(\speceff \le 1.0)$ and
Eq.~\ref{eq:noise:one:baseline:1} becomes
\begin{equation}
  \noise[K] = \frac{\Tsys}{\speceff\,\sqrt{2\,\dnu\,\tint}}.
  \label{eq:noise:one:baseline:2}
\end{equation}

\subsection{From the temperature power to the flux}

For a point source, it is more natural to express the signal, and thus the
sensitivity, in unit of flux. The flux (\flux{}) of a point source is
linked to its brightness temperature \Tas{} through
\begin{equation}
  \flux = \Jant[sd] \Tas
  \quad \mbox{with} \quad
  \Jant[sd] = \frac{2\kbolt\,\Feff}{\Aeff},
  \label{eq:conversion:0}
\end{equation}
where \kbolt{} is the Boltzman constant, and \Aeff{} is the effective area
of the antenna~\citep[eq. 3-113 in][]{kra82}, and \Jant[sd] the conversion
factor for a typical interferometer antenna. The effective area depends on
the observing wavelength when the surface rms accuracy becomes a
significant fraction of the wavelength. For NOEMA, the effective area is
close to the antenna geometrical surface at 3\,mm but significantly lower
than it at 1\,mm. Using the same conversion factor for the sensitivity, we
yield
\begin{equation}
  \noise[Jy] = \frac{\Jant[sd]\,\Tsys}{\speceff\,\sqrt{2\,\dnu\,\tint}}.
  \label{eq:noise:one:baseline:3}
\end{equation}
We have the same subtlety for \Jant[sd] as for \Tsys, ie, \Jant[sd] here
characterizes one baseline between, eg., antennas $i$ and $j$. Instead of
just \Jant[sd], we should write in all generality $\Jant[sd,ij]$ with
\begin{equation}
  \Jant[sd,ij] = \sqrt{\Jant[sd,i]\,\Jant[sd,j]},
\end{equation}
where $\Jant[sd,i]$ and $\Jant[sd,j]$ are the single dish conversion
factors for antenna $i$ and $j$. For simplicity, we will also keep the
notation \Jant[sd] hereafter.

\subsection{Collecting the measurements from all baselines}

Assuming that the atmosphere and receiving system is independent of the
pairs of antenna considered, the previous equation is valid for any
baseline of the interferometer. All the visibilities can be averaged into a
single visibility to define the point source signal. This implies that the
standard deviation (sensitivity) on the point source signal will be divided
by the square root of the number of baselines, i.e., the number of pairs of
antennas, $\nant\,(\nant-1)/2$, where \nant{} is the number of
antennas. The point source sensitivity is then
\begin{equation}
  \noise[Jy] = \frac{\average{\Jant[sd]\,\Tsys}}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}},
  \label{eq:noise:all:baselines:1}
\end{equation}
where $\average{\Jant[sd]\,\Tsys}$ is the average for the different
baseline. \jp{It is unclear to me whether this average is weighted.}

\subsection{Signal decorrelation}

\Jant[sd] characterizes the antenna hardware, \ie{} it assumes perfect
atmospheric conditions or the use of autocorrelations, as in single-dish
measurements. In interferometric mode, the phase of the turbulent
atmosphere above each antenna of a given baseline has a random part that
causes an additional ``attenuation'' of the amplitude of the correlation. A
point source of 1\,Jy flux will appear as a source of \atmeff{}\,Jy flux
(with $\atmeff \le 1.0$), if we only use the \Jant[sd] factor. This is
called atmospheric decorrelation and it depends on the weather during the
observations.

However, the last calibration step of interferometric data is to measure a
point source of known flux to deduce the actual conversion factor,
\Jant[int], taking into account the atmospheric decorrelation that happens
during the observations. By definition of \atmeff{}, we yield
\begin{equation}
  \Jant[int] = \frac{\Jant[sd]}{\atmeff}
  \quad \mbox{and} \quad
  \Jant[int] \ge \Jant[sd].
\end{equation}
It can be shown that \atmeff{} is related to the atmospheric rms phase
noise (\pharms) through
\begin{equation}
  \atmeff{} = e^{-\frac{\pharms^2}{2}} \le 1.0.
\end{equation}

\subsection{Noise vs signal}

One subtlety is that the additive noise is unaffected by the atmospheric
decorrelation, in contrast with the signal, because thermal noise is a
random process as the turbulence phase noise. This has two consquences.
\begin{enumerate}
\item As the conversion factor, \Jant[int], is applied to the data that can
  contain signal as well as noise, any attempt to measure the noise rms on
  visibilities or imaged data will thus result in a standard deviation
  larger than the one given in Eq.~\ref{eq:noise:all:baselines:1} by a
  factor \atmeff. So when we estimate the noise level of an interferometer,
  we need to take into account the interferometric conversion factor that
  depends on the typical weather conditions (i.e., the atmospheric rms
  phase noise). This gives
  \begin{equation}
    \noise[Jy] = \frac{\average{\Jant[int]\,\Tsys}}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}}.
    \label{eq:noise:all:baselines:2}
  \end{equation}
\item \jp{I still need to clarify my mind about the potential 2nd one.}
\end{enumerate}

\subsection{Summary and interpretation}

The point source sensitivity for an interferometric measurement reads
\begin{equation}
  \boxed{%
    \noise[Jy] = \frac{\average{\Jant[int]\,\Tsys}}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}},
    \quad \mbox{with} \quad
    \Jant[int] = \frac{\Jant[sd]}{\atmeff}
    \quad \mbox{and} \quad
    \atmeff{} = e^{-\frac{\pharms^2}{2}} \le 1.0,
  }
    \label{eq:noise:1}
  \end{equation}
where \noise[Jy] is the rms noise flux obtained by integration with an
interferometer of \nant{} identical antenna during the \tint{} integration
time in a frequency resolution \dnu{} with the system temperature, \Tsys{}.
\Jant[int] is the conversion factor of any given interferometer antenna
taking into account the typical amount of atmospheric decorrelation,
\atmeff, at the observed wavelength. The product $\Jant[int]\,\Tsys$ is
averaged over all independent baselines.

Equation~\ref{eq:noise:1} is true only when the source is unresolved,
\ie{}, there is no effect of beam dilution. In practice this is rarely the
case because the interferometer tries to resolve the source. Thus, this
noise formula should be used with caution when preparing the
observations. In practice, this formula is useful when one wishes to
compare the sensitivity of two different interferometer. Indeed, this point
source sensitivity is independent of the interferometer synthesized beam
that depends on the details of the observations and, in particular, the
interferometer configuration and the completeness of the Earth synthesis.

\section{The interferometric extended source sensitivity}
\label{sec:sensitivity:extended}

The name of interferometric ``extended'' source sensitivity can be
misleading. Indeed, the notion of noise for an extended source depends on
the source shape. For instance, a flat source of infinite extent would be
measured through the visibility exactly located the phase center, and all
other visibilities would only show noise. In this thought experiment the
noise level on the constant brightness would thus be the noise of a single
visibility! More generally, if we take a Gaussian source of a given FWHM,
only the visibilities located inside a $uv$ radius of a few time
$\wave/\emr{FWHM}$ would contribute the the flux measurement of that
Gaussian, and the other visibilities would just be noise.

So the only well defined ``extended'' source sensitivity is the sensitivity
for a source that exactly fills the Gaussian synthesized beam. This is what
we will derive in this section.

\subsection{Yielding the interferometric extended source sensitivity}

The point source sensitivity is well adapted to unresolved sources because
it directly delivers the estimation of the flux of these sources. For
extended sources, the point source sensitivity that is expressed in unit of
Jy/Beam, is difficult to understand because it depends on the synthesized
beam resolution in a non-trivial way. When a source is resolved (extended
compared to the expected synthesized beam), it is much easier to think in
temperature brightness. We thus convert back to a brightness temperature
scale, but we now do it at the synthesized beam resolution.

In order to generalize Eq.~\ref{eq:conversion:0} to the final product of an
interferometer, we use the fact (see
Sect.~\ref{app:reminders:effaper:antsolang}) that the beam area
(\solang[ant]) of a telescope of effective collecting surface \Aeff{} is
linked to the observing wavelength (\wave) through
\begin{equation}
  \solang[ant] \, \Aeff = \wave^2.
\end{equation}
This yields
\begin{equation}
  \flux = \Jant \Tas
  \quad \mbox{with} \quad
  \Jant = \frac{2\kbolt\,\solang[ant]\,\Feff}{\wave^2}.
  \label{eq:conversion:1}
\end{equation}
We will use this relation twice.
\begin{itemize}
\item On one hand, we have to use the solid angle of the primary beam
  $\solang[prim]$ for \Jant[sd] and \Jant[int]. Noting that this beam area
  corresponds to the main beam area of each telescope, this yields
  \begin{equation}
    \Jant[int]
    = \frac{\Jant[sd]}{\atmeff}
    = \frac{1}{\atmeff}\,\frac{\Feff}{\Beff}\,\frac{2\kbolt\,\solang[prim]}{\wave^2}.
    \label{eq:conversion:2}
  \end{equation}
\item On the other hand, we have to use the solid angle of the synthesized
  beam $\solang[syn]$ for the conversion factor that we have to apply to
  the deconvolved product (\Jant[syn]). After calibration (including the
  the impact of the atmospheric decorrelation), imaging, and deconvolution
  (including a potential phase self-calibration), an interferometer mimick
  the observation by a \textbf{perfect} telescope of angular resolution
  equal to the synthesized beam. In this case, the beam area is equal to
  the main beam solid angle (i.e., the interferometric beam efficiency is
  equal to unity or $\Beff^\emr{int} = 1$), and to the foward beam solid
  angle (i.e., $\Feff^\emr{int} = 1$). We thus yield
  \begin{equation}
    \Jant[syn]
    = \frac{2\kbolt\,\solang[syn]}{\wave^2}.
    \label{eq:conversion:3}
  \end{equation}
  Note that we don't use the decorrelation efficiency in the later
  equation. This is also due to the fact that after the data reduction, the
  deconvolved product should appear as if it was observed by a perfect
  antenna whose response is exactly a Gaussian of angular size
  $\solang[syn]$.
\end{itemize}

Combining Eq.~\ref{eq:noise:1}, \ref{eq:conversion:2},
and~\ref{eq:conversion:3}, we yield the usual
\begin{equation}
  \boxed{%
    \noise[K] = \frac{\noise[Jy]}{\Jant[syn]}
    \quad \mbox{with} \quad
    \Jant[syn] = \frac{2\pi\,\kbolt\,\beam[maj]\beam[min]}{4\,\ln{2}\,\wave^2},
  }
\end{equation}
which can be rewritten as
\begin{equation}
  \boxed{%
    \noise[K]
    = \frac{\solang[prim]}{\solang[syn]}\,\average{\frac{\Feff\,\Tsys}{\Beff\,\atmeff}}\,\frac{1}{\speceff \, \sqrt{\nant\,(\nant-1)\,\dnu\,\tint}}
    = \frac{\beam[prim]^2}{\beam[maj]\,\beam[min]}\,\average{\frac{\Feff\,\Tsys}{\Beff\,\atmeff}}\,\frac{1}{\speceff \, \sqrt{\nant\,(\nant-1)\,\dnu\,\tint}},
  }
  \label{eq:noise:2}
\end{equation}
where \noise[K] is the rms noise brightness, \beam[prim] the half primary
beam width, and \beam[maj] and \beam[min] the half beamwidth along the
major and minor axes of the synthesized beam.

\subsection{Interpretation}

Equation~\ref{eq:noise:2} clearly states that the sensitivity to extended
sources depends on the dilution of the synthesized beam in the primary
beam. This is why this formulation of the sensitivity is well adapted to
resolved sources.

For a given interferometer, the primary beamwidth is a fixed quantity while
the synthesized beam is to first order proportional to the longest baseline
in the current interferometer configuration. Hence, doubling the largest
baseline will multiply \noise[K] by a factor $4 (=2^2)$ for the same
integration time or it will multiply the integration time by a factor
$16 (=2^4)$ in order to reach the same sensitivity. This just reflects that
while the interferometer tries to mimic a single-dish antenna of same
diameter as the largest baseline, all the antenna of the interferometer
only fill a fraction of the total collecting area of the single-dish, this
fractions decreasing with a power of two as the baseline linearly
increases.

\section{The system temperature}
\label{sec:system-temperature}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[p]
  \centering %
  \includegraphics[width=0.80\linewidth]{astro-noema-sensitivity-estimator-tsys-table-sbr}
  \caption{\textbf{Top:} Summer (red) and Winter (blue) semester $\Tsys$
    for different precipitable water vapor (PWV) amount and for a source at
    zenith. The numbers indicate PWV values assumed in the
    computation. \textbf{Middle:} Assumed forward effiencies in the
    computation. \textbf{Bottom:} Assumed receiver temperatures in the
    computation. This version is for the single band mode.}
  \label{fig:tsys:table:sbr}
\end{figure}

\begin{figure}[p]
  \centering %
  \includegraphics[width=0.80\linewidth]{astro-noema-sensitivity-estimator-tsys-table-dbr}
  \caption{Same as Fig.~\ref{fig:tsys:table:sbr}, but for the dual band mode.}
  \label{fig:tsys:table:dbr}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[p]
  \centering %
  \includegraphics[width=0.775\linewidth]{astro-noema-sensitivity-estimator-tsys-minmax-sbr}
  \caption{\textbf{Top:} Minimum and maximum $\Tsys$ obtained in the
    intermediate frequency bandwidth as a function of the local oscillator
    frequency used in the tuning. \textbf{Middle:} Assumed forward
    effiencies in the computation. \textbf{Bottom:} Assumed receiver
    temperatures in the computation.  This version is for the single band mode.}
  \label{fig:tsys:minmax:sbr}
\end{figure}

\begin{figure}[p]
  \centering %
  \includegraphics[width=0.775\linewidth]{astro-noema-sensitivity-estimator-tsys-minmax-dbr}
  \caption{Same as Fig.~\ref{fig:tsys:minmax:sbr}, but for the dual band mode.}
  \label{fig:tsys:minmax:dbr}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[p]
  \centering %
  \includegraphics[width=0.80\linewidth]{astro-noema-sensitivity-estimator-tsys-cont-sbr}
  \caption{\textbf{Top:} Averaged ``continuum'' $\Tsys$ as a function of
    the local oscillator frequency used in the tuning. \textbf{Middle:}
    Assumed forward effiencies in the computation. \textbf{Bottom:} Assumed
    receiver temperatures in the computation.  This version is for the single band mode.}
  \label{fig:tsys:cont:sbr}
\end{figure}

\begin{figure}[p]
  \centering %
  \includegraphics[width=0.80\linewidth]{astro-noema-sensitivity-estimator-tsys-cont-dbr}
  \caption{Same as Fig.~\ref{fig:tsyscont:sbr}, but for the dual band mode.}
  \label{fig:tsys:cont:dbr}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\subsection{Definition}

The system temperature is a summary of the noise added by the system. This
noise comes from 1) the receiver and the optics, 2) the emission of the
sky, and 3) the emission picked up by the secondary side lobes of the
telescope. It is usual to approximate it (in the \Tas{} scale) with
\begin{equation}
  \Tsys = \frac{\paren{1+\gim}\exp\cbrace{\taus\,\Airmass}}{\Feff}
  \bracket{\Feff\,\Tatm\,\paren{1-\exp\cbrace{-\taus\,\Airmass}}+\paren{1-\Feff}\Tcab+\Trec},
\end{equation}
where \gim{} is the receiver image gain, \Feff{} the telescope forward
efficiency, $\Airmass = 1/\sin(\emr{elevation})$ the airmass, \taus{} the
atmospheric opacity in the signal band, \Tatm{} the mean physical
atmospheric temperature, \Tcab{} the ambient temperature in the receiver
cabine and \Trec{} the noise equivalent temperature of the receiver and the
optics. All those parameters are easily measured, except \taus{}, which
depends on the amount of water vapor in the atmosphere and which is
estimated with complex atmospheric models.

The \Tsys{} value is expressed so that all these terms are corrected for
the attenuation by the atmosphere, the coupling of the antenna to the sky,
and the side-band rejection. In other words, the system temperature is
given in units that assume a perfect antenna (coupling equal to 1) located
outside the atmosphere for a single-sideband signal.

\subsection{Line vs continuum system temperature}

Figure~\ref{fig:tsys:table:sbr} shows the system temperature as a function of the
radio frequency. The $\Tsys$ can vary significantly over the large
bandwitdh of the 2SB NOEMA receivers. Figure~\ref{fig:tsys:minmax:sbr} shows
the minimum and maximum system temperature inside the IF bandwidth for all
possible local oscillator tunings. As a result, for continuum estimation, a
frequency averaged $\Tsys$ is interpolated from a pre-computed table.  The
relevant frequency in that case is the LO frequency of the tuning (see
Fig.~\ref{fig:tsys:cont:sbr}).  The averaging is done such as
\begin{equation}
  \boxed{%
    \frac{1}{<\Tsys>^2} = \frac{1}{N} \sum \frac{1}{\Tsys^2}.
   }
\end{equation}

\subsection{Receiver temperature}

The receiver temperature has two main contributions.
\begin{itemize}
\item The first one comes from the receiver when used in single band
  mode. The quality of current receivers and associated optics ensures that
  this one is to first order independent of the radio-frequency for each
  receiver, except at the edges of the receiver band. We thus model it as a
  constant receiver temperature per band. This contribution is shown on the
  bottom panels of Fig.~\ref{fig:tsys:table:sbr} to~\ref{fig:tsys:cont:sbr}.
\item The second ones comes from the dichroic that splits the beam as a
  function of the wavelength when in dual band
  mode. Figure~\ref{tsys:dichroic} shows the variations of the dichroic
  receiver temperature as a function of the radio frequency. This
  contribution has a non-negligible contribution to the overall receiver
  temperature budget. We thus only add it when in dual band mode. Moreover
  it varies much as a function of the radio frequency. We thus model it as
  the sum of a linear contribution, plus a Gaussian centered around
  240\,GHz. This last bump corresponds to a physical phenomenon, named Wood
  anomaly, which occurs in the dichroic when it operates in reflection
  under oblique incidence.
\end{itemize}

\section{In practice}
\label{sec:in-practice}

\subsection{Line vs continuum system temperature}

In the online estimator (to be used for proposal preparation), the $\Tsys$
is interpolated in frequency, airmass, amount of precipitable water vapor,
and ambient temperature from tabulated values (see
Fig.~\ref{fig:tsys:table:sbr}). The airmass is estimated using the maximum
elevation of a source at the chosen Declination. The values are different
for summer and winter due to the different atmospheric
characteristics. Moreover, the chosen amount of precipitable water vapor
depends on the receiver band (in addition to the season) because the NOEMA
operation team schedule the different receiver bands according to the
actual weather (high frequency bands are scheduled only during the best
weather conditions).

The system temperature for a given LO frequency can be computed using an
atmospheric model in ASTRO (ASTRO$\backslash$ATMOSPHERE command) with
ambient temperature and precipitable water amount as input. The overall
computing time of the continuum system temperature would be prohibitive
because of the computation of the atmospheric model over the
$2 \times 8\,$GHz IF bandwidth sampled every 100\,MHz. Instead the
computation is done off-line on a grid of LO frequency sampled every
1\,GHz.

\subsection{The number of polarizations}

All NOEMA antennas are equipped with dual polarization receivers.  They
measure the signal coming from the pointed direction in two perpendicular
polarizations in the same frequency range.  For the current generation of
receiver (2006) and correlators, one or two polarizations are processed by
the correlators, depending on the project settings.  We thus have to
introduce the number of polarizations \npol{}, which can be set to 1 or 2
and insert it in the radiometer equation with
\begin{equation}
  \noise[Jy] = \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}.
  \label{eq:noise:3}
\end{equation}

\subsection{Actual computations}

The sensitivity estimator computes the relationship between \tint{} and
\noise[Jy] with
\begin{equation}
  \boxed{%
    \noise[Jy] =
    \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
    \quad \mbox{with} \quad
    \Jant[int] = \frac{\Jant[sd]}{\atmeff}
    \quad \mbox{and} \quad
    \atmeff{} = e^{-\frac{\pharms^2}{2}} \le 1.0,
  }
  \label{eq:noise:4}
\end{equation}
where \Tsys{} is interpolated in frequency and airmass from the table, and
the other parameters are defined by the observatory. It then computes the
relationship between \noise[K] and \noise[Jy] with
\begin{equation}
  \boxed{%
    \noise[K] = \frac{\noise[Jy]}{\Jant[syn]}
    \quad \mbox{with} \quad
    \Jant[syn] =
    \frac{2\pi\,\kbolt\,\beam[maj]\beam[min]}{4\,\ln{2}\,\wave^2}.
  }
\end{equation}

\subsection{Elapsed telescope time vs on-source time}

The goal of a sensitivity estimator is to find the rms noise obtained when
observing during the elapsed telescope time, \ttel{}.  The total
integration time spent on-source \ton{} is shorter than the elapsed
telescope time due to several factors.  The actual on source time is then
computed taking into account the following points:
\begin{enumerate}
\item {\bf Instrumental setup time:} At the beginning of an observing track
  a significant time ($\tsetup \sim 40$ minutes according to history of
  observations) is spent in receiver tuning and calibration observations
  before observing the actual astronomical target. This means that even for
  a very short ON source time, a project cannot be shorter than
  \tsetup{}. The setup time is computed as
  \begin{equation}
    \boxed{%
      \tsetup = \tsetupmin + (\nfreqphase-1) \, \tsetupperfreq,
    }
  \end{equation}
  where \tsetupmin{} is the minimum setup time when only one frequency is
  observed (40 minutes), \tsetupperfreq{} is the additional setup time per
  additional frequency, and \nfreqphase{} is the number of frequency
  observed in frequency cycling mode. \tsetupperfreq{} is for the moment
  also set to 40 minutes.
\item {\bf Number of tracks per science goal:} Also, for long projects
  observed in several (\ntrack) tracks the time spent for tuning and
  calibration is $\ntrack \times \tsetup$. We thus define the time spent
  for observations (i.e. without instrumental setup) \tobs{} as:
  \begin{equation}
    \boxed{%
      \tobs = \ttel - \ntrack \times \tsetup.
    }
  \end{equation}
  The number of tracks is computed as
  \begin{equation}
    \boxed{%
      \ntrack = \frac{\ttel}{\tvisible + \tsetup},
    }  
  \end{equation}
  where \tvisible\ is the typical time when the source is visible from
  Bure, which depends on the source declination:
  \begin{itemize}
  \item Sources above $0 \deg$ are observed 8 hours at most (for
    instrumental management),
  \item 8.2 hours for a declination of $0 \deg$ (truncated to 8 hours),
  \item 6.5 hours for a declination of $-10 \deg$,
  \item 3.9 hours for a declination of $-20 \deg$,
  \item 0.0 hours for a declination of $-30 \deg$.
  \item Sources below $-30 \deg$ can not be observed.
  \end{itemize}
  A linear interpolation with the declination is performed in the
  appropriate range between $-30 \deg$ and $0 \deg$.\\
  For short projects ($\ttel < \tvisible + \tsetup$), the number of tracks
  \ntrack{} is set to 1.  Otherwise, the floating value of \ntrack{} is
  used in the computation of \tobs{}. Since \tsetup{} is constant whatever
  the length of a track the use of a floating value for \ntrack{} is
  somehow unnatural but it ensures that the conversion from \ttel{} to
  \tobs{} is a monotonic function, without regular threshold effects.
\item {\bf Observing efficiency:} After the initial phase of instrumental
  setup, the observing mode does not dedicate 100\% of the time to the
  astronomical target. Part of the time is spent for calibration (pointing,
  focus, atmospheric calibration,...) and to slew the telescopes between
  useful integrations.  The time actually spent on source \ton{} is defined
  as
  \begin{equation}
    \boxed{%
      \ton = \tobs \times \obseff
    }
  \end{equation}
  where $\obseff{} < 1.0$ is the observing efficiency. The exact computation of
  \obseff{} depends on the observing mode. The observing mode can be split
  into two categories that can be combined:
  \begin{enumerate}
  \item {\bf Detection vs Imaging projects:} From the sensitivity
    estimation viewpoint, the main difference between these observing modes
    is the number of gain calibrators regularly observed: $\ngaincal = 1$
    for detection, and 2 for mapping projects.
  \item {\bf Frequency cycling mode:} In this case, the gain calibrators have to
    be observed at each frequency of the cycle.
  \end{enumerate}
  These results in a computation of the observing efficiency as
  \begin{equation}
    \boxed{%
      \obseff = \frac{1}{\overheadobs}
      \quad \mbox{with} \quad
      \overheadobs = \overheadmin + \ngaincal\,\nfreqphase\,\overheadperfreqgaincal,
      \quad 
      \overheadmin = 1.3,
      \quad \mbox{and} \quad
      \overheadperfreqgaincal = 0.3.
    }
  \end{equation}
  Note that the exact value of \overheadmin{} and \overheadperfreqgaincal{}
  actually depend on several parameters such as the distance between the
  source and the calibrator(s). In ``standard'' mode, we obtain the usual
  $\overheadobs = 1.6$ and 1.9, for detection and mapping projects,
  respectively. In frequency cycling mode with 2 frequencies, we
  yield $\overheadobs = 1.9,$ and 2.5 for detection and mapping projects,
  respectively.
\item \textbf{From \tobs{} to on-source time:} Finally, the distribution of
  observing time into the time spent on-source, \ton, actually used to
  estimate the sensitivity depends on three main observation kinds that are
  assumed exclusive from each other.
  \begin{description}
  \item[Single-source, single-field observations] where the telescope tracks
    a single source during the full integration time. This mode is used when
    the signal-to-noise ratio is the limiting factor.
  \item[Track-sharing, single-field observations] where the telescope
    regularly cycles between a few close-by sources. This mode is used when
    the sources are so bright that the limiting factor is the Earth
    synthesis, not the signal-to-noise ratio.
  \item[Single-source mosaicking] where the telescope regularly cycles
    between close-by pointings that usually follows a hexagonal compact
    pattern whose side is $\wave/(2\dprim)$, where \dprim{} is the diameter
    of the interferometer antennas. This modes is used to image sources wider
    than the primary beam field of view.
  \end{description}
  In the following, we will work out the equations needed by the sensitivity
  estimator for each of these observing modes.
\item \textbf{Overall efficiency} Finally, the overall observing
  efficiency, \toteff, is evaluated by computing the sum of the on-source
  time over the total telescope time
  \begin{equation}
    \boxed{%
      \toteff = \frac{\Sigma \ton}{\ttel}
    }
  \end{equation}
  The sum of the on-source time allows us to take into account the fact
  that 1) the telescope time may be shared between sources or frequencies,
  and 2) dual band observations are more efficient than single band ones;.
  We also stress that when frequency cycling is combined with dual band
  observations, both receiver bands are affected by the efficiency loss of
  the frequency cycling even though one of the two bands could not require
  frequency cycling at all! A warning is raised when $\toteff \le 0.25$.
\end{enumerate}

\subsection{Sensitivity for a single-source, single-field observations}
\label{sec:single-field}

That's the simplest case. The point source sensitivity in this case is
\begin{equation}
  \noise[Jy] = \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}},
\end{equation}
where \ton{} is related to the total elapsed telescope time \ttel{} through
\begin{equation}
  \boxed{%
    \ton  = \obseff \, \freqeff \, \paren{\ttel \,-\,  \ntrack \times \tsetup},
  }
  \label{eq:tel-vs-on}
\end{equation}
where \freqeff{} is the fraction of \tobs{} spent at one tuning
frequency. $\freqeff = 1.0$ in ``standard'' mode and
$\freqeff \sim 0.5$ when cycling between two frequencies. The sum of
$\freqeff$ needs to equal 1.0!

\subsection{Sensitivity for track-sharing, single-field observations}
\label{sec:track-sharing}

In this case, the telescope time is equally divided between the \nsource{}
observed sources. This yields
\begin{equation}
  \noise[Jy] = \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
\end{equation}
\begin{equation}
  \mbox{with} \quad
  \boxed{%
    \ton =  \obseff \, \freqeff \paren{\frac{\ttel \,-\,  \ntrack \times  \tsetup}{\nsource}}.
  }
\end{equation}
While it is technically feasible to observe sources in track-sharing with
different integration times, this case is not implemented (yet) in the
sensitivity estimator, and the different sensitivities should be computed
independently.

\subsection{Sensitivity for mosaicking}
\label{sec:mosaicking}

Note: No provision is made to estimate the sensitivity of a mosaic in
frequency cycling mode. This is to avoid mixing the complexity of both
modes...

Mosaicking is a particular case of wide-field imaging: The user wishes to
observe a given field of view larger than the primary beam size with a
sensitivity as uniform as possible. The targeted field (which area is
\Amap{}, define by the user) can be divided in a number of independent
resolution elements or independent (primary) beams \nbeam. We have:
\begin{equation}
  \nbeam = \frac{\Amap}{\Abeam}
\end{equation}
where \Abeam is the area of the primary beam. It is linked to the telescope
full width at half maximum (\beam{}) by
\begin{equation}
  \Abeam = \frac{0.8\,\pi\,\beam[prim]^2}{4\,\ln(2)},
\end{equation}
The 0.8 factor represents the truncation of the beam at 20\% of its
maximum, which is performed during the imaging process.

For the sensitivity estimation we assume a standard sampling of targeted
field and the on-source time is equally divided between the independent
primary beams \nbeam{} in the targeted field of view. We
thus yield at the center of a Nyquist sampled mosaic
\begin{equation}
  \noise[Jy] = \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
  \label{eq:mos}
\end{equation}
\begin{equation}
  \quad \mbox{with} \quad
  \ton = \obseff \moseff \paren{\frac{\ttel \,-\, \ntrack \times
      \tsetup}{\nbeam}},
  \quad \mbox{and} \quad
  \moseff = \frac{\tintpointpercycle}{\tintpointpercycle+\tslew},
\end{equation}
where where \tintpointpercycle{} is the integration time per pointing
during each single mosaic cycle, and \tslew{} is the time to slew between
two consecutive pointings. There are several subtleties in these
computations.
\begin{itemize}
\item \textbf{\Amap{} must be larger than 2 times \Abeam{}.} Below this we
  advise to use the track sharing mode with two independent fields.
\item \textbf{Eq.~\ref{eq:mos} is only valid at the center of a mosaic, not
    at its edges!} Indeed, the processing (imaging and deconvolution) of a
  mosaic implies a division by the primary beam of the interferometer. As
  the primary beam is to first order a Gaussian decreasing to zero, this
  implies that the noise of the mosaic will vary over the field of view. In
  particular it increases sharply at the edges of the field of view.
\item \textbf{In these equations, \nbeam{} is smaller that \npoint, the
    number of pointed positions that are observed in the mosaic.}  The
  cycling of the pointings of the mosaic should ensure Nyquist sampling of
  the observed field of view. This implies that there is an important
  redundancy between the pointings, contrary to track sharing where the
  sources are supposed to be fully independent on the sky. For instance,
  when mosaicking with a hexagonal compact pattern, each line of sight will
  be observed by 7 contiguous pointings, except at the mosaic edges. It can
  thus been shown that the number of mosaic pointings, \npoint, is related
  to the number of independent elements through
  \begin{equation}
      \npoint = \nbeam \, \paren{\frac{7}{4}}^2,
  \end{equation}
  for a correctly sampled mosaic. \textbf{Equation~\ref{eq:mos} is thus
    only valid inside a correctly sampled mosaic.}
\item \textbf{Degradation of the observing efficiency} The pointings of a
  mosaic must be observed in relatively short time cycles to ensure that
  all pointings are observed with similar weather conditions and that they
  share similar $uv$ coverage. This will minimize the shift-variant part of
  the interferometer wide-field imaging response. This calls for the
  shortest possible integration time per pointing. However, the
  interferometer takes time to slew from one pointing to the next one
  without integrating.  As a result, the observing efficiency \obseff{} is
  degraded in the cases of mosaics. We model it as a mosaic efficiency
  factor, \moseff.  Having a large integration time per pointing during one
  cycle compared to \tslew{} will decrease the mosaicking overhead. This
  requirement is in sharp contrast with the previous one, namely the need
  to homogenize the interferometer wide-field response. The best compromise
  comes from two different considerations.
  \begin{enumerate}
  \item The smallest integration time per scan is set by the acquisition
    system (for instance, the maximum achievable data rate). In pratice, we
    enforce that
    \begin{equation}
      \tintmin \le \tintperscan
      \quad \mbox{with} \quad
      \tintmin = 10\,\mbox{sec}.
    \end{equation}
  \item The distance covered by a visibility in the $uv$-plane during an
    integration should always smaller than the distance associated to
    tolerable aliasing (see Pety and Rodr\'iguez-Fern\'andez 2010 for more
    details).  This can be written as the following condition (Eq.~C.3 in
    this article)
    \begin{equation}
      \frac{\tintperscan}{1 s} << \frac{6900}{\beam[alias] / \beam[syn]},
      \label{eq:uvcond}
    \end{equation}
    where $\beam[alias]$ is the map angular size, and $\beam[syn]$ the
    angular resolution. For a given angular resolution, the interferometer
    minimum integration time corresponds to
    \begin{equation}
      \tintperscan \le \eta\,\frac{6\,900}{1\,\emr{sec}}\,
      \sqrt{\frac{\beam[maj]\beam[min]}{\Amap}},
    \end{equation}
    where \textbf{$\eta$ is a ad-hoc number set to 0.5} to ensure that the
    condition defined in Eq.~\ref{eq:uvcond} is met.
  \item For operational reasons, the duration of a scan is limited to
    45\unit{sec}.    
  \item In summary,
    \begin{equation}
        10\,\mbox{sec} \le \tintperscan = \emr{min}\paren{45\unit{sec},\eta\,\frac{6\,900}{1\,\emr{sec}}\,
          \sqrt{\frac{\beam[maj]\beam[min]}{\Amap}}}
        \quad \mbox{with} \quad
        \eta = 0.5.
    \end{equation}
  \end{enumerate}
  As the typical slew time between two pointings is $\tslew = 11\,$sec, we
  yield that
  \begin{equation}
    0.47 \le \moseff \le 1.0.
  \end{equation}  
\item \textbf{Improving the observing efficiency for small mosaics} In A
  configuration, the integration time per scan should be less than 22\,sec
  to meet the criteria of Eq.~\ref{eq:uvcond}. However, this implies a
  mosaic efficiency of only $\moseff = 0.67$! It is possible to greatly
  increase this efficiency for small mosaics by observing several time the
  same pointing in time-contiguous scans. Let's assume that we use an
  integration time of \tintmin{} per scan, and that we repeat it
  \nrepeatperpoint{} before slewing to another pointing position. This gives
  \begin{equation}
    \moseff = \frac{\tintpointpercycle}{\tintpointpercycle+\tslew}
    \quad \mbox{and} \quad
    \tcycle = \npointpertrack\,\paren{\tintpointpercycle+\tslew},
  \end{equation}
  with
  \begin{equation}
    \tintpointpercycle = \nrepeatperpoint\,\tintperscan.
  \end{equation}
  As stated above, we wish to cycle as quickly as possible through the
  different pointings in a single cycle to homogeneize the observing
  conditions over the mosaic. This implies to introduce a maximum time to
  cycle all the mosaic pointings. Let's assume that we want to homogeneize
  the conditions during the maximum time allowed between two gain
  calibrations, ie., $\tcalmax = 25\,\mbox{min}$. This yields the maximum
  number of repeats per pointing and per cycle and thus the mosaic
  efficiency as
  \begin{equation}
    \nrepeatmax = \frac{\tcalmax/\npointpertrack-\tslew}{\tintperscan},
    \quad \mbox{and} \quad
    \moseff =  1 - \frac{\npointpertrack\,\tslew}{\tcalmax}.
  \end{equation}
  The maximum number of repeats must be larger than 1. This sets the limit
  in number of pointings per track between small and large mosaics.
  \begin{equation}
    \npointpertracklarge = \emr{floor}\paren{\frac{\tcalmax}{\tintperscan+\tslew}}.
  \end{equation}
\item \textbf{Checking the feasability of a large mosaic} For large
  mosaics, it is advised to cycle all the pointings of a given mosaic in
  less than 1 hour. This ensures that the uv plane will be sampled at least
  8 times per standard track of 8 hours.  Setting the maximum time to cycle
  all the pointings, \tcyclemax, to 60 minutes, we yield that the maximum
  number of pointing per track is
  \begin{equation}
    \npointmax = \frac{\tcyclemax}{\tintmin+\tslew} \sim 150.
  \end{equation}
  Hence, if the PI wishes to observe an area that will require more that
  150 pointings per independent track, the estimator will ask to either
  increase the requested elapsed telescope time or to decrease the
  requested field-of-view area.

  Moreover, we need to adapt the optimal integration time per scan to
  ensure that all pointings per track will be cycled in \tcyclemax. This
  gives
  \begin{equation}
    \tintperscan = \emr{min}\cbrace{\tintperscan,\paren{\frac{\tcyclemax}{\npointpertrack}-\tslew}}.
  \end{equation}
  In the case of large mosaics, the minimum
  mosaic efficiency is
  \begin{equation}
    \moseffmin = \frac{\tintmin}{\tintmin+\tslew} = 0.47.
  \end{equation}
\end{itemize}
  
In summary, correctly setting up the sensitivity estimate for a mosaic requires
\begin{enumerate}
\item To compute the number of independent primary beams in the targeted
  field of view
  \begin{equation}
    \boxed{%
      \nbeam = \frac{\Amap}{\Abeam},
      \quad \mbox{where} \quad
      \Abeam = \frac{0.8\,\pi\,\beam[prim]^2}{4\,\ln(2)}.
    }
  \end{equation}
\item To compute the number of pointings in the mosaic and the number of
  pointings per track
  \begin{equation}
    \boxed{%
      \npoint = \nbeam \, \paren{\frac{7}{4}}^2,
      \quad \mbox{and} \quad
      \npointpertrack = \emr{min}\paren{\npoint,\frac{\npoint}{\ntrack}}.
    }
  \end{equation}
\item To compute the optimal integration time per scan with
  \begin{equation}
    \boxed{%
        10\,\mbox{sec} \le \tintperscan = \emr{min}\paren{45\unit{sec},\eta\,\frac{6\,900}{1\,\emr{sec}}\,
          \sqrt{\frac{\beam[maj]\beam[min]}{\Amap}}}
        \quad \mbox{with} \quad
        \eta = 0.5.
    }
  \end{equation}
  The minimum integration time per scan of 10 seconds implies the maximum
  dynamic of scales that can be observed in a single track.
\item To compute the number of points per track that define the limit
  between small and large mosaics with
  \begin{equation}
    \boxed{%
      \npointpertracklarge = \emr{floor}\paren{\frac{\tcalmax}{\tintperscan+\tslew}},
      \quad \mbox{where} \quad
      \tslew = 11\unit{sec},
      \quad \mbox{and} \quad
      \tcalmax = 25\unit{min}.
    }
  \end{equation}
\item To compute the maximum number of repeats per pointing and per cycle,
  depending on the mosaic size.
  \begin{itemize}
  \item For a small mosaic $(\npointpertrack \le \npointpertracklarge)$, it is
  \begin{equation}
    \boxed{%
      \nrepeatmax = \frac{\frac{\tcalmax}{\npointpertrack}-\tslew}{\tintperscan}.
    }
  \end{equation}
  \item For a large mosaic $(\npointpertrack > \npointpertracklarge)$, it is just set to $\nrepeatmax = 1$. 
  \end{itemize}
\item To check that the maximum number of pointings per track is not yet
  reached with
  \begin{equation}
    \boxed{%
      \npointpertrack \le \npointmax,
      \quad \mbox{where} \quad
      \npointmax = \frac{\tcyclemax}{\tintmin+\tslew} \sim 150,
      \quad \mbox{and} \quad
      \tcyclemax = 60\unit{min}.
    }
  \end{equation}
  If the PI wishes to observe an area that will require more that 150
  pointings per independent track, the estimator will ask to either
  increase the requested elapsed telescope time or to decrease the
  requested field-of-view area.
\item To adapt the optimal integration time per scan to ensure that all
  pointings per track will be cycled in \tcyclemax{} \textbf{only for a
    large mosaic} with
  \begin{equation}
    \boxed{%
      \mbox{if} \quad
      \npointpertrack > \npointpertracklarge,
      \quad \mbox{then} \quad
      \tintperscan = \emr{min}\cbrace{\tintperscan,\paren{\frac{\tcyclemax}{\npointpertrack}-\tslew}}.
    }
  \end{equation}
\item To compute the actual mosaic efficiency and time to complete a full
  cycle with
  \begin{equation}
    \boxed{%
      \moseff = \frac{\tintpointpercycle}{\tintpointpercycle+\tslew},
      \quad \mbox{and} \quad
      \tintpointpercycle = \nrepeatmax\,\tintperscan,
    }
  \end{equation}
  and
  \begin{equation}
    \boxed{%
      \tcycle = \npointpertrack\,\paren{\tintpointpercycle+\tslew}.
    }
  \end{equation}
  These two quantities should be given as feedback to the user.
\item To finally compute the sensitivity with
  \begin{equation}
    \boxed{
      \noise[Jy] =
      \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}},
      \quad \mbox{and} \quad
      \ton = \obseff \moseff \paren{\frac{\ttel \,-\, \ntrack \times
          \tsetup}{\nbeam}}.
    }
    \label{eq:mos}
  \end{equation}
\end{enumerate}


\begin{thebibliography}{9}
  \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\bibitem[{Kraus (1982)}]{kra82} Kraus, J. D., in McGraw-Hill 1982 (1966),
  Radio Astronomy.
\end{thebibliography}

\clearpage{} 

\appendix{}

\section{A few reminders: Flux vs brightness in radio-astronomy}

\subsection{Antenna power pattern, solid angles, associated efficiencies}

The power pattern, \pattern(\direction,\freq), is a measure of the response of
the antenna to radiation as a function of the angles $\theta$ and
$\phi$. It is a normalized (maximum value unity), dimensionless
quantity. It can depend on the frequency of observation.

The beam area or beam solid angle (or pattern solid angle) is
\begin{equation}
  \boxed{%
    \solang[ant](\freq) = \sflux{\solang}{4\pi}{\pattern(\direction,\freq)},
  }
\end{equation}
where
\begin{table}[h]
  \centering
  \begin{tabular}[h]{cll}
    \solang[ant] & [\emr{sr}]            & beam area, \\
    \pattern     & [\emr{dimensionless}] & normalized power pattern of antenna, \\
    \dsolang     & [\emr{sr}]            & infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$. \\
  \end{tabular}
\end{table}

The forward beam and main beam solid angles are the integral of the power
pattern over $2\pi$ and the main lobe, respectively, i.e.,
\begin{equation}
  \boxed{%
    \solang[fb](\freq) = \sflux{\solang}{2\pi}{\pattern(\direction,\freq)}, %
    \quad \mbox{and} \quad %
    \solang[mb](\freq) = \sflux{\solang}{\emr{main\,\,lobe}}{\pattern(\direction,\freq)}.
  }
\end{equation}
The forward and beam efficiencies are then defined as
\begin{equation}
  \boxed{%
    \Feff = \frac{\solang[fb]}{\solang[ant]}, %
    \quad \mbox{and} \quad %
    \Beff = \frac{\solang[mb]}{\solang[ant]}. %
  }
\end{equation}

\subsection{Source and observed flux density}

The integral of the brightness over the source extension yields the total
source flux density
\begin{equation}
  \boxed{%
    \flux[sou](\freq) = \sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)},
  }
\end{equation}
where
\begin{table}[h]
  \centering
  \begin{tabular}[h]{cll}
    \flux[sou]  & [\emr{W\,m^{-2}\,Hz^{-1}}]          & flux density of source, \\
    \brightness & [\emr{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}] & brightness as a function of position over source, \\
    \dsolang    & [\emr{sr}]                          & infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$. \\
  \end{tabular}
\end{table}

When the source is observed with an antenna of power pattern (\pattern),
the observed flux density in direction (\direction[0]) is
\begin{equation}
  \flux[obs](\direction[0],\freq) = \sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\pattern(\dist{}{0},\freq)},
\end{equation}
where
\begin{table}[h]
  \centering
  \begin{tabular}[h]{cll}
    \flux[obs](\direction[0])  & [\emr{W\,m^{-2}\,Hz^{-1}}]          & observed flux density, \\
    \brightness                & [\emr{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}] & brightness as a function of position over source, \\
    \pattern                   & [\emr{dimensionless}]               & normalized antenna power pattern, \\
    \dsolang                   & [\emr{sr}]                          & infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$. \\
  \end{tabular}
\end{table}

This integral is a correlation. We now introduce the mirror symmetric of
the antenna pattern to yield a convolution
\begin{equation}
  \flux[obs](\direction[0],\freq) = \sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)},
\end{equation}
\begin{equation}
  \mbox{where} \quad %
  \sympatt(\dist{0}{},\freq) = \pattern(\dist{}{0},\freq).
\end{equation}
The distinction of these two patterns is only important for
non-axisymmetrical cases.

\subsection{Observed brightnesses}

The brightness observed in direction (\direction[0]) is the flux density
observed in this direction divided by the typical beam area 
\begin{equation}
  \boxed{%
    \brightness[\emr{obs}](\direction[0],\freq) = \frac{1}{\solang}\,\sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)},  
  }
\end{equation}
where
\begin{table}[h]
  \centering
  \begin{tabular}[h]{cll}
    \brightness[obs](\direction[0])  & [\emr{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}] & \textbf{observed} brightness, \\
    \brightness                      & [\emr{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}] & \textbf{actual} brightness as a function of position over source, \\
    \solang                          & [sr]                                & typical beam area, \\
    \pattern                         & [\emr{dimensionless}]               & normalized antenna power pattern, \\
    \dsolang                         & [\emr{sr}]                          & infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$. \\
  \end{tabular}
\end{table}

The \textbf{typical} beam area is an ambiguous notion. Indeed, there are
three different solid angles that can be used in this formula: The antenna
beam solid angle (\solang[ant]), the forward beam solid angle
(\solang[fb]), or the main beam solid angle (\solang[mb])
\begin{align}
  \brightness[\emr{ant}](\direction[0],\freq) &= \frac{1}{\solang[ant]}\,\sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)},\\
  \brightness[\emr{fb}](\direction[0],\freq)  &= \frac{1}{\solang[fb]}\,\sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)},\\
  \mbox{and} \quad
  \brightness[\emr{mb}](\direction[0],\freq)  &= \frac{1}{\solang[mb]}\,\sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)}.
\end{align}
It is trivial to show that
\begin{equation}
  \boxed{%
    \brightness[\emr{fb}] = \frac{1}{\Feff}\,\brightness[\emr{ant}] %
    \quad \mbox{and} \quad
    \brightness[\emr{mb}] = \frac{\Feff}{\Beff}\,\brightness[\emr{fb}]. %
  }
\end{equation}
\textbf{At IRAM, the observed brightness obtained after a single-dish
  calibration are by default computed using the forward beam solid angle.}

\subsection{Basic relation between received spectral power and sky
  brightness}

The spectral power (infinitesimal power per unit frequency bandwidth)
received from a solid angle of the sky on a surface of area is
\begin{equation}
  \label{eq:spec:power}
  \dpow(\direction,\freq) = \brightness(\direction,\freq)\,\cos \theta \,\dsolang\,\darea,
\end{equation}
where
\begin{table}[h]
  \centering
  \begin{tabular}[h]{cll}
    \dpow       & [\emr{W\,Hz^{-1}}]                  & spectral power, \\
    $\theta$    & [rad]                               & angle between \dsolang{} and Zenith, \\
    $\phi$      & [rad]                               & angle between \dsolang{} and East, \\
    \freq       & [Hz]                                & frequency of observation, \\
    \brightness & [\emr{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}] & sky brightness at position (\direction) and frequency \freq{}, \\
    \dsolang    & [sr]                                & infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$, \\
    \darea      & [\emr{m^{2}}]                       & infinitesimal surface area.
  \end{tabular}
\end{table}

\subsection{Brightness temperature}

\subsubsection{Brightness temperature}

In 1928, Nyquist showed that the noise power per unit bandwidth produced by
a resistor of resistance, $R$, and physical temperature, \Temp, is
\begin{equation}
  \dpow = \kbolt\,\Temp.
\end{equation}
This relationship indicates that any spectral power can be expressed as a
temperature using this relationship. For instance, the observed main beam
temperature is defined as
\begin{equation}
  \Temp[\emr{mb}](\direction[0],\freq) = \frac{1}{\solang[mb]}\,\sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)}.
\end{equation}

\subsubsection{Interpretation}

It happens that when an astronomical source, whose angular extent fills
exactly the main lobe, emits as a black-body in the Rayleigh-Jeans regime
$(h\freq \ll \kbolt\Temp)$, the main-beam temperature is directly the
physical temperature of the emitting source. However, this interpretation
is only valid when the stated conditions are met. This almost never happens
in (sub)-millimeter radio-astronomy because the emitting sources are often
out of local thermodynamic equilibrium and $h\freq \sim \kbolt\Temp$. So
this interpretation is mostly misleading for newcomers. 

As the measured spectral power in radio-astronomy have very small values,
it is easier to express them in term of temperatures that will be orders of
magnitude larger because $\kbolt = 1.380658\times
10^{23}\unit{J\,K^{-1}}$. As a matter of fact, most observed brightness
temperatures have values close to 1 (within a factor 1000!)

So it's easier to interpret the above relationship as: expressing spectral
powers in terms of temperature is just a useful convenience.

\subsection{Effective aperture and aperture efficiency}

Let's define the effective aperture of an antenna, \Aeff, through the
relationship between the received spectral power and the sky brightness
\begin{equation}
  \dpow(\direction[0],\freq) = \frac{1}{2}\,\Aeff\sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)}.
\end{equation}
This definition is based on Eq.~\ref{eq:spec:power} with the assumption
that the power pattern is independent of infinitesimal surface area of the
receiving antenna in order to be able to factorize the integral on this
infinitesimal surface area. Moreover the factor 1/2 corresponds to the case where the radiation is incoherent
and unpolarized, and the receiver is only sensitive to one
polarization.

This effective aperture corresponds to the fraction of the power density of
a plane wave that is intercepted by the antenna. It has the unit of a
surface [\emr{m^{-2}}]. It thus resembles a cross-section in particle
physics. We thus define the aperture efficiency, \Eaper, as the ratio of
the effective aperture by the geometric aperture of the antenna
\begin{equation}
  \Eaper = \frac{\Aeff}{\Ageo} < 1, %
  \quad \mbox{where} \quad %
  \Ageo = \pi \paren{\frac{D_\emr{ant}}{2}}^2.
\end{equation}

\subsection{Relationship between effective aperture and antenna solid angle}
\label{app:reminders:effaper:antsolang}

\subsubsection{Statement}

An important relationship in antenna theory is
\begin{equation}
  \boxed{%
    \Aeff(\freq)\,\solang[ant](\freq) = \wave^2,
  }
\end{equation}
where \wave{} is the observed wavelength.

\subsubsection{Demonstration}

A generic demonstration of this statement uses a thought experiment. Let's
assume that the antenna is put in a box at a given physical
temperature, \Temp. This box can be considered an emitting black-body. If the
frequency is appropriately chosen, the Rayleigh-Jeans regime is
fulfilled. This implies that the source brightness is
\begin{equation}
  \brightness(\direction,\freq) = \frac{2\,\kbolt\Temp}{\wave^2},
\end{equation}
and the observed spectral power is
\begin{equation}
    \dpow(\direction[0],\freq)
  = \frac{1}{2}\,\Aeff\sflux{\solang}{\emr{source}}{\brightness(\direction,\freq)\,\sympatt(\dist{0}{},\freq)}
  = \frac{1}{2}\,\Aeff\,\solang[ant](\freq) \,\frac{2\,\kbolt\Temp}{\wave^2}.
\end{equation}
If we plug a matched resitor at the antenna output, it will deliver a
noise power of
\begin{equation}
    \dpow(\direction[0],\freq) = \kbolt\,\Temp,
\end{equation}
because its temperature is \Temp. As it is matched to the antenna, this
power is also the spectral power measured by the antenna. Hence, equating
the last two equations, we yield the searched relationship.

\subsubsection{Interpretation}

While this relationship was derived using a particular experimental setup,
it yields a result that only characterize the state of the antenna. As
usual in thermodynamics, the state is independent on the exact setup. This
implies that the relationship is valid independent on the way it was
yielded.

Nevertheless the demonstration is useful to understand that the solid angle
that we have to consider in this relationship is the beam area, i.e., the
coupling of the antenna to all direction in $4\pi\unit{sr}$. It is then
possible to rewrite this relationship as
\begin{equation}
  \boxed{%
    \Aeff(\freq)\,\solang[fb](\freq) = \wave^2\,\Feff(\freq) %
    \quad \mbox{and} \quad %
    \Aeff(\freq)\,\solang[mb](\freq) = \wave^2\,\Beff(\freq).
  }
\end{equation}
Using the definition of the aperture efficiency, we yield
\begin{equation}
  \Beff(\freq) = \Eaper\,\Ageo\,\frac{\solang[mb](\freq)}{\wave^2}.
\end{equation}
As
\begin{equation}
  \Ageo = \frac{\pi}{4}\,D^2,
  \quad
  \frac{\solang[mb](\freq)}{\wave^2} = \frac{\pi}{4\ln2}\,\paren{\frac{\beam[mb]}{\wave}}^2,
  \quad \mbox{and} \quad
  \beam[mb] = \coupling \frac{\wave}{D},
\end{equation}
we yield
\begin{equation}
  \boxed{%
    \Beff(\freq)
    = \frac{\pi^2}{16\ln2}\coupling^2\,\,\Eaper(\freq)
    \simeq 0.8899\,\coupling^2\,\Eaper(\freq).
  }
\end{equation}
The factor \coupling{} defines the coupling of the optics to the sky, which
can be computed with Gaussian optics. Its value is typically $\coupling \sim
1.2$. Finally, the Ruze theory indicates that the aperture efficiency is
linked to the RMS $(\sigma)$ of the deviation of the telescope surface to a
perfect parabola as
\begin{equation}
  \boxed{%
    \Eaper(\freq) = \Eaper^0 \, \exp \cbrace{- \paren{\frac{4\pi\sigma}{\wave}}^2}.
  }
\end{equation}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
