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\begin{document}

\title{IRAM Memo 2015-2\\[3\bigskipamount]
  NOEMA sensitivity estimator}%
\author{J. Pety$^{1,2}$, J. Boissier$^{1}$, E. Reynier$^{1}$, S. Bardeau$^{1}$\\[0.5cm]
  1. IRAM (Grenoble) \\
  2. Observatoire de Paris}%
\date{Feb, 12$^{th}$ 2020\\
  Version 4.0}

\maketitle

\begin{abstract}
  This memo describes the equations used in the NOEMA sensitivity
  estimator available in the online sensitivity estimator
  (\texttt{https://oms.iram.fr/tse/\#noema}), to be used for
  proposal preparation and in the \texttt{GILDAS/ASTRO} program
  (so-called detailed estimator, independent of NOEMA call for
  proposals).
\end{abstract}

\newpage{}

\tableofcontents{}

\newpage{}

\section{The interferometric point source sensitivity from first
  principles}
\label{sec:sensitivity}

\subsection{System temperature}

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 by 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{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 previous section, 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}

\subsection{Quantization efficiency}

The Analog-to-Digical-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) 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{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[K] = \frac{\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\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 \temp{} through
\begin{equation}
  \flux = \Jant[sd] \temp
  \quad \mbox{with} \quad
  \Jant[sd] = \frac{2\kbolt}{\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 1 at 3\,mm but significantly lower than 1 at 1\,mm.

Using the same conversion factor for the sensitivity, we yield
\begin{equation}
  \noise[Jy] = \frac{\Jant[sd]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}}.
  \label{eq:noise:all:baseline:1}
\end{equation}

\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 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[sd] = \atmeff\,\Jant[int].
\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}}.
\end{equation}

\subsection{Noise vs signal}

One subtlety is that the noise is unaffected by the atmospheric
decorrelation, in contrast with the signal, because noise is a random
process as the turbulence phase noise.

But 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 results in a standard deviation
larger than the one given in Eq.~\ref{eq:noise:all:baseline: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{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}}.
  \label{eq:noise:all:baseline:2}
\end{equation}

\section{The interferometric extended source sensitivity}

\subsection{Starting from the point source sensitivity}

As a summary, the point source sensitivity for an interferometric
measurement reads
\begin{equation}
  \noise[Jy] = \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\tint}},
  \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 a system temperature given by
\Tsys{}.  \Jant[int] is the conversion factor of a typical interferometer
antenna taking into account the typical amount of atmospheric decorrelation
at the observed wavelength.

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.

\subsection{We yield 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 is expressed in unit of
Jy/Beam that 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.

After calibration (including the calibration of the atmospheric
decorrelation), imaging, and deconvolution (including a potential phase
self-calibration), an interferometer mimick the observation by a telescope
of angular resolution equal to the synthesized beam. However, the notion of
effective collecting surface is ambiguous in this case. In order to
generalize Eq.~\ref{eq:conversion:0} to the final product of an
interferometer, we use the fact that the solid angle resolution $(\Omega)$
of a telescope of effective collecting surface \Aeff{} is by definition
linked to the observing wavelength (\wave) through
\begin{equation}
  \Omega \, \Aeff = \wave^2.
\end{equation}
We can thus generalize Eq.~\ref{eq:conversion:0} as
\begin{equation}
  \flux = \Jant \temp
  \quad \mbox{with} \quad
  \Jant = \frac{2\kbolt\,\Omega}{\wave^2}.
  \label{eq:conversion:1}
\end{equation}
On one hand, we have to use the solid angle of the primary beam
$\Omega_\emr{prim}$ for \Jant[sd] and \Jant[int]. This yields
\begin{equation}
    \Jant[int]
  = \frac{\Jant[sd]}{\atmeff}
  = \frac{2\kbolt\,\Omega_\emr{prim}}{\atmeff\,\wave^2}.
  \label{eq:conversion:2}
\end{equation}
On the other hand, we have to use the solid angle of the synthesized beam
$\Omega_\emr{syn}$ for the conversion factor that we have to apply to the
deconvolved product (\Jant[syn])
\begin{equation}
    \Jant[syn]
    = \frac{2\kbolt\,\Omega_\emr{syn}}{\wave^2}.
    \label{eq:conversion:3}
\end{equation}
Note that we don't use the decorrelation efficiency in the later
equation. This is 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
$\Omega_\emr{syn}$.

Combining Eq.~\ref{eq:noise:1}, \ref{eq:conversion:2},
and~\ref{eq:conversion:3}, we yield 
\begin{equation}
  \noise[K]
  = \frac{\Omega_\emr{prim}}{\Omega_\emr{syn}}\,\frac{1}{\atmeff}\,\frac{\Tsys}{\speceff \, \sqrt{\nant\,(\nant-1)\,\dnu\,\tint}},
  = \frac{\beam[prim]^2}{\beam[maj]\,\beam[min]}\,\frac{1}{\atmeff}\,\frac{\Tsys}{\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{In practice}

\subsection{Line vs continuum system temperature}

\begin{figure}[t]
  \centering %
  \includegraphics[width=\linewidth]{astro-noema-sensitivity-estimator-tsys-table}
  \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.}
  \label{fig:tsys:table}
\end{figure}

\begin{figure}[t]
  \centering %
  \includegraphics[width=\linewidth]{astro-noema-sensitivity-estimator-tsys-minmax}
  \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.}
  \label{fig:tsys:minmax}
\end{figure}

\begin{figure}[t]
  \centering %
  \includegraphics[width=\linewidth]{astro-noema-sensitivity-estimator-tsys-cont}
  \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.}
  \label{fig:tsys:cont}
\end{figure}

In the online estimator (to be used for proposal preparation), the $\Tsys$
is interpolated in frequency and airmass from tabulated values (see
Fig.~\ref{fig:tsys:table}). 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).

In the ASTRO detailed sensitivity estimator, the system temperature is
computed using an atmospheric model (ASTRO$\backslash$ATMOSPHERE command)
with ambient temperature and precipitable water amount as input.

The $\Tsys$ can vary significantly over the large bandwitdh of the 2SB
NOEMA receivers. Figure~\ref{fig:tsys:minmax} 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}).  The averaging is done such as
$1/<\Tsys>^2 = 1/N \sum 1/\Tsys^2$.

This is not implemented in the ASTRO detailed estimator, due to the
prohibitive computing cost of the atmospheric model over the
$2 \times 8\,$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}
  \noise[Jy] = \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}},
  \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}
  \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}

\section{Observing mode and elapsed telescope time}

The goal of a time estimator is to find the elapsed telescope time
(\ttel{}) needed to obtain a given rms noise, while a sensitivity estimator
aims at finding the rms noise obtained when observing during \ttel{}.  The
total integration time spent on-source \ton{} is shorter than the elapsed
telescope time due to several factors.  As of Gildas Jul17 release, the
input time of the sensitivity estimator is telescope time. The actual on
source time is then computed taking into account the following two points:


\begin{enumerate}
\item {\bf Instrumental setup time:} At the beginning of an observing track
  a significant time ($\ttune \sim 40$ 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
  \ttune{}. Also, for long projects observed in several (\ntrack) tracks
  the time spent for tuning and calibration is $\ntrack \times \ttune$. We
  thus define the time spent for observations (i.e. without instrumental
  setup) \tobs{} as:
  \begin{equation}
    \tobs = \ttel - \ntrack \times \ttune
  \end{equation}
  The number of tracks is computed as
  $\ntrack = \frac{\ttel}{\ttrack + \ttune}$ where \ttrack\ is the typical
  duration of a track, which depends on the source declination:
  \begin{itemize}
  \item Sources above $0 \deg$ are observed 8 hours at most,
  \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 ($\ttel < \ttrack + \ttune$), the number of tracks
  \ntrack{} is set to 1.  Otherwise, the floating value of \ntrack{} is
  used in the computation of \tobs{}. Since \ttune{} 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.
\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}
    \ton = \tobs \times \obseff
  \end{equation}
  where \obseff{} is the observing efficiency.
\end{enumerate}

The exact computation depends on the observing mode. There are three main
observation kinds.
\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.

\subsection{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 the time spent on-source. It is related to the total
elapsed telescope time \ttel{} through:
\begin{equation}
  \ton  = \frac{\ttel \,-\,  \ntrack \times  \ttune}{\obseff}
  \label{eq:tel-vs-on}
\end{equation}
For single source projects, \obseff{} is estimated to be about 1.6. Note
that the exact value will depend on several parameters such as the number
of calibrators and the distance between the source and the calibrator(s).

\subsection{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
  \ton =  \frac{\ttel \,-\,  \ntrack \times  \ttune}{\obseff \times \nsource}
\end{equation}
Note that 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.  As for single source projects, \obseff{} is set to 1.6 in
track-sharing mode.

\subsection{Mosaicking}
\label{sec:mosaicking}
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.

Note that \nbeam{} is not the number of pointed positions that are observed
for the mosaic ($\npoint > \nbeam$, see below).

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. To first order, we
thus yield:
\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 =  \frac{\ttel \,-\, \ntrack \times  \ttune}{\obseff \times \nbeam}
\end{equation}

There are several subtleties in this computation.
\begin{itemize}
\item \Amap{} must be larger than 2 times \Abeam{} (below this we advise to
  use the track sharing mode with two independent fields).
\item 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. In other words,
  Eq.~\ref{eq:mos} does not apply to the mosaic edges!
\item 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. Equation~\ref{eq:mos} is only valid
  inside a correctly sampled mosaic.
\item The pointings of a mosaic must be observed in short time cycles to
  ensure that all pointings are observed with similar weather conditions
  and that they share similar $uv$ coverage. This minimizes 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 and we have
  another relationship between the elapsed telescope time and the on-source
  time as:
  \begin{equation}
    \ton =  \frac{\ttel \,-\,  n_{\emr{tracks}} \times  \ttune}{\obseff \moseff \times \nbeam}
    \quad \mbox{with} \quad
    \moseff = \frac{\tint+\tslew}{\tint},
  \end{equation}
  where \tint{} is the integration time per pointing and \tslew{} is the
  time to slew between two consecutive pointings. Having a large
  integration time per pointing 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 is set by the acquisition system (for
    instance, the maximum achievable data rate). In pratice, we enforce
    that
    \begin{equation}
      \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{\tint}{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}
      \tintmin \le \frac{1}{\eta}\,\frac{6\,900}{1\,\emr{sec}}\,
      \sqrt{\frac{\beam[maj]\beam[min]}{\Amap}},
    \end{equation}
    where $\eta$ is a ad-hoc integer set to 5 to ensure the condition
    defined in Eq.~\ref{eq:uvcond}.
  \end{enumerate}
  As the typical slew time between two pointings is $\tslew = 11\,$sec, we
  yield that
  \begin{equation}
    1 \le \moseff \le 2.3
  \end{equation}
\item If the time to cycle all the pointings, \tcycle, is set to 45
  minutes, we yield that the maximum number of pointing per track is
  \begin{equation}
    \npointmax = \frac{\tcycle}{\tintmin+\tslew} = 130.
  \end{equation}
\item Finally, if the PI wishes to observe an area that will require more
  that 130 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. The computation is done as follows.
  \begin{enumerate}
  \item The number of tracks is then computed as described in
    section~\ref{sec:single-field}.
  \item The number of point per track is then
    $\npointpertrack = \paren{\frac{7}{4}}^2 \, \frac{\nbeam}{\ntrack}.$
    This value must be lower than \npointmax.
  \end{enumerate}
\end{itemize}

In summary, the sensitivity of a Nyquist sampled mosaic is
\begin{equation}
  \noise[Jy] = \frac{\Jant[int]\,\Tsys}{\speceff\,\sqrt{\nant\,(\nant-1)\,\dnu\,\npol\,\ton}}
\end{equation}
\begin{equation}
  \mbox{with} \quad
  \ton =  \frac{\ttel \,-\,  n_{\emr{tracks}} \times  \ttune}{\obseff \moseff  \nbeam}
\end{equation}

\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}

\appendix{}


\end{document}

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