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\begin{document}
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\begin{center}{\Large \textbf{Detection of extended component of 47 Tuc, due to unresolved pulsar emission}}\end{center}
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  Circulation:~~\= \today \kill
  Author: \> Tim Connors\\
  Revision: \> \today\\
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\abstract{
We present a method for statistically detecting an amount of pulsars
in a cluster (47 Tucanae) that can not be individually resolved
because of their low flux. The method involves forming a low resolution
image of the core of a cluster, that has a beam that is matched with the expected
spatial distribution of pulsars, and so will see unresolved pulsars as
an extended component, and subtracting those pulsars that are in the
process of scintillating. From this, we find typically $\sim 500
\mu$Jy contributed from easily resolved pulsars, and another $\sim
400$ from unresolved pulsars, suggesting there are 10 times as many
pulsars 10 times less luminous than the 4 visible on most days - ie, at
least 40 more pulsars in 47 Tuc.}
\section{Introduction}
Timing solutions have given the position\cite{cite:camiloetal2001} and
flux\cite{cite:camiloetal2000} of 15 pulsars in 47 Tuc (all
millisecond pulsars), out of the 20 pulsars listed in those papers, and
this represents the current level of sensitivity, with all of the
pulsars being only visible during periods of interstellar
scintillation (5 more are occasionally visible but are not yet located
because of their rare appearance or binary orbit). We can expect perhaps 200 more
pulsars\cite{cite:camiloetal2000} in 47 Tuc, if the luminosity
distribution is similar to the galactic distribution of millisecond
pulsars, which is given by
\begin{equation}
\label{eqn:lumdist}
dN=L^{-1} d\, \log L
\end{equation}
From this, for every lower decade of flux, there will be 10 times more
pulsars, up to a cutoff (since the integral of pulsar count diverges
for no cutoff). We should expect that they are spatially distributed
in the same way, independent of luminosity.

Camilo et al\cite{cite:camiloetal2000} gave mean fluxes for the 15
located pulsars, using timing measurements from the 64m Parkes
telescope, and the flux totals to 2.06 mJy. We assume that the flux of the 5 other
unlocated pulsars are going to be small compared to this, and so if
there are 200 pulsars, there will be about 4 mJy in the extended
component, and 2 mJy from the as yet undetected pulsars.

We suggest that imaging the core of the cluster with the ATCA, with a
suitably compact array (for a large synthesised beam), we should be
able to see the group of pulsars as a single extended component, and
be able to remove the effects of background sources and those bright
pulsars that may be in a process of scintillating.

\section{Observations and data analysis}
All data analysis was performed using the {\sc Miriad} package. The
observations were done at 1408 and 1708 MHz, with a bandwidth of
128MHz using the usual ATCA setup, although only the 1408 MHz data is
used here. All data was collected from the ATCA, with the timing
positions presented by Camilo et. al.\cite{cite:camiloetal2001} from
the Parkes 64m telescope being used to identify the pulsars that we
remove later.

The cluster was imaged in a number of different configurations
(table~\ref{tab:configs}), over many days.
\begin{table}[!h]
   \caption{\footnotesize Observing configurations. Field centre 0:24:06.00,-72:03:00.00 (J2000)}
%   \centering
   \label{tab:configs}
        \tiny {
   \begin{tabular}{|l||c|c|c|c|c|c|c}
      \hline
        \bf{Observation date}   & 99-Feb-12  ---  99-Dec-31 & 99-Jan-25 & 00-May-08 & 00-Dec-23 & 00-Dec-25 & 00-Dec-27 \\
      \hline
        \bf{Configuration}      & many: 1.5A, 1.5B, 6A, 6C, 6D & 750B & 375 & 750C   & 750C     & 750C     \\
      \hline
        \bf{Image type}         & Large array for high res image & \multicolumn{5}{|c|}{5 antennae for extended sources, 6th for removal of bright pulsars} \\
      \hline
        \multirow{6}{20mm}{\bf{Comments}}& \multirow{6}{35mm}{Many days combined to make a very high res image with very high S/N of $28 \mu Jy$} & \multirow{6}{30mm}{Because of maintenance, 3rd antenna missing for most of day, and many faults during observing $\Rightarrow$ only 4 antennae left. Data unused currently} & & & & \multirow{6}{15mm}{Loss of several hours because of error and thunderstorm} \\ 
         & & & & & & \\ 
         & & & & & & \\ 
         & & & & & & \\ 
         & & & & & & \\ 
         & & & & & & \\ 
      \hline
   \end{tabular}
}
\end{table}
The 1.5 and 6km array data, that are not sensitive to the extended
component of the unresolved pulsars, were mosaiced together to form a
sensitive high resolution image of the core\cite{cite:mcconnelletal2001}.
The 375 and 750m arrays were combined to form a low resolution image
of the core and was well matched with the spatial distribution of the
known pulsars.

An attempt was made to verify, using {\sc uvgen}, that a source with
characteristic size of the pulsar distribution half width of
70''\cite{cite:camiloetal2001} would not be attenuated appreciably by
the $\sim 60$'' beam size of the 375/750 data. This work is
incomplete.

A self-calibration of the low resolution image using itself as a model
would be extremely difficult, since there are only 5 antennae that are
used to form the small baselines. The individual small array images
were self calibrated (for amplitude and phase) using {\sc gpscal},
with the model being McConnell and Ables' (2000) 6km mosaic. This
might have the side-effect of decreasing the amplitude of anything
that does not appear in the 6km mosaic - which would include the
extended flux we are searching for.

Subtracting the point sources in the high resolution image should
leave us with the extended component we are looking for in the low
resolution image (as well as some sources that vary between the
observations). Because McConnell and Ables' (2000) 6km mosaic has
modelled the point sources well, we can subtract them from the UV
data, removing both the sources and their sidelobes from the image.
We do this by forming the UV data for the smaller arrays individually,
and subtracting the clean components of McConnell and Ables' (2000)
mosaic formed by using the 1.5 and 6km arrays, using {\sc
uvmodel}. Because this will remove the pulsars that appear in
McConnell and Ables' (2000) high resolution mosaic (ie, those ones
that are bright enough for a few of the observations), we mask the
clean image first, using {\sc maths}, to exclude the pulsars (and
anything else within 130'' of the core), and assume that anything that
appears within this mask and is similar in all the days observations
is a background source, so remove those manually from the low
resolution data.

Each of the pulsars scintillate, and some are quite dramatic - if one
happens to be bright on any one day, then we may falsely conclude that
we have found the extended flux, when the extra flux really belongs to
a single pulsar, and not 200 of them. Hence, we form an image using
just the antenna 6 baselines (hence the UV coverage is poor, with only
$\sim 6$km baselines) for each day, and {\sc imfit} to each of the
sources that are both above $3\sigma$, and that correspond to one of
the already known and located pulsars. The sources were both modelled
as point sources, and Gaussian (since they may vary within the day's
observation, due to scintillation), but the difference between the
two was minimal, so the point source method was chosen for
simplicity. We assume that the pulsars we do not know about never get
bright enough to skew our extended flux measurements
significantly. Since the 6km baseline image is hard to calibrate
because of the lack of good UV coverage, we assume that the two
background sources near the core are reasonably constant, and scale
the pulsars by an amount that scales the brightest of these
`calibrators' to the level obtained in the McConnell and Ables' (2000)
1.5/6km mosaic.  The scaled fluxes from the pulsars are then
subtracted from the UV data from each individual day, again with {\sc
uvmodel}, and a combined image is again formed. This image should now
only contain the extended component due to the unresolved pulsars in
the centre, with the main sources of noise being low level confusion.

\section{Results}
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The two sets of data (high resolution 1.5/6km arrays, and low
resolution 375/750m arrays) were combined and imaged, and appear on
the same scale in figures~\ref{fig:highresmos} and \ref{fig:lowresmos}. It is
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%     \label{fig:lowresmos} 
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  \begin{figure}[!htb]
    \begin{center}
      \parbox{8cm}{
%        \rotatebox[origin=cb]{-90}{
\epsfig{file=pres1.c059.contour.ps, height=7cm, width=8cm}
        \caption{\footnotesize McConnell and Ables' (2000) 1.5 and 6km array image, at 1408MHz, with a $8\times10$''
                 synthesised beam. The RMS noise level is $28\mu$Jy, the contours are at 
                 3, 12 and 48$\sigma$, and the range is from 0 - 0.4 mJy. 
                 The central circle is the size of the core at 23''.}
        \label{fig:highresmos}
      }
      \hspace{0.2cm}
      \begin{minipage}{8cm}
%        \rotatebox[origin=cb]{-90}{
\epsfig{file=pres1.gs5.contour.ps, height=7cm, width=8cm}
        \caption{\footnotesize A mosaiced 375 and 750m array image, at 1408MHz, with an approximate 
                 $63\times68$'' beam, and a RMS noise level of $35\mu$Jy. The contours 
                 are at the same level as figure~\ref{fig:highresmos}, so are at 
                 $\sim 2.5$, 10, 40$\sigma$, and the range is from 0 - 0.4 mJy.}
        \label{fig:lowresmos}
      \end{minipage}
    \end{center}
  \end{figure}
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%     \includegraphics[width=0.6\textwidth]{pres1.gs5.contour.eps}
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apparent already that there are some extended sources in the field of
the core, in particular the two bright sources near the top left
corner, and these do not become subtracted completely, later on.

From each day, the 6km only baselines were individually imaged, and one 
(2000-Dec-27) appears in figure~\ref{fig:6km}. Evident are the two
  \begin{figure}[!htb] 
     \centering
%     \rotatebox[origin=cb]{-90}{
\epsfig{file=pres1.ant6.ps, height=7cm, width=8cm}
     \caption{\footnotesize Image constructed using only antenna 6 baselines from the 750C configuration
              on the 2000-Dec-27. The RMS in the image was $50\mu$Jy, but the image quality 
              was quite poor because of the lack of UV coverage, and poorly calibrated. 
              The only contour is at $3 \sigma$, and the range is from 150 to $200 \mu$Jy. Most of the points above the 
              contour are noise, but those points that corresponded to a known 
              pulsar (within a small proportion of the synthesised beam) that were above $3\sigma$ 
              were modelled by point sources.}
     \label{fig:6km} 
  \end{figure}
constant sources (labelled 
%$\alpha$ and $\beta$
142 and 178), and 4 pulsars. The only contour is at $3\sigma$, so
most of the random points are noise, but some may be scintillating
pulsars. The central circle (radius 23'') is the size of the core of the cluster.

Table~\ref{tab:fluxes} lists each of the pulsars, and the two constant
  \begin{table}
    \centering
    \caption{\footnotesize Fluxes of each resolved pulsar. The lack of data for the  2000-Dec-25
             is because of observing problems - this data is as yet unused. The column labelled
             ``Only 6km'' is the raw fitted (for a point source) flux from the individual 6km 
             baseline map, and ``Rescaled'' is after the source 142 is scaled to be the same 
             as in McConnell and Ables' (2000) 6km mosaic ($643 \mu$Jy)}
    \label{tab:fluxes}
        \tiny{
    \begin{tabular}{|c||l|l|r|r|r|r|r|r|r|r|r|r|}
      \hline
        \multirow{3}{0.01cm}{} & \multirow{3}{0.85cm}{\bf{Source/ Pulsar name}} & \multirow{3}{0.85cm}{\bf{Camilo et.~al.\cite{cite:camiloetal2000} flux}} & \multicolumn{10}{|c|}{\bf{Observation Date}} \\
      \cline{4-13}
        & & & \multicolumn{2}{|c|}{\bf{99-Jan-25}} & \multicolumn{2}{|c|}{\bf{00-May-05}} & \multicolumn{2}{|c|}{\bf{00-Dec-23}} & \multicolumn{2}{|c|}{\bf{00-Dec-25}} & \multicolumn{2}{|c|}{\bf{00-Dec-27}} \\
      \cline{4-13}
        & & & $\!\!$Only 6km$\!\!$ & $\!\!$Rescaled$\!\!$ & $\!\!$Only 6km$\!\!$ & $\!\!$Rescaled$\!\!$ & $\!\!$Only 6km$\!\!$ & $\!\!$Rescaled$\!\!$ & $\!\!$Only 6km$\!\!$ & $\!\!$Rescaled$\!\!$ & $\!\!$Only 6km$\!\!$ & $\!\!$Rescaled$\!\!$ \\
      \hline
      \hline
        \multirow{8}{0.45cm}{Flux $\!\!$($\mu$Jy)$\;$} &  142 & & & & 564 & \bf{643} & 839 & \bf{643} & 761 & \bf{643} & 715 & \bf{643} \\
        & 178 &     & & & 285 &     & 218 &     & 270 &     & 313 &     \\
        & C   & 360 & & &     &     & 172 & 132 & 202 & 171 & 178 & 160 \\
        & D   & 220 & & & 244 & 278 & 311 & 238 &     &     & 285 & 256 \\
        & E   & 210 & & &     &     &     &     &     &     & 365 & 328 \\
        & F   & 150 & & &     &     &     &     & 318 & 268 &     &     \\
        & J   & 540 & & &     &     &     &     & 175 & 148 & 268 & 241 \\
        & Q   & 50  & & &     &     & 287 & 220 &     &     &     &     \\
      \hline
    \end{tabular}
}
  \end{table}
sources (source 142 and 178) fluxes, and the scaled value. Since source 142 is
$643\mu$Jy in the high resolution map, we scale it to be $643\mu$Jy on
all days, and all the others as appropriate. Source 178 is similarly always
left (ie, not scaled) as the $226\mu$Jy that is was in the high resolution map.

Due to persistent problems during the 1999-Jan-25 observation,
culminating in the loss of one antenna for most of the day, this data
had to be discarded.

It is seen in table~\ref{tab:fluxes}, that some of the mean fluxes
from the antenna 6 baselines are a lot lower than Camilo
et. al.\cite{cite:camiloetal2000} claim (particlularly pulsar J), and
this is as yet unexplained.

Subtracting the point sources in the masked high resolution image, the
two constant sources manually, and the 4 significant pulsars from the
2000-Dec-27 data yields the image in figure~\ref{fig:27rem}. The
  \begin{figure}[!htb] 
     \centering
%     \rotatebox[origin=cb]{-90}{
\epsfig{file=pres1.indiv.ps, height=7cm, width=8cm}
     \caption{\footnotesize Image from 2000-Dec-27 that shows both the pulsars and the two
        constant sources, 142 and 178, removed. This then forms part of the image in 
        figure~\ref{fig:remall}. The range is from 0 to $400 \mu$Jy, and the contours are 
        at 3, 6, 12, 24 \ldots $\times$ the RMS from figure~\ref{fig:highresmos}. Since the RMS 
        of this image is $50 \mu$Jy, the contours are at 1.7, 3.4, 6.8$\sigma$. The peak of the core in 
        this image is $380 \mu$Jy.}
     \label{fig:27rem} 
  \end{figure}
process was repeated for the other days, and the result was combined
to yield figure~\ref{fig:remall}. The combined image was also produced
  \begin{figure}[!htb]
    \begin{center}
      \parbox{8cm}{
%        \rotatebox[origin=cb]{-90}{
\epsfig{file=pres1.rem.known.ps, height=7cm, width=8cm}
        \caption{\footnotesize This is all the days data combined, with all point sources away from 
        the core, and the two constant sources, 142 and 178 subtracted, but the pulsars left intact. The contours are 
        at 3, 6, 12, 24 \ldots $\times$ the RMS from figure~\ref{fig:highresmos}. Since the RMS 
        of this image is $60 \mu$Jy, the contours are at 1.4, 2.8, 5.6 \ldots $\sigma$. The range of the image is 0 to $400 \mu$Jy. The peak of the core in 
        this image is $530 \pm 60 \mu$Jy/beam, and the integrated flux is $\sim 810$Jy (fitted for gaussian)}
        \label{fig:remconst}
      }
      \hspace{0.2cm}
      \begin{minipage}{8cm}
%        \rotatebox[origin=cb]{-90}{
\epsfig{file=pres1.rem.all.ps, height=7cm, width=8cm}
        \caption{\footnotesize This is all the days data combined, with all point sources away from 
        the core, the two constant sources, 142 and 178, and the pulsars, subtracted. The contours are 
        at 3, 6, 12, 24 \ldots $\times$ the RMS from figure~\ref{fig:highresmos}. Since the RMS 
        of this image is $60 \mu$Jy, the contours are at 1.4, 2.8, 5.6 \ldots $\sigma$. The range of the image is 0 to $400 \mu$Jy. The peak of the core in 
        this image is $280 \pm 60 \mu$Jy/beam, and the integrated flux is $\sim 415 \mu$Jy (fitted for Gaussian)}
        \label{fig:remall}
      \end{minipage}
    \end{center}
  \end{figure}
without subtracting the pulsars in figure~\ref{fig:remconst} - only 
subtracting the two constant sources. The integrated
flux within the core of figure~\ref{fig:remconst} is only $810 \pm 60
\mu$Jy, which is much lower than the expected 4mJy. Similarly, after
subtracting the pulsars to give figure~\ref{fig:remall}, we are left
with much lower than the 2mJy expected - only $415 \pm 60 \mu$Jy. The
errors quoted are the image RMS (fitted using {\sc imsad} from the
region surrounding the core). Note that this error is typically 3 times
larger, for all the images created in this analysis, than the
theoretical RMS quoted by {\sc invert}, and the difference is
attributed to image confusion.
\section{Discussion}
The image with all pulsars still present (figure~\ref{fig:remconst})
shows that we do not even measure all of the flux from the known
pulsars - let alone any extra from the unresolved pulsars. This is
unexplained, but once the bright pulsars are also subtracted
(figure~\ref{fig:remall}), there is a similar amount of flux from the
unresolved pulsars, to those visible on any one day.

The most obvious cause of the loss of flux, is in self calibration -
when calibrating for amplitude and phase, any object not appearing in
the model will be scaled down. The possibilities for self-calibration
are to use the 6km image as a model, and calibrate for phase only, or
to calibrate for amplitude as well, making sure that the rms gain is
not scaled to unity. A final alternative is to form a smaller image
(because of computational requirements) using the 6th antenna for each
day (or for all days combined), that will suffer because of poor UV
coverage, and using that as a model in self-calibrating each day. This
may have the benifit of including at least some more flux from the
core, because of the presence of smaller baselines than those in the
McConnell and Ables' (2000) 6km mosaic. The results from testing the
peak and approximate integrated flux from each of these methods is in
table~\ref{tab:fluxfromcal}, but since this work is incomplete, some
\begin{table}[!h]
   \centering
   \caption{\footnotesize Peak and integrated flux from different methods of self-calibration. \\
                1: no calibration. \\
                2: Selfcalibration, phase only, model: McConnell and Ables (2000). \\
                3: Amplitude and phase, {\sc miriad: options=noscale}. \\
                4: Amplitude and phase, model: smaller image containing 6th antenna}
%   \centering
   \label{tab:fluxfromcal}
        \footnotesize{
   \begin{tabular}{|l|l||c|c|}
      \hline
        \multirow{2}{4cm}{\bf Calibration and flux measurement on} & \multirow{2}{3cm}{\bf Calibration Type} & \multirow{2}{3cm}{\bf Peak flux ($\mu$Jy/beam)}   & \multirow{2}{3cm}{\bf Integrated flux($\mu$Jy)} \\
 & & & \\
      \hline
      \hline
        all days combined & 1               & 515 & 893 \\
      \hline
        all days combined & 2               & 422 & 622 \\
      \hline
        all days combined & 3               & 347 & 703 \\
      \hline
        only 2000-Dec-27  & 3               & 465 & 525 \\
      \hline
        only 2000-Dec-27  & 4               & 482 & 605 \\
      \hline
   \end{tabular}
}
\end{table}
of the methods are from a previous non-optimised mosaic (without the pulsars
subtracted), and some are from only one day (2000-Dec-27).

It is clear that we would not have so much ``missing'' flux, if a
self-calibration was performed on each day, using the full 6 antennae
from that day, and so this will be used in the future. One problem faced
is that there is a lack of good UV coverage, so this process will not
necessarily be trivial.

Incorrect flux measurements from Camilo et. al. (2000) cannot be ruled
out either - if their fluxes are systematically higher, we can not
expect to measure an extended component equal to the sum of pulsar
fluxes for the 15 measured pulsars.

Even though we do not see all the flux from the known pulsars, we do
see some extra flux once the resolved ones are subtracted, that
indicated we are seeing some of the extended component we are looking
for, although it is may not be equal in intensity to the resolved
pulsars listed in Camilo et. al, 2000, depending on the sucess of the
self-calibration method proposed above. Some possible mechanisms for
there not being the expected unresolved flux are summarised below.

Equation~\ref{eqn:lumdist} implies a power law with index -1. If this
is not correct, and the exponent is actually more negative, then 
there will not be as much flux in the lower luminosity decade, and
there will be less extended flux. This is also a more elegant solution
than having a low luminosity cutoff around $L_{400} \sim 1$ mJy
kpc$^2$\cite{cite:camiloetal2000}.

If for some reason, the fainter pulsars are systematically more spread
out than the sudden $3 r_c$ boundary for the brighter
pulsars\cite{cite:camiloetal2001}, and are distributed far enough that
synthesised beam of $\sim 60$'' attenuates the integrated flux
appreciably, then the flux due to the decade of low luminosity pulsars
will be less than the decade of higher luminosity pulsars, although a
physical reason for any possible difference in distribution has not been
discovered.

One promising sign is that it seems (within the error caused by noise
in the images) that the integrated flux of the core once the pulsars
visible on each day are subtracted, is half that of the total flux
before subtraction. If indeed, we are seeing all the flux present, and
the fluxes listed in Camilo et. al. are wrong, then there is a good
chance the prediction of roughly 10 times more pulsars (whose beam is
in our direction) being present in 47-Tuc than we currently resolve is
correct - ie perhaps more than 40, since we see up to 4. Care needs
to be taken however, since there may be other sources near the core
that are not pulsars, and weren't subtracted, that may influence the
fitting, particularly since the fit was not for a point source.
\begin{thebibliography}{99}
\bibitem{cite:camiloetal2000} Camilo, F., Lorimer, D.R., Freire, P., Lyne, A.G. and Manchester, R.N. 2000, ApJ, 535, 975
\bibitem{cite:camiloetal2001} Freire, P.C., Camilo, F., Lorimer, D.R., Lyne, A.G., Manchester, R.N. and D'Amico, N. 2000, MNRAS, submitted.
\bibitem{cite:mcconnelletal2001} McConnell, D., Deacon, R., Ables, J.G. 2000, PASA, submitted.
\end{thebibliography}
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\end{document}

