\documentstyle[psfig]{l-aa}

\begin{document}

\appendix

\section{Observations and reductions}

\subsection{Instrumental magnitudes}

The journal of the observations is reported in Table~1 of the main text.
$V$ and $I$ images have been obtained during two nights in 1993 February at
the ESO 3.5 m NTT with EMMI at the Nasmyth focus. The scale is 0\farcs44/px
and the size of the chip is 1024 $\times$ 1024 px$^2$, which gives a square
field 7\farcm5 wide.  With nine fields, we mapped a region of $\simeq$~15.5
arcmin in radius around the cluster centre, as shown in Fig.~1 of the main
text.

Two further images were secured on 1993 December 10 in order to
have a comparison background field. The frames are centred 30\arcmin~
east of NGC~1851, and cover 9.6\arcmin~ $\times$ 7.4\arcmin~, with a
scale of 0\farcs34/px.



The image processing has been performed in part with {\sc midas} and in part
with {\sc iraf}; photometry has been completed at the {\em Instituto de
Astrof\'{\i}sica de Canarias} (IAC) with {\sc daophot ii} (Stetson 1987, 1994) in
the {\sc iraf} implementation.  A quadratic dependence of the PSF with
radius was necessary to account for off-axis aberrations and the
best-fitting analytic function was found to be a Moffat ($\beta =
2.5$) function.

\subsection{Calibration of standard system} \label{cal9}


\begin{table}[t]
\caption[]{Comparison with previous photometries of Walker (1992a) and Alcaino
et al. (1990). The errors on the mean are quoted; the standard
deviation of the data are 0.05 mag in $V $and 0.06 in ($V$--$I$) for the Alcaino
sample, and 0.04 mag in $V$ for the Walker sample.}
\label{x_phot}
\begin{tabular}{rcc}
\noalign{\smallskip}
\hline\hline
\noalign{\smallskip}
Ref. & $V$--$V_{\rm ref}$  & ($V$--$I$)--($V$--$I$)$_{\rm ref}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Alcaino et al. (1990) & 0.012 $\pm$ 0.004 & 0.035 $\pm$ 0.004 \\
        Walker (1992a) & 0.001 $\pm$ 0.001 & \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}


\begin{figure}[t]
\psfig{figure=xfig.ps,width=8.8cm}
\caption[]{Plot of the photometric zero-point differences: dotted lines
are the means reported in Table \ref{x_phot}; panels (a) and (b) refer to
Alcaino et al. (1990); panel (c) refers to Walker (1992a). The differences
are in the sense {\em this paper}--{\em reference}}
\label{xfig}
\end{figure}



Let us call $m_1$ and $m_2$ the aperture magnitudes of a star in the
fields 1 and 2; we can remove the dependence on the local exposure
conditions by normalizing each magnitude according to the relation
\[
\overline{m} = m + 2.5\,\log (t + \delta\,t) - k_m \, X,
\]
where $t$ and $\delta \, t$ are the exposure time and the shutter
delay, $k_m$ is the extinction coefficient and $X$ is the air mass.
For each star in the overlap area of any two fields we expect
$\overline{m}_1 - \overline{m}_2 = 0$, mean value. In turn, we
actually have PSF magnitudes. The corrections to aperture magnitudes
have been found assuming that $m = m^{\rm PSF} + \Delta$ (Stetson 1987)
and computing the mean offsets on a set of bright isolated
objects. The difference in normalized magnitudes for the same star in
two overlapping fields can therefore be written as:
\[
\begin{array}{l}
\overline{m}_1 - \overline{m}_2 = (m_1^{\rm PSF} - m_2^{\rm PSF}) +
(\Delta_1 - \Delta_2) \\

+ 2.5\,\log (t_1+\delta\,t_1)/(t_2+\delta\,t_2)
- k_m \, (X_1 - X_2).
\end{array}
\]
The observed scatter of the mean $<\overline{m}_1 - \overline{m}_2>$
around the zero point then gives a measure of the internal
uncertainties arising from all sources of error (aperture corrections,
exposure times and delays, and air masses). Comparing the 8 external
fields to the central one we find $<\overline{v}_1 - \overline{v}_2> =
0.029 \pm 0.015$ and $<\overline{i}_1 - \overline{i}_2> = 0.009 \pm
0.018$; the central $\overline{v}$ and $\overline{i}$ magnitudes plus
the quoted offsets have then been taken as the instrumental
zero-point, with 1--$\sigma$ uncertainties of 0.015 mag in $v$ and
0.018 mag in $i$.


Next, the above $\overline{v}$ and $\overline{i}$ instrumental
magnitudes have been corrected to the standard $VI$ system, making use
of the following relations:

\[
\begin{array}{crl}
V & = & {v + 0.04  (V - I) - 0.84} \\
{I} & = & {i - 0.01  (V - I) - 1.51}
\end{array}
\]
\noindent 
The numerical coefficients have been obtained by Veronesi (1995), from
the observations of eight Landolt (1992) standard stars in the same
observing run.  The calibration uncertainty is estimated 0.03 mag in $V$
and 0.02 mag in $I$, so the total zero-point error is $\sigma_{V}$
= 0.03 mag and $\sigma_{I}$ = 0.03 mag, obtained adding the latter errors
to the internal uncertainties.

We have compared our photometry with other photometric data in the
literature.  Figure~\ref{xfig} shows the comparison with Walker
(1992a, 956 common stars) and Alcaino et al. (1990, 236 common
stars); Table~\ref{x_phot} gives the zero-point differences and it can
be seen that a general agreement exists to within a few 0.01 mag,
although there is a notable dispersion in the data.


An interesting feature of Fig.~\ref{xfig} is the stars in panel
(c) at $V$ = 16.0 $\pm$ 0.5 mag and falling outside the 3-$\sigma$
dispersion strip (dashed lines).  There are $\sim$~30 such stars:
14  are known RR~Lyr (Liller 1975; Wehlau et al. 1978;
Wehlau et al. 1982).  The remaining $\sim$~20 objects are candidate
new RR~Lyr variables or anomalous stars and are discussed in
Sect.~3.5.


Almost all the objects marked with open and filled squares in
Fig.~7 have $\Delta\,V_{\rm Walker} < 0$ mag, i.e. they are
systematically brighter in our photometry (cf. also panel (c) in
Fig.~\ref{xfig}): the figure also shows that these objects are
just above the $V$ photometric limit of the W92 diagram (dotted line),
so we suspect that the observed discrepancies are due to a systematic
error at the limit of W92 photometry. This is further confirmed by the
fact that, apart from one case, stars for which $\Delta\,V_{\rm
Walker} > 0$  mage are all above $V$ = 17.9 mag (filled squares in
Fig.~7).

A systematic error in the W92 photometry is also our suggested explanation
for the four blue objects in Fig.~7 marked with open
triangles.  These are all at the inner radial limit of the W92 CMD 
($\simeq$ 25 arcsec), i.e. where the problems of blending become
severe. Moreover, stars 8, 9 and 10 are part of the objects $\sim$~0.4
mag bluer than the BHB in the W92 CMD, while they fall at the ``right''
position within our BHB.

\subsection{Calibration of the background field}


\begin{figure}[t]

\centerline{
\psfig{figure=cback.ps,width=8.8cm}
}
\caption[]{
The quantity $(V-v_{\rm norm})$ ($V = V_{\rm  Landolt (1992)})$ as a function of
the standard-star colour $(V-I)$; the straight line represents the adopted
calibration: $ V = v_{\rm norm} + 0.029 \, (V-I) - 0.040$ with a residual
dispersion of 0.004~mag (rms). The $v_{\rm norm}$ magnitudes have been
computed from the instrumental aperture magnitudes and normalized according
to Eq.~\ref{mnorm}. }
\label{cback}
\end{figure}

Eight standard stars from Landolt (1992) were used to calibrate the
background field;
we only needed to calibrate the $V$ frames, since the calibration of the
$I$ band is available in Zoccali et al. (1997). The magnitudes determined
within an aperture of 6.8\arcsec~ have been compared to the standard
values, after the following normalization:
\begin{equation} \label{mnorm}
m_{\rm norm} =  m_{\rm local} + 2.5 \log t_{\rm exp} -k_{\rm m}
\, X.
\end{equation}
The extinction coefficient $k_V = 0.121$ has been
provided by Geneva Observatory.
Figure~\ref{cback} shows the quantity
 $(V-V_{\rm norm})$ as a function of the $(V-I)$ colour, and a linear
fit yields
\[
V = v_{\rm norm} + 0.029 \, (V-I) - 0.040
\]
with a residual dispersion of 
0.004~mag (rms). Similarly, Zoccali et al. for the $I$ band find
\[
I = i_{\rm norm} - 0.004 \, (V-I) -  0.604
\]
For the background field magnitudes, an aperture correction
$\Delta m_{\rm local}^{\rm PSF}$ of $0.20$~mag and $0.24$~mag has
been applied for the $V$ and $I$ bands, respectively. These values
have been determined using a set of $\sim 10$ isolated and bright
stars, and all aperture magnitudes have been measured within an
aperture of 6.8\arcsec~: we checked that outside this aperture the
magnitudes are seeing independent. Finally, the values of $t_{\rm
exp}$ and $X$ used for the $V$ and $I$ frames are 120~s, 180~s and
1.443, 1.403 respectively.



\subsection{Completeness and photometric errors}


\begin{table}[t]
\caption[]{Photometric errors as a function of magnitude.}
\label{photerrors}
\begin{tabular}{rrrrrrr}
\hline \hline \noalign{\smallskip}
\multicolumn{1}{c}{$V$} &
\multicolumn{1}{c}{$I$} &
\multicolumn{1}{c}{off $V$} &
\multicolumn{1}{c}{off $I$} &
\multicolumn{1}{c}{$\sigma_{V}$} &
\multicolumn{1}{c}{$\sigma_{I}$} &
\multicolumn{1}{c}{$\sigma_{V-I}$} \\
\noalign{\smallskip} \hline \noalign{\smallskip}
  16.74  &  15.78   &  0.007  &  0.000   &   0.01  &   0.03  &   0.04  \\
  17.24  &  16.32   &  0.000  &  0.000   &   0.03  &   0.03  &   0.04  \\
  17.74  &  16.84   &  0.008  &  0.015   &   0.02  &   0.03  &   0.03  \\
  18.74  &  17.86   &  0.008  &  0.013   &   0.04  &   0.03  &   0.05  \\
  19.23  &  18.53   &  0.007  &  0.002   &   0.03  &   0.03  &   0.04  \\
  19.52  &  18.79   &  0.018  &  0.019   &   0.05  &   0.06  &   0.08  \\
  19.73  &  19.05   &  0.009  &  0.008   &   0.05  &   0.04  &   0.07  \\
  19.80  &  19.10   &  0.017  &  0.015   &   0.06  &   0.05  &   0.08  \\
  20.02  &  19.31   &  0.018  &  0.016   &   0.05  &   0.05  &   0.07  \\
  20.23  &  19.52   &  0.014  &  0.015   &   0.05  &   0.07  &   0.09  \\
  20.52  &  19.78   &  0.014  &  0.010   &   0.06  &   0.05  &   0.08  \\
  20.99  &  20.23   &  0.018  &  0.010   &   0.06  &   0.08  &   0.10  \\
  21.02  &  20.24   &  0.017  &  0.017   &   0.07  &   0.07  &   0.10  \\
  21.50  &  20.57   &  0.010  &  0.095   &   0.08  &   0.11  &   0.13  \\
  22.00  &  21.07   &  0.021  &  0.028   &   0.10  &   0.12  &   0.16  \\
  22.03  &  21.07   &  0.030  &  0.023   &   0.14  &   0.12  &   0.19  \\
  22.54  &  21.43   &  0.034  &  0.022   &   0.16  &   0.14  &   0.21  \\
  23.04  &  21.74   &  0.102  &  0.042   &   0.17  &   0.15  &   0.23  \\
  23.51  &  22.23   &  0.153  &  0.041   &   0.20  &   0.18  &   0.27  \\
\noalign{\smallskip} \hline
\end{tabular}
\end{table}

Completeness at the various magnitude levels has been computed by
adding artificial stars to the original frames and checking what
percentage is recovered, after a reduction stage equivalent to that of the
original images. Completeness within the CMD is evaluated considering
only those stars identified both in the $V$ and $I$ frames. One major
complication with clusters of high concentration such as NGC~1851 is that
completeness is a function not only of the magnitude but also of the
radial position of the stars. While this could be of minor importance
for bright stars except into the very centre of the cluster, the
probability of finding faint stars in the inner regions is a strong
function of the radius.  Crowding experiments were then carried out
for a suitably fine grid of magnitudes and radii, and completeness has
been linearly interpolated where no mesh points were available.

In Fig.~\ref{comp_curves}, the dependence of completeness both on radius 
and $V$ magnitude is shown for the central field, where 
the completeness gradient is most evident: this figure clearly shows that 
the 50\% level is reached at different radii for different magnitudes (dotted 
curve). This curve is well fitted by the line 
$V_{\rm 50\%}$ = 0.036~$r$ + 16.82; similarly, it is found 
$I_{\rm 50\%}$ = 0.037~$r$ + 15.89. 




\begin{figure}[t]
\psfig{figure=field1ms.ps,width=8.8cm,angle=-90}
\caption[]{Completeness as a function of magnitude and radius for 
the central field.}
\label{comp_curves}
\end{figure}


The evaluation of completeness also allows checking the internal 
photometric errors, 
simply by comparing output magnitudes to the magnitudes of the artificial stars;
besides adding a random scatter, reduction also introduces an offset, as 
can be 
seen in Table \ref{photerrors} (offsets are labeled 
{\em off V} and {\em off I}
in this table, and correspond to {\em off m} = $m_{\rm in} - m_{\rm out}$); these are in
the sense that the recovered magnitudes are always brighter than the 
added ones, as expected (Stetson 1987). 
Errors and offsets increase vs. magnitude, but keep within
a few $0.01$ mag to the very limit of our photometry.



\section{{\it HST} data} \label{calHST}



The compactness of NGC~1851 prevents the study of its central regions
with ground-based instruments. We then searched the {\it HST} archives through {\sc 
starview} and found that ten images of NGC~1851 were available: 
two of them  (see Table~2) are WFPC/I 
160
sec exposures in the F785LP and F555W filters (which closely match the $I$ and
$V$ colours). 

Technical details about WFPC/I can be found in {\it HST} user manuals: this camera
consists of four 800 $\times$ 800 px CCDs (CCD nr. 5 to 8), which subtend an
area of 70.4 $\times$ 70.4 arcsec$^2$ on the sky (due to the gaps
between adjacent chips, the effective area is 68 $\times$ 68
arcsec$^2$). The scale is 0\farcs0439/px, the read-out noise is reported to
be 13 $e^{-}$, with a conversion factor of 7.6 $e^{-}$/ADU. Saturation takes
place below 4095 ADU, the actual value depending on the current
flat-fielding. A histogram of the intensity distribution, revealed that in
our case saturation occurs at $\sim$ 3600 ADU for the F555W filter, and at
$\sim$ 3200 ADU for the F785LP filter. The most relevant bad features are
many dead columns/rows across the surface of the chips, which have been
removed.




Stellar photometry has been performed again with {\sc daophot ii}, and we limited 
ourselves to the PC field, 
covering the inner $\sim$~30$'' \times$~30$''$ of the cluster.
First we simply tried 
to derive aperture magnitudes within a 1 px radius, i.e. within the central 
sharp peak which comprises 10\% of the energy:
it has been shown (Gilmozzi 1990; Ortolani \& Murtagh 1992) that
aperture photometry of the central 0\farcs1 of the PSF, gives satisfactory
results for the WFPC1. 
The major complication was that 
a very high detection threshold had to be set  
in order to avoid contamination from PSF artefacts: there are so many 
misidentifications
that our 
cross-matching program failed to join the two colour tables. We then performed 
PSF photometry by means of {\sc allstar}, which provides a check on the input list of 
objects and helps reject many spurious detections; we simply chose a
Gaussian PSF function, which is adequate for this purpose.
Finally, simple aperture photometry has been
performed on the resulting catalogue, which comprises $\sim$ 1900 objects. 

Reduction of the {\it HST} material has been accomplished with the main purpose of 
checking radial trends in the HB populations in the inner part of the cluster;
 we
just need to distinguish between the red and blue part of the HB, 
so absolute calibration is not required. However,
for the aim     of comparing the two data sets, we made a 
simple calibration: as there are no stars in common with the calibrated
catalogue, since the PC field falls in a region where the completeness of
the NTT/EMMI data  goes almost
to zero, we chose three significant points within the two CMDs and found the 
transformation parameters, assuming linear relations. The relations may then be 
written as follows:

\smallskip
\noindent
\begin{small}
\begin{tabular}{rl}
$V -I$  = & 0.81~(F555W - F785LP) + 0.01~F785LP + 1.40 \\
     $I$  = & 0.14~(F555W -- F785LP) + 1.00~F785LP -- 2.70
\end{tabular}
\end{small}
\smallskip

\noindent
Figure~8 (lower panel) shows the {\it HST}-calibrated CMD, 
together with
a plot of the fiducial points for the NTT/EMMI set: the overlap is quite 
satisfactory for the purposes of the present paper.




\begin{thebibliography}{}
 \bibitem{} Gilmozzi R. 1990, Space Telescope Instrument Science Report
WFPC 90-007
\bibitem{} Ortolani S., Murtagh F. 1992, {\sl ST-ECF Newsl.}, No. 17, 5
 \bibitem{} Stetson P.B. 1994, PASP 106, 250
 \bibitem{} Veronesi C. 1995, Masters Thesis in Astronomy, Univ. of
  Padova, Italy 
 \bibitem{} {Zoccali} M., {Piotto} G., Zaggia S.R., Capaccioli M. 1998,
  A\&A, accepted
\end{thebibliography}

\end{document}
