As Fig. 7 confirms, the observed (small) variation of the shape of the LF with distance from the cluster centre is fully consistent with the mechanism of mass segregation ensuing from energy equipartition as currently understood in many other clusters (Meylan & Heggie 1997). We, therefore, cannot hold internal dynamical evolution responsible for the observed inverted global MF in NGC 6712: such a mechanism, in fact, could only account for the inverted MF near the core of the cluster, but not elsewhere (see De Marchi et al. 2000).

There remain, thus, only two ways to explain the inverted global MF of this cluster, as already proposed by De Marchi et al. (1999). Namely, either NGC 6712 was born with an inverted IMF, at least for stars less massive than $\sim$ 0.8 $M_{\odot }$ , or the interaction with the tidal field of the Galaxy (and more specifically disk and bulge shocking through its frequent and repeated perigalacticon passages) has imparted a strong modification to the stellar population of this cluster during its life-time, as its orbit forces it to penetrate deeply into the bulge at its disk crossings.

Although the former hypothesis cannot be completely ruled out, it is highly unsatisfactory as it would explain one anomaly - the inverted global MF - by invoking another one, namely an inverted IMF. A careful investigation of the deep LF of a dozen GCs by Paresce & De Marchi (2000) shows no evidence of such an inverted IMF. Their sample does, admittedly, only cover $\sim$ 10% of the total population of GC, but it contains clusters in widely different orbits and dynamical states so that it can be regarded as representative of the whole Galactic GC system. Still, the hypothesis of an inverted IMF cannot be excluded, at least until the origin of the inverted LF observed long ago in E3 (McClure et al. 1985) and Pal5 (Smith et al. 1986) is understood (see Sect.4 above).

On the other hand, there seems to be more solid observational and theoretical support for the latter hypothesis, namely that the cluster has suffered severe tidal stripping which has remarkably altered its stellar population. Recent calculations of the orbit of NGC 6712 and consequent destruction rate by Gnedin & Ostriker (1997) and Dinescu et al. (1999) clearly suggest that it has one of the highest destruction rates of a large sample of Galactic GC and that it is one of the few objects for which the tidal-shock rate is higher than its two-body relaxation rate. One would, thus, expect that the strong tidal interaction during the disk crossings and the consequent tidal shocking should provoke a continuous loss of low-mass stars, especially from beyond the half-light radius, and a consequent rapid change of the stellar mass distribution. Precisely on this basis, Takahashi & Portegies Zwart (2000) have suggested that NGC 6712 has lost 99% of its mass during its life-time and that it is now obviously only a pale remnant of its initial much more massive condition.

$\begin{figure} \par\resizebox{\hsize}{!}{\includegraphics{h2459f8.eps}}\end{figure}$

Figure 8: Surface density profile of $\sim$ 0.75 $M_{\odot }$ stars. The dotted line shows a King-type profile with $r_{\rm c} = 1^\prime$ and $r_{\rm t}=5\hbox {$.\mkern -4mu^\prime $ }2$ , whereas the thick dashed line shows the superposition of the latter on a plateau of field stars of uniform surface density.

One might wonder, however, how the internal dynamics of a cluster which has suffered such a tremendous mass loss could still conform so well to the predictions of standard two-body relaxation and energy equipartition as Fig. 7 indicates. Johnston et al. (1999), on the other hand, have shown that the tidal stripping operated by the Galaxy on a GC results in a steady differential loss of stars (light stars being dislodged more easily than massive ones) with the consequent continuous decrease of the exponent $\alpha$ of the global MF and the ensuing flattening of the latter. Although the inclination of the cluster's orbit determines the rate at which the MF exponent decreases, all orbits with perigalacticon within a few kpc of the Galactic centre are exposed to this erosion. For clusters such as NGC 6712, whose orbit is mostly contained within the disk (Dauphole et al. 1996), the heating due to disk and bulge shocking is diluted over a long time, comparable with the dynamical relaxation time of the cluster itself. It is, thus, not unreasonable that two-body relaxation can proceed almost undisturbed, and it does so on a continuously varying mass spectrum.

A natural consequence of tidal stripping is the formation of tidal tails surrounding the cluster (Grillmair et al. 1995). While the latter might be relatively easy to identify around clusters on highly inclined orbits and currently well away from the Galactic plane, looking for extra-tidal populations around NGC 6712 is very difficult, because of its orbit and current location in the Galaxy, and even more so because the surface brightness of this excess of stars which might have been ejected from the interior but are still loosely bound to it is expected to be about 4 orders of magnitude lower than in the core (Johnston et al. 1999). We have, nevertheless, searched the region near the cluster's tidal radius (> $5^{\prime}$ ) but, not surprisingly, the radial density profile that we have measured does not reveal any obvious over-density near the cluster's boundary (see Fig. 8). On the other hand, we were forced to limit our investigation to stars in the range 19.5 < R < 20.5 (i.e. to $\sim$ 0.75 $M_{\odot }$ stars), so as to minimise the effects of variable photometric completeness and crowding with distance, and it is quite likely that most of these stars should today dwell preferentially in the central regions of the cluster, rather than in its periphery, as a result of mass segregation.

The radial density profile that we show in Fig. 8, however, has allowed us to define a more reliable tidal radius for this cluster. The thick dashed line marks a typical King-type profile with $r_{\rm c} = 1^\prime$ and $r_{\rm t}=5\hbox {$.\mkern -4mu^\prime $ }2$ , superimposed on a plateau of field stars. A tidal radius of $\sim$ $5^{\prime}$ is fully consistent with our finding of a statistically null cluster LF in annulus A5 (which extends to $r=5\hbox{$.\mkern-4mu^\prime$ }1$ ). Although we have limited our analysis to stars for which photometric completeness is always >85%, severe crowding and the concentration of many saturated stars in the innermost regions could make our determination of the core radius $r_{\rm c}$ uncertain. Using shorter FORS1 exposures of the central $\sim$ $2^\prime$ radius of this cluster, however, Paltrinieri et al. (2001) also find a value of $r_{\rm c}\simeq 1^\prime$ , in excellent agreement with that estimated here.

Thus, assessing whether NGC 6712 was indeed much more massive in the past than it is now, as suggested by the work of Takahashi & Portegies Zwart (2000), would require a more accurate search for tidal tails surrounding the cluster, using a large field of view and sophisticated reduction techniques such as those developed by Grillmair et al. (1995) and, more recently, by Leon et al. (2000). On the other hand, the severe field contamination would necessarily limit the effectiveness of this technique. Moreover, even revealing the presence of tidal tails would not provide strong constraints on the original cluster mass. To be sure, tidal stripping has taken place throughout the whole life of the cluster and the majority of the stars lost in this way should be today totally unbound and dispersed elsewhere in the Galaxy.

A more precise, quantitative estimate of the original cluster mass could, however, come from a census of the WD population in its core. If NGC 6712 was indeed originally as massive as 10⁷ $M_{\odot }$ , then a large number of WDs should now populate its core. Even ignoring the effects of mass segregation (which would further increase the WD population in the core by draining them from the periphery and forcing them to drift there by virtue of their higher mass), the prescriptions of Renzini (1985) suggest that $\sim$ 2000 WDs brighter than $M_V \simeq 14$ ( $V \simeq 28.5$ ) should dwell in the central $100^{\prime\prime}$ radius of the cluster, if the latter had its present mass $\sim$ 5Gyr ago. Clearly, if its mass that long ago was even only ten times as much as it is now, we would expect of order $20\,000$ WDs within a $100^{\prime\prime}$ radius. Already on the basis of the available data we can conclude that a large fraction ( $\simeq$ 60% or more) of the cluster's mass must be in the form of heavy remnants (see Sect.4). Whether the cluster was originally only a few times more massive than it is now or whether it was one of the most massive in the Galaxy cannot be determined with certainty at present, although the N-body simulations of Vesperini & Heggie (1996) seem to suggest the latter option. With a powerful instrument such as the Advanced Camera for Surveys soon to be installed on board the HST, however, this scenario can easily be tested observationally and the suspected ongoing dissolution of NGC 6712 reliably characterised.

5 Discussion and conclusions