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Otherwise, we are likely to estimate the rate of mass loss poorly as the debris along the orbit of the satellite confuses the calculation. The solid squares, open squares and stars show the results when viewed from along the x -, y - and z -axis respectively. The solid line shows where the estimated and known mass-loss rates agree, and the dotted lines are for factor-of-2 discrepancies.

The solid symbols show the analysis for the bound stars, and the open symbols show the results for all stars. The line of sight for each column of panels is labelled in the top row see Section 4. Repetition of panel x -axis view of Fig. The average velocities are zero except at the viewpoints that are sensitive to the orbital motion of the cluster. In the panels corresponding to the x -axis view of Figs 18 and 19 , the dotted lines indicate the expected velocity gradient if the cluster were rigidly corotating with its orbit.

The bound stars follow this line fairly closely, with some indication of a smaller gradient in velocities towards the centres of the clusters. In our analysis, two effects are clearly contributing to the measured rotation of the cluster — the intrinsic rotation of the cluster in our simulations, roughly corresponding to corotation with the tidal field , and the velocity gradient in stripped material.

Unfortunately, we cannot assume that tidal torquing of a real satellite necessarily results in corotation. The rotation seen in our simulations is likely to be an artefact of the near-circular orbits we have chosen. In models, run along eccentric orbits, of the disruption of the Sagittarius dwarf galaxy, Johnston et al. Rotation has been detected in several observational surveys of Galactic globular clusters.

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Merritt et al. Drukier et al. In both these studies, rotation was found to be less important towards the edge of the cluster, suggesting that in these cases it is intrinsic and not caused by tidal torquing or stripping of the bound system. However, the observed results can be confusing when looking along the orbit y -axis view panels of Fig.

This problem may not be so severe in reality if the sample can be selected to exclude stars beyond a few tidal radii from the cluster in the figure, all stars at the projected separation were included.

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An observer would also see an apparently enhanced velocity dispersion when looking at the outskirts of the cluster from the centre of the Galaxy. In particular, note that the dispersion in the unbound material roughly corresponds to the dispersion of the bound material within the point where stripping occurs. This suggests that satellites that are being more violently stripped will have larger dispersions in their debris trails, though not in excess of the maximum dispersion of the bound material.

The Sagittarius dwarf spheroidal galaxy is an example of a system in which it can be plausibly argued that this behaviour is observed: its highly distorted surface density contours suggest that it is likely to be surrounded by a cloud of unbound stars and its velocity dispersion is roughly constant along the entire length of its major axis Ibata et al. In their analysis of the velocities in the outskirts of M15, Drukier et al. They suggest that this deviation from the behaviour expected for an isolated cluster could be caused by tidal heating, and our simulations confirm this interpretation.

They comment that the radial position of this minimum is much smaller than the tidal radius found by G95 by fitting King models to star count profiles. This is also consistent with our analyses — the star counts and velocity analyses become contaminated by unbound stars well within the outermost radius of the bound system.

Hence the minimum in the velocity dispersion profile is a good indicator of where this contamination becomes important, but does not necessarily correspond to the edge of the system. In Figs 20 and 21 we mimic the observational analyses recently applied to the globular clusters NGC Zoccali et al. The solid line in Fig. The points indicate what x would be measured to be at different points along the orbit and at different positions in the cluster if viewed from the centre of the Galaxy — at the centre of the cluster filled squares , at the core radius open squares and at the half-light radius filled hexagons.

Each of these points is calculated using several thousand stars. In Fig. The dashed lines show the final analysis for the same model but for a simulation that included relaxation effects.

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Mass function index x as a function of time for all stars bound to model 0a, p3 solid line , and as measured at various points in the cluster. Initial solid lines and final mass and luminosity functions for model 0a, p3 simulated with dashed lines and without dotted lines diffusion effects. Suppose we have estimated the fractional mass-loss rate from a cluster using its observed population of extra-tidal stars. We test this idea with our simulations by using the methods outlined in Section 4. We then simply estimate the IMF solid squares to be. The mass functions have been normalized to the IMF of each model represented by solid horizontal lines and multiplied by arbitrary constants.

The solid squares show the IMF reconstructed from the estimated mass-loss rate, global and local mass functions. The open squares show the result of the same reconstruction, but using the outermost stars to find the local PDMF. The open squares show the result if the PDMF of the outermost stars is used. The first approximation can be addressed if the luminosity function of the cluster is known as a function of radius, as is the case in our simulations — Fig. Despite these uncertainties, this method provides a new approach — directly based on observations rather than using complex dynamical models — to exploring the question of whether the IMF in globular clusters was universal or environment-dependent.

All these quantities are shown in Table 4. They do not correlate well with previous calculations that attempted to determine the robustness of each object from the ratio either of the expected to observed tidal radius where the former was calculated given a dynamical estimate of the mass of the satellite or of the external tidal field to internal field see IH Nevertheless, these mass-loss rates clearly indicate that the satellite system of the Milky Way could easily be diminished by several members in the next 10 Gyr, which in turn suggests that there may have been several more satellites of the Milky Way in the past.

If these dSph galaxies do indeed have such large mass-loss rates, why have we not detected tidal streamers?

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In the cases of Ursa Minor and Sculptor this question was addressed by Johnston , using a semi-analytic technique. She found that, if each galaxy was losing mass at the rate of 10 per cent per Gyr, the local number count densities along the streamers i. Hence, even increasing these mass-loss rates to the tabulated values would not make the streamers striking features in the sky. Finally, we should qualify this discussion with a cautionary note. If the implied dark matter halo of a dSph galaxy is much more spatially extended than the visible matter this could affect the appearance of tidal debris see Moore and hence the validity of our simple estimates for mass loss.

Table 5 presents the results of mass-loss estimation for the 12 globular clusters analysed by G95 using their tables 3—14, kindly made available to us by C. In the other cases, only the latter estimate is made as an upper limit on the mass-loss rate. The estimated limits for the mass-loss rates range from a few to over per cent in the next Gyr, again implying that the Galaxy's globular cluster system will evolve substantially in the next Hubble time.

This provides observational support for the many purely theoretical studies that have reached the same conclusion using semi-analytic models Aguilar et al. However, the upper limit we calculate is only in direct contradiction with the theoretical calculation in the cases of NGC and However, the original profiles were made either using star counts directly, or with additional photometry to subtract off some of the background.

We have also, for the sake of simplicity, restricted our discussion to annularly averaged surface densities. Clearly, more information is contained in two-dimensional surface density maps see e. However, these will be more sensitive to the orbital phase and the mass-loss history of the satellite and would require more detailed analytic modelling to interpret. The method proposed in Section 4. Of course it is non-trivial to find the global PDMF, the local mass function in the exterior, and to detect extra-tidal stars around a cluster.

However, there are currently two examples in the literature where this has already been done — M15 and M55 see G95; Piotto et al. In the case of M15, Piotto et al. Unfortunately, in the absence of mass segregation our method would find no evolution of the IMF, because it does not model differential mass loss resulting from relaxation effects, but only steady stripping of the most weakly bound stars.

As Piotto et al. In the case of M55, Zaggia et al. Despite these problems, these studies show that it is currently feasible to design future observations that can address these issues. Observations of extra-tidal stars and enhanced velocity dispersions in the outskirts of clusters can be attributed to tidally stripped material.

The star counts become dominated by unbound stars at the point where the slope of the surface density profile changes or the dispersion reaches a minimum. This should not be identified with the tidal radius of the cluster, because the edge of the bound population can still lie significantly beyond this radius. The mass-loss rate from a Galactic satellite can be directly estimated from the population of extratidal stars within a few of its tidal radii, the orbital time period and our line of sight. Using current observations we calculated the mass-loss rate from dwarf spheroidal satellites and globular clusters and found that both systems will undergo significant evolution in the next Hubble time.

Future observations should be able to place much stronger limits on these destruction rates. Our review of the literature suggests that it is currently observationally feasible to carry out such a programme.

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We thank Mike Irwin and Carl Grillmair for providing data used for our analysis, and Tad Pryor for his careful and valuable referee's report. Aguilar L. Hut P. Ostriker J. Bahcall J. Soneira R. Bergbusch P. VandenBerg D. Binney J. Tremaine S. Princeton Univ. Press , Princeton NJ. Google Scholar. Google Preview. Capaccioli M. Ortolani S. Piotto G. Stiavelli M.