The accretion history of the Milky Way – I. How it shapes globular clusters and dwarf galaxies Free

ABSTRACT

1 INTRODUCTION

The stellar system content of the Milky Way (MW) halo has been used to determine its merger history, the understanding of which has been considerably improved with the orbital constraints derived from the all-sky Gaia EDR3 observations. Since the dynamical probes of the MW gas disc are limited by 20–25 kpc in radial distance, the halo inhabitants are also used as dynamical probes to constraint the mass of our Galaxy at large radii, and ultimately, its total mass.

However, which type of orbital equilibrium with the MW potential has been reached by the halo stellar systems? Are their orbits ruled by the Jeans equation meaning that they can be used to probe the MW mass? Are they at pseudo equilibrium as, e.g. regularly affected by tidal shocks such as globular clusters (GCs) when passing near the inner bulge, or when crossing the disc (Aguilar, Hut & Ostriker 1988; Gnedin & Ostriker 1999)? Alternatively, could some of them be near their first pericentric passage having recently lost their gas and being mostly out of equilibrium (Yang et al. 2014)? Comparative studies of their orbits could bring some answers to these questions.

Thanks to the Gaia astrometric mission, considerable efforts have been made to reveal the history of the ancient merger events having affected the Milky Way (Kruijssen et al. 2019, 2020; Massari, Koppelman & Helmi 2019; Malhan et al. 2022). This has generated a large variety of events often nicknamed on the basis of various mythologies. Perhaps, the most achieved effort to date has been made by Malhan et al. (2022) who used specific search tools that compares angular momenta, total energy of 170 globular clusters, 41 stellar streams, and 46 dwarf galaxies, providing some homogeneity in characterizing previous events. Apart from Sagittarius (Sgr) and Willman I, their classification of ancient mergers does not include dwarf galaxies, which are likely newcomers due to their considerably higher energy and angular momentum (Hammer et al. 2021).

Section 2 presents the data from Gaia EDR3, and indicates how they have been analysed to produce orbital quantities, and it is accompanied by Appendix  C containing the result for GCs. Section 3 reveals the strong correlation between intrinsic (r_half) and orbital (R_peri) parameters, and provides an interpretation of its origin. Section 4 suggests that the epoch of infall of stellar systems in the MW halo can be approximated by the total energy, which is discussed and explained in Section 5.1 and implies that dwarfs are newcomers. In Section 5, we also show that stellar systems (GCs and dwarfs) with large eccentricity are actually affected by MW tides, and then how dwarfs have different nature depending on whether they belong to the Vast Polar structure or not.

2 DATA AND DERIVED ORBITAL PARAMETERS

The goal of this paper is to investigate whether the intrinsic properties of MW halo inhabitants are consistent with self-equilibrium or determined by their orbital properties. Despite the importance of previous theoretical investigations, few has been done on the observational side, and this paper aims at filling this gap.

Here we consider all stellar systems as a single class aiming not to distinguish between globular clusters and dwarfs. Our motivation is two-fold: first, these systems have a similar range in stellar mass, and second, as pointed out by Marchi-Lasch et al. (2019), there are similarities between structural properties of ultra faint dwarfs and low-surface brightness GCs. This leads us to distinguish three different populations, the high-surface brightness (HSB-GCs, with log (SB/L_⊙pc⁻²) > 2), the low-surface brightness (LSB-GCs, with log (SB/L_⊙ pc⁻²) < 2) globular clusters (see Fig. 1), and the dwarfs for which the surface brightness (SB) is similar or smaller than that of LSB-GCs. The classification by surface brightness is possibly more restrictive than that of Marchi-Lasch et al. (2019) in selecting LSB-GCs. It is also much more accurate, because it comes from the work of Baumgardt, Sollima & Hilker (2020) who significantly improve the determination of absolute luminosity, providing corrections that can be as high as two magnitudes in V-band. To do so, they counted individual stars using HST photometry, selecting them from their proper motion, colour-magnitude diagram, and radial velocity.

Figure 1.

Distribution of globular cluster surface brightness in logarithmic scale. Surface-brightness in the V-band have been averaged within the half-light radius. The 127 HSB-GCs (log (SB/|${\rm L}_{\odot}\,{\rm pc}^{-2}$|⁠) > 2) distribute almost along a gaussian with a peak at log (SB/|${\rm L}_{\odot}\,{\rm pc}^{-2}$|⁠) = 3.3. The 29 LSB-GCs include the GCs represent the low-surface brightness tail of the GC distribution, and they are on average 315 times fainter than HSB-GCs.

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In the following, we have used the data¹ for 156 GCs from Baumgardt (2017), Baumgardt & Hilker (2018), Baumgardt et al. (2020), Baumgardt & Vasiliev (2021), Sollima & Baumgardt (2017), and their proper motions from Gaia EDR3 (Vasiliev & Baumgardt 2021). Tables C1 and C2 in Appendix  C display the resulting orbital parameters (velocities and orbital radii), and their error bars for a MW model following Eilers et al. (2019, see also more descriptions in Jiao et al. 2021). For dwarfs, we are using the data from Gaia EDR3 (Li et al. 2021) using the same MW model, and same prescriptions for deriving parameters. The Eilers et al. (2019) model of the MW rotation curve includes a bulge, a thick and a thin disc following Pouliasis, Matteo & Haywood (2017). The halo is represented by a NFW model (Navarro, Frenk & White 1996) that requires a cut-off radius fixed at R_vir = 189 kpc to avoid infinite mass; this reduces the effect of the mass exterior to a given radius in the potential calculation by +4|$\pi$|Gρ₀|$R_{\rm s}^{2}$|/(1+R_vir/R_s) after combining equations (2.28) and (2.64) of Binney & Tremaine (2008), where ρ₀ = 0.0106 |${\rm M_{\odot } \, pc^{-3}}$| is the central density of the halo and R_s = 14.8 kpc is the halo scale radius.

3 IS THERE A CORRELATION BETWEEN INTRINSIC AND ORBITAL PARAMETERS?

3.1 Correlation between half-light radius and pericentre

Pioneering studies of van den Bergh (1994), van den Bergh (2011), van den Bergh (2012) revealed the existence of a correlation between GC half-light radii and their galactocentric distances, R_GC. Fig. 2 shows that it is still present despite significant progresses made in estimating GC distances, as well as their photometric parameters (Baumgardt 2017; Sollima & Baumgardt 2017; Baumgardt & Hilker 2018; Baumgardt et al. 2020; Baumgardt & Vasiliev 2021). For the 156 GCs listed in Table C1, it leads to a correlation² with ρ = 0.63, and the only difference with van den Bergh (2011) is that the slope of the relation is slightly less steep (0.59 instead of 0.66).

Figure 2.

Half-light radius versus their galactocentric distances (R_GC) for HSB-GCs (crosses) and LSB-GCs (filled circles). Red symbols identify GCs with tidal tails, which has been based on the compilation made by Zhang, Mackey & Da Costa 2022.

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Such a correlation prompted van den Bergh (2011) to suggest that core-collapsed clusters have the tendency to be near the Galactic Centre. This is not unexpected from the theoretical side, since the inner GCs suffer stronger tidal forces from the MW. Within such a context, the most energetic GC stars may leave the system after gaining energy from tidal shocks, then lowering the system binding energy, finally leading to an adiabatic contraction when leaving the pericentre.

Since the half-light radius is a fundamental structural parameter, we investigate its dependency onto a pure orbital parameter specifically the pericentre, R_peri. This has been firstly investigated on the basis of Gaia DR2 data by de Boer et al. (2019), but the error bars on pericentres were too large for providing robust conclusions. Using Gaia EDR3 data, we find a stronger correlation when replacing the galactocentric distance by the pericentre. Fig. 3 reveals that the half-light radius of 156 GCs correlates well with their orbital pericentres, with r_half following |$R_{\rm peri}^{1/3}$| with ρ = 0.70, t = 12, with an extremely low probability that it occurs by chance, P = 1.3 × 10⁻²³.

Figure 3.

Half-light radius versus their orbital pericentre (R_peri) for HSB-GCs (crosses), LSB-GCs (filled circles), and dwarf galaxies (triangles). The dotted line (slope = 0.33) indicates the correlation for the 156 GCs, which are shown in the top panel together with 30 dwarfs, while in the bottom panel only the 77 GCs and 12 dwarfs with orbital eccentricity larger than 0.6. Few stellar systems very offset from the correlation and suspected to be strongly tidally disrupted are labelled.

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• The correlation becomes even more significant (ρ = 0.79 and P = 9 × 10⁻¹⁸) for half of the sample, i.e. the 77 GCs with high eccentricity (ecc > 0.6, see bottom panel of Fig. 3);

• The above is true even after accounting for the few GCs that are fully off from the correlation, especially Pal 14, Pal 15, and Pal 4, and which are all with very eccentric orbits, while two of them possess tidal tails (see red dots);

• Dwarf galaxies from Li et al. (2021) show a much steeper slope but with a less tight correlation (ρ = 0.55, P = 2 × 10⁻⁴ for 37 dwarfs).

The significant correlation shown in Fig. 3 suggests that MW tides could be at work, and that they are even more efficient for GCs with eccentric orbits. The strongest tidal shocks on GCs are generated as the cluster crosses the Galactic disc or passes through pericentre on a highly eccentric orbit (Binney & Tremaine 2008). However, one may wonder whether the observed correlation could be a hard-to-disentangle combination of star formation processes, and subsequent dynamical evolution in the hierarchical build-up of the Milky Way halo. Former studies have indicated how the MW may have filtered the different populations of stellar systems to establish the present-day GC population (Fall & Rees 1977, 1985; Gnedin & Ostriker 1997). In particular, Fall & Rees (1985, see their fig. 4) predicted that the combination of tides and thermal instabilities may lead to a decreasing density with the galactocentric distance, which seems quite consistent with the (r_half, R_GC) correlation shown in Fig. 2, and then with the (r_half, R_peri) correlation shown in Fig. 3.

A major drawback with this possibility is that the Fall & Rees (1985) scenario assumed a monolithic collapse for the initial formation of the MW. In the recent literature the favored formation scenario for spiral galaxies is through the hierarchical scenario, i.e. mostly through gas-rich major mergers (Hammer et al. 2005; Yang et al. 2008), which are essential in providing enough angular momentum into Galactic discs (Hammer et al. 2009; Hopkins et al. 2009, 2010; Stewart et al. 2009). This also applies to the MW, as evidenced by the finding of GSE imprints in the Energy-Angular momentum diagram (see Malhan et al. 2022 for imprints related to GCs Belokurov et al. 2018; Haywood et al. 2018). If GCs were formed initially as proposed by Fall & Rees (1977) and Fall & Rees (1985), their density-galactocentric distance relation would be rapidly diluted after one or more major merger.

3.2 Is the (r_half, R_peri) correlation due to MW tides?

Because r_half is much more correlated to R_peri than to R_GC, one may conclude that (1) the correlation shown in Fig. 2 is generated by that in Fig. 3, and (2) it is a strong indication that GC structural parameters are impacted by pericentre passages. Simulations are necessary to go beyond that understanding, and we have used those made by Martinez-Medina et al. (2022), who tested how the GC half-light radii with different concentration parameter values can be affected by pericentre passages. Interestingly, they found that half-light radii of concentrated GCs have shrunk by ∼3 per cent after one pericentre passage, while conversely, the least concentrated GCs show an expansion of their half-light radii by ∼7 per cent. In the following, we investigate a very simple model based on the Martinez-Medina et al. (2022)’s simulations, and accordingly to the latter, we have considered equal the impact of tidal shocks due to a point mass and to disc passages.

We have verified from the Harris (2010) compilation of 144 GCs that HSB-GCs have an average concentration parameter of W₀ = 7.2, which is quite similar to that of compact clusters chosen by Martinez-Medina et al. (2022, W₀ = 8). We have further assumed a linear interpolation of the half-light radius evolution with pericentric passage (see details in Appendix  A), from W₀ = 8 GCs (Δr_half/r_half = −3 per cent) to W₀ = 2 GCs (Δr_half/r_half = +7 per cent). To test how tidal shocks may explain the (r_half, R_peri) correlation, we have adopted a single initial half-light radius for all GCs to verify how it evolves in Fig. 4. The latter shows that the correlation is mostly retrieved including its significance (ρ = 0.76) and its slope (0.39). It indicates that the modest loss of half-light radius at each pericentre is compensated by the numerous pericentre passages (up to 100 for a 2 Gyr duration) experienced by the closest GCs to the centre.

Figure 4.

Predicted half-light radius (in parsec) versus orbital pericentre (R_peri, in kpc) for HSB-GCs (crosses) and LSB-GCs (filled circles) after 2 Gyr evolution of GCs that were assumed having an initial half-light radius of 5 pc. The dotted line (slope = 0.33) indicates the correlation for the 156 GCs defined in Fig. 3. The inset shows the histograms of present-day GC r_half at present (black) and 2 Gyr ago (red).

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The evolution of GC half-light radii is also illustrated in the inset of Fig. 4. Assuming the present-day GC r_half and an evolution driven by tidal shocks, one finds that the peak of the distribution evolves to r_half = 2.5 pc nowadays from 3.5 pc 2 Gyr ago. One may also notice that the correlation is better retrieved after considering eccentric GCs. When including the less eccentric GCs (ecc < 0.6), the correlation significance and the slope drop to 0.54 and 0.22, respectively. We have also tried larger look back times, such as e.g. 6 instead of 2 Gyr ago. It has led to larger slopes, which are due to extremely small theoretical GCs due to their extremely large number of pericentre passages. The above results show the limits of our exercise: the most eccentric GCs are expected to be the most affected in the last 2 Gyr, while extending our test to much larger elapsed periods leads to unrealistic values for r_half since the pericentre is likely evolving with time.

We also notice that the points with log (R_peri) > 0.5 in Fig. 4 do not follow the correlation, and that the corresponding GCs have essentially kept their initial r_half value. They are mostly LSB-GCs, and we find that they have experienced only a single pericentre passage during the 2 Gyr elapsed time, i.e. they have not yet been reshaped by tides. Given that actual LSB-GC (as well as dwarf) half-light radii are mostly above the r_half ∼ |$R_{\rm peri}^{1/3}$| line in Fig. 3, this indicates that the their structural parameters are mostly fixed by their star formation processes, or alternatively by another mechanism (see section 5). Our model is surely over-simplified, and a more sophisticated model is necessary, though beyond the scope of the present paper. Such a more detailed study should account for the full evolution of GCs, including their orbital history, as well as for the MW mass evolution, and this would very useful for disentangling the different effects on half-mass radii from tides and from star formation. On the other hand, it is unlikely that the present-day distribution of GC half-light radii can be attributed to their initial distribution function, while accounting for existing simulations (Martinez-Medina et al. 2022) provides sufficient evidence that GCs are shaped by MW tides.

Figs 3 and 4 show that at present-day, eccentric GCs with small pericentres are the most impacted by tides. Appendix  A details how the correlation between GC surface brightness and concentration parameter has been established. We have also determined that the few GCs (especially Pal 14, Pal 15, and Pal 4) that escape the correlation shown in Fig. 3 are heavily tidally stripped or shocked at present-time (see Appendix  B).

4 CALIBRATING THE EPOCH OF GC EMERGENCES IN THE MW HALO

The correlation between an intrinsic quantity, the half-light radius, with an orbital parameter, the pericentre, is suggestive of the important role played by the MW potential in shaping GCs. However, to compare its impact on the various inhabitants of the halo, we need to calibrate their relative epochs of infall, or alternatively, the time when they have emerged into the MW halo. This can be approximated from numerical simulations (Boylan-Kolchin et al. 2013), which show that the total energy is decreasing with infall look-back time, similarly to the onion skin model initiated by Gott (1975) for explaining the outer density profile of elliptical galaxies. This does not come at a surprise, since the more orbits made by a stellar system in the halo, the more dissipated its energy would be,³ e.g. by encountering other subsystems including giant molecular clouds, or at passages through the MW disc, or near the bulge.

We estimate the epochs of infall with the help of the identifications of substructures in the Galactic halo made by Malhan et al. (2022), which have been assumed to be linked to past merger events. By studying the associations of these events with GCs, Malhan et al. (2022) have been able to identify substructures associated to the bulge to the disc to a novel substructure called Pontus, and also to Gaia Sausage Enceladus (GSE), LMS1-Wukong, and Sgr. To this we have added the Kraken event identified by Kruijssen et al. (2020), because it could be the earliest merger that can be identified in the MW halo. We also notice that Kruijssen et al. (2020) associated to GSE some of the GCs associated to Pontus by Malhan et al. (2022).

Fig. 5 presents the relation between the total energy and the angular momentum for eccentric (bottom) and non-eccentric (top) stellar system. First, it is very similar to that between the total energy and the pericentre, simply because the angular momentum correlates extremely well with the pericentre. Second, the total energy appears to be well associated to the time occurrence of the merger events. In most halo substructures, GCs show a narrow range in energy suggesting that the total energy is a proxy of the infalling time.

Figure 5.

Total energy versus the angular momentum, h = R_GC ×V_tan in logarithmic scale, for HSB-GCs (crosses), LSB-GCs (filled circles), and dwarfs (triangles). Identified structures by Malhan et al. 2022 and Kruijssen et al. 2020 are added by different colours indicated in the figure. VPOS dwarfs and GCs are also shown in blue, while the few dwarfs associated to the LMC are shown with cyan. The top panel shows 79 GCs and 20 dwarfs with eccentricities smaller than 0.6, while the bottom panel shows the 77 GCs and 15 dwarfs with eccentricities larger than 0.6. The dotted line shows the limit that cannot be passed by any orbits, as it is fixed for V_rad = 0 and a circular orbit (lowest possible energy for a given angular momentum). Note that few dwarfs having a positive energy are not represented, including Leo I in the VPOS, Carina II and III and Phoenix II, all related to the LMC, and Hydra II and Leo V.

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If correct, it would assume a very early epoch for the bulge GCs as indicated by their ages (Kruijssen et al. 2019). This is indeed corroborated by the fact that most bulge GCs have their orbits circularized, as indicated by the comparison between the two panels of Fig. 5 showing that most bulge GCs have low-eccentricity orbits. We have followed Kruijssen et al. (2020) by indicating Kraken as being the most ancient detected merger in the MW halo, this followed by Pontus, and that by GSE with an approximated age of 8–10 Gyrs, and for which ages have been derived from that of their associated GCs (Kruijssen et al. 2019). The methodology of Kruijssen et al. (2020) is to compare the observed GC age distribution from Kruijssen et al. (2019) to that derived from cosmological simulations. They associate a merger with an epoch defined as ‘as the moment at which SUBFIND can no longer find a bound subhalo, and the subhalo is therefore considered to have merged into the halo of the central halo’. In principle, this corresponds to the time when the merger is close to completion, though it appears quite uncertain since, in their subsequent work, Kruijssen et al. (2020) assumed instead that it corresponds to the entrance of the system into the halo. On the other hand, during merging encounters, star formation usually reaches its peak at the time of merger (see their fig. 4 Puech et al. 2012) rather than at first pericentre (Di Matteo et al. 2008). Such star formation at merging is likely accompanied with the formation of proto-GCs.

However, the above uncertainty has small consequences in the case of major merger events such as Kraken or GSE, because due to dynamical friction, the time between halo entrance and final merger is relatively small and similar to their quoted uncertainties, i.e. ΔT ∼ 2 Gyr. The situation for Sgr is very different, because (1) the merger is far from being completed, and (2), it is likely a minor merger. The Sgr merger event has been considerably studied and modelled, since conversely to former merger events, the Sgr core, associated GCs, and stream are still easily recognizable.⁴ It is likely that many Sgr GCs have taken their birth into the Sgr surrounding material, i.e. before its entrance into the MW halo. Because we are interested in the location of Sgr GCs into the (E_tot, h) diagram, their total energy should be quite similar to that at the Sgr infall time, or its first entrance into the halo.

To estimate the Sgr infall time, we have used the most accurate analyses of the star formation history (SFH) of Sgr. We further assume that before entering the MW halo, the Sgr progenitor should have been considerably gas-rich as suggested by the distribution of gas-rich dwarfs in the Local Group (Grcevich & Putman 2009). One can then estimate its first infall and gas depletion times using the Sgr SFH. Sgr hosts populations of different ages as seen in deep HST photometry (Siegel et al. 2007) that clearly displays several turn-offs in their colour-magnitude diagrams. From the photometry and detailed abundances for a sample of giant stars within 9′ from the centre of M54, Mucciarelli et al. (2017) found a double-peaked metallicity distribution function, with peaks at −1.5 and −0.5. They also claimed that the chemical evolution of Sgr implies a strong gas loss occurring between 2.5 and 7.5 Gyr ago, which they presume occurring at the first passage at pericentre of Sgr following its infall into the Milky Way. More recently, Alfaro-Cuello et al. (2019) using deep, albeit low resolution, spectroscopic data that covers only the inner region of M 54 (2|${_{.}^{\prime}}$|⁠) radius, to be compared to 7|${_{.}^{\prime}}$|4 of the tidal radius of the cluster) claimed the detection of three different populations a young one (2.3 Gyr old), an old population (12 Gyr old), and an intermediate-age population (4.3 Gyr old) that is more widespread. The young population can coincide with the last event of gas depletion that probably occurs in the very central Sgr region, while the intermediate-age population is suggestive of a longer event during which ram-pressure progressively removes the gas from Sgr, while star formation is sustained by gas pressurization from ∼6 to 3 Gyr ago (see fig. 7 of Alfaro-Cuello et al. 2019).

Using SDSS photometry and spectroscopy de Boer, Belokurov & Koposov (2015) found that the SFH of the Sgr Stream and found that the SFH increases 6 Gyr ago with a peak 5 Gyr ago and then a rapid decline 4 Gyr ago (see their fig. 6). Interestingly, the most recent models of Sgr (Vasiliev, Belokurov & Erkal 2021; Wang et al. 2022a) assumed an infall from 3 to 4.7 Gyr ago. Finally an infall more recent than 6 Gyr ago is necessary to account for the age of Whiting 1 (6.5 Gyr old). Putting all the available evidence together we conclude that the first infall of Sgr must have occurred between 4 and 6 Gyr ago.

Besides GCs, Fig. 5 also shows how other dwarfs than Sgr are distributed in the (E_tot, h) plane (see also Hammer et al. 2021). Most of the dwarfs belonging to the Vast Polar Structure perpendicular to the disc (VPOS, see Pawlowski et al. 2014; Pawlowski 2018; Li et al. 2021) show small eccentricities and follow the line with V_rad = 0 (or V_3D = V_rot), i.e. a minimal energy for a given angular momentum (see blue triangles). This has been attributed by Hammer et al. (2021) to the fact that most of them are newcomers and have recently lost their gas due to ram-pressure, which affects more their radial velocities, leading to the observed anisotropic distribution of velocities (Cautun & Frenk 2017; Li et al. 2021). However, there are other dwarfs showing high-energy and high-orbital eccentricity, including dwarfs associated to the LMC on the basis of their relative Gaia motions (Erkal & Belokurov 2020; Patel et al. 2020). Because of that, we have considered two substructures of the VPOS, one still called the VPOS (blue triangles), the other called LMC for associated dwarfs to this galaxy (cyan triangles).⁵ Dwarfs that are not in the VPOS are shown in Fig. 5 as black triangles, and many of them have significant radial velocities, as indicated by their offset from the V_3D = V_rot line.

5 DISCUSSION

5.1 Relating total energy to the epoch when stellar systems entered the MW halo

The estimated lookback times of entry into the MW halo of different populations of GCs of Kruijssen et al. (2019) can be compared to their orbital binding energies (see Fig. 5 and section 4). This is shown in Fig. 6, which has been built on the basis of the narrow total energy range shown by each identified structures in the MW halo.

Two simple mechanisms could explain such a relation for which stellar systems coming the earlier in the MW halo have smaller energy:

• The MW mass increase with time, which imposes a low energy for early-coming stellar systems;

• Early-coming stellar systems have experienced several orbits, and then have large chances to encounter other systems to lose further their energy.

Fig. 5 shows the distribution of previous mergers in the MW, illustrating also that dwarfs possess higher total energy, and angular momentum than Sgr and that of most GCs. The time elapsed between GSE and Sgr events is ∼4 Gyr, corresponding to a difference of energy of 0.25 dex (a factor ∼1.8). Extrapolating the same energy difference towards higher energies would lead to an energy value right at the average value for dwarfs as indicated in Fig. 6. Even if the total energy cannot be taken as a very precise clock, it suggests that most dwarfs are new or recent (<3 Gyr) comers into the MW halo (see also Hammer et al. 2021). Another clue for that is the very large scatter presented by dwarfs in the (E_tot, h) plane. This is because newcomers may have different origins, and then possess different initial energies and angular momenta.

Figure 6.

Lookback time of stellar system entry in the Milky Way halo as a function of current binding energy for different families of globular clusters, and for the 25 dwarf galaxies that do not belong to the low energetic Sgr system (excluding then Sgr, Segue II, Tucana III, and Willman I) or to the high-energetic LMC system (Carina II, Carina III, Horologium I, Hydrus I, Phoenix II, and Reticulum II). The blue line is a quick linear fit. A naïve interpretation is that dwarf satellites with |$\log (-E_{\rm tot}/[\rm km^2\, s^{-2}]) < 4.17$| are on their initial approach, a value very close to the logarithm of the average energy (4.14) of 25 dwarfs (see text), whose scatter provides an upper limit of −E_tot = 4.34. The latter combined with the linear fit suggests a lookback time of halo entry smaller than 3 Gyr.

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This contradicts several studies making the assumption that many dwarfs are long-lived satellites of the MW for nearly a Hubble time. Such an assumption neglects, e.g. the impact of the MW mass evolution. For example, in ΛCDM cosmology, the mass growth of the Milky Way since the epoch when bulge GCs were accreted (z = 4.5) is roughly a factor 50 (from fig. 5 of van den Bosch 2002). With such a small MW mass, only bulge GCs with the largest binding energies could be bound, while it is unlikely that low-binding energy dwarfs that lie at large distances can be captured at the same time.

5.2 The evolution of half-light radius with total energy and look-back time

The top panel of Fig. 7 shows how the half-light radius of the 156 GCs correlates with the total energy, i.e. smaller the energy, smaller the half-light radius. From the slope of the relation one finds that r_half follows (−E_tot)^−1.0, with a correlation of ρ = −0.69 associated to an extremely low probability that it occurs by chance, P = 1.2 × 10⁻²³. This just indicates that bulge GCs (average r_half = 2.5 pc) are smaller than GSE GCs (average r_half = 3.6 pc), which are smaller than Sgr GCs (average r_half = 9.7 pc). Since the total energy gives a proxy of the look-back time (see Figs 5 and 6), it implies that the half-light radius of GCs is decreasing with the time spent into the MW halo.

Figure 7.

Total energy versus half-light radius in logarithmic scale, for HSB-GCs (crosses), LSB-GCs (full dots), and dwarfs (triangles). Identified structures by Malhan et al. 2022 and Kruijssen et al. 2020 are represented by the colour codes used in Fig. 5. VPOS dwarfs and GCs are also shown in blue colour, while the few dwarfs associated to the LMC are shown with cyan colour. The top panel shows 156 GCs and 35 dwarfs (those with ecc < 0.6 are labelled), while the bottom panel shows the 77 GCs and 15 (labelled) dwarfs with eccentricities larger than 0.6. The dotted line shows the fit of the GC points in the top panel, and of both GCs and dwarfs in the bottom panel, and the significance of the correlation is given on the bottom-left of each panel.

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The correlation between half-light radius and total energy followed by the GCs is not similar for dwarfs, which are far much more scattered in the (E_tot, r_half) plane. In particular, many dwarfs (and all the classical dwarfs) lie well above the correlation. However, the bottom panel of Fig. 7 illustrates a different behaviour for GCs and dwarfs with eccentric orbits (ecc > 0.6). First, eccentric GCs show a more robust correlation between their half-light radii and their total energies (77 GCs, ρ = −0.82, P = 1.7 × 10⁻²⁹) also with a slope of −1.0. Second, dwarfs appear to fall well within this correlation, thought with a larger scatter. It suggests that all eccentric stellar systems in the MW halo follow the same (E_tot, r_half) relation. The average half-light radii of eccentric GCs associated to the Bulge, to GSE, to Sgr, and to dwarfs are 1.95, 3.6, 9.8, and 83.5 pc, respectively, implying a decrease with the time elapsed into the halo. Such a decrease has been theoretically investigated for GCs (Aguilar et al. 1988; Gnedin & Ostriker 1999) since eccentric systems passing near their pericentres are likely affected by MW tidal shocks. This increases the internal energy of their stars, leading the least bound to be expelled. Then, after GCs have lost mass, they contract adiabatically when leaving the pericentre towards the apocentre.

This in turn enables the formation of tidal structures formed from stars tidally ejected from the cluster. This is well illustrated by the occurrence of tidal tails in GCs from Zhang et al. (2022, see their Table 3): only 9 per cent (7 among 79) of ecc < 0.6 GCs possess tidal tails, which contrasts with 22 per cent (17 among 77) of eccentric GCs (ecc > 0.6). Appendix  B shows that three GCs (Pal 4, Pal 14, and Pal 15) are tidally disrupted since they have their tidal radii smaller than r_half. Two of them possess strong tidal tails: Pal 14 (Sollima et al. 2011) and Pal 15 (Myeong et al. 2017). Despite the observational efforts (Ibata et al. 2021; Zhang et al. 2022), there is still a possibility that some tails have not been discovered yet, and we would like to suggest deep observations of Pal 4.

The most tidally disrupted GCs by the MW that are labelled in Fig. 3 are LSB-GCs. LSB-GC have sufficiently small densities to be strongly affected by MW tides, despite their large pericentres (average R_peri: 13.8 kpc). The latter value is six times larger than that for HSB-GCs, but this is largely compensated by the 300 times lower densities of LSB-GC compared to HSB-GCs. LSB-GCs have been suspected by Marchi-Lasch et al. (2019) to be recent comers into the MW halo, and we confirm this (see Figs 5 and 7), simply because their average total energy (E_tot = −7.9 × 10⁴|$\rm km^{2} s^{-2}$|⁠) is significantly higher than that of HSB-GCs (E_tot = −15.7 × 10⁴|$\rm km^{2} s^{-2}$|⁠).

5.3 Dichotomy between eccentric and non-eccentric dwarfs

Almost all VPOS dwarfs have low eccentricity, while most other dwarfs (including the five LMC related dwarfs) have eccentric orbits. The latter are almost all ultra-faint dwarfs, except Sextans and Leo I.⁶ Bottom panel of Fig. 7 shows that they follow the (r_half, E_tot) correlation made by eccentric GCs, which suggests that they could be affected by MW tides as well, as it was proposed by Hammer et al. (2019), Hammer et al. (2020). This is also supported by the tidally disrupted properties of Tucana III (Drlica-Wagner et al. 2015), and perhaps of Triangulum II (Martin et al. 2016). This may further apply to Hercules, which could be also tidally disrupted (Küpper et al. 2017; Garling et al. 2018). In the near future, we intend to verify whether all eccentric and energetic ultra-faint dwarfs have morphologies and kinematics consistent with being tidally disrupted.

The dichotomy between low-eccentric VPOS dwarfs and other dwarfs with high eccentricity, including those apparently linked to the LMC is also reflected in Fig. 5. VPOS dwarfs lie near the V_rad = 0 line in Fig. 5, while almost all other dwarfs have still a significant radial velocity and lie much further from this line. It suggests that dwarfs could have been shaped by different combinations of tides and ram pressure effects:

• VPOS dwarfs with low eccentricities include almost all classical dwarfs (but Sextans and Leo I), and one may conjecture that it takes more time for ram-pressure to remove gas from more massive dwarfs; if correct, it means that ram pressure is particularly efficient to slow them down especially along their radial motion, explaining the observed velocity asymmetry for dwarfs (Cautun & Frenk 2017; Hammer et al. 2021), as it comes from especially from massive dwarfs of the VPOS;

• Non-VPOS (and LMC-associated) dwarfs are mostly eccentric and ultra-faint dwarfs, and their gas could have been almost instantaneously removed by ram pressure, which would have preserve their orbit eccentric, while their half-light radii could be affected by both tides and ram pressure;

• Because of their relatively recent infall, both VPOS and non-VPOS dwarfs may have lost their gas at relatively recent epochs (<3 Gyr), leading to an expansion of their residual stars due to a loss of gravity (Hammer et al. 2019, 2020). Such a mechanism could explain why their half-light radii is much larger than what is expected from the (r_half, R_peri) correlation in Fig. 3.

The above confirms that Gaia EDR3 results are sufficiently robust to avoid specific corrections to dwarf tangential velocities as proposed by Correa Magnus & Vasiliev (2022) and Pace, Erkal & Li (2022), even if this could be necessary for a few extremely faint dwarfs having proper motions with very large error bars.

5.4 Some lessons about the MW mass assembly

Fig. 5 provides useful informations about the origin of each merger having affected the MW. Pontus and GSE GCs are all with high eccentricity, while bulge and disc GCs are mostly made of systems with low eccentricities. This indicates that GSE and Pontus were likely related to a very eccentric infall of their progenitors. Part of the eccentricity of the GSE GC orbits may come from an orbit radialization (Vasiliev, Belokurov & Evans 2022), but this could apply only to the GSE progenitor, and not to the associated GCs, because Vasiliev et al. (2022) found that radialization is not efficient for very large mass ratios, i.e. GC masses are tiny in comparison to that of the MW or of its bulge. Interestingly, the eccentric GSE GCs appear to be more compact than expected from the relation shown in the top panel of Fig. 7, which is consistent with the fact that they are actively tidally shocked.

At increasing energies, one finds that the Sgr system eccentricity is in between that of bulge and GSE GCs while the Sgr eccentricity is 0.66. However, the situation of the Sgr dwarf appears to be rather exceptional: it is very offset from the (r_half, E_tot) correlation of the bottom panel in Fig. 7, while it possesses the most prominent system of tidal tails. Perhaps this is just due to the fact that despite its large eccentric orbit, the initial Sgr was so massive that it took more time for the MW to strip it efficiently, and have it joining the correlation of Fig. 7. The simulations by Wang et al. (2022b) show that after 1.5–2 Gyr from now, almost all the stars in the extended Sgr will be removed, letting only the nuclear GC NGC 6715 that lie on to the correlation. One can also wonder how a dwarf like Triangulum II can share the same location in the (E_tot, r_half) plane as Eridanus, while the luminosity of the latter is 40 times larger. The same applies to Tucana III and Pal 15, which both display tidal tails. If both type of stellar systems are recent comers, this just means that MW tides may have affected similarly systems with different masses, and indeed tidal theory shows little effect on host to satellite mass ratio.

6 CONCLUSIONS

This paper shows that an intrinsic structural parameter such as the half-light radius depends on orbital parameters such as pericentre, orbital eccentricity, angular momentum, and total orbital energy. Orbital parameters have been derived from Gaia EDR3 with an unprecedented accuracy for both GCs (this paper, Appendix  C) and dwarfs (Li et al. 2021). It results that the intrinsic GC sizes tightly depend on the orbital parameters, the smaller they are, the smaller their pericentre, angular momentum, and orbital energy. We have shown that this is likely due to the impact of MW tidal shocks that progressively shrink GC half-light radii.

We have also used the archaeological studies of the former mergers that occurred into the MW (Kruijssen et al. 2020; Malhan et al. 2022). Here, we have been able to characterize the infall time of each event from its total orbital energy. We have shown that the latter can be used as a proxy of the epoch when stellar systems have emerged in the MW halo. This provides a relatively simple interpretation of the properties of MW halo inhabitants:

• Bulge, Kraken, GSE, and Sgr GCs are inhabitants of the MW halo since ∼12–13, 11–12, 8–10, and 4–6 Gyr ago, respectively;

• Most dwarfs are newcomers into the MW halo less than 3 Gyr ago, which is confirmed by their large total energy and angular momentum, as well as by the large scatter of their distribution in most fundamental relations;

• VPOS dwarfs that include the most massive ones have a different behaviour than other dwarfs in both (E_tot, h) and (r_half, E_tot) planes; first they have a minimal energy for their angular momentum meaning small eccentricity orbits, and second, most of their half-light radii are larger than expectations from the strong correlation between r_half and E_tot established for GCs, all properties consistent with a recent lost of their gaseous content due to ram-pressure from the MW halo;

• Many ultra-faint dwarfs unrelated to the VPOS or related to the LMC possess high-orbital energy, suggesting that they came very late in the MW halo. Since their orbits are highly eccentric, and that they share the same dependency between r_half and E_tot than eccentric GCs, they could be affected as well by strong tidal effects. Such a behaviour is not unexpected, because even if late arrivals do not favor strong tidal effects, this could be compensated by their very low surface or volume densities.

This paper is the first one to investigate in detail the relation between structural and orbital parameters of MW inhabitants since the pioneering work of van den Bergh (2011). It also consolidates all the work made by Gaia EDR3 in predicting robust proper motions and tangential velocities, including their error bars. This is true also for most dwarf galaxies, because otherwise we would have not observed the dichotomy presented by VPOS and non-VPOS dwarf properties.

The correlations shown in this paper are made using a MW mass model following Eilers et al. (2019), assuming a total mass of 8.2 × 10¹¹|${\rm M}_{\odot }$|⁠, which has been purposely made for reproducing the MW rotation curve with a NFW halo. An advantage of this choice is that all GCs and almost all dwarfs have negative total orbital energies. One may wonder whether the results depend on this choice. We have investigated the three other models of the MW made by Jiao et al. (2021) that cover the largest possible mass range (from 2.3 to 15 × 10¹¹|${\rm M}_{\odot }$|⁠) able to reproduce the MW rotation curve. Main results of this paper are unaffected, because they depend on the relative positions of the stellar systems along the total energy axis, the latter being a proxy of the infall time.

We conclude that the MW tides and ram-pressure appear to be quite preponderant in shaping the structure of the halo inhabitants, which opens the road for studying their morphologies and their kinematics, to verify whether these properties can be influenced as well by the host, our Galaxy. This work may help the understanding of GC and dwarf origins, as well as testing sophisticated physical models of MW GCs such as that of Reina-Campos et al. (2022). It would be interesting to verify whether they could reproduce the (r_half and R_peri) correlation found in Fig. 3. In principle, if accounting for tidal and ram-pressure effects due to the MW, the intrinsic dynamical evolution of the stellar systems can be traced back with a good accuracy in time, based on their present-day orbital energy.

ACKNOWLEDGEMENTS

We are very grateful to Frederic Arenou and Carine Babusiaux for their advices, namely to use Gaia EDR3 data and their statistics. We warmly thank Holger Baumgardt for his help in using his remarkable work in gathering, and producing very advanced data for globular clusters. We thank the referee for the very useful remarks that have substantially improved the manuscript. We are grateful for the support of the International Research Program Tianguan, which is an agreement between the Centre National de la Recherche Scientifique (CNRS) in France, Nathional Astronomical Observatories of China (NAOC), Institute of High Energy and Particles (IHEP), and the Yunnan University in China. Marcel S. Pawlowski acknowledges funding of a Leibniz-Junior Research Group (project number J94/2020) and a KT Boost Fund by the German Scholars Organization and Klaus Tschira Stiftung.

DATA AVAILABILITY

All necessary data used in this paper are available in the Tables of the Appendix  C and in Li et al. (2021).

Footnotes

REFERENCES

APPENDIX A: RELATING CONCENTRATION PARAMETER AND SURFACE BRIGHTNESS

Figure A1.

The left-hand panel shows the relationship between King profile compactness c and central concentration parameter W₀ based on fig. 4.9 of Binney & Tremaine (2008). The right-hand panel shows the correlation between the logarithm of the averaged surface brightness inside r_half and the King profile compactness c.

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Fig. 4 illustrates how eccentric GCs are affected by tides, following the scheme from Martinez-Medina et al. (2022), and deriving the number of pericentre passages from the orbital parameters given in Appendix C.

APPENDIX B: TIDAL RADIUS AND SEARCH FOR GCS PRESENTLY DOMINATED BY TIDAL EFFECTS

However, the tidal formula of Innanen et al. (1983), as those of King (1962) and Bertin & Varri 2008, add a centrifugal term, which is only meaningful if the GCs where phase locked with the Milky Way (they are insufficiently rigid to be phase locked). Furthermore, their formulae are only relevant for quasi-circular orbits, where the tidal effects are long-lived. For eccentric orbits, the instantaneous tide is short lived and changes direction, and one should instead treat the tide as a shock and orbit average it. Following Mamon (2000) and its refinement by Tollet et al. (2017), a star of a MW satellite should suffer a velocity impulse equal to the tidal acceleration at pericentre times the duration R_peri/V_peri of the pericentric passage. Mamon (2000) showed that eccentric orbits suffer tides that limit the system to a radius, where the mean density is not the mean density of the host at pericentre, but lower by roughly the square of the ratio of pericentric to circular velocity at pericentre. Therefore at fixed pericentre, the more eccentric orbits (with higher pericentric velocities) suffer less from tides. This is contrary to the instantaneous formulae of King (1962), Innanen et al. (1983), and Bertin & Varri (2008), which predict, at given pericentre, a too small tidal radius for eccentric orbits, and are thus only applicable for quasi-circular orbits and not for eccentric orbits.

The few GCs (especially Pal 14, Pal 15, and Pal 4) that escape the correlation shown in Fig. 3 have average density inside r_half close to that of the MW at their pericentres, even after accounting for the corrective factor introduced by Mamon (2000). It may indicate that low-density GCs on high-eccentric GCs are actively shaped by the MW tides, which perturb them sufficiently to move them off the correlation between (r_half, R_peri) shown in Fig. 3.

Because for all GCs, the half-light radius decreases with the pericentre following a tight 1/3 power law relation, it further suggests that for all GCs, their radii could have been settled by MW tides.

APPENDIX C: ORBITAL PARAMETERS OF 156 GCS

Description of the Table C1 content: Column 1 lists the globular cluster name; Column 2–4 is the galactocentric distance, angle with respect to the North Galactic Pole and azimuthal angle; Column 5–7 gives the velocities in three dimensions; Column 8 provides the galactocentric tangential velocity, and Column 9 lists the total velocity in the Galactic rest frame.

Description of the Table C2 content: Column 1 lists the globular cluster name; Column 2 and 3 give the pericentre and the apocentre of the orbit; Column 4 is the eccentricity of the orbit; in Column 5, we provide the probability of the object being unbound. Column 6 gives the orbital phase.