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A&A
Volume 636, April 2020
Article Number A106
Number of page(s) 12
Section Galactic structure, stellar clusters and populations
DOI https://doi.org/10.1051/0004-6361/202037558
Published online 27 April 2020

© ESO 2020

1. Introduction

Within the Λ-cold dark matter (ΛCDM) theoretical framework (White & Rees 1978) Milky Way (MW)-like galaxies are surrounded by several satellite galaxies. The ΛCDM framework supports the hierarchical scenario of structure formation, according to which smaller dark matter (DM) haloes form in the earlier stages of the Universe, and later merge to form higher mass structures, with the central massive host galaxies being the end product of this process (Blumenthal et al. 1984). The satellite galaxies around the MW are the remnants of its past accretors.

A stream of gas, the Magellanic Stream, is observed in the MW environment (Wannier & Wrixon 1972; Mathewson et al. 1974; Nidever et al. 2010). This stream is attributed to the tidal interaction between the Large and Small Magellanic Clouds (LMC and SMC), with a possible contribution of the MW environment to this process (Diaz & Bekki 2011; Besla et al. 2016; D’Onghia & Fox 2016; Wang et al. 2019). Observations of the northern Galactic hemisphere give evidence of stellar overdensities in the shape of two elongated streams (Ibata et al. 1994; Newberg et al. 2007). It is believed that these streams are the debris of a progenitor (identified with the Sagittarius dwarf galaxy) that has been interacting with the MW in the last gigayears, has thereby dissolved in the environment, and has lost matter (Martínez-Delgado et al. 2004). Other observations include the discovery of the Monoceros-Canis Major streams, with a progenitor of estimated mass equal to that of the Sagittarius dwarf, around 109 M⊙ (Newberg et al. 2002).

Observations show that streams from dwarf galaxies are currently being destroyed in the MW environment; signs of the past accretion of satellites can also be detected. From the abundance analysis of Gaia Data Release 2 (DR2, Gaia Collaboration 2018), recent work from Helmi et al. (2018) suggests that most of the inner halo of our Galaxy originated from a past major impact with the so-called Gaia Enceladus progenitor around 10 Gyr ago. Using DR2 photometry data combined with spectroscopic information from other surveys (Wilson et al. 2010; Cui et al. 2012; Kunder et al. 2017; Abolfathi et al. 2018; Marrese et al. 2019), Koppelman et al. (2019) has identified new members of the Helmi stellar stream, originally identified in the solar neighbourhood by Helmi et al. (1999). Combining photometry and metallicity measurements with N-body experiments, they found low-metallicity populations and predicted that the Helmi stream originates from a dwarf galaxy of mass M ∼ 108 M⊙, accreted onto the MW 5 Gyr ago.

Different observational and theoretical studies have focused on the spatial distribution of the MW satellites and of their debris, evidencing the presence of the so-called Vaste Plane of Satellites (VPOS), but also aligned streams of disrupted satellites (Pawlowski et al. 2012). By means of numerical simulations, Santos-Santos et al. (2019) found that the inertia tensor related to the spatial distribution of the bulk of satellites around the MW has a flatness c/a ranging from 0.1 for less than 10 satellites around the MW to 0.2 for more than 25 satellites around the MW. Buck et al. (2016) showed however that even when forming these spatially coherent planes, satellites seem to be kinematically incoherent and these planar structures do not last a long time. Lisanti & Spergel (2012) focused on the DM debris in the Via Lactea II simulation and found evidence for the spatial homogeneity of its distribution. These results seem to be in contrast, in the perspective of understanding the distribution of the MW satellites and of their stellar and DM debris.

When considering the origin of the MW halo, the theoretical work of Pillepich et al. (2015) predicts that 70% of MW stellar content is formed in situ, that the majority of the ex situ stars come from infalling satellites and characterise most of the stellar halo, and that the mass of these accreting satellites is relevant to the formation process of the MW halo. The additional study of Deason et al. (2016) suggests that the main contribution to the MW halo build-up (around 109 M⊙) comes from larger satellites such as the LMC and SMC, whereby the contribution from ultra-faint dwarf galaxies (UFDs, with stellar masses below 105 M⊙) is minimal.

Large numerical investigations such as the Aquarius (Aq; Springel et al. 2008) and Via Lactea II (Diemand et al. 2008) cosmological simulations successfully described the clustering properties of Cold DM in large portions of the Universe (∼100 Mpc) down to the scales of subhaloes and the main MW halo core (∼10−100 pc). These projects made use of state-of-the-art gravitational solvers (GADGET 3 after GADGET 2, and PKDGRAV; see Springel 2005; Stadel 2001), but lacked prescriptions for baryonic physics, which affects the final properties of the galactic haloes (Pontzen & Governato 2012), and in the case of galaxies like the MW influences the process of matter stripping from their satellite galaxies (Garrison-Kimmel et al. 2017). More recently, big simulation projects were dedicated to combine gravity and baryonic physics with refined numerical recipes (see AREPO, Springel 2010) in order to obtain more accurate results in better agreement with observations of galaxies in the Universe (Vogelsberger et al. 2014a,b; Genel et al. 2014; Sijacki et al. 2015) or more specifically with the observational properties of dwarf galaxies (Wetzel et al. 2016, Latte simulation).

The zoom-in simulations Auriga (Grand et al. 2017) and Eris (Guedes et al. 2011) focus on the detailed properties of MW-like objects with a higher resolution (6 × 103 M⊙ for gas particles and 4 × 104 M⊙ for DM in Auriga, and 9.8 × 104 M⊙, 2 × 104 M⊙, and 6 × 103 M⊙ for DM, gas, and stars in Eris). For their study on the environmental effects on MW satellites, Buck et al. (2019) adopted a similar mass resolution, with stellar particles reaching a resolution of 6.7 × 103 M⊙ and DM particles having a resolution of up to ∼105 M⊙. For comparison, in the IllustrisTNG (The Next Generation) project, the typical resolution of particles in TNG50 (its highest resolution simulation box; Pillepich et al. 2019; Nelson et al. 2019) is 8.5 × 104 M⊙ for stars and 4.5 × 105 M⊙ for DM. In other projects like APOSTLE (A Project Of Simulating The Local Environment, Sawala et al. 2016, 2017), which focuses on reproducing the kinematics and dynamics of the Local Group, the best resolutions for stellar and DM particles are 104 M⊙ and 5 × 104 M⊙, respectively.

While keeping in line with the best available resolution of DM particles (M = 3.4 × 103 M⊙ at best), Wang et al. (2015) made an advancement in resolving baryons for their NIHAO (Numerical Investigation of a Hundred Astrophysical Objects) cosmological simulations, with adopted masses as low as M = 6.2 × 102 M⊙ for gas particles. Their high-resolution simulations were used to study the evolution and properties of galaxies in a range of masses going from dwarf DM haloes of mass M ∼ 109 M⊙ to MW-like haloes with masses M ∼ 1012 M⊙. Higher resolutions were also employed in cosmological simulations of isolated dwarf galaxies by Macciò et al. (2017, hereafter M17), using the N-body code GASOLINE (Wadsley et al. 2004) with Smoothed Particle Hydrodynamics (SPH) prescriptions for gas dynamics (Lucy 1977; Gingold & Monaghan 1977). For their dwarf galaxy models, M17 reached resolutions as high as M = 1.2 × 102 M⊙, M = 4 × 10 M⊙, and M = 6 × 102 M⊙ for gas, star, and DM particles, respectively. M17 evolved their dwarf galaxy models in isolation (i.e. without any host MW exerting gravitational and hydrodynamical effects on them) in the redshift range [100 <  z <  1]. The position-velocity (i.e. phase-space) distribution of stars and DM at z = 1 in these objects reflects the additional hydrodynamics, gas cooling, and stellar feedback recipes implemented in the M17 simulations. Having switched on star formation (SF) in their simulations, M17 obtained satellites consisting of a realistic combination of DM particles, gas particles, and star particles.

Given the above picture, in this work we present our study on the general properties of the MW satellites debris via numerical simulations. To do this we adopted a hybrid approach, for which we combined high-resolution MW models from the literature (with parameters extracted from cosmological data) with M17 high-resolution dwarf galaxy models (employing them as satellites of the MW). Furthermore, the initial distribution of the satellites around the MW comes from the results of the Aq cosmological simulations.

Our approach is similar to what was employed in Moetazedian & Just (2016, hereafter MJ16). Following the numerical prescriptions of Yurin & Springel (2014), MJ16 combined cosmological initial set-up and high-resolution numerical simulations. They modelled six MW numerical realisations, each consisting of live disc, bulge, and halo, and each with halo parameters extracted from the Aq-A2 to Aq-F2 cosmological simulations at redshift z = 0. They combined each MW model with a number of N-body satellites for which they extracted the parameters from the corresponding Aq snapshot. The mass resolution of their discs is 3.4 × 103 M⊙, higher than the stellar resolution in cosmological simulations. With this hybrid approach they were able to study the effect of a cosmologically motivated set of satellites on a high-resolution MW disc. Since we wanted to address the distribution of both the stellar and DM satellite debris in our study, in contrast to MJ16 we decided to employ a selection of satellites made of baryons and DM (hereafter hybrid satellites) that we extracted from the sample of dwarf galaxies described and studied in M17.

This paper is organised as follows. In Sect. 2 we show the selection of satellite models for our simulations and we discuss the properties of the satellites. At the end of Sect. 2 we show the numerical set-up of our simulations. In Sect. 3 we show our results, addressing the stripping of satellite matter, the radial distribution of the debris, and its shape. In Sect. 4 we also investigate the surviving fraction of satellites at the end of our simulations and the final DM-to-stellar mass ratio inside the surviving satellites. In Sect. 5 we conclude and we discuss our results.

2. Numerical simulations

We ran a set of N-body simulations of the MW interacting with its satellites to address the properties of the stripped satellite debris. For each simulation we took the MW model used by MJ16 for the corresponding Aq set-up. The initial MW data were originally extracted by MJ16 as Navarro-Frenk-White (NFW) haloes (Navarro et al. 1995a) and with a best-matching numerical profile, with total mass equal to the NFW virial mass M200 (Navarro et al. 1995b) of the original NFW model and with the same inner density profile, following the prescriptions of Springel et al. (2005). The discs of these numerical models have an exponential profile with the distance of the Sun set to R0 = 8 kpc from the Galactic centre, while the bulge has a Hernquist density profile (see Hernquist 1990).

For each MW halo selected in the corresponding Aq simulation, MJ16 employed a subhalo mass cut of 108 M⊙ and required the subhaloes to have a pericentre passage within 25 kpc of the MW within 2 Gyr. They resimulated these systems as higher resolution DM-only N-body spheroids (50 K particles each), for a total number of satellites per simulation ranging from a minimum of 12 (Aq-E2) to a maximum of 24 (Aq-F2). Their satellites have a range in mass that spans from 108 M⊙1 to 6 × 1010 M⊙. They modelled each satellite as an NFW profile, and each satellite has particles of equal mass. They placed the satellites in the positions indicated from the corresponding Aq simulations. In Table 1 we show the main properties of the MW models.

Table 1.

Simulation set-up for the MW models employed in the MJ16 simulations and the corresponding number of satellites.

2.1. Selecting the satellite galaxies

For the selection of the satellite galaxies to be used in our simulations in place of the DM-only ones from MJ16, we extracted the best dwarf galaxies from the sample of M17.

The two samples of satellites come from different simulations (DM-only versus full N-body-SPH) run up to different final redshifts (zfinal = 0 versus zfinal = 1). Therefore, we chose to match them by minimising the distances of the two satellite samples in the log(M200)–log[(vmax/rmax)2] space. Here, vmax/rmax is the ratio of the maximum circular velocity of the satellite to the radius of maximum circular velocity. This ratio is an indicator of the inner density (and therefore of the depth of the potential well) of each satellite. In fact,

( v max r max ) 2 = G × M ( < r max ) r max × 1 r max 2 = 4 π G 3 ρ ¯ ( < r max ) , $$ \begin{aligned} \Bigg (\frac{v_{\rm max}}{r_{\rm max}}\Bigg )^2 = \frac{G \times M({<}r_{\rm max})}{r_{\rm max}} \times \frac{1}{r_{\rm max}^2} = \frac{4 \pi G}{3} \bar{\rho }({<}r_{\rm max}), \end{aligned} $$(1)

where ρ ¯ ( < r max ) $ \bar{\rho}({ < }r_{\mathrm{max}}) $ is the average density within the radius of maximum circular velocity.

Due to the intrinsic differences in properties of the two samples of objects, we do not expect the matching sample to be as homogeneously distributed in the log-log space as the MJ16 satellites. Consequently, five of the M17 satellites fall in the relevant parameter range and so each candidate dwarf matches many satellites of MJ16. We also note that the selected objects tend to be lower in central density than the ones from MJ16. In Table 2 we show the number of particles, the masses of the gas, DM, and stellar component for each of the five selected dwarfs.

Table 2.

Properties of the five selected satellites before they are cut in tidal radii.

In the next step we cut the total masses of the selected dwarfs to their initial tidal radii. In order to do this mass cut we first calculated the tidal radius of each satellite of MJ16, as in Ernst & Just (2013):

r tid ( r ) = ( G M tid ω 2 − d 2 Φ ( r ) d r 2 ) 1 3 · $$ \begin{aligned} r_{\rm {tid}}(r) = \left(\frac{GM_{\rm tid}}{{\omega }^2 -\frac{d^2\Phi (r)}{\mathrm{d}r^2}}\right)^{\frac{1}{3}}\cdot \end{aligned} $$(2)

Here Mtid is the tidal mass of the satellite, r is its Galactocentric distance (GCd), ω is its orbital angular speed, and Φ is the gravitational potential of the MW. In order to obtain the tidal radius for each M17 satellite, we used the approximation

r tid , M 17 = r tid , MJ 16 ( M 200 , M 17 M 200 , MJ 16 ) 1 3 · $$ \begin{aligned} r_{\rm tid,M17} = r_{\rm tid,MJ16} \Bigg (\frac{M_{\rm 200,M17}}{M_{\rm 200,MJ16}} \Bigg )^{\frac{1}{3}}\cdot \end{aligned} $$(3)

The resulting distribution of satellite masses is shown as full coloured symbols for the case of Aq-E2 in Fig. 1. Since vmax and rmax are not altered by the tidal cutting, the satellites are shifted horizontally in the figure. For Aq-E2 only the two satellites sat1 and sat4 of M17 were relevant.

thumbnail Fig. 1.

Distribution of the candidate best-matching dwarf galaxies in the log(M200)−log(vmax/rmax) plane. Empty grey circles represent all the five best-matching dwarf galaxies in the M17 sample, which are used for the six Aq simulations (see Table 2 for their properties). The tessellation with grey segments represents the division in regions where the satellite galaxies of MJ16 are closest to any of the five best-matching dwarf galaxies of M17. For the case of Aq-E2 the satellites of MJ16 are shown as empty triangles according to their M200 and vmax/rmax values. Filled triangles are the same satellites after tidal cutting, i.e. with their Mtid values. The filled circles show the corresponding M17 satellites after tidal cutting. The colours of the filled symbols represent the matched pairs. For Aq-E2 only sat1 and sat4 have triangles that fall in their regions.

2.2. Numerical dwarf galaxies as candidate satellites: properties

We show in this section that, thanks to our hybrid approach, we achieve a mass resolution which is an order of magnitude better than what is possible in current self-consistent cosmological simulations; that the density profiles of their stellar and DM components have different slopes, thereby allowing us to treat them as two distinct populations; and that the mass distribution of DM and stars as a function of specific energy returns two distinct populations that we cannot naturally derive from the DM-only satellites. To do this, we show the properties of satellite 3 (particle mass distribution, density profile, and mass distribution as a function of specific energy) to represent all five satellites since they all have similarities in these aspects.

2.2.1. Mass resolution

In the top panel of Fig. 2 we can see that DM and stellar particle masses are orders of magnitude below the masses of the corresponding particles in the cosmological simulations TNG50, Eris, and Latte. The disc of our MW models, which is made of collisionless stellar particles, has a better resolution than the stars in the Eris simulation. This puts our satellite models and the disc models in a better position in terms of resolution. The fact that the stellar and DM debris particle masses are at least 2 orders of magnitude below the corresponding best cosmological simulation resolutions allows a more accurate statistical investigation of the stellar and DM debris in our simulations, avoiding low number noise in the calculations.

thumbnail Fig. 2.

From top to bottom: particle mass distributions, radial density profiles, and specific energy distributions for the three components (stars, gas, and DM) of satellite 3. In all figures, red is for stars, green for gas, and blue for DM. Cyan is for one corresponding DM-only satellite from MJ16. Top panel: the additional vertical dashed red line is the stellar mass resolution for the TNG50 simulation, the red dot-dashed vertical line is for Latte, and the red thick line is for Eris. The purple thick line is for the disc resolution in the MW model of MJ16. Central panel: the density profiles are given as a function of the distance r from the density centre of the satellite. For comparison, the thick grey line represents an NFW inner profile (with radial dependence ∝r−1). The black dashed vertical line indicates the tidal radius position of satellite 3 after tidal radius rescaling from the corresponding DM-only satellite. Bottom panel: the dashed and thick histograms represent the specific energy distribution in terms of mass per energy bin for each component of satellite 3 before and after applying the tidal cut to the satellite, respectively. The cyan line shows the same for the DM-only satellite.

2.2.2. Density profiles

The central panel of Fig. 2 shows the density profiles of gas, DM, and stars in satellite 3, as well as the density profile of a DM-only satellite that is substituted by satellite 3 in Aq-B2. The grey straight line shows the inner slope ∝r−1 of a NFW profile for comparison. We can see that while DM and gas have a slope of their inner density profiles close to the NFW case, the stellar component has a significantly steeper inner density profile. This is already a hint that there is no simple recipe to select a realistic stellar component based on a DM-only simulation.

2.2.3. Specific energy distribution

For each satellite particle we calculated its total specific binding energy (i.e. binding energy per unit mass) as

E bin = − ϵ tot , $$ \begin{aligned} E_{\rm {bin}} = - \epsilon _{\rm tot}, \end{aligned} $$(4)

with

ϵ tot = ϵ pot + ϵ kin , $$ \begin{aligned} \epsilon _{\rm tot} = \epsilon _{\rm pot} + \epsilon _{\rm kin}, \end{aligned} $$(5)

where ϵtot, ϵpot, and ϵkin are the specific total, potential, and kinetic energy of the particles. For the gas particles, an additional term, the specific thermal energy (ϵtherm) was counted in the sum. The specific thermal energy was calculated as

ϵ therm = 3 2 n ρ k B T , $$ \begin{aligned} \epsilon _{\rm therm} = \frac{3}{2} \frac{n}{\rho } k_{\rm B} T, \end{aligned} $$(6)

where kB is the Boltzmann constant and

n ρ = 1 0.6 m H $$ \begin{aligned} \frac{n}{\rho } = \frac{1}{0.6 m_{\rm H}} \end{aligned} $$(7)

is the number of particles per unit mass for fully ionised gas, with the term mH at the denominator being the hydrogen mass, mH = 1.67 × 10−27 kg.

Since the particle masses of the M17 satellites vary over a wide range, we weighted the particle distributions in specific energy by their masses. In the bottom panel of Fig. 2 we plot the specific energy distributions of each component for the satellite 3 at z = 1 before and after tidal cutting. For comparison the specific energy distribution function (also weighted by particle mass) of a corresponding DM-only satellite of MJ16 is shown in cyan. Even after tidal cutting, the gas and DM particles of M17 dwarfs and the DM particles of MJ16 satellites have specific energy distribution functions with a similar slope. In contrast, the shape of the distribution function of the stellar component is very different to the distribution of the DM component of the M17 and the MJ16 satellites. This confirms the need for baryonic physics in the satellite models in order to obtain a realistic stellar component.

2.3. Numerical set-up and simulation properties

We ran a total of 12 simulations, two for each corresponding Aq set-up from MJ16. For the first set of six simulations, one for each Aq set-up from MJ16, we put the five best-matching satellites in place of the corresponding MJ16 satellites at the same initial positions and velocities. For the second set of six simulations, again one for each Aq set-up from MJ16, we used the same five best-matching satellites in place of the corresponding MJ16 satellites, this time rotating the MW disc by 90° to obtain a control set of simulations to check the final distribution of the satellite debris.

Each satellite contains hundreds of thousands of DM particles and thousands of star particles. This, considering the number of satellites per simulation, allows a good statistical investigation of the properties of the DM and stellar debris.

We made use of the N-body code GADGET-4 (Springel et al., in prep.). GADGET-4 is a tree-code (Barnes & Hut 1986) with additional SPH (Springel & Hernquist 2002; Hopkins 2013) and SF (Springel & Hernquist 2003) recipes; it is parallelised following Message Passing Interface (MPI) prescriptions.

Typically, gas is stripped from satellites as they enter the MW environment (Grebel et al. 2003; Frings et al. 2017; Simpson et al. 2018). Additionally, considering the median case, the surviving gas in satellites forms the greatest fraction of their present-day stellar component before z = 1 (Weisz et al. 2014). Therefore, as a first approximation, we can switch off the hydrodynamics and the SF recipes and focus only on running N-body simulations.

The M17 satellites are made of DM, stars, and gas. In this work we are only interested in the tidal debris of the DM and the stellar component. Since the gas component contributes to the depth of the satellite potential wells, we kept it in the simulations as N-body particles. On the other hand, the gas particles represent only a tiny fraction at each specific energy compared to the DM component (see Fig. 2), which also results in a negligible contribution to the total debris. Therefore, for simplicity we added the gas particles to the DM component in the following analysis.

In all our simulations we employed a softening ϵ = 200 pc for the MW halo (in order to minimise spurious scatter effects on the DM halo particles) and ϵ = 50 pc for the disc and bulge (for a higher resolution in the force calculation). The satellite DM and star particles have softenings ϵ = 25 pc and ϵ = 10 pc, respectively. Given that for the employed gravitational softening kernel (Monaghan & Lattanzio 1985) the Newtonian force radial dependence ∝r−2 is exactly reproduced at distances r >  2.8ϵ (Springel 2005), the smaller softening choice adopted for the satellite particles allows a very high resolution of the forces, and hence a more accurate description of the tidal forces acting on the satellites. The MW disc, bulge, and halo particles have masses of 3.4 × 103 M⊙, 3.8 × 104 M⊙ and 4.4 × 105 M⊙, respectively. MJ16 also chose to use 10M particles in the disc of their MW models in order to have a high-resolution disc. This is useful because the higher resolution of the disc means less spurious scattering of the satellite debris particles that approach the inner MW halo, dominated by the disc.

Each simulation was run on 96 parallel CPUs on the computer cluster bwForCluster. We ran each of the six simulations for a total of 2 Gyr. For our analysis, we focused mainly on the final snapshot of each simulation. However, we generated outputs every 50 Myr in order to track the process of mass stripping from the satellites.

2.4. Impact of satellites on MW disc thickening and heating

As a cross-check of the quality of our selection of satellites, we first compare the impact of our satellites on the MW disc with the corresponding results from MJ16 data. MJ16, addressing the dynamical impact of the MW satellites on the Galactic disc, focused on the vertical thickening of the disc and on its heating (i.e. on the increase in time of the vertical velocity dispersion of its stars). We compare these results to check whether a different distribution of satellites (the distribution of hybrid satellites) has a different impact on the thickening and heating of the disc, and hence whether it produces a different dynamical effect on the disc.

For each simulation we calculated the disc vertical thickness zrms and the squared vertical velocity dispersion σ z 2 $ \sigma _z^2 $ for each ring-like bin of radius R in the disc. The vertical thickening and heating were calculated as Δzrms = zrms, 2 − zrms, 0 and Δ σ z 2 = σ z , 2 2 − σ z , 0 2 $ \Delta \sigma_{z}^{2} = \sigma_{z,2}^2 - \sigma_{z,0}^2 $, where the subscripts 2 and 0 indicate that the given quantity is calculated at time t = 2 Gyr and t = 0 Gyr, respectively.

In Fig. 3 we show the radial profile of the disc vertical thickening and heating for the two simulation sets. We note that, even if the disc thickening and heating are higher on average for MJ16 data, within the rms scatter of the six simulations our results are consistent with those from MJ16.

thumbnail Fig. 3.

Top and bottom panels: radial profiles of disc thickening and disc heating, respectively. The cyan thick line represents the data from MJ16 simulations; the orange dashed line represents the data from our simulations.

The implications are that the MW disc is similarly excited if a population of DM-only satellites or hybrid satellites is used. Since the selection of satellites from MJ16 was made based on their dynamical effect on the MW disc (i.e. based on a mass cut), having a population of satellites with similar mass range (see Fig. 1) implies that the effects on the disc are similar and weakly dependent on the presence of DM-only or of additional stars as well. The thicker scatter in MJ16 data is due to the F2 case, where a massive satellite of more than 1010 M⊙ is strongly interacting with the MW disc.

3. Results

3.1. Stripping of matter

In this subsection we show how much stellar and DM debris is stripped by the MW and their distribution in the Galactic environment. For each satellite we calculated its density centre at every snapshot in order to define its position in time. We used the tidal radius calculation of Eq. (2) and at each snapshot we checked what fraction of total satellite stars and DM was found outside of the tidal radii of the satellites. We show the result in Fig. 4. The fraction of tidally stripped stellar debris increases to a maximum of 30% of the total initial stellar satellite mass. Instead, the satellite DM is stripped up to 80% of the total initial mass. We also plot the results of our analysis on the data from MJ16. DM-only satellites also lose most of their DM, up to 70% of the initial DM mass.

thumbnail Fig. 4.

Fraction of the satellite matter stripped out of the tidal radii of all the satellites as a function of time. Colour-coding and lines: dashed red for stars, thick blue for DM, and dot-dashed cyan for DM-only satellites from the MJ16 simulations. For each colour, the lines are the average over the six simulations, and the shaded areas represent the rms scatter.

In order to interpret these results, we focus on the properties and orbital distribution of the satellites in our simulations. These satellites were modelled as N-body objects made of star particles, mostly found in their core region, and DM particles, which have shallower and more extended density profiles. Thus, DM can be stripped more easily and larger fractions of its mass, originally residing in the satellites, can end up as debris in the MW environment. Instead, stars are mostly confined in the inner regions of the satellites and significant stripping of this matter only occurs when the satellites closely approach the MW.

3.2. Radial distribution of the debris

We wanted to know whether there is any difference in the final distribution of the stellar and DM debris in the MW environment. We addressed this by calculating the probability distribution function Pstripped of the debris (normalised for the total debris mass within the virial radius of the MW; all the debris out of this radius is considered lost) that ends at a given GCd from the MW centre, as a function of the GCd. The radial bins have a width of ∼6 kpc. Pstripped is a measure of how much mass out of the total debris ends in a given spherical shell. In the top panel of Fig. 5 we plot the results of this calculation. The distribution of the DM debris is quite smooth and uniform throughout all radii up to the MW virial radius with some deficit in the innermost region. In the stellar distribution there are some prominent peaks in the inner 100 kpc which correspond to more confined stellar streams of satellites close to pericentre passage. In the outer MW halo, significantly lower fractions of stellar debris are found. For comparison, the MW halo scale lengths in these MW models range between 20 kpc and 30 kpc (see Table 1). We note that DM-only satellites debris and hybrid satellites DM debris suffer similar tidal stripping and have similar values of Pstripped.

thumbnail Fig. 5.

Top panel: radial profile of Pstripped for the stripped debris mass deposited at a given spherical shell, normalised to the total debris mass of stars and DM, respectively. Lines and colour-coding as in Fig. 4. Bottom panel: radial matter density profile of the debris in log-scale. Lines and colour-coding as in Fig. 4. The DM-only satellite data from MJ16 are not plotted here. The thick black and dashed black line are the best-fit profiles for the DM and stellar debris, respectively. All profiles are normalised to the corresponding density values at r ∼ 25 kpc.

In the bottom panel of Fig. 5 we show the radial density profiles of the DM and stellar debris in the MW halo, and their best-fitting functions, for the simulations with the hybrid satellites. The radial density profiles were calculated dividing the values of Pstripped by the spherical-shell volumes of the corresponding bins. To fit the data, we chose the generalised NFW profile as a function of GCd:

ρ ( r ) = ρ 0 ( r r s ) α ( r r s + 1 ) β − α · $$ \begin{aligned} \rho (r) = \frac{\rho _0}{\bigg (\frac{r}{r_{\mathrm{s} }}\bigg )^{\alpha } \bigg (\frac{r}{r_{\mathrm{s} }} + 1 \bigg )^{\beta -\alpha }}\cdot \end{aligned} $$(8)

Here, ρ0 is the scale density, rs is the scale radius, α controls the inner slope, and β controls the outer slope. The best-fit parameters are given in Table 3. For both debris components we found that the inner slope is positive (α <  0) describing the mass deficit inside ∼5 kpc for DM and for stars. The profile scale radius is higher for stars, reaching ∼45 kpc against the ∼15 kpc of the DM debris profile. The outer slope of the DM debris is very close to that of a standard NFW profile. In contrast, the stellar debris shows a much steeper drop at large radii.

Table 3.

Best-fit parameters for the fitting curves of the density profiles of DM and stellar debris in Fig. 5.

This picture points to the fact that at large radii fewer stars are stripped, and the stars stripped in the inner halo are on more circular orbits. This also explains the inner peaks of stellar mass fractions in Fig. 5. DM debris can be released at any distance and the contribution to its radial distribution comes from both inner and outer satellites, thus the probability distribution function of the DM debris is radially more uniform and its density profile is less steep. Calculating the cumulative fraction of the stellar debris as a function of GCd, we find that ∼30% of the total stellar debris is inside 30 kpc and ∼50% is within 60 kpc of GCd.

3.3. Shape of the debris

We now consider the geometrical distribution of the tidal debris in the inner MW region. We focused on the inner 25 kpc of GCd in order to see what the impact of the local MW environment is on the debris. We wanted to understand the following points: (a) Does the debris finally show a spherical or flat geometry? (b) What is the orientation of the debris distribution? (c) Is there any difference between the DM and stellar debris?

In order to answer these questions, we introduced the Second Order Momenta Tensor (hereafter SOMT) in our analysis. The SOMT is a rank-2 tensor for which the jk entry is calculated as

I jk = ∑ i = 1 N m i x i , j x i , k , $$ \begin{aligned} I_{jk} = \sum _{i=1}^{N}{m_i x_{i,j} x_{i,k}}, \end{aligned} $$(9)

where i indicates the ith particle (out of N particles), mi is its mass, and xi, j and xi, k are its cartesian coordinates (Binney & Tremaine 2008; Polyachenko et al. 2016). In our case N is the total number of particles that fall in a given sphere centered on the MW Galactic centre. The definition employed here for the SOMT addresses the global geometrical distribution of the debris, no matter how sub-structured it is. An indicator of the flattening of the matter distribution is represented by the ratio c/a between the SOMT semi-minor and semi-major axes c and a, whereby c and a are the square roots of the eigenvalue with the smaller and larger magnitude, respectively.

We focused on the inner 25 kpc in the MW halo since we were interested in the behaviour of the debris in the local halo environment. Specifically, for each simulation we calculated the tensor for the debris falling in each sphere centered on the Galactic Centre. We chose a radial binning of 5 kpc to minimise the noise due to low number statistics and to smooth the contribution of single streams. For each simulation we calculated c/a at each sphere, and then for each sphere we averaged the ratio over the six simulations. In Fig. 6 we show the radial profile of c/a for the stellar and DM debris together with the root mean square (rms) scatter. From Fig. 6 it is evident that the stellar and DM debris are both geometrically flat, but with a significant scatter. For the DM debris we found c/a ∼ 0.55, whereas the stellar debris is flatter by a factor of two. The flattening depends only weakly on the size of the sphere.

thumbnail Fig. 6.

Radial profile of the c/a ratio for the SOMT of the stellar (red) and DM (blue) debris after 2 Gyr simulation. Lines and colour-coding as in Fig. 4. For each profile, the central line is the average profile over the six simulations. The shaded areas represent the rms scatter.

We now discuss the spatial orientation of the flattening, which is defined by the direction of the shortest eigenvector ec of the SOMT corresponding to the eigenvalue c2. We calculated this orientation in the θ − ϕ (latitude-azimuth) angles-space for the DM and stellar debris at each radius and for each simulation separately. In Fig. 7 we plot the distribution of the directions of ec in the angles-space at different GCds, using a Mollweide projection. If ec was defined at any point with negative z corresponding to θ <  0, we changed its sign in order to have it with positive values of z. This was done to avoid ambiguities in the interpretation of the direction of the minor axis in the angles-space since every eigenvector allows taking both orientations along its direction, with a difficult interpretation in the Mollweide projection2.

thumbnail Fig. 7.

Mollweide projection of the distribution of the orientation of the semi-minor axis c. The azimuth ϕ changes horizontally, while the latitude θ changes vertically. Squares represent DM, triangles represent stars. Smaller markers are for inner radii, larger markers are for outer radii. Colour-coding as in Fig. 1; it corresponds to the simulations with different initial conditions. The horizontal dot-dashed line indicates the Galactic plane.

The orientations of the short axis in the different simulations (different colours) are distributed over a wide region in the angles-space. This suggests that there is not a huge impact of the disc on the final distribution of the debris. Additionally, there is a large variation in the orientation with increasing GCd (size of symbols) as a sign of strong substructures in the debris. Furthermore, squares (DM) and triangles (stars) of the same simulation (same colour) do not occupy the same region in the angles-space. We interpret this as an indication that the DM and stellar debris show significantly different distributions and structures.

We wanted to quantify how big the differences in ϕ and θ are between the short axis of the stellar and DM debris tensor. To do this, we calculated the great-circle distance (GCircD) Δα at each sphere between DM and stars as

Δ α ( DM , ∗ ) = arccos [ sin θ DM sin θ ∗ + cos θ DM cos θ ∗ cos Δ ϕ ] , $$ \begin{aligned} \Delta \alpha ({\mathrm{DM} },*) = \arccos {\Big [ \sin {\theta } _{\mathrm{DM} } \sin {\theta }_* + \cos \theta _{\mathrm{DM} } \cos {\theta }_*\cos \Delta \phi \Big ]}, \end{aligned} $$(10)

where ϕDM, θDM are the angles of ec for the DM debris SOMT as a function of the radius R of the sphere; ϕ*, θ* are the same angular quantities for the stellar debris; and Δϕ = ϕDM − ϕ*. We show these results in Fig. 8. The average difference in angular distribution between DM and stars is no more than 10−20° in the central part of the halo (the first few kpc), whereas it reaches up to 40° going out towards 15−20 kpc, where the scatter around the average is very high. This indicates that it is not possible to find a simple systematic correlation between the DM and stellar debris orientations in the environment around the MW. This is reinforced by the different c/a and the different radial distribution of the debris seen in Figs. 5 and 6. The DM and stellar debris do not share the same geometry once they are stripped from their satellite progenitors. Observationally, this means that it is not possible to track the distribution of the DM debris from the distribution of the stellar debris directly.

thumbnail Fig. 8.

Average radial profile of the GCircD between DM and stellar debris as a function of spherical radius r. The shaded area represents the rms scatter.

It is important at this point to understand if the disc can have any impact at all on the final distribution of the debris, or if this depends on the initial distribution of the satellites only. To test this, we ran another set of six simulations, each corresponding to one of the six previous simulations. For each simulation set-up, we rotated the initial relative position and velocity of the disc particles by 90° around the y-axis resulting in a disc in the y − z plane rotating around the x-axis3. We ran these six new simulations for 2 Gyr.

At the end of the simulation, we analysed the final angular distribution of the debris. If the disc were mainly responsible for the flattening of the satellite debris, we would expect the minor axis distribution of the SOMT to be rotated approximately similarly to the disc by ∼90°. As can be seen in Fig. 9, the final Mollweide projection of the debris is mostly overlapping the same region of the original debris distribution and shows no systematic rotation. This confirms that the initial conditions of the satellites have a larger effect than the disc orientation on the final debris distribution.

thumbnail Fig. 9.

Mollweide projection of the distribution of the c axis in the angles-space, for the rotated and non-rotated disc. Data from Fig. 7 are re-plotted as empty markers. Solid markers are for rotated disc simulations.

We then quantified the GCircD at each sphere between the debris in the new set of simulations and the debris from the original set of simulations

Δ α ( rot , 0 ) = arccos [ sin θ rot sin θ 0 + cos θ rot cos θ 0 cos Δ ϕ ] , $$ \begin{aligned} \Delta \alpha ({\mathrm{rot} },0) = \arccos {\Big [ \sin \theta _{\mathrm{rot} } \sin \theta _0 + \cos \theta _{\mathrm{rot} }\cos \theta _0\cos \Delta \phi \Big ]}, \end{aligned} $$(11)

where in this case ϕrot, θrot are the angles of the c minor axis for the SOMT within the given spherical radius r for the debris from the second set of simulations with rotated disc orientation; ϕ0, θ0 are the same angular quantities for the original set of simulations; and Δϕ = ϕrot − ϕ0.

In Fig. 10 we can see that on average Δα(rot, 0)∼10° for DM, while at the outer radii it reaches values of up to Δα(rot, 0)∼20° for stars. So, the rotation of the disc initial conditions produces a limited, yet not completely negligible rotation of the final distribution of the debris, which implies that the disc may have a moderate impact on the final distribution of the local satellite debris.

thumbnail Fig. 10.

Radial profile of the GCircD Δα of the rotated disc simulations compared to the corresponding original simulations, for the short axis orientation of the stellar and DM debris. Lines and colour-coding as in Fig. 4. The shaded areas are the rms scatter.

4. Additional investigation: fraction of surviving satellites and DM/stellar mass ratio

In addition to the final distribution of the debris in the MW halo, we checked the fraction of satellites that survived after 2 Gyr. To determine whether a satellite survived stripping, we chose a threshold fraction of 10% of its total initial mass. A satellite that has a final tidal mass higher than 10% of its initial mass is considered to have survived. Otherwise, it is counted as dissolved. In Fig. 11, top panel, we show the fractions of the surviving satellites for our six simulations and for the MJ16 simulations. The error bar around each data point is the scatter between the six simulations of the same set.

thumbnail Fig. 11.

Top panel: fraction of surviving satellites, with 10% of their initial mass adopted as the threshold to determine their survival. Colour-coding as in Fig. 3. The orange and cyan error bars are the rms scatter for our set of simulations and for the MJ16 simulations, respectively. Bottom panel: logarithm of the ratio of total mass to stellar mass in the satellites in the Aq-A2 to Aq-F2 simulations, log(Mtot/M*), for the initial and final snapshot. Purple pentagons are the average of each corresponding simulation for the initial data at t = 0 Gyr; pink diamonds are instead for the final data at t = 2 Gyr. The purple and pink error bars are the rms scatter for all the satellites in each corresponding simulation (initial and final snapshot, respectively). The dot-dashed black line is the density ratio Ωm/Ωbar of all matter to baryonic matter. The pink dashed line represents the best fit for the final ratios (Mtot/M*)best. Only the satellites that survive the stripping process are employed to calculate the average and the standard deviation in each simulation.

In all simulations more than ∼65% of all satellites survive after 2 Gyr. The averages of the six simulations for both cases are reported as a thick cyan line (DM-only) and a dashed orange line (hybrid). We also plot the area of the rms scatter between them with the same colour-coding. We found fsurvived ∼ 0.8 ± 0.1 for both sets of simulations, thus we confirm the similarity of results between them with no systematic differences.

We also looked at the ratio between DM and stellar mass inside the satellites at the beginning and at the end of the hybrid simulations, to understand the evolution of the matter content inside the satellites. This result is plotted in the bottom panel of Fig. 11 in logarithmic values. The best-fit value ( M tot / M ∗ ) best = 768 − 301 + 217 $ (M_{\mathrm{tot}}/M_{*})_{\mathrm{best}} = 768^{+217}_{-301} $ at the final epoch is also shown. For all the satellites that survived, the ratio of DM to stellar matter, though still much higher than the cosmic ratio, decreased in time. This is a consequence of what was shown in Fig. 4 where more DM than stars is stripped from the satellites.

For comparison, we give the cosmic ratio of all matter to baryons from the Wilkinson Microwave Anisotropy Probe (WMAP)-7 results (see Komatsu et al. 2011), Ωm/Ωbar ∼ 6.0 being much lower than the initial ratio in our simulations. It appears that the DM-to-star ratio is subject to two phases. As argued in M17, in the first phase, before they strongly interact with the MW, the satellite galaxies lose large quantities of gas; therefore, their total-matter–to–baryons ratio increases (so they start in our simulations with a high total-matter–to–stars ratio). In a second phase (i.e. in more recent epochs), strong tidal interactions with the MW deplete the satellite dwarf galaxies of more DM than stars and drive the total-matter–to–stars ratio towards lower values by a factor of 2–4 on average.

Assuming a stellar mass-to-light ratio of two in solar units, the satellites in our simulations fall in the regime 1.2 × 103 L⊙ − 4 × 106 L⊙ corresponding to UFDs (Simon 2019, hereafter S19)4. The mass-to-light ratios, which correspond to the tidal mass-to-light ratios, of all our surviving satellites are shown as a function of luminosity in Fig. 12. The tidal mass-to-light ratio was estimated for the first time by Faber & Lin (1983) for seven dwarf spheroidal galaxies, finding a much lower value than our mass-to-light ratio of (M/L) best ∼ ( 1.5 × 10 3 ) − 606 + 434 $ _{\mathrm{best}} \sim (1.5 \times 10^3)^{+434}_{-606} $M⊙ L ⊙ − 1 $ L_{\odot}^{-1} $. S19 showed that M/L inside the half-light radius Rh decreases with increasing luminosity from ∼1000 (with a large scatter) for L = 103L⊙ to ∼10 for L = 107L⊙, which is similar to the values of Errani et al. (2018) inside 1.8 Rh. We found the same trend of mass-to-light ratios as a function of luminosity, but our values are systematically higher by a factor of a few. This offset is expected because the tidal radius is larger than 2 Rh in most cases.

thumbnail Fig. 12.

Distribution of the surviving satellites in the L–M/L plane for the six simulations adopting a stellar mass-to-light ratio of two. Colour-coding as in Fig. 1.

5. Conclusions and discussion

5.1. Conclusions

We have performed a set of N-body simulations to investigate the general properties of the satellite debris distribution in the MW environment. For the first time, we took advantage of a set of satellite galaxy models taken from cosmological high-resolution hydrodynamical simulations, placed in initial orbits derived from the results of N-body cosmological simulations of structure formation, and of a full high-resolution N-body MW host model taken from the literature, consisting of live disc, bulge, and halo. For a statistical analysis, the initial conditions of six Aquarius simulations were used. Following the approach of MJ16, all satellites with tidal masses higher than 108 M⊙ and with a pericentre passage closer than 25 kpc were taken into account. We ran each simulation for 2 Gyr.

We investigated the general properties of the debris of satellite galaxies in the global and local MW environment. We focused on the differences in the distribution of stellar and DM debris. Based on our findings, we can state the following:

  • The stellar component in the satellite galaxies is much more tightly bound than the DM component and cannot be deduced from DM-only simulations;

  • The stripping process acting on satellite DM is more efficient than on satellite stars and releases more DM debris than stellar debris in the MW environment (80% compared to 30%);

  • The radial density profile of the DM debris covers the whole host halo and shows a standard NFW slope in the outer region; the stellar debris is confined to the inner 50 kpc with a steep cutoff outside; both profiles show a deficit in the inner 5 kpc;

  • The stellar debris shows more prominent peaks of the radial probability distribution function than the DM debris, pointing to a more confined structure of individual streams in the inner part of the MW halo;

  • The debris of DM and stars distribute with some degree of flatness (c/a ∼ 0.55 and 0.3, respectively); the orientation of the minor axis is very different for the different simulations and shows no obvious correlation to the MW disc plane; the orientation of the DM and the stellar debris is not well correlated in each simulation, thus it is not possible to reconstruct DM debris and streams directly from observed stellar streams;

  • Changing the orientation of the disc by 90° has a small effect on the distribution of the satellite debris; this confirms that the structure of the DM and stellar debris of satellite galaxies is mainly determined by the initial conditions of the satellites; the flattened potential of the disc only plays a minor role;

  • The tidal total-to-stellar mass ratio of the satellites decreases by a factor of 2–4 during the simulations and shows at the end mass-to-light ratios that are consistent with observations of the Local Group UFDs.

In conclusion, our work states that the stellar and DM components of the satellites are subject to different stripping efficiencies and different redistributions in the MW environment, thus they are not strongly correlated. As a consequence, observed stellar streams cannot directly be converted to the distribution of the DM debris. Furthermore, it shows the importance of cosmological initial conditions as well as the realistic structure of satellite galaxies in determining the final distribution of the satellite streams around the MW.

5.2. Discussion

The fact that the debris (particularly the stellar debris) has some degree of flatness shows that with a cosmologically motivated initial set-up (e.g., in our case, from Aq simulations data) it is possible to obtain a flat spatial distribution of the debris. On the other hand, our debris does not show any distribution on a unique plane, as can be seen from Fig. 7, where within different radii the debris tensor seems to occupy different regions in the angles-space in the same simulation and for each individual component (DM and stars). This contrasts with what is stated by Pawlowski et al. (2012), that favours the formation of a plane of debris. Furthermore, since our results predict no systematic orientation of the stellar and DM debris, this means that other techniques may be needed to trace the distribution of DM streams other than addressing the distribution of the stellar streams alone.

Regarding the mass loss of the satellites in our simulations, S19 stated that tidal stripping affects the luminosity but not the metallicity of the stellar populations of satellites, and since the luminosity-metallicity relation is satisfied for the observed UFDs, then the stripping of stars acting on satellites must be moderate. This conclusion is in line with what we obtained for the stripping fraction of stars, which is not high for stellar satellite matter. S19 found the result in agreement with previous estimates, such as from Kirby et al. (2013).

When considering the fraction of surviving satellites, regardless of whether the satellites are DM-only or hybrid, this is not the feature that determines their survival. In fact, no systematic differences were found between the MJ16 simulation sample and our simulation sample. The inner properties of the satellites, as previously stated, differ in the shape of their density profiles, shallower for DM in hybrid satellites than in DM-only satellites. However, the difference in slope of these profiles is limited, and the energy distribution of DM particles is similar for DM-only and hybrid satellites. Instead, it seems that other drivers, such as the satellite initial conditions of our simulations, determine the survival probability of the satellites; therefore, this is related to their orbital parameters and intrinsic structure.

The fact that the tidal stripping exerted by the MW is not strongly efficient on its satellite galaxies seems to be compatible with what Peñarrubia et al. (2008) found in their simulations. In particular, they found that even after 99% of the stars are stripped from a satellite, the King profile (King 1966) of the remaining stellar component remains unchanged. Furthermore, they found that the stripping of stars is efficient only when the satellites approach their orbital periapsis, i.e. when they get closer to the MW centre. Previous work from Bullock & Johnston (2005) that has investigated the effect of MW–satellite interactions in the past epochs has underlined the importance of these mergers to form the stellar halo. However, they adopted a combination of N-body methods and semi-analytical models to address their study. The MW, for instance, was added only analytically as a time-evolving potential. Here we have taken advantage of full N-body simulations, and the live evolution of the gravitational potential of the MW, with effects such as dynamical friction (Chandrasekhar 1943) being naturally incorporated, however without secular long-term growth. Furthermore, our analysis is different from that of Bullock & Johnston (2005) in the sense that it focuses on the current distribution of the residual debris coming from recently accreted satellite structures rather than focusing on the build-up of the MW halo from past accretion processes.

1

This mass cut in MJ16 was done to address the impact of satellite galaxies on the MW disc kinematics and dynamics, and in this case they show that the impact of satellites increases with satellite mass as ∝M2.

2

In fact, the opposite of a vector in this kind of projection is not simply represented as the symmetric opposite in the map with respect to (ϕ, θ)=(0, 0).

3

This effective rotation transforms the x-axis to the −z-axis, the z-axis to the x-axis, and leaves the y-axis unchanged.

4

There is one satellite in our simulation from the Aq-F2 set-up that ends with L = 50 L⊙, but it is below the range considered in Fig. 4 of S19.

Acknowledgments

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 (“The Milky Way System”, subproject A02). The authors acknowledge support by the state of Baden-Württemberg through bwHPC. Part of this research was carried out on the High Performance Computing resources at New York University Abu Dhabi. We thank J. Frings for providing data for the selection of dwarf galaxies employed for the simulations. M.M. thanks V. Springel and his collaborators for support using the code Gadget-4. M.M. thanks also M. Arca Sedda, A. Borch, A. Pasquali, B. Avramov, B. Bidaran, A. Savino, T. M. Jackson and M. Donnari for further discussion.

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All Tables

Table 1.

Simulation set-up for the MW models employed in the MJ16 simulations and the corresponding number of satellites.

In the text
Table 2.

Properties of the five selected satellites before they are cut in tidal radii.

In the text
Table 3.

Best-fit parameters for the fitting curves of the density profiles of DM and stellar debris in Fig. 5.

In the text

All Figures

thumbnail Fig. 1.

Distribution of the candidate best-matching dwarf galaxies in the log(M200)−log(vmax/rmax) plane. Empty grey circles represent all the five best-matching dwarf galaxies in the M17 sample, which are used for the six Aq simulations (see Table 2 for their properties). The tessellation with grey segments represents the division in regions where the satellite galaxies of MJ16 are closest to any of the five best-matching dwarf galaxies of M17. For the case of Aq-E2 the satellites of MJ16 are shown as empty triangles according to their M200 and vmax/rmax values. Filled triangles are the same satellites after tidal cutting, i.e. with their Mtid values. The filled circles show the corresponding M17 satellites after tidal cutting. The colours of the filled symbols represent the matched pairs. For Aq-E2 only sat1 and sat4 have triangles that fall in their regions.
In the text
thumbnail Fig. 2.

From top to bottom: particle mass distributions, radial density profiles, and specific energy distributions for the three components (stars, gas, and DM) of satellite 3. In all figures, red is for stars, green for gas, and blue for DM. Cyan is for one corresponding DM-only satellite from MJ16. Top panel: the additional vertical dashed red line is the stellar mass resolution for the TNG50 simulation, the red dot-dashed vertical line is for Latte, and the red thick line is for Eris. The purple thick line is for the disc resolution in the MW model of MJ16. Central panel: the density profiles are given as a function of the distance r from the density centre of the satellite. For comparison, the thick grey line represents an NFW inner profile (with radial dependence ∝r−1). The black dashed vertical line indicates the tidal radius position of satellite 3 after tidal radius rescaling from the corresponding DM-only satellite. Bottom panel: the dashed and thick histograms represent the specific energy distribution in terms of mass per energy bin for each component of satellite 3 before and after applying the tidal cut to the satellite, respectively. The cyan line shows the same for the DM-only satellite.
In the text
thumbnail Fig. 3.

Top and bottom panels: radial profiles of disc thickening and disc heating, respectively. The cyan thick line represents the data from MJ16 simulations; the orange dashed line represents the data from our simulations.
In the text
thumbnail Fig. 4.

Fraction of the satellite matter stripped out of the tidal radii of all the satellites as a function of time. Colour-coding and lines: dashed red for stars, thick blue for DM, and dot-dashed cyan for DM-only satellites from the MJ16 simulations. For each colour, the lines are the average over the six simulations, and the shaded areas represent the rms scatter.
In the text
thumbnail Fig. 5.

Top panel: radial profile of Pstripped for the stripped debris mass deposited at a given spherical shell, normalised to the total debris mass of stars and DM, respectively. Lines and colour-coding as in Fig. 4. Bottom panel: radial matter density profile of the debris in log-scale. Lines and colour-coding as in Fig. 4. The DM-only satellite data from MJ16 are not plotted here. The thick black and dashed black line are the best-fit profiles for the DM and stellar debris, respectively. All profiles are normalised to the corresponding density values at r ∼ 25 kpc.
In the text
thumbnail Fig. 6.

Radial profile of the c/a ratio for the SOMT of the stellar (red) and DM (blue) debris after 2 Gyr simulation. Lines and colour-coding as in Fig. 4. For each profile, the central line is the average profile over the six simulations. The shaded areas represent the rms scatter.
In the text
thumbnail Fig. 7.

Mollweide projection of the distribution of the orientation of the semi-minor axis c. The azimuth ϕ changes horizontally, while the latitude θ changes vertically. Squares represent DM, triangles represent stars. Smaller markers are for inner radii, larger markers are for outer radii. Colour-coding as in Fig. 1; it corresponds to the simulations with different initial conditions. The horizontal dot-dashed line indicates the Galactic plane.
In the text
thumbnail Fig. 8.

Average radial profile of the GCircD between DM and stellar debris as a function of spherical radius r. The shaded area represents the rms scatter.
In the text
thumbnail Fig. 9.

Mollweide projection of the distribution of the c axis in the angles-space, for the rotated and non-rotated disc. Data from Fig. 7 are re-plotted as empty markers. Solid markers are for rotated disc simulations.
In the text
thumbnail Fig. 10.

Radial profile of the GCircD Δα of the rotated disc simulations compared to the corresponding original simulations, for the short axis orientation of the stellar and DM debris. Lines and colour-coding as in Fig. 4. The shaded areas are the rms scatter.
In the text
thumbnail Fig. 11.

Top panel: fraction of surviving satellites, with 10% of their initial mass adopted as the threshold to determine their survival. Colour-coding as in Fig. 3. The orange and cyan error bars are the rms scatter for our set of simulations and for the MJ16 simulations, respectively. Bottom panel: logarithm of the ratio of total mass to stellar mass in the satellites in the Aq-A2 to Aq-F2 simulations, log(Mtot/M*), for the initial and final snapshot. Purple pentagons are the average of each corresponding simulation for the initial data at t = 0 Gyr; pink diamonds are instead for the final data at t = 2 Gyr. The purple and pink error bars are the rms scatter for all the satellites in each corresponding simulation (initial and final snapshot, respectively). The dot-dashed black line is the density ratio Ωm/Ωbar of all matter to baryonic matter. The pink dashed line represents the best fit for the final ratios (Mtot/M*)best. Only the satellites that survive the stripping process are employed to calculate the average and the standard deviation in each simulation.
In the text
thumbnail Fig. 12.

Distribution of the surviving satellites in the L–M/L plane for the six simulations adopting a stellar mass-to-light ratio of two. Colour-coding as in Fig. 1.
In the text

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