MNRAS , 1–15 (2021) Preprint Compiled using MNRAS LATEX style file v3.0

Structure and kinematics of tidally limited satellite galaxies inLCDM

Raphaël Errani1,2,★, Julio F. Navarro2, Rodrigo Ibata1, Jorge Peñarrubia31 Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France2 Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada3 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK

preprint, submitted November 10 2021

ABSTRACTWe use 𝑁-body simulations to model the tidal evolution of dark matter-dominated dwarfspheroidal galaxies embedded in cuspy Navarro-Frenk-White subhalos. Tides gradually peeloff stars and dark matter from a subhalo, trimming it down according to their initial bindingenergy. This process strips preferentially particles with long orbital times, and comes to anend when the remaining bound particles have crossing times shorter than a fraction of theorbital time at pericentre. The properties of the final stellar remnant thus depend on the energydistribution of stars in the progenitor subhalo, which in turn depends on the initial densityprofile and radial segregation of the initial stellar component. The stellar component mayactually be completely dispersed if its energy distribution does not extend all the way to thesubhalo potential minimum, although a bound dark remnant may remain. These results implythat “tidally-limited” galaxies, defined as systems whose stellar components have undergonesubstantial tidal mass loss, neither converge to a unique structure nor follow a single tidal track,as claimed in earlier work. On the other hand, tidally limited dwarfs do have characteristic sizesand velocity dispersions that trace directly the characteristic radius (𝑟mx) and circular velocity(𝑉mx) of the subhalo remnant. This result places strong upper limits on the size of satelliteswhose unusually low velocity dispersions are often ascribed to tidal effects. In particular, thelarge size of kinematically-cold “feeble giant” satellites like Crater 2 or Antlia 2 cannot beexplained as due to tidal effects alone in the Lambda Cold Dark Matter scenario.

Key words: dark matter; galaxies: evolution; galaxies: dwarf; Local group; methods: numerical

1 INTRODUCTION

The Lambda ColdDarkMatter (LCDM) scenariomakes well-definedand falsifiable predictions for the radial density profile of dark matterhalos, as well as for their abundance as a function of virial1 mass.These predictions are particularly relevant for the study of dwarfspheroidal (dSph) galaxies, dark matter-dominated systems whosestellar components act as simple kinematic tracers of the structureof their surrounding dark halos (see; e.g., Mateo et al. 1993; Walkeret al. 2007).

These studies may be used to test the expected density profilesof cold dark matter halos, which are well approximated by theNavarro-Frenk-White formula (hereafter, NFW; Navarro et al. 1996b,1997). This issue has received much attention in the past few years,

★ errani@unistra.fr1 We shall define “virial” quantities as thosemeasuredwithin spheres ofmeandensity equal to 200× the critical density for closure, 𝜌crit = 3𝐻 20 /8𝜋𝐺,where 𝐻0 = 67 km s−1Mpc−1 is the value of Hubble’s constant (PlanckCollaboration et al. 2020). Virial quantities are designated with a “200”subscript.

albeit withmixed results, with some authors arguing that the structureof dwarf galaxy halos is consistent with the “cuspy” NFW shape andothers claiming that the data suggest density profiles with a sizableconstant-density “core” (for reviews see, e.g., Gilmore et al. 2007;Bullock & Boylan-Kolchin 2017).

The interpretation of these studies is further complicated bythe fact that the assembly of the baryonic component of a galaxymay induce changes in the dark matter density profile, perhaps evenerasing the expected cusp and imprinting a core (Navarro et al. 1996a;Read & Gilmore 2005; Governato et al. 2010; Pontzen & Governato2012). However, such baryon-induced effects should be minimal invery faint dark matter-dominated galaxies, simply because the totalfraction of mass in baryonic form is too small to be able to affectgravitationally the dark matter component (see; e.g., Peñarrubia et al.2012; Di Cintio et al. 2014; Benítez-Llambay et al. 2019).

Dwarf galaxies may also be used to probe another robustLCDM prediction, concerning the mass function of low-mass halos.In LCDM these halos are so numerous, and their mass functionso steep, that accommodating the comparatively scarcer number ofdwarfs and their much shallower stellar mass function requires thatgalaxy formation efficiency should decline steadily with decreasing

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halo mass, effectively restricting dwarf galaxy formation to halos ina narrow range of virial mass (Guo et al. 2010; Ferrero et al. 2012;Sales et al. 2017; Bullock & Boylan-Kolchin 2017).

A firm lower limit to that narrow range (𝑀200 ∼ 109 𝑀�)is suggested by the minimum virial temperature needed to allowhydrogen to cool efficiently, after accounting for the effects ofan ionizing UV background (the “hydrogen cooling limit”, HCL;Efstathiou 1992; Quinn et al. 1996; Gnedin 2000; Okamoto &Frenk 2009; Benitez-Llambay & Frenk 2020). In quantitative terms,this implies that essentially all dwarfs with stellar mass, 𝑀★ <

107 𝑀� , no matter how faint they may be, should form in halos withcharacteristic circular velocity2 somewhere between 20 km s−1 and40 km s−1 (Fattahi et al. 2018).

Together with the NFW profile mentioned above, the HCLminimum mass sets a velocity dispersion “floor” for dark matter-dominated dwarfs of given radius. This velocity “floor” is, to firstorder, simply a fraction of the circular velocity (at that radius) of ahalo at the HCL boundary. For example, assuming that the maximumcircular velocity of an HCL halo is 𝑉mx ≈ 20 km s−1 (reached ata radius 𝑟mx = 3.4 kpc), such halo would have a circular velocityof ≈ 11 km s−1 at 300 pc. This implies that a dwarf with 3D half-light radius, 𝑟h, comparable to that size should have a line-of-sight(i.e., 1D) velocity dispersion, 𝜎los, in excess of 11/

√3 ≈ 6 km s−1.

The same argument results in 𝜎 & 9 km s−1 for 𝑟h ∼ 1 kpc, and𝜎los & 4 km s−1 for 𝑟h ∼ 100 pc (these numbers assume a halo ofaverage concentration at 𝑧 = 0.)

There are a number of dwarfs in the Local Group that appearto violate these limits (see, e.g., the compilation maintained byMcConnachie 2012), a result which, taken at face value, would call fora review of some of the basic assumptions on which these predictionsare based. Unfortunately, much of the wealth of available kinematicdata on dwarfs concerns satellite galaxies in the Local Group, mainlythose orbiting the Milky Way (MW) and the Andromeda (M31)galaxies (for a review, see Simon 2019). The predictions describedabove do not apply to satellites, as tides arising from the MW andM31 may strip a significant fraction of the total dark matter contentof a dwarf while leaving the stellar component relatively undisturbed(Peñarrubia et al. 2008; Errani et al. 2015; Sanders et al. 2018).Translating the LCDM predictions described above to the realm ofsatellite galaxies thus requires a good understanding of how the darkand stellar components of dSphs evolve as a result of tidal effects.

This is an issue that has been studied in earlier work in thecontext of LCDM, resulting in a number of conclusions and sug-gestions about how to interpret kinematic and photometric data onLocal Group satellites in the context of LCDM. Two highlights ofthat work include the suggestion that (i) the stellar mass, size, andvelocity dispersion of dSphs evolve along well-specified “tidal tracks”which depend only on the total amount of mass lost from withinthe stellar half-light radius (Peñarrubia et al. 2008), and that (ii) thestellar components of “tidally limited” satellites, defined as thosewhose stellar mass has been substantially reduced by tides, approacha “Plummer-like” density profile shape roughly independent of theinitial distribution of stars (Peñarrubia et al. 2009).

These conclusions, however, were based on simulations ex-ploring a rather limited set of initial conditions, in terms of the

2 It is customary to use the maximum circular velocity of a halo, 𝑉mx, as aproxy for virial mass, 𝑀200. These two measures are strongly correlated inLCDM via the mass-concentration relation (see; e.g., Ludlow et al. 2016).The radius at which the circular velocity peaks is usually denoted 𝑟mx. Thischaracteristic radius and circular velocity fully specify the structure of anNFW halo.

assumed initial dSph structure, and also of the assumed radial se-gregation between stars and dark matter. Considering a broaderrange of possibilities for either may result in revised predictions thatcould impact, in particular, the properties of “tidally limited” dwarfs.This is important, especially in light of the recent discovery of apopulation of dwarfs with unusually large sizes and low velocitydispersions, well below the limits mentioned earlier (see, e.g., theCrater 2 and Antlia 2 dSphs; Torrealba et al. 2016; Caldwell et al.2017; Torrealba et al. 2019).

Although it is tempting to associate such systems with tidally-limited dwarfs (Frings et al. 2017; Sanders et al. 2018; Amorisco2019), it is important to realize that the extreme estimated tidallosses put them in a regime hitherto unexplored in previous work.For example, Fattahi et al. (2018) argue that Crater 2 may be theresult of a system that has lost more than 99% of its stars and hasseen its velocity dispersion reduced by a factor of ∼ 5. As discussedby Errani & Navarro (2021), exploring this regime with N-bodysimulations requires resolving the progenitor subhalo with over 107particles, well beyond what has been achieved with cosmologicalhydrodynamical simulations.

As a result of these limitations, many questions regardingthe structure and survival of tidally-limited galaxies remain un-answered. Recent work, for example, has argued convincingly thatNFW subhalos are only very rarely fully disrupted by tides, almostalways leaving behind self-bound dark remnants that are missed incosmological simulations of limited resolution (Kazantzidis et al.2004; Goerdt et al. 2007; Peñarrubia et al. 2010; van den Bosch et al.2018). How do these results affect the structure and survival of thestellar components of such systems? Do the stellar components ofdSphs also survive to some extent (giving rise to “micro-galaxies”, asargued in Errani & Peñarrubia 2020), and, if so, what are their prop-erties? Should we expect them all to have similar density profiles?Do their evolution follow well-defined “tidal tracks” in luminosity,size, and velocity dispersion? Shoud we expect a large population oftidally-limited dwarfs of extremely low surface brightness awaitingdiscovery?

We explore some of these issues here using a large suite of𝑁-body simulations designed to study the tidal evolution of NFWhalos in the gravitational potential of a much more massive system,with particular emphasis on the regime of extreme tidal mass loss.These simulations extend our earlier work on the subject (Errani &Navarro 2021), where we focussed on the survival and structure ofthe dark component of the tidal remnant. Our emphasis here is onthe evolution of a putative stellar component, assumed to be gravit-ationally unimportant compared to the dark matter. In the interestof simplicity, we only consider spherical, isotropic models in thiscontribution, but our approach should be relatively straightforwardto extend to include modifications to these assumptions.

2 NUMERICAL METHODS

We use the set of high-resolution 𝑁-body simulations of the tidalevolution of NFW dark matter subhalos introduced in Errani &Navarro (2021) and briefly summarized in this section.

2.1 Subhalo model

We model subhalos hosting dSph galaxies as 𝑁-body realisations ofNFW (Navarro et al. 1996b, 1997) density profiles,

𝜌NFW (𝑟) = 𝜌s

(𝑟/𝑟s) (1 + 𝑟/𝑟s)2, (1)

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where 𝜌s and 𝑟s are a scale density and scale radius, respectively. TheNFW profile has a circular velocity curve 𝑉c (𝑟) =

√𝐺𝑀 (< 𝑟)/𝑟

that reaches a peak velocity of 𝑉mx ≈ 1.65 𝑟s√𝐺𝜌s at a radius

𝑟mx ≈ 2.16 𝑟mx. TheNFWdensity profile is fully specified by the twoparameters, which may be taken to be {𝑟mx, 𝑉mx}. We shall hereafterrefer to them as the “characteristic radius” and “characteristic velo-city” of the subhalo, respectively. For future reference, we also intro-duce at this point the subhalo characteristic mass,𝑀mx = 𝑟mx𝑉2mx/𝐺,and characteristic timescale, 𝑇mx = 2𝜋 𝑟mx/𝑉mx.

As the density profile of Eq. 1 leads to a diverging cumulativemass for 𝑟 → ∞, we truncate the profile exponentially at 10 𝑟s.We generate 107-particle equilibrium 𝑁-body realisations of NFWhalos with isotropic velocity dispersion, drawn from a distributionfunction computed numerically using Eddington inversion, followingthe implementation3 described by Errani & Peñarrubia (2020).

2.2 Host halo

We study the tidal evolution of the 𝑁-body subhalo in an analytical,static, isothermal spherical host potential,

Φhost = 𝑉20 ln(𝑟/𝑟0) , (2)

where 𝑉0 = 220 km s−1 denotes the (constant) circular velocity, and𝑟0 is an arbitrary reference radius. The circular velocity curve isflat, with a value 𝑉0 chosen to match roughly that of the MilkyWay (see, e.g., Eilers et al. 2019). These parameters correspondto a virial mass at redshift 𝑧 = 0 of 𝑀200 = 3.7 × 1012M� , and avirial radius of 𝑟200 = 325 kpc, respectively. Note that gravitationaleffects are scale-free, so all dimensional quantities are just quotedfor illustration, and may be rescaled as needed.

2.3 Orbits

We consider the evolution of subhalos on eccentric orbits with peri-to-apocentre ratio of 1:5. This value is consistent with the averageperi-to-apocentre ratio of Milky Way satellite galaxies (Li et al.2021, table 1), and close to the peri-to-apocentre ratio for dark mattersubstructures determined by van den Bosch et al. (1999) for isotropicdistributions. As discussed by Errani & Navarro (2021), the orbitaleccentricity affects mainly the rate of tidal stripping, but not theremnant structure. We shall only explore the properties of subhalos attheir orbital apocentres, where they spend most of their orbital time,and when they are closest to equilibrium. Large deviations fromequilibrium are expected near pericentre, as discussed by Peñarrubiaet al. (2009), but we defer their study to future contributions.

2.4 Subhalo tidal evolution

All subhalos have initial characteristic velocities substantially smallerthan that of the host halo, i.e.,𝑉mx0/𝑉0 . 0.02. This choice allows usto neglect the effects of dynamical friction, and should be appropriatefor studying the evolution of the subhalos of faint and ultrafaintsatellites of galaxies like the MW or M31.

The simulations explore a wide range of initial characteristicradii, 𝑟mx0, chosen so that the ratio of initial subhalo crossing time,𝑇mx0, to the circular orbital time at at the pericentre of the orbit,𝑇peri = 2𝜋 𝑟peri/𝑉0, lies in the range 2/3 < 𝑇mx0/𝑇peri < 2.

3 The code to generate 𝑁 -body models and corresponding stellar taggingprobabilities is available online athttps://github.com/rerrani/nbopy.

As discussed by Errani & Navarro (2021), this choice ensuresheavy mass losses due to tidal stripping. After a few orbits, tidaleffects gradually slow down and eventually become negligible oncethe remnant approaches a characteristic timescale, 𝑇mx ≈ 𝑇peri/4.The structure of the remnant approaches asymptotically that of anexponentially truncated cusp,

𝜌asy (𝑟) =𝜌cut 𝑒−𝑟/𝑟cut

(𝑟/𝑟cut), (3)

with 𝜌cut and 𝑟cut given by the total mass of the bound remnant,𝑀asy = 4𝜋 𝜌cut 𝑟3cut.

As a subhalo is tidally stripped, its characteristic radius andvelocity decrease along well-defined tidal tracks (Peñarrubia et al.2008), which may be parameterized by

𝑉mx/𝑉mx0 = 2𝛼 (𝑟mx/𝑟mx0)𝛽 [1 + (𝑟mx/𝑟mx0)2]−𝛼, (4)

with 𝛼 = 0.4, 𝛽 = 0.65 (Errani & Navarro 2021).

2.5 Simulation code

The 𝑁-body models are evolved using the particle-mesh code super-box (Fellhauer et al. 2000). This code employs a high- and a medium-resolution grid of 1283 cells each, co-moving with the subhalo centreof density, with cell size Δ𝑥 ≈ 𝑟mx/128 and ≈ 10 𝑟mx/128, respect-ively. A third, low-resolution grid is fixed in space with a cellsize of ≈ 500 kpc/128. The time integration is performed using aleapfrog-scheme with a time step of Δ𝑡 = min(𝑇mx, 𝑇peri)/400. Thischoice of time step ensures that, for NFW density profiles, at radiicorresponding to the cell size of the highest-resolving grid, circularorbits are still resolved by (at least) ≈ 16 steps.

The convergence tests of Errani & Navarro (2021, appendixA) for the set of simulations used in the present work suggest thatthe structural parameters {𝑟mx, 𝑉mx} of the 𝑁-body models may beconsidered unaffected by resolution artefacts as long as 𝑟mx > 8Δ𝑥(grid resolution being the main limitation for this set of simulations),which translates to a minimum remnant bound mass fraction of𝑀mx/𝑀mx0 ≈ 1/300.

2.6 Stellar tracers

We embed stars as massless tracers using the distribution function-based approach of Bullock& Johnston (2005) in the implementation3of Errani & Peñarrubia (2020). For spherical, isotropic models (theonly kind we consider here), the distribution function depends on thephase-space coordinates only through the energy 𝐸 = 𝑣2/2 +Φ(𝑟).We associate to each 𝑁-body dark matter particle in the initialconditions a probability

P★(𝐸) = (d𝑁★/d𝐸) /(d𝑁DM/d𝐸) (5)

of being drawn from a (stellar) energy distribution d𝑁★/d𝐸 . Usingthese tagging probabilities, P★(𝐸), as appropriately normalizedweights, the properties of different stellar tracer components can beinferred directly from the 𝑁-body particles. As stars are modelledas massless tracers of the underlying dark matter potential in thiswork, we make no distinction between stellar mass 𝑀★ and stellarluminosity 𝐿.

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Figure 1. Surface brightness profiles of stars and dark matter correspondingto the illustrative case shown in Fig. 2. The relative density normalizationbetween stars and dark matter is arbitrary. Initial profiles are shown with solidlines; those after 10 orbital periods with dashed lines. Projected half-lightradii, 𝑅h, as well as the characteristic radius of the underlying dark mattercomponent are highlighted using filled circles and filled triangles, respectively.Two different stellar tracers are shown, one with 𝑅h0/𝑟mx0 = 3/4, and a moresegregated one with 𝑅h0/𝑟mx0 = 1/4. After ten orbital periods, both stellarprofiles have similar half-light radii, 𝑅h ≈ 𝑟mx, as highlighted by the greyshaded band.

3 RESULTS

3.1 Tidal evolution of NFW subhalos

To illustrate some of the general features of the evolution of thestellar and dark matter components of NFW subhalos, we discusshere the results of a simulation of a system on an orbit with apericentre-to-apocentre ratio of 1:5. We assume that the initial stellarcomponents may be modelled as exponential spheres,

𝜌★(𝑟) = 𝜌★0 exp (−𝑟/𝑟★𝑠) , (6)

where 𝜌★0 is the initial central stellar density and 𝑟★s ≈ 𝑅h/2 is thescale radius, with 𝑅h denoting the projected stellar half-light radius.

We use the same simulation to follow the evolution of twoindependent stellar tracerswith different degrees of radial segregationrelative to the dark matter. In one tracer, the initial star-to-darkmatter radial segregation is 𝑅h0/𝑟mx0 = 3/4; the second tracer ismore deeply embedded inside the halo, with 𝑅h0/𝑟mx0 = 1/4. Theprojected density profiles of these two systems, normalised to thesame central density value, are shown in Fig. 1 with solid red andgreen lines, respectively. The dark matter is shown as well, witha blue line. (Note that the relative normalization of the density isarbitrary, as the stellar components are assumed to be “massless”.)

Figure 2 shows snapshots of the dark matter (top row) andof each of the two embedded stellar tracers (middle and bottomrows) in the initial conditions (left-most column), and at differentapocentric passages, chosen after 5, 10, 20 and 30 orbits. Theapocentre and pericentre radii are 200 kpc and 40 kpc, respectively,and 𝑇mx0/𝑇peri ∼ 0.9.

As discussed in Sec. 2.4, tides gradually strip the subhalo,causing 𝑉mx and 𝑟mx to decrease, quite rapidly at first, but slowingdown as the subhalo approaches the asymptotic remnant state, with𝑇mx ≈ 𝑇peri/4. The effects of tidal stripping on this orbit are quitedramatic: the mass within 𝑟mx declines to 13%, 6.3%, 2.6%, and1.4% of the initial value at each of the selected snapshots.

The evolution of the stellar tracers is qualitatively similar; theirhalf-light radii decrease gradually as a result of tidal stripping, and thereduction in size decelerates as the tidal remnant stage is approached.Interestingly, the half-light radii of both stellar tracers appear toconverge to the same value, despite the fact that they initially differedby a factor of 3. After 30 orbital periods, the difference in half-lightradius is just ∼ 10%. Furthermore, these half-light radii (shownby solid circles in the middle and bottom rows of Fig. 2) becomecomparable to the characteristic radius, 𝑟mx, of the dark matter (solidcircles in the top row of Fig. 2).

This is apparent in Fig. 1, where the dashed lines show thedensity profiles after 10 orbital periods. Solid circles on each curvemark the current value of the half-light radius. At the final time theradii are quite similar (as highlighted by the grey band), as discussedabove. The total mass lost is, however, quite different. Only ∼ 6%of the initial characteristic dark matter mass remains bound at thattime, compared with 𝐿/𝐿0 ≈ 0.24 and 0.01 for the more and lessdeeply embedded stellar component, respectively.

This discussion foretells some of our main results: “tidallylimited” systems, regardless of initial size, converge to a size set bythe characteristic radius of the remnant subhalo. As we shall seebelow, these remnants also converge to the same velocity dispersion,which itself traces the characteristic velocity of the remnant. Thisimplies that, in the “tidally limited” regime, the properties of thestellar component are poor indicators of the initial mass/size/velocityof its progenitor: stars, unlike the dark matter, follow no unique“tidal tracks” as they are stripped.

3.2 Dark Matter Evolution

As discussed in the previous section, the evolution of stars and darkmatter are closely coupled. We study the evolution of the dark matterin a bit more detail next, before discussing the evolution of the stellarcomponents in Sec. 3.3.

3.2.1 Circular velocity profile

As discussed by Errani & Navarro (2021), tides lead to a gradualchange in the shape of the mass profile of an NFW subhalo as well asto a reduction in its characteristic size and circular velocity. We showthis in Fig. 3, where we plot the evolution of the circular velocityprofile 𝑉c (𝑟) = [𝐺𝑀 (< 𝑟)/𝑟]1/2 of the subhalo shown in Fig. 2.

Curves are colour coded according to the remaining boundmass fraction,𝑀mx/𝑀mx0. The subhalo evolves from an initial NFWprofile towards a density profile well-described by an exponentiallytruncated cusp (Eq. 3) with corresponding circular velocity profile,

𝑉asy (𝑟) = 𝑉cut

[1 − (1 + 𝑟/𝑟cut) 𝑒−(𝑟/𝑟cut)

𝑟/𝑟cut

]1/2, (7)

with 𝑟cut ≈ 0.56 𝑟mx, 𝑉cut =√𝐺𝑀asy/𝑟cut ≈ 1.83𝑉mx and total

mass 𝑀asy = 4𝜋𝑟3cut𝜌cut.The distance between adjacent circular velocity curves de-

creases with decreasing remnant mass, as the subhalo asymptoticallyapproaches the asymptotic remnant state, where its crossing timeis roughly a quarter of the host halo crossing time at pericentre,𝑇mx ≈ 𝑇peri/4.

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Figure 2. Tidal evolution of an NFW subhalo (top row, with 𝑀mx0 = 106M� , 𝑟mx0 = 0.48 kpc and 𝑉mx0 = 3 km s−1) and two embedded exponential stellarcomponents of different initial size (middle and bottom row for 𝑅h0/𝑟mx0 = 3/4 and 1/4, respectively) on a 1:5 eccentric orbit in an isothermal potential. Eachpanel corresponds to a simulation snapshot taken at apocentre and shows the projected surface density. The surface density is normalised by the instantaneousmean density enclosed within 𝑟mx (or 𝑅h). As tides strip the subhalo, its characteristic size 𝑟mx decreases, and the relative change in size decreases withsubsequent pericentre passages: the tidal evolution slows down and a stable remnant state is asymptotically approached. Similarly, the half-light radii of embeddedstellar components decrease during tidal stripping. Crucially, for the later stages of the tidal evolution, the size of the half-light radius 𝑅h appears to followclosely the characteristic size 𝑟mx of the dark matter subhalo they are embedded in, independent of their initial extent.

3.2.2 Binding energy

The tidal evolution is particularly simple when expressed in termsof the initial binding energy4 of the subhalo particles (see; e.g.,Choi et al. 2009). This is shown in Fig. 4, where we show theenergy distribution of particles in the remnant at various times,again coloured by remaining bound mass fraction. We define initialbinding energies in dimensionless form, E, using the potentialminimum Φ0 of the initial subhalo (this is a well-defined quantityfor an NFW profile, which has a finite central escape velocity),Φ0 ≡ Φ(𝑟 = 0) = −4𝜋𝐺𝜌s𝑟2s ≈ −4.63𝑉2mx,

E ≡ 1 − 𝐸/Φ0, (8)

and 𝐸 = 𝑣2/2 + Φ(𝑟). The most-bound state is E = 0, and theboundary between bound and unbound lies at E = 1. NFW energydistributions are well approximated by a power-law for E → 0. Thedefinition used in Eq. 8 is particularly appropriate when describing

4 For simplicity, we compute binding energies in the rest frame of the subhaloand do not include the gravitational effects of the main halo. Recall that ouranalysis focusses on the subhalo remnant structure at apocentre, where thesubhalo spends most of its orbital time and where the effect of the main halois minimal.

highly-stripped systems, where the least bound particles are the mostlikely to be stripped away by tides.

Indeed, as shown in Fig. 4, as tides strip the subhalo the energydistribution is gradually trimmed off, leaving only the initially mostbound particles in the final remnant (see also Drakos et al. 2020;Stücker et al. 2021; Amorisco 2021). The truncation is quite sharp,and may be approximated by a “filter function”,

d𝑁/dE|i,t =d𝑁/dE|i

1 +(𝑎 E/Emx,t

)𝑏 , (9)

with 𝑎 ≈ 0.85, 𝑏 ≈ 12, where d𝑁/dE|i denotes the initial NFWenergy distribution and Emx,t the peak of the truncated energydistribution (see arrow in Fig. 4 marking the location of the peak forthe final snapshot).

As tides strip the system, the peak energy Emx,t shifts towardsmore and more bound energies in the initial conditions. We thereforeexpect a strong correlation between the remnant bound mass, andEmx,t. This is shown in the top panel of Fig. 5, where Emx,t is seento decrease with decreasing bound mass fraction, 𝑀mx/𝑀mx0. Therelation may be approximated by a power law,

Emx,t ≈ 0.77 (𝑀mx/𝑀mx0)0.43 , (10)

applicable for 𝑀mx/𝑀mx0 < 1/3. Similarly, the middle and bottompanels of Fig. 5 show the corresponding dependence of Emx,t on

MNRAS , 1–15 (2021)

6 Errani et al.

−1

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mx0

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20T

orb

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Figure 3. Circular velocity profiles 𝑉c (𝑟 ) of the subhalo shown in Fig. 2.Each curve corresponds to a snapshot at a subsequent apocentric passage. Astides strip the subhalo, the characteristic size, 𝑟mx, and velocity, 𝑉mx, of thesubhalo decrease along well-defined tidal tracks (dashed black curve). Thedistance between subsequent circular velocity curves appears to decreasewith decreasing remnant bound mass 𝑀mx/𝑀mx0 (see colour-coding), andtidal evolution virtually ceases once a remnant state is reached where thecharacteristic crossing time of the subhalo,𝑇mx = 2𝜋𝑟mx/𝑉mx, is determinedby the orbital time at pericentre, 𝑇mx ≈ 𝑇peri/4. For reference, the grid sizeof the particle mesh code is shown using a grey vertical line, and a time scalecorresponding to 10 times the simulation time step Δ𝑡 is shown with a dottedblack line.

−5

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Initial conditions

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/dE

log10 Ei

NFWfilter

−2

−1.5

−1

−0.5

0

log 1

0M

mx/M

mx0

Emx,t

Figure 4. Initial binding energy distribution of particles that remain boundto the subhalo after subsequent pericentric passages. Initial energies aremeasured relative to the potential minimum, E = 1 − 𝐸/Φ0, where forNFW halos Φ0 ≈ −4.63𝑉 2mx (for the 𝑁 -body realisation of an NFW haloused in this work, exponentially truncated at 10 𝑟s, see Sec. 2.1, we findΦ0 ≈ −4.32𝑉 2mx). The characteristic bound mass fraction, 𝑀mx/𝑀mx0, ofthe remnant is used for colour-coding. As tides strip the system, the remnantconsists of particles of increasingly bound initial energies. The maximumof the initial energy distribution corresponding to a specific bound remnantis denoted by Emx,t and indicated for the most-stripped remnant using anarrow. The empirical filter function (Eq. 9), applied to the initial NFW energydistribution, is shown for the snapshots of Fig.2, and matches well the energydistribution of the bound particles.

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x0 fit + tidal track

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0

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log 1

0V

mx/V

mx0

log10 Emx,t

Figure 5. The top panel shows the tidal truncation energy scale Emx,t in theinitial conditions (see arrow in Fig. 4) as a function of bound mass fraction𝑀mx/𝑀mx0 of the relaxed system, which may be well approximated bya power law (for 1/100 < 𝑀mx/𝑀mx0 < 1/3, fit Eq. 10, dashed curve).The middle and bottom panels show the corresponding correlations for thecharacteristic size, 𝑟mx/𝑟mx0, and characteristic velocity, 𝑉mx/𝑉mx0, of theremnant, respectively. The dashed curves in the middle and bottom panel areobtained by combining the fit (Eq. 10) with the tidal tracks (Eq. 4).

the characteristic size, 𝑟mx/𝑟mx0, and velocity, 𝑉mx/𝑉mx0, of theremnant.

We show in Appendix A that similar results are obtained whenusing initial orbital times instead of binding energy.

We have so far described the remnant using the distributionof initial binding energies. Of course, as mass is lost, the remnantrelaxes and binding energies change. In particular, as the shape of theremnant density profile converges to an exponentially truncated cuspthe energy distribution of the remnant converges to a well-definedshape depending only on the current potential minimum, Φ0 (𝑡). Foran exponentially truncated cusp (Eq. 3), the potential may be writtenas

Φasy (𝑟) = Φ0,asy1 − 𝑒−𝑟/𝑟cut

𝑟/𝑟cut, (11)

with Φ0,asy = −𝐺𝑀asy/𝑟cut ≈ −3.35𝑉2mx.The energy distribution of an isolated, exponentially truncated

NFW cusp with isotropic velocity dispersion may be computedanalytically, and is shown by a black curve in Fig. 6. As is clear fromthis figure, the remnant final energy distribution quickly approachesthat of an exponentially truncated cusp. In contrast to the initialNFW profile (shown, for reference, as a dashed curve with arbitrarynormalisation), which has most mass at low binding energies, theexponentially truncated cusp has a well defined peak energy, beyondwhich the energy distribution drops steeply.

Once the subhalo has been stripped to less than 10 per centof its initial mass, the energy distribution corresponding to anexponentially truncated density cusp provides a good description forthe energy distribution measured for the N-body remnant. Deviationsat small values of Ef are expected due to limitations in numericalresolution introduced by the finite code grid size, Δ𝑥; energy values

MNRAS , 1–15 (2021)

Tidally limited satellite galaxies 7

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Relaxed system

log 1

0dN

/dE

log10 Ef

NFWtruncated cusp

−2

−1.5

−1

−0.5

0

log 1

0M

mx/M

mx0

Figure 6.Energy distribution of the subhalo remnant at subsequent apocentricpassages. Energies are defined as E = 1 − 𝐸/Φ0, with Φ0 ≈ −3.35𝑉 2mx asexpected for exponentially truncated NFW cusps. Curves are colour codedby the bound mass fraction, and normalized to the same total mass, for easeof comparison. For remnant masses of 𝑀mx/𝑀mx0 . 1/10, the shape ofthe energy distribution evolves only weakly, as the density profile convergesto its asymptotic shape. Note that the energy distribution as measured inthe presence of the host halo tidal field and extra-tidal material does deviatefrom that of an (isolated, isotropic) exponentially truncated cusp, shown assolid black line. Energies likely compromised by numerical resolution (i.e.𝐸 < Φ(8Δ𝑥)) are shown in grey.

log 1

0E f

log10 Ei

median16th, 84th percentile

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0

log 1

0d2 N

/dE

idE f

E f∝E i

Emx,t

10 Torb

Figure 7. Initial (Ei) vs “final” (Ef) energy after 10 orbital periods forthe bound remnant shown in Fig. 2. The colour-coding corresponds to thenumber of bound particles in each {Ei, Ef } pixel. For particles with bindingenergies below the tidal energy-cut Emx,t (indicated by an arrow), the medianrelation of the mapping is approximately linear (Ef ∝ Ei). The 16th and 84thpercentiles of the scatter around the median relation are shown using dashedcurves.

corresponding to energies smaller than Φ(8Δ𝑥) are shown in greyin Fig. 6.

3.2.3 Initial-to-final energy mapping

The initial and final binding energies of particles that remain boundare closely related. We illustrate this dependence in Fig. 7, whichshows final energies as a function of their initial values for the

log 1

0E f

log10 Ei/Emx,t

fit±0.03dexunresolved

−1

−0.5

0

−1 −0.5 0 0.5−2

−1

0

log 1

0M

mx/M

mx0

Figure 8.Median relation shown in Fig. 7 for all snapshots with bound massfraction 𝑀mx/𝑀mx0 > 1/100 (see colour coding), but for initial energies,E𝑖 , scaled to the tidal truncation energy scale, Emx,t. The relation betweenfinal and initial energies is approximately independent of tidal mass loss inthese scaled units. The fit proposed in Eq.12 is shown by the solid black line.

subhalo highlighted in Fig. 2, after 10𝑇orb. The relation is almostlinear for most energy values, except for particles initially less boundthan the “peak” energy Emx,t.

Expressing initial binding energies in terms of this peak res-ults in a relation between initial and final energies that is almostindependent of the degree of tidal stripping. This is shown in Fig. 8,where each curve shows the median relation between between Efand Ei/Emx,t for all of our remnants. The relation is nearly inde-pendent of the remaining bound mass, especially for heavily strippedsystems with 𝑀mx/𝑀mx0 < 0.1. The median behaviour may beapproximated by a simple empirical fit,

Ef =[1 + (𝑐 Ei/Emx,t)𝑑

]1/𝑑, (12)

with 𝑐 = 0.8 and 𝑑 = −3.

3.3 Evolution of the stellar component

The results of the previous subsection enable a thorough descriptionof the tidal evolution of the dark matter component of an NFWsubhalo. Under the assumption that stars may be treated as tracersof the gravitational potential, their evolution would just follow thatof a subset of the dark matter, defined mainly by their initial bindingenergy distribution.

Let us consider again the example considered in Fig. 2. Theinitial energy distributions of stars and dark matter are shown withsolid curves in Fig. 9. For exponential profiles, the most boundregions are well approximated by power-laws that peak at E★ (acharacteristic value which depends on the radial segregation of starsand dark matter) and are sharply truncated beyond E★.

The initial energy distribution of particles that remain boundafter 10 orbital periods are shown by the lines connecting circlesof the corresponding color. The black dashed curves overlappingeach of these lines indicate the result of applying the “filter function”introduced in Sec. 3.2.2 (Eq. 9). Note that the same filter, appliedto either stars or dark matter, yields an excellent description of theinitial energy distribution of the particles that remain bound after10𝑇orb.

This has a number of important implications. It suggests that

MNRAS , 1–15 (2021)

8 Errani et al.

−4

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dark matter

Rh/rmx

=1/4

R h/rmx

=3/

4

E?

E?

Rh/rmx = 1/8

Emx,t

log 1

0dN

/dE

log10 E

initial10 Torbfilter

Figure 9. Initial binding energy distribution of dark matter (blue) and stars(red and green for 𝑅h0/𝑟mx0 = 1/4 and 3/4, respectively) for the simulationshown in Fig. 2. The initial distribution of all particles is shown using solidlines, and the distributions of those particles which remain bound after tenorbital periods are shown using dotted lines. After ten orbital periods, thetidal energy cut reaches well within both stellar distributions, and truncatesthem at approximately the same energy. The result of applying the empiricalfilter function (Eq. 9; see also Fig. 4) is shown using black dashed curves,which match the initial energy distributions of dark matter and stars thatremain bound after ten orbital periods remarkably well.

(i) in practice the only parameter that matters when determining theprobability that a given particle will remain bound (or be stripped) isits initial binding energy. It also implies that (ii) stellar componentswhose initial energy distributions peak beyond the truncation energy,Emx,t, will share the same outermost energy distribution as the darkmatter remnant. Finally, (iii) it suggests that the final structure of thebound stellar remnant will depend on how stars populate energiesmore bound than Emx,t in the initial system; in other words, the“tidal tracks” and final structure of the stellar remnant should dependsensitively on the stars’ initial density profile and radial segregationrelative to the dark matter. We explore these implications in moredetail next.

3.3.1 Initial and final stellar remnant structure

As mentioned above, our results suggest that the structure of thestellar remnants will be linked to its initial energy distribution andto the degree of stripping undergone by the subhalo. We adopt aflexible parameterization of the stellar energy distribution in order toexamine the relation between the initial and final density profiles ofthe stars. In particular, we explore initial binding energy distributionsof the form:

d𝑁★/dE =

{E𝛼 exp

[−(E/Es)𝛽

]if 0 ≤ E < 1

0 otherwise,(13)

where, as usual, E = 1−𝐸/Φ0. The distribution behaves like a power-law towards the most-bound energies (E → 0), is exponentiallytruncated beyond some scale energy Es, and it peaks at E★ =

Es (𝛼/𝛽)1/𝛽 < 1.Fig. 10 shows two examples, one with 𝛼 = 𝛽 = 3 (shown in red)

and another with 𝛼 = 𝛽 = 6 (shown in green). The value of Es ischosen so that both models have the same initial 2D half-light radius,𝑅h0/𝑟mx0 = 1/4 (isotropic solutions for different initial half-lightradii are discussed in Appendix C). As 𝛼 = 𝛽, the scale energy andpeak energy coincide; i.e., E★ = Es ≈ 1/3. The 𝛼 = 𝛽 = 3 model

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Exp.

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mer

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= 3

α=β

=6

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/dE

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Eddington inversionfit

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)/ρ(0

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Exp.Plummerα=β =3α=β =6

−2

−1

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−2 −1 0

2D1/21/2

Rh

log 1

0Σ

(R)/

Σ(0

)

log10 R/rmx

Figure 10. Density (bottom left) and surface density profiles (bottom right)corresponding to the energy distribution of Eq. 13 (with 𝛼 = 𝛽 = 3 shownin magenta, and 𝛼 = 𝛽 = 6 in orange), for stellar tracers with radius𝑅h0/𝑟mx0 = 1/4 embedded in an NFW dark matter halo. For reference,the corresponding profiles for exponential and Plummer spheres are shownas grey solid and dashed curves, respectively. The top panel shows theenergy distributions corresponding to the density profiles shown below. Theenergy distribution for exponential and Plummer spheres is computed usingEddington inversion and shown as grey dots; grey solid curves show fits ofEq. 13 to these distributions, highlighting that the functional form of Eq. 13is sufficiently flexible to enable the modeling of a representative range ofdifferent stellar systems.

spans a much wider range of energies than the 𝛼 = 𝛽 = 6 model (seetop panel), and thereby also samples more energies that probe deeperinto the dark matter density cusp. As a consequence, the resultingsurface brightness profile of the 𝛼 = 𝛽 = 3 model has a smaller coreradius (defined as the radius where the projected stellar density dropsby a factor of 2 from its central value; i.e., Σ(𝑅c) = Σ(0)/2) thanthe 𝛼 = 𝛽 = 6model. The corresponding 2D and 3D density profilesare shown in the bottom panels of Fig. 10. For reference, the ratio𝑅c/𝑅h between core and half-light radius in the initial conditions is0.5 and 0.9 for 𝛼 = 𝛽 = 3 and 𝛼 = 𝛽 = 6, respectively.

Many different density profiles may be approximated by varying𝛼 and 𝛽 in Eq. 13. For example, a Plummer model may be approx-imated quite well by 𝛼 = 14, 𝛽 = 0.45, whereas exponential profilesare well fit by 𝛼 = 3.3, 𝛽 = 4.0. Indeed, the density profile for the𝛼 = 𝛽 = 3 case is almost indistinguishable from an exponentialprofile over two decades in radius, as may be seen in the bottompanels of Fig. 10.

As expected, the final and initial energy distribution of thestars are highly dependent on each other. For example, we show inFig. 11 the final energy distributions for the 𝛼 = 𝛽 = 3 (magenta)and 𝛼 = 𝛽 = 6 (orange) stellar tracers in the “tidally limited”regime; i.e., after substantial mass loss. Various snapshots areshown at apocentric passages and span a large range of bound mass

MNRAS , 1–15 (2021)

Tidally limited satellite galaxies 9

−1

0

1

−0.5 −0.4 −0.3 −0.2 −0.1 0

1/10 > Mmx/Mmx0 > 1/100

log 1

0dN

/dE

log10 Ef

α= β

= 3

α=β

=6

dark matter

Figure 11. Energy distributions of dark matter (blue) and stars (magenta andorange for initial 𝛼 = 𝛽 = 3 and 𝛼 = 𝛽 = 6 profiles, respectively) in therelaxed remnant system (arbitrary normalisation). Only snapshots where thetidal energy truncation reaches well into the initial energy distribution ofthe stars are shown. All three energy distributions peak at approximately thesame energy (log10 Ef ≈ −0.15). The energy distribution for an isolated,exponentially truncated cusp with isotropic velocity dispersion is shown as ablack dashed curve. For the stellar models, energy distributions computedusing the empirical model of Appendix E are shown as black solid curves.

fraction: 1/100 < 𝑀mx/𝑀mx0 < 1/10. Energies Ef = 1−𝐸/Φ0 arenormalised by their instantaneous potential minimum (approximatedasΦ0 ≈ Φ0,asy ≈ −3.35𝑉2mx). The shape of the energy distributionsconverges rapidly (as shown in Fig. 6 for dark matter alone), andlittle evolution in the energy distribution shape is seen for remnantmasses smaller than 0.1𝑀mx0.

The final energy distributions of the remnants may also becomputed using the “filter” function of Eq. 9 to select bound particlesfrom the initial conditions, combined with the initial-to-final energymapping given by Eq. 12. The result of this calculation is shownby the solid black lines in Fig. 11 and are seen to approximate wellthe results of the simulations. A step-by-step description of how toimplement this calculation is given in Appendix E.

The different initial energy distributions give rise to differentdensity profiles for the stellar remnants, depending on the valuesof 𝛼 and 𝛽 chosen. We show this in Fig. 12, where we plot the 3Ddensity profiles corresponding to the energy distributions shownin Fig. 11. Radii are scaled to the instantaneous characteristic sizeof the underlying dark matter halo, 𝑟mx (𝑡). All profiles are sharplytruncated beyond 𝑟mx, and have similar outer slopes (consistent withthe findings of Peñarrubia et al. (2009), who studied tidal strippingof stellar King models deeply embedded within NFW subhalos).The inner regions, however, differ, retaining clean memory of theirinitial profiles. As in the initial conditions, the 𝛼 = 𝛽 = 6 remnantprofile has a larger ratio of core to half-light radius (𝑅c/𝑅h ∼ 0.8)than the 𝛼 = 𝛽 = 3 profile (𝑅c/𝑅h ∼ 0.6).

3.3.2 Tidal tracks

The results presented in Fig. 9 suggest that luminosity and size ofthe stellar components should remain largely unaffected by tidesuntil the truncation in energy starts to overlap the energy distributionof the stars. Using the notation illustrated in that figure, this occurswhen Emx,t becomes comparable or smaller than E★. We showthis in Fig. 13, where we plot the evolution of the stellar projected

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−1 −0.5 0 0.5

dark matter

stars α = β = 3

stars α = β = 6

3D

log 1

0ρ

log10 r/rmx(t)

truncated cuspmodel

Figure 12. Density profiles corresponding to the models discussed in Fig. 11(vertical normalisation is arbitrary). While the characteristic size of all threeprofiles is set by the tides (𝑅h ∼ 𝑟mx), the inner regions of the stars retainmemory of the initial profile and energy distribution. The tidally limiteddensity profile of the NFW subhalo is well-described by an exponentiallytruncated cusp (dashed curve). For the stellar models, density profileshave been computed from the empirical energy distribution discussed inAppendix E, and are shown as solid black curves. Radii which are likelyaffected by the limited numerical resolution are shown in grey (see Sec. 2.5).

half-light radius 𝑅h, as well as of the central line-of-sight velocitydispersion 𝜎los and luminosity (i.e., stellar mass), 𝐿, of the twostellar populations highlighted in the example shown in Fig. 2.

The evolution is plotted as a function of the tidal truncationenergy, Emx,t, which is a proxy for the total mass loss due to tides(see Fig. 5). The smaller Emx,t the higher the mass loss, so tidalevolution runs from right to left in Fig. 9. The location of the peakE★ for each of the two initial stellar energy distributions is indicatedby the vertical arrows.

As expected, if the tidal truncation energy does not overlap thestellar energy distribution (i.e., if Emx,t > E★), the size of the stellarcomponent remains largely unaffected by tides. However, once Emx,tdrops below E★, the stellar components rapidly lose mass, decreasein size, and gradually become kinematically colder. As expected,the more extended stellar component (𝑅h0/𝑟mx0 = 3/4) is morereadily affected than the more deeply segregated component with𝑅h0/𝑟mx0 = 1/4.

Interestingly, once Emx,t becomes much smaller than E★, thestellar half-light radius converges to the dark matter characteristicradius, 𝑟mx, regardless of its initial value. In addition, the velocitydispersion becomes directly proportional to the dark matter charac-teristic velocity, 𝑉mx. In other words, the stellar components of these“tidally limited systems” become direct tracers of the characteristicparameters of the tidal remnant, almost regardless of their initialradial segregation (see, also, Kravtsov 2010).

This is a general result, which applies to all systems in the“tidally limited” regime. We illustrate this in Fig. 14 where we plotthe “tidal tracks” of 𝛼 = 𝛽 = 3 systems with different initial radialsegregation; i.e., 𝑅h0/𝑟mx0 = 1/2, 1/4, 1/8, and 1/16, each shownwith different symbols (Appendix D shows the same tidal tracks inunits of remnant bound mass fraction 𝑀mx/𝑀mx0). As the subhalogets stripped, its characteristic radius and velocity (𝑟mx and 𝑉mx)follow the track indicated by the dashed line. The 3D half-lightradius and velocity dispersion of the stellar components (the latter

MNRAS , 1–15 (2021)

10 Errani et al.

multiplied by√3 to convert it to a circular velocity) evolve as well,

generally getting smaller and colder as stripping progresses.Only the most embedded tracers “puff up” initially, increasing

their size5 as they get kinematically colder, but the net increase issmall. Interestingly, once the stellar component parameters reachthe tidal track of the dark matter remnant, they follow it closely.Qualitatively, this result applies to all the values of 𝛼 and 𝛽 we havetried, and it therefore appears of general applicability: the size andvelocity dispersion of a tidally-limited stellar system traces closelythe 𝑟mx and 𝑉mx of the dark remnant6.

This result has a couple of interesting implications. One isthat there is no unique “tidal track” describing the evolution of astellar system in the size-velocity plane, as had been suggested byPeñarrubia et al. (2008), i.e., the tidal track depends on the initialdistribution of stars within the subhalo. With hindsight, it is easy totrace that conclusion to the fact that those authors tested a singlestellar model (a King profile) and a rather limited range of radialsegregation. In fact, Fig. 14 shows clearly that the same stellartidal remnant may be produced from very different initial radialsegregations, each following different tracks.

A second implication is that the tidal track of a subhalo (i.e., thedashed line in Fig. 14) sets a firm lower limit on the velocity dispersionof an embedded stellar remnant of given size. (Or, equivalently, anupper limit to the size of an embedded system of given velocitydispersion.)

We discuss next an application of this argument to the inter-pretation of the sizes and velocities of dwarf galaxies in the LocalGroup.

3.4 Application to Local Group dwarfs

We now use the results of the previous subsections to interpretavailable structural data on Local Group dwarfs, in the contextof LCDM. As discussed in Sec. 1, the assumption of a minimummass for galaxy formation, such as that expected from the hydrogencooling limit, together with the mass-concentration relation expectedfor NFW halos in LCDM, yield strong constraints on the size andvelocity dispersion of dwarf galaxies.

Following Fattahi et al. (2018), the grey band in Fig. 15 indicatesthe mass profiles of NFW halos with 𝑉mx in the range 20-40 km/s,the virial mass range expected to host all isolated dwarfs with𝑀★ < 107 𝑀� . The solid orange line indicates the 𝑟mx-𝑉mx relationexpected from the LCDM mass-concentration relation (Ludlowet al. 2016). The accompanying error band delineates the one- andtwo-sigma scatter inferred from cosmological N-body simulations.

For dSphs, the velocity dispersion of the stars may be used toestimate the circular velocity, 𝑉h, at the 3D stellar half-light radius,𝑟h, using the Wolf et al. (2010) mass estimator7. While not exact,

5 Appendix B discusses in more detail the slight expansion of stellar tracersbefore the onset of strong tidal stripping, as well as the drop in velocitydispersion even before a reduction in size is observed.6 For stellar models with initial 𝛼 = 𝛽 = 3 and 𝛼 = 𝛽 = 6 energydistributions, in the tidally limited regime, we measure 𝑅h ≈ 0.93 𝑟mx,𝜎los ≈ 0.70𝑉mx, and 𝑅h ≈ 1.26 𝑟mx, 𝜎los ≈ 0.76𝑉mx, respectively, seeAppendix D.7 In contrast to the Walker et al. (2009) and the Errani et al. 2018 massestimators, which are calibrated to estimate the mass enclosed within the 2Dprojected half-light radius 𝑅h (and within 1.8𝑅h, respectively), the Wolfet al. (2010) estimator is tailored to estimate the mass enclosed within the3D deprojected half-light radius, which facilitates the comparison to circularvelocity curves.

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E?

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Rh0/rmx0 = 3/4Rh0/rmx0 = 1/4tidal track

rmx

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mx0

log10 Emx,t

Vmx

−3

−2

−1

0

−1 −0.5 0

log 1

0L/L

0log10 Emx,t

Figure 13. Evolution of projected stellar half-light radius, 𝑅h, line-of-sightvelocity dispersion, 𝜎los, and luminosity (stellar mass), 𝐿, as a function ofthe tidal energy cut Emx,t, for the simulation shown in Fig. 2. Half-mass radiiand velocity dispersions are normalised to the initial characteristic radius,𝑟mx0, and characteristic velocity, 𝑉mx0, of the dark matter halo, respectively.The evolution of 𝑟mx and𝑉mx is shown using black solid curves, and proceedsfrom right to left in these panels. Before the tidal energy cut reaches thecharacteristic energy of the stellar tracer (i.e., while Emx,t > E★), the stellarhalf-light radius and total luminosity change only slightly. Once Emx,t . E★,the half-light radius decreases rapidly and approaches the characteristicradius 𝑟mx of the dark matter halo. Similarly, for Emx,t . E★, the line-of-sight velocity dispersion closely traces the subhalo characteristic velocity,𝜎los ≈ 𝑉mx/2, and the total luminosity of the bound stellar component dropsrapidly.

we expect these estimates to be accurate to about 0.1 dex (see alsoWalker et al. 2009; Campbell et al. 2017; Errani et al. 2018). Dwarfsunaffected by tides should lie on the grey area if they follow LCDMpredictions. Encouragingly, Local Group isolated (field) dwarfs,shown by magenta symbols in Fig. 15, seem to be in reasonableagreement with that prediction.

Systems below the grey band are, in the LCDMcontext, typicallyinterpreted as satellites whose velocity dispersions and sizes havebeen affected by tides, bringing them below the grey band. Indeed, nodwarf with𝑀★ < 107 𝑀� should fall above the grey band, accordingto LCDM. The ones that seem to deviate from that precept, such asBootes 2 and Carina 3, have fairly large uncertainties in their circularvelocity estimates, mainly due to the extremely small number of starswith measured radial velocities. Furthermore, Ji et al. (2016) arguethat the velocity dispersion of Bootes 2 is likely over-estimated dueto the presence of binaries, and that the reported velocity dispersionis best seen as an upper limit. It remains to be seen whether thesegalaxies are actually in conflict with LCDM once their velocitydispersions are better constrained.

Galaxies below the grey band are all satellites, a requisite fora tidal interpretation of their location in the size-velocity plane.However, tides cannot shift them to arbitrarily low velocities withoutaffecting their size, as shown by our discussion of the tracks inFig. 14. Indeed, the tidal track of the halo with the minimum massneeded to form a galaxy should provide firm limits on the size and

MNRAS , 1–15 (2021)

Tidally limited satellite galaxies 11

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Figure 14.Three-dimensional half-light radii, 𝑟h, as well as circular velocitiesat that radius, 𝑉h ≡ 𝑉c (𝑟h) measured from the simulations described inSec. 2. The evolution of four different stellar tracers with initial segregations𝑅h0/𝑟mx0 = 1/2, 1/4, 1/8, 1/16 are shown using different symbols. Symbolsare colour-coded by bound mass fraction. The circular velocity profiles ofthe remnants for𝑀mx/𝑀mx0 = 1, 1/10 and 1/100 are shown using colouredsolid lines. Extended stellar tracers rapidly decrease in size once tides truncatethe underlying subhalo, but more segregated tracers first expand slightlybefore being truncated by the tides. The parameters of embedded stellarsystems stay always close or above the dark matter subhalo tidal track (blackdashed curve).

velocity of equilibrium “tidally limited” dwarfs. This may be seenas a lower limit on the velocity at given size, or, equivalently, as anupper limit to the size given a velocity.

This limit is indicated by the blue line in Fig. 15, which outlinesthe tidal track of a 𝑉mx = 20 km/s halo of average concentration.This restricts the loci of tidally limited dwarfs to the region shadedin blue: no dwarf should populate the region below the blue line.

Interestingly, there are a number of satellites that seem toviolate this condition. These are galaxies of unusually low velocitydispersion for their size, like the MW “feeble giant” satellites Crater2 and Antlia 2, as well as the M31 satellites And 19, And 21 and And25. Our results imply that tidal effects are not a viable explanationfor these galaxies in the context of LCDM.

This conclusion echoes the conclusions of some earlier work,such as that of Sanders et al. (2018), who reported “tension” whentrying to account for the size of Crater 2 in tidally limited NFWhalos, and argued that the tension may be alleviated if the progenitorsubhalo had a sizable constant-density “core”.

On the other hand, Frings et al. (2017) reports simulations thatcan apparently produce satellites as large and as kinematically coldas Crater 2. However, these results are likely affected by numericallimitations. For example, the closest “Crater 2 analogue” that theyreport corresponds to a halo whose maximum circular velocity hasbeen reduced to nearly 1/10th of its original value. As discussedby Errani & Navarro (2021), adequately resolving the tidal remnantin that case is extremely difficult, and would require many moreparticles and much better spatial resolution than used by Frings et al.(2017). In addition, some of the Frings et al. (2017) satellites havedensity profile cusps somewhat shallower than NFW, hindering adetailed comparison with our results.

Finally, Amorisco (2019) argued that transients induced by

tides may be the cause of the large radii and low velocity dispersionsof these objects. This interpretation could in principle be checkedby looking for other signs of ongoing tidal disturbance, such as avelocity gradient across the system (Li et al. 2021), extended tidaltails, or “bumps” in the surface density profile caused by escapingstars (Peñarrubia et al. 2009). We have not considered such transientsin the analysis presented here, which only considers the (equilibrium)structure of galaxies near apocentre.

To summarize, galaxies below the blue tidal track shown inFig. 14 present a serious challenge to the tidal interpretation suggestedby models based on the LCDM scenario. Possible resolutions ofthis challenge include: (i) revisions to the observed photometric orkinematic parameters of these galaxies, many of which are extremelychallenging to study due to their extremely low surface brightness;(ii) the possibility that these galaxies are not bound, equilibriumsystems, but are in the process of being tidally stripped, (iii) thatthese systems formed in halos with masses substantially below thehydrogen cooling limit, or (iv) that the dark matter halo of thesegalaxies deviates from the NFW shape, perhaps signalling the effectsof baryons on the inner cusp of the halo even in ultrafaint galaxies,or, perhaps more intriguingly, effects associated with the intimatenature of dark matter, such as finite self-interactions, or other suchdeviations from the canonical LCDM paradigm.

4 SUMMARY AND CONCLUSIONS

We have studied the tidal evolution of dark matter-dominated satel-lites embedded in LCDM dark matter subhalos as they orbit inthe potential of a massive galaxy. Our study assumes that subhalosmay be approximated by spherically symmetric, isotropic, cuspyNavarro-Frenk-White density profiles on eccentric orbits. We furtherassume that subhalo masses are much smaller than the host halomass, so as to neglect the effects of dynamical friction. Our mainfindings may be summarized as follows.

(i) Tides lead NFW subhalos to evolve following “tidal tracks”that describe the changes in characteristic size and velocity as tidalmass losses accumulate. These tidal tracks lead to a well definedasymptotic tidal remnant with characteristic crossing time set by theorbital time at pericentre, 𝑇mx ≈ 𝑇peri/4. The asymptotic structure ofthe tidal remnant is well approximated by an exponentially truncatedcusp.(ii) Tidal evolution can be conveniently modelled in terms ofbinding energy; tides gradually truncate subhalos energetically,systematically removing particles with low binding energies. Thebound remnant consists of particles initially more strongly boundthan the energy threshold imposed by tides.(iii) For the majority of particles, the initial binding energy alone issufficient to determine when and whether a particle will be strippedor not. Initial and final binding energies are strongly correlated, ina way that allows the structure of the remnant to be inferred as afunction of tidal mass loss.(iv) If gravitationally unimportant, the stellar components of tidallystripped NFW subhalos evolve according to how stars populate theinitial energy distribution of the subhalo. The same gradual energytruncation applies to dark matter and stars, independent of the initialdensity structure or radial segregation of the stellar component.(v) The structure of stellar remnants retain memory of their initialstructure, and, in particular, of how stars populate the most boundenergy levels of the inner dark matter cusp. Their evolution is thussensitively dependent on their initial structure and radial segregation.

MNRAS , 1–15 (2021)

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Figure 15. Three-dimensional half-light radii, 𝑟h, and circular velocities 𝑉h, for 𝐿 < 107 L� Milky Way- and Andromeda satellites, as well as Local Group fieldgalaxies (left panel). Observational values are estimated using the Wolf et al. (2010) mass estimator (𝑟h ≈ 4/3𝑅h, 𝑉h ≈

√3𝜎los) from the half-light radii and

velocity dispersions listed in McConnachie (2012) (version January 2021, and updated velocity dispersions for Antlia 2, Crater 2 (Ji et al. 2021), Tucana (Taibiet al. 2020), And 19 (Collins et al. 2020) and And 21 (Collins et al. 2021)). The cosmological 𝑟mx-𝑉mx (i.e., mass-concentration) relation at redshift 𝑧 = 0 isshown using a solid red curve, with yellow bands corresponding to 0.1 dex scatter in concentration (Ludlow et al. 2016). A grey band indicates NFW halos withmaximum circular velocity in the range 20 km s−1 < 𝑉mx < 40 km s−1, and delineates the regions where all isolated dwarfs should lie (see,e.g., Fattahi et al.2018). A tidal track from the lowest-mass subhalo of that band (blue solid curve) puts a lower bound on the region where tidally-stripped dwarf galaxies could befound. Most satellite galaxies shown fall in either the grey shaded or blue shaded region and are consistent with these constraints. Dwarfs outside the blue andgray regions are not easily explained in this framework. The panel on the right shows an alternative view of the same data, but cast in terms of mean density ��hwithin the half-light radius 𝑟h. The dwarf galaxies that fall below the grey and blue shaded regions have densities too low to be consistent with stripped NFWsubhalos. See text for further discussion.

Unlike the dark matter, stellar components do not follow unique“tidal tracks” in stellar mass, size, and velocity dispersion.(vi) The stellar component may be completely dispersed by tidesif the stars don’t populate the most bound energy levels of thedark matter cusp, making the potential existence of “micro-galaxies”critically dependent on how stars populate the most bound energylevels of an NFW subhalo.(vii) “Tidally limited” satellites, defined as those which have losta substantial fraction of their initial stellar mass, have radii andvelocity dispersions that trace directly the characteristic radius andvelocity of the subhalo remnant.

Most Local Group dwarfs have structural parameters consistentwith them being embedded in cuspy NFW halos, but there are also anumber whose dynamical masses are below what is expected fromLCDM. The properties of such systems, all of which are M31 orMW satellites, are usually assumed to result from the effects of tidalstripping. Our work sheds further insight into this interpretation.

Indeed, coupled with the assumption of a minimum halo mass,such as that suggested by hydrogen cooling limit considerations, ourresults place strong constraints on the size and velocity dispersionof tidally limited systems embedded in the remnants of cuspy NFWsubhalos. These constraints may be expressed as a firm lower limiton the velocity dispersion of an embedded stellar remnant of givensize, or, equivalently, as an upper limit to the size of an embeddedsystem of given velocity dispersion. Inspection of available data fordwarf galaxies in the Local Group, however, reveals a number ofultrafaint satellites that breach these limits.

The findings we report here imply that such systems cannot beunderstood as equilibrium systems embedded in the tidal remnantsof cuspy NFW subhalos. This presents a serious challenge to ourunderstanding of the formation and evolution of these unusual

galaxies in LCDM. Reconciling systems like Crater 2, Antlia 2,or And 19 with LCDM seems to require substantial revision toat least one of the assumptions of this work. Either the reportedobservational parameters of such systems are in substantial error,or the systems are far from dynamical equilibrium, or they inhabitsubhalos whose density profiles differ significantly from NFW.

Neither alternative is particularly attractive in the context ofLCDM. It may indicate that baryons may affect the inner halo cuspeven in extremely faint dwarfs or, more intriguingly, may indicateeffects associated with the intimate nature of the dark matter, suchas finite self-interactions or other such deviations from the canonicalLCDM paradigm. Resolving this challenge will likely require aconcerted numerical and observational effort designed to improveour understanding of the formation and evolution of some of thefaintest galaxies known.

ACKNOWLEDGEMENTS

RE acknowledges support provided by a CITA National Fellowship.Both RE and RI acknowledge funding from the European ResearchCouncil (ERC) under the European Unions Horizon 2020 researchand innovation programme (grant agreement No. 834148). This workused the DiRAC@Durham facility managed by the Institute for Com-putational Cosmology on behalf of the STFC DiRAC HPC Facility(www.dirac.ac.uk). The equipment was funded by BEIS capitalfunding via STFC capital grants ST/K00042X/1, ST/P002293/1,ST/R002371/1 and ST/S002502/1, Durham University and STFCoperations grant ST/R000832/1.

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Tidally limited satellite galaxies 13

DATA AVAILABILITY

The data underlying this article will be shared on reasonable requestto the corresponding author.

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APPENDIX A: DISTRIBUTION OF ORBITAL PERIODS

As an NFW subhalo loses mass through tides, the least-bound particlesget stripped first: the initial energy distribution d𝑁 /dE is sharpy truncatedbeyond some energy Emx,t which depends on the remnant bound mass (seeFig. 4). For isotropic, spherical NFWprofiles, and for the majority of particles,energy is the only parameter determining whether it will be stripped, or not.Fig. A1 shows the distributions of initial radii 𝑟 (left panel) and radial orbitalperiods𝑇𝑟 (right panel) of those particles which will form the bound remnantat subsequent apocentre snapshots. Orbital periods are computed from theinitial energy 𝐸 and angular momentum 𝐿 of each 𝑁 -body particle in theinitial NFW potential (see, e.g., Binney & Tremaine 1987, eq. 3-16),

𝑇𝑟 = 2∫ 𝑟peri

𝑟apod𝑟

{2 [𝐸 −ΦNFW (𝑟 ) ] − 𝐿2/𝑟2

}−1/2, (A1)

and are normalized by the initial crossing time at the radius of maximum cir-cular velocity,𝑇mx0 = 2𝜋𝑟mx0/𝑉mx. The remnant mass fraction𝑀mx/𝑀mx0is colour-coded.

As 𝑀mx/𝑀mx0 decreases, the bound remnant consists of particleswhich in the initial conditions were located at decreasing distance to thesubhalo centre. Radii alone, however, do not determine whether a particlewill remain bound or not. Indeed, at fixed radius in the left panel of Fig. A1,some particles get stripped, but others remain bound.

In contrast, the truncation in radial orbital period𝑇𝑟 imposed by tides issteep (see right panel of Fig. A1), selecting most particles with radial orbitalperiods shorter than the truncation period. Tidal evolution stalls once therelaxed remnant has a crossing time of roughly one quarter of the host halocrossing time at pericentre, 2𝜋𝑟mx/𝑉mx ≈ 𝑇peri/4. In the initial conditions,this corresponds to a selection of particles with orbital periods shorter than≈ 𝑇peri/16.

APPENDIX B: RADIUS-DEPENDENT ENERGYDISTRIBUTION

Depending on how deeply embedded a stellar tracer is within its dark matterhalo, the peak of its energy distribution shifts. The deeper embedded a stellartracer is, the more its peak energy shifts towards strongly-bound energies.This has qualitative consequences for the tidal evolution of stellar structuralparameters, which are discussed in this appendix. Figure B1 shows, in theinitial conditions, the number of dark matter particles (left panel) and stellar

MNRAS , 1–15 (2021)

14 Errani et al.

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Figure A1.Radii 𝑟/𝑟mx0 (left panel) and radial orbital periods𝑇r/𝑇mx0 (rightpanel) of particles in the initial conditions which remain bound at subsequentapocentre snapshots. Radii 𝑟 are normalised by the radius of maximumcircular velocity 𝑟mx0, while periods 𝑇r are normalised by the period 𝑇mx0of a circular orbit at 𝑟mx0. The remnant bound mass fraction 𝑀mx/𝑀mx0 iscolour-coded. Tides preferentially strip particles with larger orbital periods𝑇r and larger radii 𝑟 first. The truncation in orbital periods imposed by tidesis much steeper than the truncation in radius. Particles with initial orbitalperiods smaller than . 𝑇peri/16 form the asymptotic tidal remnant.

particles (left and right panel for tracers with 𝑅h0/𝑟mx0 = 1/16 and 1/2,respectively) for a given radius 𝑟 and energy E = 1 − 𝐸/Φ0. The probability𝑃 (𝑟 ) of a dark matter particle to be located at a radius 𝑟 follows directlyfrom the NFW density profile,

𝑃 (𝑟 ) = 4𝜋𝑟2𝜌NFW (𝑟 ) , (B1)

while the probability for a particle to have an energy 𝐸 at a fixed radius 𝑟equals

𝑃 (𝐸 |𝑟 ) = {2 [𝐸 −Φ(𝑟 ) ] }1/2 𝑟2 𝑓 (𝐸) , (B2)

where 𝑓 (𝐸) is the NFW distribution function, obtained through Eddingtoninversion (see, e.g., Errani & Peñarrubia 2020). The probability for a darkmatter particle to be located at a radius 𝑟 and with energy 𝐸 is then𝑃 (𝐸, 𝑟 ) = 𝑃 (𝑟 )𝑃 (𝐸 |𝑟 ) . For stellar particles, this probability needs to bemultiplied by the stellar tagging probability of Eq. 5.

No particles located at a radius 𝑟 may have an energy lower than thepotential 𝜙 (𝑟 ) = 1 − Φ(𝑟 )/Φ0 at that radius, shown as a solid black line.The dark matter (NFW) particles span a wide range in radii and energies,with substantial mass located at high binding energies within the centraldark matter density cusp. In contrast, the initial energy distribution of stellartracers is more localized, i.e. the range of radii and energies where mostparticles are located is narrower.

The average energy at a fixed radius for dark matter particles is shownas a dashed curve in all panels, while the average energy of each stellar traceris shown using a dotted line. For the more deeply embedded stellar system, atthe half-light radius 𝑅h (corresponding approximatively to the radius wheremost particles are located, see colour coding), stellar particles are on averagemore strongly bound than dark matter. Hence tidal stripping will initiallypreferentially remove dark matter. This causes the mass enclosed within thestellar half-light radius to drop (and thereby the stellar velocity dispersionto decrease), and the half-light radius to expand during relaxation. On theother hand, for the more extended stellar tracer, at the half-light radius, starsand dark matter have similar binding energies. As tides strip the system,both dark matter and stars are hence lost from the beginning, and 𝑅h drops,consistent with the models shown in Fig. D1.

APPENDIX C: ISOTROPIC STELLAR MODELS

This appendix presents the structural properties of spherical stellar systemswith isotropic velocity dispersion, embedded in NFW dark matter halos, withenergy distribution d𝑁★/dE given by Eq. 13. Figure C1 shows the energydistributions (top panel), logarithmic slopes of the surface brightness profiles(2nd row), surface brightness profiles (3rd row) and line-of-sight velocitydispersion profiles (bottom row) for models with 𝛼 = 𝛽 = 3 (left column),

and 𝛼 = 𝛽 = 6 (right column). Profiles for four different segregations,𝑅h/𝑟mx = 1/2 (red), 1/4 (orange), 1/8 (light blue) and 1/16 (dark blue), areshown.

Structural properties are computed directly from the distribution func-tion 𝑓 (E) ∝ (d𝑁★/dE)/𝑝 (E) , where 𝑝 (E) is the phase-space volumeaccessible in an NFW potential to a particle with energy E.

The more deeply embedded the stellar profile is within its dark matterhalo, the further the maximum of the energy distribution is shifted towardshigh binding energies. The range of energies selected by the 𝛼 = 𝛽 = 3model is much wider than that of the 𝛼 = 𝛽 = 6 model. For equal valuesof 𝑅h/𝑟mx, the energy E★ at which the distribution peaks is near identicalbetween the two models.

The exact shape of the stellar density profile depends on where it isembedded inside the dark matter halo. 𝛼 = 𝛽 = 3 models approximatelyresemble exponential density profiles, while 𝛼 = 𝛽 = 6 profiles have largercore radius (i.e., a radius 𝑅c so that Σ(𝑅c) = Σ(0)/2) at equal half-lightradius.

The more deeply embedded a stellar tracer is within its dark matter halo,the lower is its central velocity dispersion relative to the subhalo characteristicvelocity 𝑉mx (bottom panel). A constant density tracer of infinite extent has a3D velocity dispersion profile equal to 𝑉esc (𝑟 )/

√2, hence, for reference, the

escape velocity profile of an NFW subhalo is shown in the bottom panel ofFig. C1 (re-scaled by a factor of

√3 to compare against the 1D line-of-sight

velocity profiles shown).

APPENDIX D: TIDAL TRACKS

Structural parameters of dwarf spheroidal galaxies evolve differently undertidal stripping depending on how deeply embedded they are within the darkmatter halo, and which energies they populate within the halo. Fig. D1shows the evolution of half-light radius 𝑅h (top panel), line-of-sight velocitydispersion 𝜎los (2nd panel from top), luminosity 𝐿 (3rd panel from top) andmass-to-light ratio averagedwithin the half-light radiusΥ = 𝐿/[2𝑀 (< 𝑅h) ],as a function of remnant darkmatter mass fraction𝑀mx/𝑀mx0. The evolutionis shown for stellar tracers with initial half-light radii of 𝑅h0/𝑟mx0 = 1/2(red), 1/4 (orange), 1/8 (light blue), 1/16 (dark blue). Each point correspondsto one apocentre snapshot, taken from the set of simulations described inSec. 2. The evolution of the dark matter characteristic radius 𝑟mx and velocity𝑉mx are shown using solid black curves. Stellar tracers with 𝛼 = 𝛽 = 3 and𝛼 = 𝛽 = 6 (Eq. 13) energy distribution are shown in the left column andright column, respectively. The full initial profiles for these tracers are shownin Fig. C1.

In the “tidally limited” regime, the stellar half-light radii 𝑅h trace thedark matter characteristic size 𝑟mx (for the 𝛼 = 𝛽 = 3 and 𝛼 = 𝛽 = 6models,we find in the tidally limited regime 𝑅h ≈ 0.93 𝑟mx and 𝑅h ≈ 1.26 𝑟mx,respectively). Similarly, the stellar velocity dispersion 𝜎los evolves parallelto the dark matter characteristic velocity 𝑉mx (for 𝛼 = 𝛽 = 3 and 𝛼 = 𝛽 = 6,we find 𝜎los ≈ 0.70𝑉mx and 𝜎los ≈ 0.76𝑉mx). Once a stellar tracer hasbeen stripped to the size of 𝑅h ≈ 𝑟mx, it becomes impossible to reconstructits original extent from structural properties alone.

The rate at which the stellar luminosity 𝐿 decreases depends cruciallyon the energy distribution of the stars within the dark matter halo: luminositydrops faster for stellar tracers with energy distributions that have a steeperslope towards the most-bound energies (i.e., larger 𝛼 in Eq. 13). The totalluminosity of the remnant is given by all stars with energies more tightlybound than the tidal truncation. As the same “filter function” (Eq. 9) can beapplied to both dark matter and stars to select those particles which remainbound for a given energy truncation Emx,t, the remnant luminosity may becomputed through the integral

𝐿 =

∫ 1

0dE d𝑁★/dE |i1 +

(𝑎 E/Emx,t

)𝑏 , (D1)

with 𝑎 ≈ 0.85, 𝑏 ≈ 12 (as in Eq. 9). In the above equation, d𝑁★/dE |i denotesthe initial stellar energy distribution (normalised to give, when integrated,the initial luminosity), and Emx,t may be computed from the remnant boundmass fraction 𝑀mx/𝑀mx0 through Eq. 10. The result of this calculation isshown as a dashed curve (“model”) in the third row of Fig. D1.

MNRAS , 1–15 (2021)

Tidally limited satellite galaxies 15lo

g 10E

log10 r/rmx

φ

〈ENFW〉−2

−1.5

−1

−0.5

0

−2 −1 0 1

dark matter (NFW)

log10 r/rmx

〈Estars〉

−2 −1 0 1

Rh/rmx = 1/16

Rh

stars (α = β = 3)

log10 r/rmx

〈Estars〉

−2 −1 0 1−4

−3

−2

−1

0

d2 N/dE

dr

Rh/rmx = 1/2

Rh

stars (α = β = 3)

Figure B1. Number of NFW dark matter (left panel) and stellar particles (middle and right panels) in the initial conditions as a function of radius 𝑟 anddimensionless energy E = 1 − 𝐸/Φ0. The dimensionless potential 𝜙 ≡ 1 − Φ/Φ0 is shown as a black solid line, while the average energies 〈E〉 at fixedradius of dark matter and stellar particles are shown using dashed and dotted curves, respectively. The middle panel shows a deeply segregated stellar tracer(𝑅h/𝑟mx = 1/16). There, at the half-light radius 𝑅h (see arrow), stars are more bound than dark matter. Hence, if the system is exposed to a tidal field, initially,dark matter is stripped predominantly. In contrast, for the more extended stellar tracer (𝑅h/𝑟mx = 1/2) shown in the right panel, at the half-light radius, stars anddark matter are similarly strongly bound, and both stars and dark matter may be stripped equally.

.

For order-of-magnitude estimates, a simple power-law approximationto this integral may be used for heavily-stripped systems. For NFW (Eq. 1)density profiles and those of an exponentially truncated cusp (Eq. 3), theenergy distribution d𝑁DM/dE is well-approximated by a power-law of slope≈ 1 towards the most-bound energies, i.e. d𝑁DM/dE ∝ E for E � 1.Assuming that the truncation happens sharpy at an energy Emx,t � 1, andthat the density profile has converged to that of an exponentially truncatedcusp, this gives 𝑀mx ∝ 𝑀 ≈

∫ Emx,t0 dE d𝑁DM/dE ∝ E2mx,t. Similarly,

for stellar tracers with an energy distribution parametrised through Eq. 13and Emx,t � E★, 𝐿 ≈

∫ Emx,t0 dE E𝛼 ∝ Emx,t𝛼+1. Consequently 𝐿 ∝

𝑀 (𝛼+1)/2 for highly stripped stellar systems.

APPENDIX E: EMPIRICAL MODEL FOR EVOLVEDSTELLAR PARAMETERS

In this appendix, we discuss how to construct the relaxed energy distributionof a tidally stripped (stellar) tracer in a cold dark matter subhalo. For this, wemake use of the observation that stellar energy distributions are subject tothe same tidal truncation as the underlying NFW dark matter halo (Sec. 3.3),and of the energy mapping between the initial NFW halo and the and thestripped, relaxed system discussed in Sec. 3.2.3.

Given an initial stellar energy distribution, d𝑁★/dE |i, as well as thedesired remnant mass fraction of the underlying NFW dark mater halo,𝑀mx/𝑀mx0, the procedure involves the following steps:(i) For the desired remnant bound mass fraction 𝑀mx/𝑀mx0 of the under-lying dark matter subhalo, the tidal truncation energy Emx,t is computedfrom Eq. 10.(ii) The energy distribution d𝑁★/dE |i,t of those particles in the initialconditions which remain bound at the given tidal truncation energy Emx,tfollows from applying the filter of Eq. 9 to the initial energy distribution.(iii) Finally, we need to model the relaxation process to pass from the initialtruncated energy distribution d𝑁★/dE |i,t to the relaxed, equilibrium systemd𝑁 /dE |f . To take into account the scatter8 in the energy mapping shown in

8 Ignoring any scatter around the average energy mapping of Eq. 12, thefinal energy distribution of the relaxed system would now follow simply fromapplying the average energy mapping to the truncated distribution in theinitial conditions, i.e. d𝑁★/dE |f = d𝑁★

(E−1f (Ef )

)/dE

���i,t

|dE−1f /dEf |.

Here, E−1f (Ef ) is the inverse of Eq. 12, and |dE−1

f /dEf | is the absolutevalue of the corresponding Jacobian.

the top panel of Fig. 7, we compute the final energy distribution through theconvolutiond𝑁★

dE

����f=

∫ 1

0dEi

d𝑁★

dE

����i,tLognormal

(Ef (Ei) , 0.03 dex

). (E1)

where Lognormal(Ef (Ei) , 0.03 dex

)is a base-10 log-normal of width 0.03

dex modelling the scatter around the average relation of Eq. 12.An implementation of this procedure is made available online9, taking

as input the (stellar) initial energy distribution (parametrised through Eq. 13)as well as a desired dark matter remnant mass fraction 𝑀mx/𝑀mx0, andreturning the evolved stellar energy distribution.

9 https://github.com/rerrani/tipy

MNRAS , 1–15 (2021)

16 Errani et al.

−4

−3

−2

−1

0

1

−1.5 −1 −0.5 0

α = β = 3 NFW

log 1

0dN

/dE

log10 log10 E−1.5 −1 −0.5 0

α = β = 6

log10 log10 E

-1

0

1

2

−1 0 1

Rh/rmx

2D

Γ=−

dln

Σ/d

lnR

log10 R/Rh

1/21/41/8

1/16

−1 0 1

2D

log10 R/Rh

Exp.Plummer

-2

-1

0

−2 −1 0 1

1/2

2D

log 1

0Σ

(R)/

Σ(0

)

log10 R/rmx

−2 −1 0 1

1/2

2D

log10 R/rmx

Exp.

-1

0

−2 −1 0 1

2D

log 1

0σ

los(R

)/V

mx

log10 R/rmx

−2 −1 0 1

Vesc(r)/ √6

2D

log10 R/rmx

Figure C1. Exploration of the structure and kinematics of spherical, isotropicstellar models that follow the energy distribution of Eq. 13 (with 𝛼 = 𝛽 = 3in the left column, and 𝛼 = 𝛽 = 6 in the right column), embedded inan NFW dark matter halo. The energy distributions of stellar tracers withhalf-light radii of 𝑅h/𝑟mx = 1/2 (red), 1/4 (orange), 1/8 (light blue), 1/16(dark blue), where 𝑟mx is the radius of maximum circular velocity of theunderlying dark matter halo, are shown in the top panel. The logarithmicslope Γ = d lnΣ/d ln𝑅 of the surface brightness profiles is compared againstthat of Plummer and exponential models (grey dashed curves), showing thatthe 𝛼 = 𝛽 = 3 model approximatively resembles an exponential profile forradii smaller than the 2D half-light radius 𝑅h (2nd panel from top). Thecorresponding surface brightness profiles are compared against exponentialprofiles of the same core radius (3rd panel from top). The line-of-sightvelocity dispersion profiles are shown in the bottom panel in units of themaximum circular velocity 𝑉mx of the underlying dark matter halo.

−1.5

−1

−0.5

0

rmxα = β = 3

log 1

0R

h/r m

x0 α = β = 6

-0.8

-0.6

-0.4

-0.2

0Vmx

log 1

0σ

los/V

mx0

-5

-4

-3

-2

-1

0

log 1

0L/L

0model

-1

0

1

2

3

-2 -1.5 -1 -0.5 0

Υ/Υ

0

log10 Mmx/Mmx0

Rh0/rmx0 = 1/21/41/8

1/16

-2 -1.5 -1 -0.5 0log10 Mmx/Mmx0

Figure D1. Evolution of stellar structural parameters as a function of remnantdark matter bound mass fraction 𝑀mx/𝑀mx0 for stellar tracers with initialstructure as shown in Fig. C1. Different initial segregations are shown usingdifferent colours. As tides strip the dark matter halo, the stellar half-lightradii 𝑅h asymptotically approach the dark matter characteristic radius 𝑟mx(black solid curve, top panel), and their line-of-sight velocity dispersions𝜎los evolve in parallel to the dark matter characteristic velocity 𝑉mx (blacksolid curve, 2nd panel from top). The luminosity 𝐿 drops, and the rate ofstellar mass loss depends significantly on the inner slope 𝛼 of the underlyingenergy distribution (3rd panel from top). As 𝐿 drops, the mass-to-light ratioΥ (averaged here within the 2D half-light radius) increases (bottom panel).

MNRAS , 1–15 (2021)