Deadly dark matter cusps vs faint and extended star clusters: Eridanus II and Andromeda XXV

TL;DR

This study investigates the survival of faint, extended star clusters in dwarf galaxies and finds they can only persist in dark matter halos with shallow central density cusps, providing a method to probe dark matter profiles.

Contribution

It demonstrates that fragile star clusters can survive only in cored or shallow cusped dark matter halos, offering a new way to test dark matter density profiles in dwarf galaxies.

Findings

Clusters are disrupted in steep cusps ($ ho \,\sim\, r^{-\,\alpha}$ with $\alpha \gtrsim 0.2$).

Clusters survive in halos with shallow cusps ($\alpha \lesssim 0.2$).

Proper velocity measurements can distinguish halo density profiles.

Abstract

The recent detection of two faint and extended star clusters in the central regions of two Local Group dwarf galaxies, Eridanus II and Andromeda XXV, raises the question of whether clusters with such low densities can survive the tidal field of cold dark matter haloes with central density cusps. Using both analytic arguments and a suite of collisionless N-body simulations, I show that these clusters are extremely fragile and quickly disrupted in the presence of central cusps $\rho\sim r^{-\alpha}$ with $\alpha\gtrsim 0.2$. Furthermore, the scenario in which the clusters where originally more massive and sank to the center of the halo requires extreme fine tuning and does not naturally reproduce the observed systems. In turn, these clusters are long lived in cored haloes, whose central regions are safe shelters for $\alpha\lesssim 0.2$. The only viable scenario for hosts that have preserved their primoridal cusp to the present time is that the clusters formed at rest at the bottom of the potential, which is easily tested by measurement of the clusters proper velocity within the host. This offers means to readily probe the central density profile of two dwarf galaxies as faint as $L_V\sim5\times 10^5 L_\odot$ and $L_V\sim6\times10^4 L_\odot$, in which stellar feedback is unlikely to be effective.

Tables

Table 1. Table 1: Suite of simulations with live, α = 1 𝛼 1 \alpha=1 halos. The table lists the initial properties of the clusters and the final fate of the cluster, where D 𝐷 D corresponds to tidal disruption, while S 𝑆 S is a surviving cluster. All surviving cluster manage to sink to the center of the halo.

$(\log M_{c,i}/M_{\odot },j_{c,i},r_{h,c,i}/\mathrm{pc})$	outcome
$(\{3.9,4.1\},1.0,10.0)$	$\{S,S\}$
$(4.3,\{0.3,0.5,0.7,0.8,0.9,0.95,1.0\},10)$	$\{D,D,D,D,D,D,D,S,S\}$
$(\{4.5,4.55,4.575,4.577,4.579,4.582,4.588,4.6\},0.3,10)$	$\{D,D,D,S,S,S,S,S\}$
$(4.7,0.3,\{10,12,12.8,13.5,15\})$	$\{S,S,S,D,D\}$
$(\{4.7,4.8\},0.9,20.0)$	$\{D,S\}$
$(4.8,0.9,22.5)$	D
$(4.8,1.0,25.0)$	D
$(4.9,0.3,\{15,20\})$	$\{S,D\}$
$(4.9,0.6,\{15,20,22.5\})$	$\{S,S,D\}$
$(5.0,0.3,\{10,20,21,22,23,25\})$	$\{S,S,D,D,D,D\}$
$(5.0,\{0.8,0.85,0.9\},25)$	$\{D,S,S\}$
$(5.0,1.0,\{10,25,30,35\})$	$\{S,S,S,D\}$
$(5.3,0.3,\{25,30,35,40\})$	$\{S,S,S,D\}$
$(5.3,0.6,\{30,35\})$	$\{S,S\}$

Table 2. Table 2: Suite of simulations with live, α = 1 𝛼 1 \alpha=1 halos resulting in surviving, sunk clusters. Column 1 lists the initial conditions; Columns 2 and 3 list the mass and half light radius of the cluster once it has reached the halo center; Column 4 lists the time it takes for it to sink.

$(\log M_{c,i}/M_{\odot },j_{c,i},r_{h,c,i}/\mathrm{pc})$	$\log M_{c,f}/M_{\odot }$	$R_{h,c,f}/$pc	$t_{fr}/$Gyr
$(\{3.9,4.1\},1.0,10.0)$	$\{3.4,3.9\}$	$\{4.7,6.1\}$	$\{14.9,9.2\}$
$(4.3,\{0.95,1.0\},10)$	$\{4.0,4.1\}$	$\{6.4,6.8\}$	$\{6.4,5.6\}$
$(\{4.577,4.579,4.582,4.588,4.6\},0.3,10)$	$\{3.8,3.8,3.9,3.9,3.9\}$	$\{6.1,6.7,6.7,7.0,7.0\}$	$\{9.4,8.2,4.9,4.4,4.1\}$
$(4.7,0.3,\{10,12,12.8\})$	$\{4.4,4.0,3.2\}$	$\{8.3,8.1,5.3\}$	$\{2.3,2.8,4.3\}$
$(4.8,0.9,20.0)$	4.4	10.0	2.2
$(4.9,0.3,15)$	4.5	9.3	1.6
$(4.9,0.6,\{15,20\})$	$\{4.6,4.4\}$	$\{9.6,10.2\}$	$\{1.5,1.8\}$
$(5.0,0.3,\{10,20\})$	$\{4.9,4.4\}$	$\{8.8,10.4\}$	$\{1.1,1.5\}$
$(5.0,\{0.85,0.9\},25)$	$\{4.5,4.5\}$	$\{11.7,11.9\}$	$\{1.4,1.5\}$
$(5.0,1.0,\{10,25,30\})$	$\{5,4.7,4.6\}$	$\{8.5,12.9,12.4\}$	$\{1.1,1.4,1.5\}$
$(5.3,0.3,\{25,30,35\})$	$\{5.1,5.0,4.8\}$	$\{14.0,15.7,15.7\}$	$\{0.6,0.6,0.7\}$
$(5.3,0.6,\{30,35\})$	$\{5.1,5.0\}$	$\{15.5,15.5\}$	$\{0.6,0.6\}$

Equations

$$L_{V,c}\approx 2\times 10^3{L}_{\odot }$$

$$R_{h,c}\approx 13\mathrm{pc}.$$

$$L_{V,c}\approx 8\times 10^3{L}_{\odot }$$

$$R_{h,c}\approx 25\mathrm{pc},$$

$$7.0<\log \left[M_{tot}(<1.77R_h)/M_{\odot }\right]<7.6$$

$$6.2<\log \left[M_{tot}(<1.77R_h)/M_{\odot }\right]<7.3,$$

$$\rho (r)=\frac{{\rho }_s}{{\left(\frac{r}{r_s}\right)}^{\alpha }{\left(1+\frac{r}{r_s}\right)}^{3-\alpha }},$$

$$\left\{\begin{array}[]{rcr}r_{s}~{}{\rm kpc}^{-1}&=&1.58\\ \log\rho_{s}~{}M_{\odot}^{-1}{\rm kpc}^{3}&=&7.07\end{array}\right.\ .$$

$$r_t=\left(\frac{G{M}_c}{{\Omega }^2(r)-d^2\Phi /d{r}^2}\right),$$

$$r_t=2R_{h,c},$$

$$\frac{d{v}_c}{dt}=-4\pi G^2\frac{M_c\rho }{v_c^2}\log \Lambda f_{v<v_c},$$

$$\left\{\begin{array}[]{rcl}t_{fr}&\propto&G^{-1/2}\ {{\rho_{s}^{1/2}r_{s}^{3}}\ M_{c}^{-1}}\ \left({r_{i}/r_{s}}\right)^{3-\alpha/2}\ g(\alpha)\\ g(\alpha)&=&(6-\alpha)^{-1}(4-\alpha)(3-\alpha)^{3/2}\end{array}\right.\ .$$

$$r_t(r_{stall})\approx r_{stall}.$$

$${\rho }_c(r)=\frac{3M_c}{4\pi r_c^3}{\left(1+\frac{r^2}{r_c^2}\right)}^{-5/2}.$$

$$F_{bkg}(r)=F_{tot}(r)\left[1-\frac{M_{live}(<r)}{M_{tot}(<r)}\right].$$

$$g(E,J)=\left\{\begin{array}[]{lc}0&r_{peri}(E,J)<r_{min}\\ f(E,J)&r_{peri}(E,J)\geq r_{min}\end{array}\right.\ .$$

$$\delta v_{LOS}\equiv v_{LOS}-v_{LOS,sys},$$

$$\sigma_{c}\approx\left\{\begin{array}[]{lc}0.53~{}{\rm kms}^{-1}&{\rm in~{}EriII}\\ 0.85~{}{\rm kms}^{-1}&{\rm in~{}AndXXV}\end{array}\right.\ .$$

$$\left\{\begin{array}[]{lcr}d&=&0\\ \delta v_{LOS}&=&0\end{array}\right.\ .$$

$$\delta v_{LOS}\lesssim v_{circ}(r_{stall}\approx 50\mathrm{pc})=1.7{\mathrm{kms}}^{-1}.$$

Full text

dark matter – galaxies: halos – galaxies: structure – galaxies: star clusters – galaxies: individual (Eridanus II, Andromeda XXV) – Local Group

††software: Gadget-2 (Springel, 2005)

1 Introduction

The distribution of matter on cosmological scales is very successfully reproduced by the standard $\Lambda$ cold dark matter (DM) model: the agreement with measurements of both the cosmic microwave background and the baryonic acoustic oscillation feature (e.g., Planck Collaboration et al., 2014; Anderson et al., 2014; Frenk & White, 2012) is impressive. However, these tests only probe the DM linear power spectrum at scales $\gtrsim$10 Mpc. At the scales of galaxies and below alternative DM models make different predictions, which provides means to differentiate among them.

Cold, non relativistic DM particles virialize in haloes characterized by a central density distribution which diverges as $\rho\sim r^{-1}$ (Dubinski & Carlberg, 1991; Navarro et al., 1996), and containing a fraction of about 10% of their mass in substructure, in the form of bound sub-haloes (Diemand et al., 2008; Springel et al., 2008). Warm(-er) DM particles allow for less power at small scales: the subhalo mass function is suppressed below some model-dependent minimum mass and the total mass fraction in substructure is lowered (e.g., Bode et al., 2001; Menci et al., 2012; Lovell et al., 2014; Bose et al., 2016); halo concentration, additionally, becomes a non-monotonic function of halo mass (e.g., Ludlow et al., 2016). Models allowing for self-interactions also imply a reduction in the small-scale power, but additionally feature haloes with centrally cored density profiles (e.g., Spergel & Steinhardt, 2000; Vogelsberger et al., 2012; Elbert et al., 2015), with the size of the core depending on the strength of the interaction itself (Zavala et al., 2013; Lin & Loeb, 2016). Central density cores are also predicted in the ‘fuzzy’ DM scenario (Press et al., 1990; Hu et al., 2000; Hui et al., 2017), in which DM is made of light scalar particles that manifest their quantum properties at astrophysical scales. Cores sizes are dictated by the mass of the axion-particle and recent numerical studies are beginning to provide definite predictions for the process of cosmological structure formation within this model (Schive et al., 2014, 2016; Mocz & Succi, 2015; Du et al., 2017).

Establishing sound astrophysical tests for dark matter models on dwarf galaxy scales and below has proven especially hard so far. Recently, probing the mass function of halo substructure, with either strong lensing (e.g. Kochanek & Dalal, 2004; Vegetti et al., 2014; Hezaveh et al., 2016) or thin stellar streams (e.g., Johnston et al., 2002; Ibata et al., 2002; Erkal et al., 2016) is establishing itself as a promising venue. Here, however, I concentrate on those complimentary tests based on the detailed properties of the density profile of low-mass, dark matter dominated galaxies (virial mass $M_{200}\lesssim 10^{11}M_{\odot}$).

On the theoretical side, the predictions for the halo density profile mentioned above do not take into account the impact of baryons, and the complex hydrodynamical processes that accompany galaxy formation. Radiation and winds from young stars and supernovae, often globally referred to as stellar feedback, are an important ingredient in the formation of dwarf galaxies (e.g., Dekel & Silk, 1986; Navarro et al., 1996; Mashchenko et al., 2006; Pontzen & Governato, 2014). However, a consensus has yet to be reached on how feedback can or cannot sculpt cores into the central regions of originally cuspy dwarf galaxy haloes (e.g., Governato et al., 2012; Zolotov et al., 2012; El-Badry et al., 2016; Sawala et al., 2016; Fattahi et al., 2016). Simple energetic arguments (Peñarrubia et al., 2012; Amorisco et al., 2014) and some sub-grid implementations of the feedback processes in hydrodynamical simulations (e.g., Di Cintio et al., 2014; Oñorbe et al., 2015) suggest that core-creation is suppressed in faint enough galaxies ($L\lesssim 10^{5.5}L_{\odot}$). Dwarfs with these luminosities should preserve their primordial cusps and represent perfect targets to test the nature of dark matter. However, a different numerical implementation (Read et al., 2016) suggests that cores can emerge also in the faintest galaxies. This motivates even more strongly the need for reliable measurements of the inner density profile of low mass haloes. Even before probing the nature of DM, these measurements are crucial to understand the feedback processes themselves.

On the observational side, however, such measurements remain extremely challenging. Decades of debate have shown that this is the case for galaxies supported by rotation (e.g., Persic & Salucci, 1991; Flores & Primack, 1994; de Blok et al., 2008; Oh et al., 2011; Adams et al., 2014; Oman et al., 2015). In addition, systems that are faint enough to have possibly preserved a pristine cusp are pressure supported systems. Close enough dwarfs are the satellite galaxies of the Local Group; these include the ‘classical’ dwarf Spheroidals (dSphs, $L\gtrsim 10^{5}L_{\odot}$) and the Ultrafaints (UFs, $L\lesssim 10^{5}L_{\odot}$, e.g., McConnachie, 2012). As for all pressure supported systems, their kinematic modelling is plagued by marked degeneracies, which manifest themselves when line-of-sight kinematics alone is available. These degeneracies make it impossible to measure the galaxy’s density profile, and only allow for the determination of a mass scale (Walker et al., 2009; Wolf et al., 2010; Amorisco & Evans, 2011; Agnello et al., 2014), substantially complicating the analysis of datasets collected with painstaking effort (e.g., Battaglia et al., 2008; Walker et al., 2009). This difficulty was partially overcome by the realization that chemo-dynamically distinct stellar subpopulations can occur in Local Group dSphs (e.g., Tolstoy et al., 2004; Battaglia et al., 2006; Walker & Peñarrubia, 2011; Kordopatis et al., 2016) and can be used jointly to constrain the gravitational potential in which all stars reside. This kind of analysis can break the degeneracy between mass and anisotropy typical of pressure supported systems. Two classical dSphs, Sculptor and Fornax, could be studied with this technique. In both cases, this is found to weakly disfavor a $\rho\sim r^{-1}$ cusp (Battaglia et al., 2008; Walker & Peñarrubia, 2011; Amorisco & Evans, 2012; Agnello & Evans, 2012; Amorisco et al., 2013; Zhu et al., 2016), but the statistical significance of this result has been contended (e.g. Breddels & Helmi, 2013; Richardson & Fairbairn, 2014; Strigari et al., 2017). Alternative methods proposed in the literature to probe the central dark matter profile of local dwarf galaxies include: (i) the survival time of unbound kinematic substructure (Kleyna et al., 2003; Sánchez-Salcedo & Lora, 2010); (ii) the dynamical friction timescale of massive Globular Clusters (e.g., Hernandez & Gilmore, 1998; Sánchez-Salcedo et al., 2006; Goerdt et al., 2006; Cole et al., 2012); (iii) the internal kinematics of dwarf galaxy streams (Errani et al., 2015); (iv) the survival of loosely bound binary stars (Peñarrubia et al., 2016).

Motivated by the recent discovery of extended, low-mass star clusters in two Local Group dwarfs, in this paper I seek to establish what are the constraints that their survival to the present day puts on the DM profile of their host haloes. Extended clusters have been observed before in M31 (e.g., Huxor et al., 2011), where they are though to have been deposited by disrupted dwarfs (e.g., Mackey et al., 2010; Hurley & Mackey, 2010; Huxor et al., 2013). The two recently discovered systems are particularly extreme: as I will show, the combination of their stellar mass and size, together with the short orbital times and current projected locations make for an almost inescapable threat to their survival. This represents the main difference from a previous study, Peñarrubia et al. (2009), dedicated to the dynamical evolution of the star clusters of the Fornax and Sagittarius dSphs. Together with dense clusters, both these dwarf galaxies currently harbor diffuse star clusters, namely F1 in Fornax, Arp2 and Ter8 in Sagittarius. As shown by Peñarrubia et al. (2009), these would easily be disrupted by the tides if they were to orbit close enough to the center. However, such fragile GCs are observed to lie at significant projected distances, where at the same time (i) they are currently safe and (ii) dynamical friction is inefficient. As a consequence, as recognized by Peñarrubia et al. (2009), they are long lived. This is not the case for the systems I explore in this paper.

The dwarf galaxies considered here are Eridanus II (EriII, Koposov et al., 2015; Bechtol et al., 2015) in the periphery of the Milky Way, and Andromeda XXV (AndXXV, Richardson et al., 2011; Cusano et al., 2016), around M31. Each of them contains an extended, low mass star cluster which, given its structural properties, is extremely susceptible to the tidal field. Still, in both systems, the cluster is observed to reside – in projection – in the central regions of the galaxy, where the tides are strongest in cold DM haloes. Using both analytical arguments and collisionless numerical simulations I systematically explore the evolution scenarios that could allow the extended star clusters in EriII and AndXXV to survive to the present day. Predictions for the internal kinematics of the clusters are also discussed, providing means to to distinguish between the different scenarios, and therefore to infer the density profiles of the two host galaxies. Section 2 presents the two systems; Section 3 puts the dynamical problem into context; Section 4 describes the numerical setup; Section 5 describes possible scenarios for cuspy haloes; Section 6 concentrates on cored haloes; Section 7 discusses results and presents the Conclusions.

2 The clusters and their hosts

2.1 Eridanus II

At a distance of $D\approx 370$ kpc, EriII is a UF satellite of the MW, with a luminosity of $L_{V}\approx 6\times 10^{4}L_{\odot}$, a projected half-light radius of $R_{h}\approx$280 pc, a quite high ellipticity $\epsilon\approx 0.48$ and no evidence for the presence of gas (Crnojević et al., 2016). Using Magellan/IMACS spectroscopy Li et al. (2017) have recently targeted EriII and confirmed 28 member stars. They find a velocity dispersion of $\sigma=6.9^{+1.2}_{-0.9}$ kms*-1*, a mean metallicity of $\left[{\rm Fe/H}\right]\approx-2.4$, and can exclude the presence of young stars in the system.

A round overdensity of stars near the center of EriII was already spotted in the discovery data (Koposov et al., 2015), and subsequent deep imaging (Crnojević et al., 2016) has confirmed the presence of an extremely faint and quite extended star cluster. This has a luminosity of

[TABLE]

and a projected half-light radius of

[TABLE]

As such, EriII is the least luminous galaxy known to posses a stellar cluster, which has prompted the suggestions that other distant Milky Way GCs might in fact be hosted by yet undiscovered low surface brightness galaxies (Zaritsky et al., 2016). The stellar clusters is projected very close to the inferred center of the stellar distribution of EriII, but has a measurable offset of $d\approx 45$ pc. The resolved stars in the cluster would suggest its stellar population is old, and consistent with the old population of EriII ($\approx 10$ Gyr).

2.2 Andromeda XXV

AndXXV sits at a projected distance of $\approx 90$ kpc from the center of M31 and has a luminosity of $L_{V}\approx 5\times 10^{5}L_{\odot}$ (Richardson et al., 2011; McConnachie, 2012; Martin et al., 2016), approaching the lower edge of the range of ‘classical’ dSphs. Its projected half light radius is quite large for its luminosity, but also quite uncertain as a consequence of a chip gap in the available imaging data (Richardson et al., 2011; Martin et al., 2016), with literature values ranging between 550 and 700 pc. The internal kinematics of AndXXV was probed by Collins et al. (2013), who measured a peculiarly low velocity dispersion of $\sigma=3.1^{+1.2}_{-1.1}$ kms*-1*.

By visual inspection of stacked images, Cusano et al. (2016) have recently unveiled the presence of a concentration of stars near the central regions of AndXXV. The color magnitude diagram of the few resolved stars is compatible with that of old stars at the distance of AndXXV, which suggests that the cluster and the dwarf are indeed physically associated (Cusano et al., 2016). Further support for this can be obtained by considering the probability of chance alignments, of either a foreground GC belonging to the MW or of a GC associated with M31. The first is very small: even assuming the MW has has many as 500 (yet undetected) GCs, distributed isotropically, the probability of a chance projection within an angle of $\approx 2R_{h}/750$ kpc is of only111A similarly small probability is obtained using the same argument for the case of Eri II. $\approx 3\times 10^{-4}$. The total number of GCs in M31 is uncertain, probably as high as $\sim 450$ (Veljanoski et al., 2013), with as many as $\sim 1000$ candidates (Galleti et al., 2004). If I assume the latter figure and a 3-dimensional number density profile that declines as slowly as $r^{-2}$ (truncated for example at $r=200$ kpc), the probability of a chance alignment is of about $5\%$. This indicates that the cluster and AndXXV are very likely associated with each other.

The cluster has a luminosity of

[TABLE]

and a projected half light radius of

[TABLE]

making it significantly extended. As for EriII, the projected position of the cluster does not coincide with the centroid of AndXXV main stellar component. The precise value of the offset is uncertain due to the mentioned chip gap, but appears comparable to what is seen in EriII, $d\approx 46$ pc.

3 Dynamical ingredients

3.1 Limits from the dwarf galaxy kinematics

The observed values of the stellar velocity dispersion in EriII and AndXXV provide constraints on the host halos. Since a split in multiple stellar subpopulations is not available for these two systems, the only constraint posed by present data is on the total mass within the half-light radius (Walker et al., 2009; Wolf et al., 2010; Amorisco & Evans, 2011; Agnello et al., 2014). I use the mass estimator proposed by Campbell et al. (2016) (their eqn. (17) and Table 3). With a form similar to the estimator proposed by Amorisco & Evans (2011, 2012), this was tested on cosmological simulations and it appears to minimize bias and scatter. Taking into account both observational and systematic uncertainties, the constraints on the total mass are

[TABLE]

for EriII, and

[TABLE]

for AndXXV, where $R_{h}$ are the galaxies’ half light radii. I assume the halo has a parametric form

[TABLE]

in which the classical Navarro-Frenk-White density profile (NFW, Navarro et al., 1996) corresponds to the case $\alpha=1$. For such cuspy haloes, the 1-sigma regions shown in panel $a$ of Figure 1 illustrate the constraints (5,6). Here $\rho_{s}$ and $r_{s}$ are respectively the characteristic density and scale radius of the density profile, as in eqn. (7). Full black dots in the same panel represent mean cosmological haloes at $z=0$, satisfying the mass-concentration relation of cold DM haloes (as compiled by Ludlow et al., 2016). Points range between a virial mass $M_{200}$ of $10^{7}$ and $10^{11}~{}M_{\odot}$, in steps of 0.5 dex. Using the same steps, smaller points illustrate the range allowed by the scatter in the mass-concentration relation, for the same set of masses.

EriII appears most compatible with a cold DM halo of $\log M_{200}/M_{\odot}=9$, having

[TABLE]

This value of the virial mass is in good agreement with what would be inferred based on the dwarf’s luminosity using abundance matching (e.g., Garrison-Kimmel et al., 2017; Jethwa et al., 2016). Haloes with different inner slopes, $\alpha<1$, and compatible with the same mass constraint can be obtained as follows. The scale radius $r_{s}$ is kept fixed as in (8), as it would be if the central density cusp is removed by feedback or scoured by the orbital evolution of gaseous massive clumps (e.g., El-Zant et al., 2001; Ma & Boylan-Kolchin, 2004; Cole et al., 2011; Nipoti & Binney, 2015; Del Popolo & Pace, 2016). The characteristic density $\rho_{s}$ is adjusted as a function of $\alpha$, so that the enclosed mass $M_{tot}(<1.77R_{h})$ remains constant.

As shown by the blue lines in panel $a$ of Fig. 1, the low value of the velocity dispersion of AndXXV would suggest an unexpectedly low virial mass. Collins et al. (2013) have already identified the peculiar properties of AndXXV, which is an outlier in the population of Local Group dwarfs. They concluded that AndXXV is likely to have been recently affected by tides (see also Collins et al., 2014). Indeed, the value of the virial mass obtained above assuming dynamical equilibrium appears exceedingly low. This is especially true when compared to the dwarf’s brightness, which would instead suggest a halo at least as massive as the one in EriII.

3.2 Instantaneous tidal radii

For a cluster with mass $M_{c}$ orbiting within the spherically symmetric potential $\Phi(r)$ and instantaneously at a galactocentric distance $r$, the nominal tidal radius is given by (see e.g., Renaud et al., 2011; Amorisco, 2015)

[TABLE]

where $\Omega^{2}(r)=r^{-1}d\Phi/dr$. For a Keplerian gravitational potential generated by a mass $M$, Eqn (9) returns the classical $r_{t}=r(M_{c}/3M)^{1/3}$. However, the shape of the density profile should also be taken into account.

Panels $b$ and $c$ of Fig. 1 display contours for the instantaneous tidal radius (9), measured in pc, in the plane of orbital distance $r$ versus slope of the density profile $\alpha$, as from eqn. (7). Following the analysis of Sect. 3.1, for EriII I have assumed that, when $\alpha=1$, the halo has the properties of a mean cold DM halo with $\log M_{200}/M_{\odot}=9$, as in eqn. (8). For different values of $\alpha$, the dimensional scales are obtained as described in Sect. 3.1, i.e. so to satisfy the kinematic constraint (5). For AndXXV, the arguments in Sect. 3.1 are inconclusive. Given the dwarf’s luminosity, the value $\log M_{200}/M_{\odot}=9$ is likely a lower bound to the original virial mass of the system, and therefore represents a conservative choice with respect to the strength of the tidal field. For this reason, panel $c$ of Fig. 1 also adopts $\log M_{200}/M_{\odot}=9$. As to the clusters themselves, Fig. 1 assumes that the cluster in EriII has a mass of $M_{c}=5\times 10^{3}M_{\odot}$ and the one in AndXXV has $M_{c}=25\times 10^{3}M_{\odot}$, corresponding to a mass to light ratio of $M/L_{V}\approx 2.5~{}M_{\odot}/L_{\odot}$.

The red and blue lines in panels $b$ and $c$ of Fig. 1 display the contours

[TABLE]

where $R_{h,c}$ are the observed projected half light radii of the two clusters. These lines are approximate divides between configurations in which the cluster experiences substantial tidal loss, if $r_{t}\ll 2\ R_{h,c}$, and the opposite regime in which the cluster does not fill the Roche lobe and is safe against the tidal field, $r_{t}\gg 2\ R_{h,c}$. Cuspy density profiles result in threatening tidal fields, with a sharp demarcation between profiles with $\alpha<0.2$ and $\alpha\gtrsim 0.2$. In order to survive for a time comparable with its age in a $\alpha\gtrsim 0.2$ cusp, this simplified analysis suggests that the cluster in EriII should remain at galactocentric distances $r\gtrsim 500$ pc. The one in AndXXV at distances $r\lesssim 700$ pc. Furthermore, in the presence of a density cusp and a non-circular orbit, the addition of tidal shocking at pericenter is likely to significantly facilitate the disruption of the cluster. This is especially true since the dynamical time in the central regions of the haloes in object is short, $t_{dyn}\approx 0.1$ Gyr, resulting in repeated injections of energy into the cluster. As a consequence, clusters are very likely to be quickly destroyed if they happen to orbit at radii where $r_{t}\ll 2\ R_{h,c}$. At face value, this result is at odds with the observation that both clusters have a projected distance $d\lesssim 50$ pc from the center. I will return on this aspect in a quantitative manner in Sect. 5.1.

On the other hand, Fig. 1 suggests that if hosts are cored or have very shallow density slopes, $\alpha\lesssim 0.2$, both clusters are free to orbit at any galactocentric distance. In fact, for $\alpha\lesssim 0.2$, panels $a$ and $b$ in Fig. 1 show that the instantaneous tidal radius $r_{t}$ is a non monotonic function of the orbital radius $r$: the very central regions are safer than when $r\approx R_{h}$. This follows from the sign inversion of the eigenvalues of the tidal tensor, which mark the transition to a compressive tidal field when in a constant density environment (e.g., Chandrasekhar, 1942; Renaud et al., 2011). This simple analysis suggests that, despite their low mass and large sizes, the clusters in EriII and AndXXV would be safe against the tides in haloes with a very shallow cusp or a large core. More in general, clusters even more fragile than those considered here could survive indefinitely if, helped by dynamical friction, they can manage to cross the region $r\approx R_{h}$ and reach the sheltered inner core region.

3.3 Dynamical friction

It is well known that dynamical friction is an important ingredient in the evolution of GCs in dwarf galaxies (e.g., Tremaine et al., 1975; Oh & Lin, 2000; Lotz et al., 2001; Sánchez-Salcedo et al., 2006; Goerdt et al., 2006; Hartmann et al., 2011; den Brok et al., 2014; Brockamp et al., 2014). The scope of this section is limited to providing estimates for the sinking times tailored on the problem at hand. These estimates are useful to guide the identification of viable evolution scenarios for the clusters, to be explored numerically in Sect. 5.

The standard understanding of dynamical friction is crystallized in Chandrasekhar’s analytic formula (Chandrasekhar, 1943; Binney & Tremaine, 2008):

[TABLE]

where $M_{c}$ is the cluster mass, $v_{c}$ is the norm of its velocity, $\rho$ is the background density, $\log\Lambda$ is the usual Coulomb logarithm and $f_{v<v_{c}}$ is the fraction of the background density with velocities $v<v_{c}$. Here, I seek to estimate the dynamical friction timescale $t_{fr}$ of clusters with different masses in haloes with different density profiles. To this end, I consider the simplified scenario in which:

• •

the cluster mass $M_{c}$ remains constant during sinking, i.e., the cluster is not contemporarily affected by tides;

• •

the orbit of the cluster evolves loosing both energy and angular momentum but conserving its circularity $j=1$, i.e. it remains a circular orbit;

• •

variations in the factor $\log\Lambda f_{v<v_{c}}$ during the orbital evolution are secondary and can be neglected;

• •

the cluster orbits in the central regions of the density profile (7), where $\rho\sim{\rho_{s}}\left(r/r_{s}\right)^{-\alpha}$.

For the sake of clarity, $j$ is the orbital circularity $j\equiv J/J_{circ}(E)$, where $J$ and $E$ are the orbital angular momentum and the energy of the cluster, and $J_{circ}(E)$ is the angular momentum of a circular orbit with energy $E$. Under the model assumptions above, the dynamical friction timescale $t_{fr}$ scales as follows:

[TABLE]

This is the time it takes for dynamical friction to bring a massive object from the radius $r_{i}$ to $r=0$. I calibrate eqn. (12) on the result of an N-body simulation in which a massive particle, $\log M_{c}/M_{\odot}=4.7$, is put on a circular orbit with $r_{circ,i}=250$ pc in a cold DM halo as in eqn. (8). This sinks in a time $t_{fr}\approx 1.9$ Gyr. An analogous massive particle the same initial orbital energy, but on a very eccentric orbit, $j=0.3$, sinks in a very similar time, $t_{fr}\approx 1.85$ Gyr (see Sect. 4 for details on the numerical setup).

Panel $a$ in Fig. 2 shows the sinking time $t_{fr}$ in Gyr for clusters of mass $M_{c}$ in mean cold DM haloes ($\alpha=1$) with virial mass $M_{200}$. The starting radius $r_{circ,i}$ is kept fixed at a physical distance of $250$ pc. At fixed $M_{c}$, $t_{fr}$ increases with the virial mass of the host $M_{200}$: while $\rho(r)$ increases with $M_{200}$, so does the orbital velocity $v_{circ}(r)$, at the denominator in eqn (11). The dependence on $M_{c}$ is instead more marked: while dynamical friction can be ignored for clusters with $M_{c}\lesssim 10^{3.7}~{}M_{\odot}$, while massive clusters with $M_{c}\approx 10^{5}~{}M_{\odot}$ sink to the center very quickly. Clusters with masses $M_{c}\approx 10^{4.5}~{}M_{\odot}$ would also sink towards the center in a fraction of the Hubble time. However, as the numerical explorations of Sect. 5 show, the interplay between dynamical friction and mass loss is significant in this regime and the estimates obtained here are lower limits for $t_{fr}$. As in Fig. 1, the red and blue lines in panel $a$ of Fig. 2 display the curves $r_{t}(r)=2R_{h,c}$, here assuming $r=r_{circ,i}=250$ pc. Models close and above these lines are unlikely to survive all the way to the center of haloes with $\alpha=1$.

Panel $b$ explores the dependence of $t_{fr}$ with $\alpha$. Similarly to panels $b$ and $c$ of Fig. 1, this uses the parameters (8) for $\alpha=1$, and adjusts $\rho_{s}(\alpha)$ so to satisfy the mass constraint (5) for $\alpha<1$. The dependence of $t_{fr}$ on $\alpha$ is rather mild. This is again due to the competing dependences on the local density $\rho(r)$ and local circular velocity $v_{circ}(r)$ in eqn (11), which both decrease with $\alpha$.

Depending on the cluster mass and on the details of the host density profile, dynamical friction can only bring the cluster to a finite distance from the center, $r_{stall}$, where it becomes strongly suppressed and the sinking process ‘stalls’. This behavior is due to the formation of a central core in the host density profile, on the scale of $r_{stall}$, as a consequence of the energy and angular momentum transfer from the cluster itself (e.g., El-Zant et al., 2001; Goerdt et al., 2010). In shallow density profiles, dynamical friction is suppressed by the emergence of orbital resonances (e.g., Hernandez & Gilmore, 1998; Read et al., 2006; Inoue, 2009; Cole et al., 2012; Petts et al., 2016), which effectively halt the sinking process. This behavior is not captured by the Chandrasekhar approximation (11), and therefore not taken into account in the estimates above. Analytical arguments and numerical studies (see e.g., Goerdt et al., 2010; Petts et al., 2016) show that ‘core stalling’ happens at the radius $r_{stall}$ where

[TABLE]

Panel $c$ shows the stalling radii predicted by this equation, in pc, as a function of $M_{c}$ and $\alpha$. By definition, these are also the values of $r_{t}(r_{stall})$. The red and blue lines show the contours $r_{t}(r_{stall})=2R_{h,c}$ for the clusters in EriII and AndXXV. Models above these lines are likely to experience significant tidal mass loss on their way to the center of the host.

4 Numerical setup

In this Section, I describe the setup used for the N-body experiments presented in this paper. All simulations are collisionless N-body simulations, executed with Gadget-2 (Springel, 2005). Each features a spherically symmetric host and a spherically symmetric star cluster.

Initial conditions for the star cluster are generated assuming an isotropic Plummer phase space distribution function (e.g., Plummer, 1911; Dejonghe, 1987), for a density profile

[TABLE]

The total mass $M_{c}$ and the core size $r_{c}$ are free parameters. The projected half light radius for a Plummer model is $R_{h}=r_{c}$, while the spherical half light radius is $r_{h}\approx 1.31r_{c}$. In the runs described in Sect. 3.3 and used to calibrate the dynamical friction timescale (12), the star cluster is represented by a single massive particle.

The difficulty presented by these simulations is that both cluster and host should be resolved with live particles, and that the individual masses of these particles should be comparable. This makes for a significant computational challenge, by bringing the total number of particles to $N_{tot}\sim 10^{9}$. A live cluster is necessary to capture tidal stripping; a live halo for dynamical friction. In a static background potential, dynamical friction should be included by hand. However, a genuine accounting of the sinking process is to be preferred here, as the interplay between sinking and mass loss decides the fate of the cluster (see Sect. 5). Particles in host and cluster should have comparable masses to avoid artificial dynamical heating of the cluster (see e.g., Brandt, 2016; Koushiappas & Loeb, 2017), which would substantially facilitate its disruption.

I circumvent this computational issue by combining the use of a static potential and live particles for the host halo. The clusters inhabit the central regions of the halo, therefore resolving the host’s internal dynamics at large radii is unnecessary. I mimic the technique of particle tagging (often adopted in studying the assembly of the stellar halo of galaxies, e.g., Bullock et al., 2001; Cooper et al., 2010, 2013; Amorisco, 2017) and order the host’s mass by binding energy. Only a fraction $f_{live}$ of most bound mass is resolved in live particles, the remainder of the host mass is replaced by a static background potential. As this selection in based on an integral of the motion, the live particles are in equilibrium within the combination of their own potential and of the associated static background. The latter contributes a force

[TABLE]

where $F_{tot}$ is the total force and $M_{tot}(<r)$ is the total enclosed mass. Here, ‘total’ refers to the complete target density profile, while $M_{live}(<r)$ is the enclosed mass in live particles alone.

The minimum necessary fraction $f_{live}$ is fixed by the requirement that the mass responsible for dynamical friction in the regions of interest should be resolved in live particles. As captured by eqn (11), this mass is represented by the ‘slow’ fraction of the local density: $\rho(r)f_{v<v_{c}}$. Assuming $\alpha=1$, the red line in Fig. 3 quantifies the fraction of slow particles that are retained as live for $f_{live}=3\%$. In more detail, this is the ratio between

• •

$M_{live}\left[<r/r_{s},v<v_{circ}(r/r_{s})\right]$, i.e. the mass of the host (i) resolved in live particles; (ii) enclosed in the radius $r/r_{s}$; (iii) having orbital velocities smaller than the circular velocity at the enclosing radius, $v_{circ}(r/r_{s})$.

• •

$M_{tot}\left[<r/r_{s},v<v_{circ}(r/r_{s})\right]$: the total mass of the host satisfying (ii) and (iii).

A fraction $f_{live}=3\%$ is sufficient to fully describe dynamical friction at radii $r/r_{s}\lesssim 0.6$. This can be compared with the ratios $R_{h}/r_{s}$, $\approx 0.18$ for EriII and $\approx 0.4$ for AndXXV, ensuring that a choice of $f_{live}=3\%$ is appropriate. For this value, the black line in Fig. 3 shows the fraction $M_{live}(<r/r_{s})/M_{tot}(<r/r_{s})$, appearing in the equation for the background force component (15). Initial conditions for the live particles are generated using the phase space distribution function of the target density profile, which is obtained under the assumption of orbital isotropy through the standard Eddington inversion (e.g., Eddington, 1916; Widrow, 2000). This strategy entirely eliminates the computational issue described above. At the same time it allows me to genuinely capture the effects of dynamical friction.

5 Cuspy haloes

In this Section I assume that EriII and AndXXV have cuspy haloes, and numerically explore consequent evolution scenarios for the star clusters.

5.1 The clusters formed as they are

I first consider the case in which the clusters formed with properties similar to those currently observed: $\log M_{c}/M_{\odot}=3.7$ and $R_{h,c}=13$ pc in EriII, $\log M_{c}/M_{\odot}=4.4$ and $R_{h,c}=25$ pc in AndXXV. Fig. 4 shows what would happen if these clusters are put in orbit within a host halo with $\alpha=1$ and parameters like in eqn (8). Top panels refer to EriII, bottom panels to AndXXV. In both, the orbit has a very high circularity, $j=0.9$, which keeps the cluster away from the central regions. Orbital energies are motivated from the analysis of Sect. 3.1 and and correspond to $r_{circ}(E)=0.5$ kpc for EriII and $r_{circ}(E)=0.7$ kpc for AndXXV. Here $r_{circ}(E)$ is the radius of a circular orbit with energy $E$, and the chosen values satisfy $r_{t}\approx 2R_{h,c}$.

Fig. 4 confirms the findings of Fig. 1: a cuspy halo represents an unavoidable threat for star clusters that are as faint and as extended as in EriII and AndXXV. Both clusters disrupt completely within a Gyr. The haloes in Figure 4 have $\alpha=1$. However, Fig. 1 shows that the location of the contour $r_{t}\approx 2R_{h,c}$ is insensitive to the density slope for $\alpha\gtrsim 0.2$. For them to survive for longer times in cuspy haloes the clusters should constantly orbit at larger radii, with higher orbital energies. This result shows that the observed locations of the clusters in EriII and AndXXV are due to projection effects in this scenario.

What is the likelihood of this? In other words, what is the the probability to observe the clusters so close to the center if they are forced to orbit at $r>r_{min}$? A rough estimate can be obtained by considering the phase space of a system that has been fully voided within the loss cone corresponding to $r_{peri}<r_{min}$:

[TABLE]

Orbits that bring the cluster to a pericenter $r_{peri}<r_{min}$ are not viable as they result in quick disruption.

I take $f$ so that it describes the galaxy’s stars: when integrated over the entire phase space $f$ generates an approximately Plummer density profile with the correct half-light radius. In particular, I take $f$ to have an isotropic lowered isothermal form, which has been shown to provide a good description of both density and kinematic profiles of dSphs (e.g., Amorisco & Evans, 2011, 2012). I take $r_{min}=0.5$ kpc for EriII and and $r_{min}=0.7$ kpc for AndXXV and assume that all orbits with $r_{peri}\geq r_{min}$ are equally viable. This is equivalent to requiring that the lifetime of the clusters is $\gtrsim$1 Gyr.

The fraction of mass the phase space density $g$ generates within a projected distance $\delta_{LOS}$ from the center represents the probability of observing the cluster at an instantaneous projected location $d<\delta_{LOS}$. Such probability is shown in the top panel of Fig. 5, in red for EriII and in blue for AndXXV. The yellow shaded area shows the range selected by the observations: $\delta_{LOS}<60$ pc. For either systems individually, the probability of observing the cluster at such small radii is very small: $p\lesssim 0.4$%. As the two systems are independent, the probability of observing both this close to the center is entirely negligible. This excludes the possibility that the clusters in EriII and AndXXV formed in cuspy haloes with structural properties similar to those currently observed and have a lifetime $\gtrsim 1$ Gyr. A possible way to escape this is that the clusters formed at the center of their respective host haloes, which is examined in Sect. 5.1.1.

It is interesting to consider kinematic predictions of the different scenarios, providing means to test them. This is especially the case for EriII, since the dwarf’s mass is substantially more secure than in the case of AndXXV, as discussed in Sect. 3.1. The red histogram in the bottom panel of Fig. 5 shows the probability distribution of the cluster’s LOS velocity, relative to the systemic velocity of EriII $v_{LOS,sys}$,

[TABLE]

as obtained from the phase space distribution function $g$, for $d<60$ pc. In gray, the same probability distribution is shown for all EriII stars at $d<60$ pc, as generated by $f$. The distributions are similar, but large LOS velocities are somewhat disfavored for the cluster. In this scenario, the internal velocity dispersion of the clusters is as implied by their stellar mass. For a mass to light ratio $M/L_{V}=2.5~{}M_{\odot}/L_{\odot}$, the internal velocity dispersions are

[TABLE]

Finally, in this scenario, it is fair to ask whether DM subhaloes could represent an additional threat for the clusters. If I assume the population of bound substructures in cold DM haloes with $\log M_{200}/M_{\odot}=9$ is a scaled version of the one in Milky Way sized haloes (Diemand et al., 2008; Springel et al., 2008), close encounters that could seriously disturb the clusters are extremely rare. This is due to the combination of the high relative velocities and of the preferentially higher orbital energies of subhaloes, which are rare in the central regions (Springel et al., 2008).