MNRAS 000,1–11 (2017) Preprint 28 February 2018 Compiled using MNRAS LATEX style file v3.0 Tidal disruption of dwarf spheroidal galaxies: the strange case of Crater II Jason L. Sanders,1? N. W. Evans,1 and W. Dehnen2. 1Institute of Astronomy, Cambridge, CB3 0HA, UK 2Department of Physics and Astronomy, University Rd, Leicester, LE1 7RH, UK Accepted XXX. Received YYY; in original form ZZZ ABSTRACT Dwarf spheroidal galaxies of the Local Group obey a relationship between the line-of-sight velocity dispersion and half-light radius, although there are a number of dwarfs that lie be- neath this relation with suppressed velocity dispersion. The most discrepant of these (in the Milky Way) is the ‘feeble giant’ Crater II. Using analytic arguments supported by controlled numerical simulations of tidally-stripped flattened two-component dwarf galaxies, we inves- tigate interpretations of Crater II within standard galaxy formation theory. Heavy tidal dis- ruption is necessary to explain the velocity-dispersion suppression which is plausible if the −1 proper motion of Crater II is ¹µα∗; µδº = (−0:21 ± 0:09; −0:24 ± 0:09ºmas yr . Furthermore, we demonstrate that the velocity dispersion of tidally-disrupted systems is solely a function of the total mass loss even for weakly-embedded and flattened systems. The half-light radius evolution depends more sensitively on orbital phase and the properties of the dark matter pro- file. The half-light radius of weakly-embedded cusped systems rapidly decreases producing some tension with the Crater II observations. This tension is alleviated by cored dark mat- ter profiles, in which the half-light radius can grow after tidal disruption. The evolution of flattened galaxies is characterised by two competing effects: tidal shocking makes the central regions rounder whilst tidal distortion produces a prolate tidally-locked outer envelope. After ∼ 70 per cent of the central mass is lost, tidal distortion becomes the dominant effect and the shape of the central regions of the galaxy tends to a universal prolate shape irrespective of the initial shape. Key words: galaxies: dwarf – galaxies: structure – Local Group – galaxies: fundamental parameters – galaxies: kinematics and dynamics – galaxies: evolution 1 INTRODUCTION discrepant of these (in the Milky Way) is the recently-discovered ‘feeble giant’ dwarf spheroidal galaxy (dSph) Crater II (Rh ≈ The dwarf spheroidal galaxies (dSphs) of the Local Group lie at the 5 1:1 kpc, L ≈ 1:5 × 10 L , Torrealba et al. 2016). This dSph extreme low mass end of all known galaxies. As such they present V appears problematic to fit into standard galaxy formation theory a number of challenges for any theory of galaxy formation (Wein- due to its extreme kinematic coldness (σ ≈ 2:6 km s−1, Cald- berg et al. 2015). Their typically high dynamical-mass-to-light ra- los well et al. 2017) compared to an expected σ ≈ 11 km s−1. The tios (Mateo 1998) indicate the presence of massive dark matter los observations of Crater II are entirely in line with the predictions arXiv:1802.09537v1 [astro-ph.GA] 26 Feb 2018 haloes. The velocity dispersion and half-light radius of dSphs are −1 of Modified Newtonian Dynamics (MOND, σlos ≈ 2 km s , Mc- / 1/2 approximately related as σlos Rh which can be interpreted as Gaugh 2016) in which a low velocity dispersion (anticipated for all dSphs residing in a universal dark matter halo (Strigari et al. low luminosity dwarfs) is further suppressed by the external field 2008; Walker et al. 2009; Wolf et al. 2010; Sawala et al. 2016) cor- effect from the Galaxy (expected when the internal acceleration is responding to the smallest dark matter halo that can form and retain significantly smaller than the external, Milgrom 1983; McGaugh & stars in ΛCDM galaxy formation theory (Okamoto & Frenk 2009). Milgrom 2013). From the classical dSphs, this smallest dark matter halo has peak −1 Cosmological simulations of dSphs (Sawala et al. 2016; circular velocity Vmax ∼ 18 km s with an uncertainty / natural scatter of ∼ 3 km s−1 (Sawala et al. 2016). Frings et al. 2017; Fattahi et al. 2017) demonstrate that tidal strip- However, a number of dSphs (in both the Milky Way and ping (e.g. Hayashi et al. 2003; Peñarrubia et al. 2008; van den M31) fall significantly under this universal halo relation. The most Bosch et al. 2017) significantly suppresses the velocity disper- sion producing objects similar to Crater II. A further considera- tion is possible geometric effects (e.g Sanders et al. 2016; Sanders ? E-mail: jls,[email protected] (JLS, NWE) & Evans 2016, 2017), as significant flattening along the line of © 2017 The Authors 2 J. L. Sanders et al. sight (of both light and dark matter) can suppress the line of sight dSphs. When applied to Crater II (using Rh = 1:1 kpc), this for- velocity dispersion. Thirdly, simulations show natural variance in mula gives a velocity dispersion suppression of ∼ 0:24 relative to the concentration and maximum circular velocity of haloes hosting the universal relation. dSphs. Here, we investigate whether Crater II can be fit into the standard ΛCDM galaxy formation picture through a combination of these effects. 2.2 Shape We begin our investigation by detailing in Section2 the prop- We also consider the possibility of scatter about these relation- erties of dSphs and discussing the approximate effect of shape and ships due to geometric effects. For simplicity, we consider ‘equally- tides. In Section3, we use these properties to inspire controlled flattened’ dwarf galaxies i.e. those for which the stars and dark two-component N-body simulations to thoroughly inspect the re- matter are stratified on concentric self-similar ellipsoids with axis lationship between mass loss, velocity dispersion suppression and ratios p = b/a and q = c/a. There is some evidence for such a shape. By calibrating a simple probabilistic model using the N- simplification from simulations (Knebe et al. 2010). When viewed body simulations, we explore a wider range of possibilities for at standard spherical polar angles θ and φ relative to the Carte- Crater II putting constraints on its orbital history in Section4. In sian coordinate system defined by the principal axes, the observed Section5, we investigate the sensitivity of our conclusions to the velocity dispersion and on-sky half-light radius can be computed dark and light mass profiles, and in particular study the effects of a using simple analytic expressions (Roberts 1962; Sanders & Evans weak dark matter cusp or core. In Section6, we present conclusions 2016). For the current application, it is worth considering the oblate and thoughts on future observational tests that could be performed spheroidal case, as the observed ellipticity of Crater II is consis- in light of our work. tent with being round ( . 0:1). When viewing an oblate spheroid ‘face-on’ (θ = 0), the measured velocity dispersion is smaller than the spherical average. For instance, if the axis ratio is q = 0:3, the ∼ 2 DWARF SPHEROIDAL PROPERTIES AND suppression factor is 0:67. In a more extreme case, if q = 0:1, ∼ EVOLUTION the factor is 0:41. Additionally, the measured half-light radius is larger than the spherically-averaged half-light radius as the galaxy 2.1 Universal relation has been stretched on the sky (and compressed along the line-of- sight). For q = 0:3, the half-light radius is larger by a factor ∼ 1:5. dSphs follow a trend between the line-of-sight velocity dispersion Both of the effects produce scatter away from any universal rela- σ (luminosity-averaged over the whole dSph) and the half-light los tion between velocity-dispersion and radius in the same sense as major-axis length R . Walker et al.(2010) demonstrates that the h the discrepancy of Crater II. However, only very extreme flattening dSphs of the MW and M31 fall on the low-mass end of the re- and a fortuitous viewing down the minor axis can produce the de- lationship between circular velocity and radius traced by galaxies gree of velocity dispersion suppression and shape of Crater II. We over a wide mass range. Converting this relationship into σ gives los require a further mechanism to contribute to the suppression. ¹ +0:15º log10 σlos = 1:271−0:19 + 0:5 log10 Rh. Another suggestion (Walker et al. 2009; Wolf et al. 2010) is that all the dSphs live within a universal dark matter halo (the low- 2.3 Tidal disruption est mass halo that admits star formation and subsequently retains the stars). We adopt the results from Sawala et al.(2016) compar- Tidal disruption removes mass from the centre of a dwarf galaxy ing subhaloes in the APOSTLE simulations to the classical dSphs producing a suppression in the velocity dispersion. The tidal radius −1 and measure a mean peak circular velocity Vmax = 17:6 km s of a spherical galaxy with mass profile M¹rº with instantaneous −1 orbital frequency Ω = dφ/dt at radius r in a spherical Galactic with scatter ∆Vmax = 3:2 km s (accounting for the uncertainties and using the dark-matter-only uncertainties for Draco and Fornax potential Φ is (King 1962; van den Bosch et al. 2017) for which there are insufficient APOSTLE analogues to reliably es- 1 GM¹rtº 3 timate the uncertainty in Vmax). Converting the circular velocity to rt¹rº = : (2) p Ω2 − @2Φ velocity dispersion via Vc = 2:5σlos (Walker et al. 2010), we have r the relation For a galaxy on a circular orbit in a gravitational field with mass h r r i profile M ¹rº and logarithmic density slope −γ, this equates to σ2 = 1:85V2 sDM ln¹1 + ¹r/r ºº − ; (1) G los max r sDM r + r sDM M¹r º 1 ¹ º t 3 1 rt r = r; (3) which for small r (assuming a concentration c = 20 halo (Mac- γMG¹rº −1 ciò et al.