The effect of tides on the Sculptor dwarf spheroidal galaxy

TL;DR

This study uses N-body simulations to assess whether tidal forces from the Milky Way significantly affect the Sculptor dwarf spheroidal galaxy, concluding that its stellar component remains unaffected and its dark matter halo experiences partial stripping.

Contribution

The paper provides a detailed simulation-based analysis demonstrating that Sculptor's stellar kinematics are robust indicators of its internal dynamics despite tidal interactions.

Findings

Stellar component of Sculptor is not directly influenced by tides.

Approximately 30-60% of the dark matter halo mass is stripped by tides.

Dark-to-luminous mass ratio is about 6 within the half-light radius.

Abstract

Dwarf spheroidal galaxies (dSphs) appear to be some of the most dark matter dominated objects in the Universe. Their dynamical masses are commonly derived using the kinematics of stars under the assumption of equilibrium. However, these objects are satellites of massive galaxies (e.g.\ the Milky Way) and thus can be influenced by their tidal fields. We investigate the implication of the assumption of equilibrium focusing on the Sculptor dSph by means of ad-hoc $N$-body simulations tuned to reproduce the observed properties of Sculptor following the evolution along some observationally motivated orbits in the Milky Way gravitational field. For this purpose, we used state-of-the-art spectroscopic and photometric samples of Sculptor's stars. We found that the stellar component of the simulated object is not directly influenced by the tidal field, while $\approx 30\%-60\%$ the mass of the more diffuse DM halo is stripped. We conclude that, considering the most recent estimate of the Sculptor proper motion, the system is not affected by the tides and the stellar kinematics represents a robust tracer of the internal dynamics. In the simulations that match the observed properties of Sculptor, the present-day dark-to-luminous mass ratio is $\approx 6$ within the stellar half-light radius ($\approx0.3$ kpc) and $>50$ within the maximum radius of the analysed dataset ($\approx1.5^\circ\approx2$ kpc).

Tables

Table 1. Table 1: Sculptor’s observational parameters. From the top to the bottom: Galactic coordinates ( ℓ ℓ \ell , b 𝑏 b ), distance D 𝐷 D , total luminosity in the V 𝑉 V band L ∗ subscript 𝐿 L_{*} , stellar mass-to-light ratio in the V 𝑉 V band Υ ∗ subscript Υ \Upsilon_{*} , half-light radius R h subscript 𝑅 h R_{\text{h}} , best-fit Plummer scale length b ∗ subscript 𝑏 b_{*} , ellipticity ϵ italic-ϵ \epsilon , position angle P ​ A 𝑃 𝐴 PA (see Sec.  2.2 ); line-of-sight (los) systemic velocity V sys subscript 𝑉 sys V_{\text{sys}} , los velocity dispersion σ los subscript 𝜎 los \sigma_{\text{los}} estimated considering all the Sculptor’s stars within the half-light radius and 1.5 ∘ superscript 1.5 1.5^{\circ} (roughly the radial extent of our dataset, see Sec.  2.3.2 ). References: 1) McConnachie ( 2012 ) ; 2) Pietrzyński et al. ( 2008 ) ; 3) Mateo ( 1998 ) ; C) Cicuéndez, private communication; T) this work. † † \dagger Assuming the distance in the table the angular to physical scale conversion is f scale = 25.02 ± 1.75 ​ pc / ′ subscript 𝑓 scale plus-or-minus 25.02 1.75 pc ′ f_{\text{scale}}=25.02\pm 1.75\ \,{\rm pc}/\arcmin .

Parameter	Value	Reference
($\ell$, $b$)	(${287.53}^{\circ }$, $-{83.16}^{\circ }$)	1
$D$	$86\pm 6^{†}$ kpc	2
$L_{\ast }$	$2.3\times {10}^6{L}_{\odot }$	1
${{\rm Y}}_{\ast }$	$2$	3
$R_{\text{h}}$	$11.30\prime \pm 1.60\prime$ ($283\pm 45{\mathrm{pc}}^{†}$)	1
$b_{\ast }$	$11.15\prime \pm 0.18\prime$ ($279\pm 4.5{\mathrm{pc}}^{†}$)	C
$ϵ$	$0.226\pm 0.014$	C
$PA$	${97.4}^{\circ }\pm {1.2}^{\circ }$	C
$V_{\text{sys}}$	$111.6\pm 0.2\mathrm{km}{s}^{-1}$	T
	$8.4\pm 0.2\mathrm{km}{s}^{-1}$	T
	$8.9\pm 0.2\mathrm{km}{s}^{-1}$	T

Table 2. Table 2: Orbital parameters estimated for different proper motions of Sculptor: H18 , F18 , M18 and S17 (Cols. 1a/1b). For each of them, we integrated 10 5 superscript 10 5 10^{5} orbits varying all the Sculptor (Tab.  1 ) and Milky Way parameters accordingly with their errors (see the text). The columns report the statistics of the posterior distributions obtained for the radius of the apocentre (Cols. 2a/3a) and of the pericentre (Cols. 2b/3b) assuming the Milky Way models B14 and J95 . The numbers indicate the values obtained considering the fiducial value for the Sculptor and Milky Way parameters, the superscript and the subscript represent the interval that contains the 68 % percent 68 68\% ( 1 ​ σ 1 𝜎 1\sigma ) of the distribution. The values inside the square brackets indicate the interval containing the 99.7 % percent 99.7 99.7\% ( 3 ​ σ 3 𝜎 3\sigma ) of the distribution. † † \dagger The errors are the quadratic sum of the nominal errors reported in H18 and F18 and the 0.028 ​ mas ​ yr − 1 0.028 mas superscript yr 1 0.028\ \text{mas}\ \text{yr}^{-1} Gaia systematic error (Lindegren et al., 2018 ) .

			Milky Way model
	Proper motions		B14		J95
Model	${\mu }_{\alpha ,\ast }$ ($\text{mas}{\text{yr}}^{-1}$) (1a)	${\mu }_{\delta }$ ($\text{mas}{\text{yr}}^{-1}$) (1b)	$r_{\mathrm{apo}}$ [kpc] (2a)	$r_{\mathrm{peri}}$ [kpc] (2b)	$r_{\mathrm{apo}}$ [kpc] (3a)	$r_{\mathrm{peri}}$ [kpc] (3b)
\rowcolor[HTML]CCCCCC    \cellcolor[HTML]CCCCCCH18		$0.082\pm {0.028}^{†}$	$-0.131\pm {0.028}^{†}$	${145}_{-23}^{+17}$ $[99,246]$	${65}_{-7}^{+6}$ $[45,83]$	${105}_{-7}^{+8}$ $[84,132]$	${54}_{-4}^{+6}$ $[36,74]$
F18	$0.085\pm {0.029}^{†}$	$-0.133\pm {0.029}^{†}$	${143}_{-22}^{+17}$ $[99,244]$	${64}_{-6}^{+6}$ $[45,83]$	${105}_{-8}^{+7}$ $[83,130]$	${54}_{-7}^{+6}$ $[36,74]$
\rowcolor[HTML]CCCCCC    \cellcolor[HTML]CCCCCCM18		$-0.200\pm 0.140$	$-0.330\pm 0.110$	${361}_{-273}^{+425}$ $[81,2428]$	${81}_{-9}^{+7}$ $[21,101]$	${141}_{-55}^{+49}$ $[73,811]$	${78}_{-11}^{+11}$ $[18,101]$
S17	$0.030\pm 0.020$	$-0.140\pm 0.020$	${152}_{-21}^{+17}$ $[111,266]$	${67}_{-11}^{+5}$ $[54,85]$	${108}_{-8}^{+8}$ $[87,135]$	${60}_{-6}^{+5}$ $[45,76]$

Table 3. Table 3: Parameters that give rise to the two orbits used in the N 𝑁 N -body simulations. (1) distance of the Sun from the Galactic centre; (2) proper motion of the Sun in the Galactic frame of reference (see text); (3) Sculptor’s los systemic velocity; (4) Sculptor’s distance from the Sun; (5) Sculptor’s proper motion.

	$R_{\odot }$ [kpc] (1)	${\overrightarrow{v}}_{\odot }$ $(V_x,V_y,V_z)$ [$\mathrm{km}{s}^{-1}$] (2)	$V_{\text{sys}}$ [$\mathrm{km}{s}^{-1}$] (3)	$D$ [kpc] (4)	$\overrightarrow{\mu }$ $({\mu }_{\alpha ,\ast },{\mu }_{\delta })$ [$\text{mas}{\text{yr}}^{-1}$] (5)
FH18	$8.0$	$(-11.1,230.2,7.3)$	$111.6$	$86.0$	$(0.085,-0.133)$
EH18	$8.2$	$(-10.9,213.3,7.9)$	$111.8$	$82.3$	$(0.125,-0.164)$

Table 4. Table 4: Summary of simulation properties. For each analysed orbit (FH18 and EH18, see Tab.  3 ), we report data for the first and the last simulations in the iterative cycle (Sec.  5.3 ) as indicated by the subscript. (1)(2) stellar mass within R ∗ = 1.5 ∘ subscript 𝑅 superscript 1.5 R_{*}=1.5^{\circ} and core radius of the initial 3D stellar density profile (Eq.  2 and Eq.  9 ); (3) total stellar mass ( 5 × 10 4 5 superscript 10 4 5\times 10^{4} stellar particles); (4)(5)(6) viral mass, core radius and concentration parameter of the initial 3D DM density profile (see Sec.  4.1.2 , Eq.  5 and Eq.  10 ); (7) DM total mass ( 4 × 10 6 4 superscript 10 6 4\times 10^{6} DM particles); (8) total mass enclosed within r = 1 𝑟 1 r=1 kpc at t = 0 𝑡 0 t=0 ; (9) DM-to-stellar particle mass ratio; (10) initial dynamical time (it is used to set the simulation time step, see Sec.  5.2.1 ); (11) simulation integration time; (12) fraction of the initial stellar mass retained within (1, 2) kpc in the last snapshot ( t = t sim 𝑡 subscript 𝑡 sim t=t_{\text{sim}} ); (13) same as Col. (12), but for the DM mass.

	$M_{\ast }$ [${10}^6{\text{M}}_{\odot }$] (1)	$b_{\ast }$ [kpc] (2)	$M_{\ast ,\text{tot}}$ [${10}^6{\text{M}}_{\odot }$] (3)	$M_{\text{200}}$ [${10}^9{\text{M}}_{\odot }$] (4)	$r_{\text{c}}$ [kpc] (5)	$c$ (6)	$M_{\text{DM},\text{tot}}$ [${10}^9{\text{M}}_{\odot }$] (7)	[${10}^8{\text{M}}_{\odot }$] (8)	$f_{\text{p}}$ (9)	$t_{\text{dyn}}$ [Myr] (10)	$t_{\text{sim}}$ [Gyr] (11)	$f_{\ast }$ (12)	$f_{\text{DM}}$ (13)
\rowcolor[HTML]C0C0C0 FH180	4.60	0.28	4.68	2.45	0.14	14.89	1.01	1.04	2.69	53.8	6.85	(96%, 98%)	(65%, 55%)
FH1814	4.62	0.26	4.69	6.31	0.13	13.80	1.95	1.49	5.20	44.69	6.79	(97%, 99%)	(69%, 60%)
\rowcolor[HTML]C0C0C0 EH180	4.60	0.27	4.68	2.57	0.15	14.83	0.84	1.05	2.23	52.41	7.19	(89%, 91%)	(37%, 23%)
EH1817	4.66	0.22	4.71	22.39	0.12	12.38	2.94	2.23	7.81	35.76	7.07	(95%, 98%)	(45%, 35%)

Equations

$$N_{\ast }\propto {\left(1+\frac{R_{\text{ell}}^2}{b_{\ast }^2}\right)}^{-2},$$

$${\rho }_{\ast }(r)={\rho }_0{\left(1+\frac{r^2}{b_{\text{*}}^2}\right)}^{-2.5}.$$

$${\rho }_0=M_{\ast }\frac{3}{4\pi b_{\ast }^3}\left(\frac{b_{\ast }^2+R_{\ast }^2}{R_{\ast }^2}\right),$$

$${\Sigma }_{\ast }(R)={\rho }_0{\int }_R^{\infty }\text{d}r{\left(1+\frac{r^2}{b_{\text{*}}^2}\right)}^{-2.5}\frac{r}{\sqrt{r^2-R^2}}={\Sigma }_0{\left(1+\frac{R^2}{b_{\ast }^2}\right)}^{-2},$$

$${\rho }_{\text{DM}}(r)={\rho }_{\text{crit}}{\delta }_c{\left(\frac{r_{\text{c}}^2+r^2}{r_{\text{s}}^2}\right)}^{-0.5}{\left(1+\frac{r}{r_{\text{s}}}\right)}^{-2}.$$

$${\delta }_c=\frac{200}{3}\frac{c^3}{\ln (1+c)-c/(1+c)}$$

$${\sigma }_{\text{los}}^2(R)=2G{\Sigma }_{\ast }^{-1}(R){\int }_R^{\infty }\frac{\text{d}r}{r}{\rho }_{\ast }(r)M_{\text{tot}}(r)\sqrt{1-\frac{R^2}{r^2}},$$

$$\ln \mathcal{L}=-0.5\sum _{i=1}^{N_{\text{Scl}}}\left(\frac{{\left(V_{\text{los,i}}-V_{\text{sys}}\right)}^2}{\delta V_{\text{los,i}}^2+{\sigma }_{\text{los}}^2(R_i;\vec{\theta })}+\ln \left(\delta V_{\text{los,i}}^2+{\sigma }_{\text{los}}^2(R_i;\vec{\theta }\right)\right),$$

$${\rho }_{\ast ,\text{Nb}}(r;M_{\ast },b_{\ast })={\rho }_{\ast }(r;M_{\ast },b_{\ast })\exp \left[-{\left(\frac{r}{r_{\text{t}}}\right)}^2\right],$$

$${\rho }_{\text{DM},\text{Nb}}(r;M_{200},r_{\mathrm{c}})={\rho }_{\text{DM}}(r;M_{200},r_{\mathrm{c}})\exp \left[-{\left(\frac{r}{r_{\text{t}}}\right)}^2\right],$$

$$t_{\text{dyn}}=\sqrt{\frac{1}{G{\bar{\rho }}_{\text{h}}}}$$

$$t_{\text{f}}(r_{\text{g}})=\frac{1.17}{\ln \Lambda }\frac{M_{\text{MW}}(<r_{\text{g}})}{M_{\text{Scl}}}{t}_{\text{cross}}(r_{\text{g}}),$$

$$\begin{gathered} b_{\ast ,\text{i+1}}=b_{\ast ,\text{i}}\frac{R_{\text{h}}}{R_{\text{h,sim}}}, \\ {M}_{\ast ,\text{i+1}}=M_{\ast ,\text{i}}\frac{M_{\ast }}{M_{\text{*,sim}}}, \\ {r}_{\text{c},\text{i+1}}=r_{\text{c},\text{i}}\frac{{\sigma }_{\text{los,sim}}(<R_{\text{h,sim}})}{{\sigma }_{\text{los}}(<R_{\text{h}})}, \\ {M}_{\text{vir},\text{i+1}}=M_{\text{vir},\text{i}}\frac{{\sigma }_{\text{los}}(<R_{\ast })}{{\sigma }_{\text{los,sim}}(<R_{\ast })}, \end{gathered}$$

$$t_{\text{sim,i+1}}=t_{\text{sim,i}}-\Delta t.$$

$$\left[\begin{array}{ccc}{V}_b& V_{\ell }& V_{\text{los}}\end{array}\right]=\left[\begin{array}{ccc}{V}_x& V_y& V_z\end{array}\right]{\text{R}}_{\text{rot}},$$

$${\text{R}}_{\text{rot}}=\left[\begin{array}{ccc}-\cos (\ell )\sin (b)& -\sin (\ell )& \cos (\ell )\sin (b)\\ -\sin (\ell )\sin (b)& \cos (\ell )& \sin (\ell )\cos (b)\\ \cos (b)& 0& \sin (b)\end{array}\right].$$

Full text

††pubyear: 2019††pagerange: The effect of tides on the Sculptor dwarf spheroidal galaxy–A

1 Introduction

Dwarf galaxies are the most numerous galaxies in the Universe (Ferguson & Binggeli, 1994): around the Local Group, about 70% of galaxies are dwarfs (McConnachie, 2012). The fraction remains this high also considering the Local Volume within 11 Mpc from the Milky Way (Dale et al., 2009). Understanding their structure, formation and evolution is a fundamental goal for astrophysics and cosmology (Klypin et al., 1999; Moore et al., 1999; Bullock et al., 2000; Grebel et al., 2003; Boylan-Kolchin et al., 2011; Mayer, 2011; Anderhalden et al., 2013; Battaglia et al., 2013; Walker, 2013; Read et al., 2017). Among them, the dwarf spheroidal galaxies (dSphs), satellites of the Milky Way, are close enough to measure the kinematics of individual stars (e.g.  Kleyna et al., 2002; Tolstoy et al., 2004; Battaglia et al., 2008; Muñoz et al., 2006a; Koch et al., 2007; Walker et al., 2009a; Massari et al., 2018). The analysis of kinematic dataset with equilibrium models leads to the finding that these galaxies are some of the most dark matter (DM) dominated object known to date (Mateo, 1998; Gilmore et al., 2007), with dynamical mass-to-light ratios up to several 100s (e.g. Mateo et al. 1993; Muñoz et al. 2006b; Battaglia et al. 2008; Koposov et al. 2011; Battaglia et al. 2011; see Mateo 1998 and Battaglia et al. 2013 for further information on the mass-to-light ratios of dSphs). Therefore, these objects are ideal laboratories to test theories on DM physics and cosmology (Navarro et al., 1996; Read et al., 2016; Macciò & Fontanot, 2010; Walker, 2013; Kennedy et al., 2014; Kubik et al., 2017).

However, since these dSphs are orbiting in the gravitational field of the MW, it is possible that the stellar tracers used to infer the dynamical state of these objects are influenced by tides and the assumption of equilibrium could introduce severe biases in the mass estimate (Kroupa, 1997; Fleck & Kuhn, 2003; Metz & Kroupa, 2007; Dabringhausen et al., 2016). For example, stars in undetected tidal tails (too faint or along the line-of-sight) could “pollute” the kinematic samples inflating the velocity dispersion, with a consequent overestimate the DM content, if inferred using equilibrium dynamical models (see e.g. Klimentowski et al., 2009; Read et al., 2006).

Several works have focused on the effects of the host gravitational field on the internal properties of dwarf galaxies by means of $N$-body simulations (e.g. Read et al., 2006; Peñarrubia et al., 2008, 2009a; Frings et al., 2017; Fattahi et al., 2018). The effect of tides on these galaxies strongly depends on their orbital history and the degree of pollution of tidally perturbed stars depends on the relative orientation of the internal part of the tides with respect to the line of sight. Therefore, the natural step forward is to perform similar analysis focusing on the properties of specific dSphs orbiting around the Milky Way. These works are based on $N$-body simulations tuned to reproduce the observed properties of the analysed dSph after their orbital evolution in the Milky Way tidal field. Works of this kind have been applied to Carina (Ural et al., 2015), Leo I (Sohn et al., 2007; Mateo et al., 2008), Fornax (Battaglia et al., 2015) and to the recently discovered Crater II (Sanders et al., 2018). However, one of the biggest source of uncertainty is the estimate of the proper motions that are indispensable to put the objects in realistic orbits to measure the actual effect of tides on the observables. The choice of a well-motivated orbit is essential to consider the right amount of tidal disturbance suffered by the dSphs and to rightly evaluate the degree of tidally perturbed stars contaminating the observed dataset. Luckily, at the moment we are living in the golden era of proper motion measurements thanks to the revolutionary Gaia DR2 catalogue (Gaia Collaboration et al., 2018a; Gaia Collaboration et al., 2018b; Fritz et al., 2018) and the very long baseline supplied by the HST measurements (e.g. Sohn et al., 2017).

Following the work of Battaglia et al. (2015), focused on the Fornax dwarf spheroidal, the aim of this paper is to investigate and quantify the effects of the tides on the structure and kinematics of the Sculptor dwarf spheroidal. Battaglia et al. (2015) found that in the case of Fornax tides have an almost negligible effect on the observed structure and kinematics of the stellar component, even considering the most eccentric orbits compatible with the observational constraints. Similarly to Fornax, the Sculptor dSph is one of the best studied Milky Way satellites (see e.g. Battaglia et al., 2008; Peñarrubia et al., 2009a; Walker et al., 2009b; Breddels & Helmi, 2014; Tolstoy, 2014; Strigari et al., 2018; Massari et al., 2018), so we can rely on high-quality spectroscopic data against which to compare our models (Walker et al., 2009a; Battaglia & Starkenburg, 2012). Moreover, the predicted Sculptor orbit is less external (smaller pericentre) than that of Fornax, so we expect that it could be more influenced by tides. For this aim, we ran a set of $N$-body simulations following the evolution of the structural and kinematic properties of Sculptor-like objects on orbits around the Milky Way that are consistent with the most recent observational constraints. Then, simulating “observations” in the last snapshot of the simulations, we investigated whether the observed kinematics of the stars is a robust tracer for the current dynamical state of Sculptor or the tides are introducing non-negligible biases.

The paper is organised as follows. In Sec. 2 we summarise Sculptor’s observational properties and we present a clean sample of Sculptor’s member stars that we use to analyse the kinematics and dynamics of Sculptor. In Sec. 3 we determine the possible Sculptor’s orbits around the Milky Way making use of the most recent and accurate Sculptor’s proper motion estimates. In Sec. 4, we use the information obtained in the previous sections to define a dynamical model of Sculptor compatible with the observable properties. Then, in Sec. 5 we describe the set-up of $N$-body simulations, whose results are presented in Sec. 6. Finally, in Sec. 7 we summarise and discuss the main results of this work.

2 Observed properties of the Sculptor dSph

2.1 Structure

The Sculptor dwarf spheroidal galaxy is a satellite of the Milky Way located close to the Galactic South Pole at Galactic coordinates ($\ell$, $b$)=($287.53^{\circ}$, $-83.16^{\circ}$). The distance to Sculptor has been measured with different methods: photometry of variable stars (RR Lyrae, e.g. Martínez-Vázquez et al. 2015; Miras, e.g. Menzies et al. 2011), of horizontal branch stars (e.g. Salaris et al. 2013), using the tip of the red giant branch (e.g. Rizzi et al. 2007) and analysing the colour magnitude diagram of the stars in the galaxy (e.g. Weisz et al. 2014). We decided to take as a reference the value $D=86\pm 6\ \,{\rm km\,s}^{-1}$, adopting the distance modulus obtained by Pietrzyński et al. (2008) using the near-infrared photometry ($J$ and $K$ bands) of a sample of 76 RR Lyrae stars. This value is compatible with the recent estimate by Garofalo et al. (2018).

There is evidence that Sculptor contains two distinct distinct stellar populations: a more concentrated metal rich population with low velocity dispersion ($\approx 6-7\ \,{\rm km\,s}^{-1}$) and a more extended metal poor population with high velocity dispersion ($\approx 10-12\ \,{\rm km\,s}^{-1}$); see e.g. Tolstoy et al. (2004); Battaglia et al. (2008); Walker & Peñarrubia (2011). As well known, the existence of two populations can be exploited to constrain the mass distribution of the galaxy (e.g. Battaglia et al. 2008; Amorisco & Evans 2012; Walker & Peñarrubia 2011; Strigari et al. 2017). However, since the purpose of this work is not a detailed study of the internal dynamics of Sculptor, we assume for simplicity the presence of a single stellar component. We discuss the possible implications of this assumption in Sec. 7.

The properties of the light distribution have been derived employing the same state-of-the-art technique used by Cicuéndez et al. (2018) for the Sextans dwarf spheroidal applying it on a publicly available VST/ATLAS $g$-band and $r$-band photometric catalogue of stellar sources covering $\approx$ 4 deg2 on the line of sight to Sculptor (Cicuéndez, private communication). The best-fit (constant) ellipticity is $\epsilon=0.226\pm 0.014$ (axial ratio $q\approxeq 0.77$) and the best fit position angle is $PA=97.4^{\circ}\pm 1.2^{\circ}$, in agreement with the values of Battaglia et al. (2008). The best fit to the observed surface number-density profile is a (projected) Plummer profile

[TABLE]

where $R_{\text{ell}}$ is the elliptical radius corresponding the to major axis of the elliptical iso-density contours and the best fit scale length is $b_{*}=11.15\arcmin\pm 0.18\arcmin=279\pm 5\ \text{pc}$. The main observed properties of Sculptor are summarised in Tab. 1.

2.2 Proper motions

There are several direct estimates of the systemic proper motion of Sculptor from astrometric measurements (Schweitzer et al., 1995; Piatek et al., 2006; Sohn et al., 2017; Massari et al., 2018; Gaia Collaboration et al., 2018b; Fritz et al., 2018) and indirect estimates by Walker et al. (2008); Walker & Peñarrubia (2011) using the apparent velocity gradient along the direction of the proper motion. The currently available estimates of the Sculptor’s proper motion are summarised in Fig. 1. While the first measurements were quite uncertain and not all in agreement among each other, in the last couple of years the long time baseline of the HST data and the high-quality astrometric data from Gaia (Gaia Collaboration et al., 2016; Gaia Collaboration et al., 2018a) have drastically improved the proper motion estimate for Sculptor. In particular, those obtained using the Gaia DR2 astrometry (Gaia Collaboration et al. 2018b; Fritz et al. 2018) and the one obtained from the HST with a $\approx 11$ yr long baseline (Sohn et al., 2017) converge toward a likely “definitive” value. In the rest of the paper we consider only the four most recent estimates: Sohn et al. (2017) (S17), Massari et al. (2018) (M18), Gaia Collaboration et al. (2018b) (H18) and Fritz et al. (2018) (F18). We assumed the errors for the H18 and F18 proper motions (see Fig. 1 and Tab. 3.1) as the quadratic sum of the nominal errors reported in the corresponding papers and the Gaia DR2 proper motions systematic error $\delta\mu_{\text{s}}\approx 0.028\ \text{mas}\ \text{yr}^{-1}$. This last value has been estimated from Eq. 18 in Gaia Collaboration et al. (2018a) considering a spatial scale of $1^{\circ}$ roughly corresponding to the extension of the bulk of the Gaia DR2 Sculptor’s member analysed in H18 (see Sec. 7 for a discussion about the assumed systematic errors). Considering the older proper motion estimates it is interesting to compare the two non-astrometric measurements (Walker et al., 2008; Walker & Peñarrubia, 2011) with the most recent ones. In these older works the Sculptor’s proper motion was estimated assuming that any possible line-of-sight velocity gradient is entirely caused by the relative motion between Sculptor and the Sun (see Sec. 2.3.3). However, if a significant intrinsic rotation is present, the results of this analysis can be heavily biased. Therefore, a significant difference with respect to the astrometric proper motion measurements could be an indication of an intrinsic rotation. We found that the non-astrometric measurement are consistent within $\approx 1\sigma$ with M18 and within $\approx 2\sigma$ with S17, H18 and F18. Even if the differences are not significant, we note that these results are mainly driven by the large errors reported in Walker et al. (2008); Walker & Peñarrubia (2011). Hence, a strong claim in favour or against the presence of intrinsic rotation cannot be made from a simple comparison of astrometric and non-astrometric proper motion measurements.

2.3 Internal kinematics

2.3.1 The sample

We derive the line-of-sight (los) velocity dispersion profile of Sculptor using a state-of-the-art dataset obtained combining the MAGELLAN/MIKE catalogue presented in Walker et al. (2009a) (1541 stars, W09 hereafter) and the VLT/FLAMES-GIRAFFE catalogue Battaglia & Starkenburg (2012) (1073 star, BS12 hereafter). The BS12 catalogue contains stars from the original catalogues by Tolstoy et al. (2004), Battaglia et al. (2008) and Starkenburg et al. (2010).

2.3.2 Member selection

The two catalogues contain both genuine Sculptor’s members and Milky Way interlopers. To start with, for the stars in BS12, we applied the cleaning of contaminants using the equivalent width of the Mg I line at $8806.8$ Å as described in BS12. For the stars in W09, we removed all the objects with a member probability lower than 0.5 (see W09 for details on the Sculptor’s member probability estimate). Then, we cross-matched the spectroscopic sample with the Gaia DR2 catalogue (Gaia Collaboration et al., 2018a) to complement it with proper motion and parallax estimates. We removed all the stars that are not compatible with the H18 Sculptor proper motion and with zero parallax within three times the error. For the parallax, the error corresponds to the uncertainty on the individual parallax measurement; for the proper motion the error is a quadratic sum of the uncertainty in the individual measurement, the error on the systemic proper motion and the systematic error of 0.028 mas/yr (see Lindegren et al., 2018, see also Sec. 2.2). Finally, we removed all the stars with los velocities $V_{\rm los}$ differing at least by $50\ \,{\rm km\,s}^{-1}$ from $111\ \,{\rm km\,s}^{-1}$ (roughly the Sculptor systemic velocity). After the selection cuts, we are left with 540 stars from BS12 and 1312 stars from W09.

In order to identify and analyse duplicate objects, we cross-matched the two catalogues considering a search-window of $1\arcsec$ and we found 276 duplicated stars. For each of them we estimate the velocity difference $\Delta V_{\rm los}=V_{\rm los,BS12}-V_{\rm los,W09}$ and the related uncertain $\delta\Delta V_{\rm los}=\sqrt{\delta V^{2}_{\rm los,BS12}+\delta V^{2}_{\rm los,W09}}$. We considered that all the stars with $|\Delta V_{\rm los}/\delta\Delta V_{\rm los}|>3$ are unresolved binary systems and we filtered them out from the two catalogues. Among the duplicated stars, only 17 have been removed, resulting in a $6\%$ fraction of unresolved binaries. Considering the los velocity distribution of the 259 remaining stars, we found and correct a velocity offset of $\simeq-1\,{\rm km\,s}^{-1}$ of the stars in BS12 with respect the stars in W09. Finally, these stars are combined in a single dataset estimating the $V_{\rm los}$ as the error-weighted mean of the single velocity measurements.

The joined Sculptor’s stars catalogue contains 1559 stars ($\approx 66\%$ from W09, $\approx 17\%$ from BS12 and $\approx 17\%$ from both catalogues). As a final conservative cleaning criterion, we applied an iterative $3\sigma$ clipping removing 16 stars with large deviation from the systemic velocity. Fig. 2 shows that most of these stars are located in the central part of Sculptor, hence it is unlikely that they are Milky Way contaminants survived to the selection cuts. Rather, we considered these stars (with single observations in BS12 or W09) as likely unresolved binaries. The fraction of the removed objects ($\approx 1\%$) is roughly compatible with the binary fraction estimated analysing the stars duplicated among the two catalogues. We stress that the $\sigma$ clipping method used to filter binary systems is capable to select only short-period binaries (few days; see Minor 2013; Spencer et al. 2018) for which the typical orbital velocities are significantly higher than the los velocity errors ( $\langle\delta V_{\text{los}}\rangle\approx 2\ \,{\rm km\,s}^{-1}$) and/or than the dwarf velocity dispersion ($\sigma_{\text{los}}\approx 9\,{\rm km\,s}^{-1}$, see Fig. 3). Given the expected distribution of binary periods (Duquennoy & Mayor, 1991; Minor, 2013), the short-period systems represent only a tiny fraction of the binaries. Therefore, our estimated binary fraction represents a strong lower limit (Minor 2013 estimated a 0.6 binary fraction for Sculptor). However, even if most of the stars in our final sample are in fact part of binary systems, their expected radial velocities due to the orbital motions are smaller or compatible with the velocity errors and much smaller than the intrinsic velocity dispersion of the system. Therefore, the kinematic bias introduced by the binary stars on the estimate of the velocity dispersion profile (Sec. 2.3.3) and on the dynamical mass (Sec. 4.2) is negligible (see e.g. Hargreaves et al., 1996; Olszewski et al., 1996).

The final catalogue contains 1543 objects. The systemic velocity of Sculptor, estimated as the mean velocity of its members, is $V_{\rm sys}=111.6\ \pm 0.2\ \,{\rm km\,s}^{-1}$; the velocity dispersion is $\sigma_{\text{los}}=8.9\ \pm 0.2\ \,{\rm km\,s}^{-1}$ considering the whole catalogue and $\sigma_{\text{los}}=8.4\ \pm 0.2\ \,{\rm km\,s}^{-1}$ considering only stars within the half-light radius. The errors on these values have been estimated using the bootstrap technique (Feigelson & Babu, 2012).

2.3.3 Velocity dispersion profile

The relative motion of Sculptor with respect to the Sun causes an artificial los velocity gradient along the proper motion direction (see e.g.  Walker et al., 2008). Therefore, we corrected the measured los velocity for this perspective effect through the methodology described in the Appendix A of Walker et al. (2008) using the systemic proper motion estimated in H18 and the systemic los velocity reported in Tab. 1. The errors on the corrected $V_{\text{los}}$ have been estimated with a Monte Carlo simulation considering the observational uncertainties on $V_{\text{los}}$, the errors on $V_{\text{sys}}$ and distance (Tab. 1), and the error on the proper motions (Tab. 3.1).

The radial profile of los velocity dispersion $\sigma_{\text{los}}(R)$, has been obtained grouping Sculptor’s members in (circular) radial bins. In order to have the best compromise between good statistics and spatial resolution (bin width), we decided to select the bin edges so that each bin contains 150 stars. Since the outermost bin remains with 41 stars only, we merged it with the adjacent bin, obtaining a wider outermost bin with 192 stars. For each of the final 10 bins, we estimated the mean los velocity $\overline{V}_{\text{los}}$ and the velocity dispersion $\sigma_{\text{los}}$ fitting a Gaussian model and taking into account the $V_{\text{los}}$ uncertainties through an extreme-deconvolution technique (Bovy et al., 2011). Similarly, we estimated the representative radius of each bin taking the mean of the circular radius of the stars. The errors on these values have been estimated using the bootstrap technique (Feigelson & Babu, 2012). The resultant $\sigma_{\text{los}}(R)$ profile is shown and compared with the profiles from Walker et al. (2009b) and Breddels & Helmi (2013) in Fig. 3. The two profiles are compatible with our profile within the half-light radius. Beyond this radius the $\sigma_{\text{los}}$ estimated in this work are systematically lower with respect to the values of Breddels & Helmi (2013), but compatible with the Walker et al. (2009b) estimates. This discrepancy is not due to the predominance of stars from W09 in our joined catalogue (see Sec. 2.3.2), because the two outermost bins are dominated by stars taken from BS12 (see bottom bars in Fig. 3). The most likely explanation is that the velocity dispersion profile reported in Breddels & Helmi (2013) is slightly inflated by residual Milky Way interlopers.

The assumptions we made to obtain the final binned profile are discussed below.

• •

$N_{\text{star}}$ per bin. We repeated the analysis described above changing the number of stars per bin from 100 to 350 in steps of 50. The increase of the number of stars per bin reduces the errors in $\sigma_{\text{los}}$ but it decreases the spatial resolution. The final profiles are all compatible, but the chosen value of 150 represents the best compromise between the two kinds of uncertainties.

• •

Velocity gradient. The correction for the perspective velocity gradient due to the relative motion between the Sun and Sculptor has been applied using the H18 proper motion. We repeated the correction procedure considering also the M18 and S17 proper motions. The differences on the final velocity dispersion profiles are negligible ($<1\%$). Even if some residual shallow velocity gradient should still be present, e.g. due to an intrinsic rotational motion (e.g. Battaglia et al. 2008, see also Sec. 2.2; Zhu et al. 2016, see also Sec. 2.2), any possible bias in the velocity dispersion should be important only in the outermost bins where the ratio between the velocity dispersion and the rotation velocity is larger. Considering as a test a rotation velocity of $7\ \,{\rm km\,s}^{-1}$ and a velocity dispersion of $10\ \,{\rm km\,s}^{-1}$ and a sample of 150 stars uniformly distributed along the azimuthal angle, we found that the increase of the velocity dispersion is expected to be lower than $10\%$. This value is “safely” within the error bars of the outermost points of the estimated velocity dispersion profile.

• •

Circular vs elliptical bins. We estimated new velocity dispersion profiles as a function of the elliptical radius $R_{\text{ell}}=\sqrt{x^{2}+y^{2}/q^{2}}$ and of the circularised radius $R_{\text{circ}}=\sqrt{x^{2}q+y^{2}/q}$, where $q$ is the observed Sculptor’s axial ratio (see Sec. 2.1) and $(x,y)$ are the Cartesian coordinates in the reference aligned with the Sculptor’s principal axes. The $\sigma_{\text{los}}(R_{\text{ell}})$ and $\sigma_{\text{los}}(R_{\text{circ}})$ profiles are practically coincident and both the normalisation and the radial trend are compatible with the los velocity dispersion profile shown in Fig. 3.

We stress that the aim of this paper is not the detailed study of the velocity dispersion profile in Sculptor, but rather we want to have a self-consistent way to infer the velocity dispersion profile both in observations and in simulations. The performed analysis and the comparison of our profile with results in the literature (Fig. 3) ensures that our “simple” technique and our assumptions do not cause severe biases in the final estimate of $\sigma_{\text{los}}$.

3 The orbit of Sculptor

3.1 Milky Way model