Universal Dark Halo Scaling Relation for the Dwarf Spheroidal Satellites

The Λ-dominated cold dark matter (ΛCDM) theory is most successful in explaining cosmological and astrophysical observations on spatial scales larger than about 1 Mpc, including the temperature fluctuations of the cosmic microwave background (e.g., Komatsu et al. 2011; Planck Collaboration et al. 2016) and the clustering of galaxies (e.g., Tegmark et al. 2004), which plays a crucial role in the formation of structure in the universe.

Nevertheless, on spatial scales smaller than 1 Mpc, i.e., galactic and sub-galactic mass scales, there are outstanding issues regarding the predictions from this standard theory that are significantly in disagreement with some observational facts. These representative issues include the so-called missing satellite (Klypin et al. 1999; Moore et al. 1999), core/cusp (Moore 1994; Burkert 1995; de Blok et al. 2001; Gilmore et al. 2007), and too-big-to-fail problems (Boylan-Kolchin et al. 2011, 2012). While there are two main aspects that could resolve these issues, baryonic physics and alternative dark matter theories, there still remain uncertainties in actual dark halo structure on small mass scales that have been inferred from currently available data.

In this context, dwarf spheroidal (dSph) galaxies in the Milky Way (MW) and Andromeda (M31) galaxies are ideal sites for studying the nature of dark matter through internal stellar kinematics. This is because these galaxies are the most dark-matter-dominated systems, and line-of-sight velocities for their resolved stars can be observed precisely by high-resolution spectroscopy (e.g., Walker et al. 2009a, 2009b). Moreover, these satellites have drawn special attention as the most promising targets in the indirect detection for particle dark matter though γ-ray stemmed from dark matter annihilation (e.g., Geringer-Sameth et al. 2015; Hayashi et al. 2016). For this reason, implementing extensive dynamical analysis for those galaxies should be of great importance for shedding light on the nature of dark matter on small mass scales.

From the dynamical analysis of the galaxies including the dSphs, several studies have derived dark halo structures for various kinds of galaxies with luminosities over ~14 magnitudes based on cored dark matter density profiles with core radii (r₀) and densities (${\rho }_{0}$), such as the Burkert and pseudo-isothermal profiles, and found that the central surface density, ${\rho }_{0}{r}_{0}$, is nearly constant for all of these galaxies (e.g., Donato et al. 2009; Gentile et al. 2009; Salucci et al. 2012; Kormendy & Freeman 2016). By contrast, Boyarsky et al. (2010) defined the average column density of a dark halo derived by integrating line-of-sight, and estimated it for about 300 objects ranging from dSphs to galaxy clusters. Then they concluded that the column density weakly depends on dark halo mass except for dSphs, and that this trend is in agreement with the prediction of ΛCDM N-body simulations. Several subsequent studies have clearly shown that the dark halo surface density is not constant (Cardone & Del Popolo 2012; Del Popolo et al. 2013; Saburova & Del Popolo 2014).

More recently, Hayashi & Chiba (2015a, hereafter HC15a) proposed another common scale for dark halos. They defined the surface density inside a radius of the maximum circular velocity, ${{\rm{\Sigma }}}_{{V}_{\max }}$, and found that ${{\rm{\Sigma }}}_{{V}_{\max }}$ shows a very weak trend or is almost constant over a wide range of ${V}_{\max }$ from dwarf galaxy to giant spiral/elliptical galaxy scales, irrespective of different dark halo profiles and galaxy types. In addition, Hayashi & Chiba (2015b) showed that ${{\rm{\Sigma }}}_{{V}_{\max }}$ is also constant with respect to B-band luminosity of galaxies over ~14 magnitudes from ${M}_{B}=-8$ to −22 mag.

Following this work, Okayasu & Chiba (2016) investigated the evolution of baryonic components of the dSphs in the MW and M31, under the constraint of a constant ${{\rm{\Sigma }}}_{{V}_{\max }}$ for their dark halos, and found that the models well reproduce star formation histories as derived from both observations and simulations. Thus, the properties of the surface density ${{\rm{\Sigma }}}_{{V}_{\max }}$ may play an important role in understanding star formation histories of dSph satellites and provide important constraints on the nature of dark matter and galaxy formation.

However, our knowledge for theoretically derived ${{\rm{\Sigma }}}_{{V}_{\max }}$ is yet largely uncertain due to several nonlinear effects, including the shift in the inner slope of a density profile, difference in tidal effects from the host halo, and scatters in merging histories, even though the uncertainties of ${{\rm{\Sigma }}}_{{V}_{\max }}$ from observations still largely remain. In particular, calculating ${{\rm{\Sigma }}}_{{V}_{\max }}$ on the dwarf galaxy mass scales in HC15a merely extrapolates from the mass-concentration relation estimated by using massive dark halos with heavier than $\sim {10}^{10}{M}_{\odot }$ in the cosmological simulations (Klypin et al. 2016). Thus, in this work, we investigate the evolution of dark matter subhalos associated with an MW-like dark halo from cosmological N-body simulations to understand the properties of this surface density in more detail. In particular, we focus on less-massive subhalos associated with MW-sized dark halos because the dark halo surface density at these halo mass scales plays a key role in understanding the nature of dark matter and formation histories of low-mass galaxies.

This paper is organized as follows. In Section 4, we briefly introduce a dark halo surface density defined by HC15a. In Section 3, we describe the properties of our cosmological simulations and selection criteria of MW-sized host halos and their subhalos in this work. In Section 4, we compare dark matter surface densities calculated from observations and simulations and then present the universal scaling relation of subhalos. In Section 5, we discuss the origin of this universal relation by analyzing orbital evolutions of subhalos. We summarize our results in Section 6.

In this work, we adopt the mean surface density of a dark halo defined by HC15a to compare this density estimate from observational data with those from pure N-body simulations. Given any of the parameters of a dark halo (e.g., scale length, scale density, and any slopes of dark matter density profiles), this surface density within a radius of the maximum circular velocity, ${V}_{\max }$, is given as

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{{\rm{V}}}_{\max }}=\displaystyle \frac{M({r}_{\max })}{\pi {r}_{\max }^{2}},\end{eqnarray} \tag{ 1 }$$

where ${r}_{\max }$ denotes a radius of maximum circular velocity of a dark halo, and its enclosed mass within ${r}_{\max }$ is given as

$$\begin{eqnarray}&&M({r}_{\max })={\int }_{0}^{{r}_{\max }}4\pi {\rho }_{\mathrm{dm}}({r}^{\prime }){r}^{\prime 2}{{dr}}^{\prime }\end{eqnarray} \tag{ 2 }$$

where ${\rho }_{\mathrm{dm}}$ denotes a dark matter density profile. Under the axisymmetric assumptions, the variables of the spherical radius, ${r}^{\prime }$, are changed to those of the elliptical radius, ${m}^{\prime }$, and then Equation (2) can be rewritten by

$$\begin{eqnarray}&&M({m}_{\max })={\int }_{0}^{{m}_{\max }}4\pi {\rho }_{\mathrm{dm}}({m}^{\prime }){Q}^{2}{m}^{\prime 2}{{dm}}^{\prime },\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{m}^{\prime 2}={R}^{2}+\displaystyle \frac{{z}^{2}}{{Q}^{2}},\end{eqnarray} \tag{ 4 }$$

where ${m}^{\prime }$ is described by cylindrical coordinates (R, z) and the dark halo axial ratio, Q, respectively. This surface density is fundamentally proportional to the product of a scale density, ${\rho }_{* }$, and radius, r_*, for any of the density profiles of a dark halo, where the definition of ${\rho }_{* }$ and r_* depends on an assumed density profile (in contrast to ${{\rm{\Sigma }}}_{{V}_{\max }}$), such as the so-called Navarro–Frenk–White (NFW) profile (Navarro et al. 1996) and Burkert profile (Burkert 1995).

Calculating ${{\rm{\Sigma }}}_{{V}_{\max }}$ and ${V}_{\max }$ of dark halos in the dSph galaxies, we apply our axisymmetric mass models (e.g., Hayashi & Chiba 2012, 2015b) to their available kinematic data to obtain more reliable and realistic limits on their dark halo structures. This is motivated by the fact that the observed light distributions of the dSphs are actually non-spherical shapes (e.g., McConnachie 2012) and that ΛCDM theory predicts non-spherical virialized dark subhalos (e.g., Jing & Suto 2002; Kuhlen et al. 2007; Vera-Ciro et al. 2014). Thus, relaxing the assumption of spherical symmetry in the mass modeling should be valuable in evaluating the dark halo structures of the dSphs. In the axisymmetric mass models we use here, we employ the axisymmetric Jeans equations for the velocity anisotropy of tracer stars, ${\beta }_{z}=1-\overline{{v}_{z}^{2}}/\overline{{v}_{R}^{2}}$, where $\overline{{v}_{R}^{2}}$ and $\overline{{v}_{z}^{2}}$ are the velocity second moments toward R and z directions, respectively. We should bear in mind that there is a degeneracy between the shape of the dark halo Q and ${\beta }_{z}$, as shown in Cappellari (2008) and Hayashi & Chiba (2015b). The stellar surface densities of dSphs are assumed by an oblate Plummer profile (Plummer 1911) with an inclination angle of a dSph. For the dark matter distribution, on the other hand, we adopt an inner slope of the dark halo density profile as a free parameter as well as the parameterization of the central density and scale length of a dark halo (Hayashi & Chiba 2015b; Hayashi et al. 2016). Finally, we adopt six free parameters: the axial ratio, the central density, the scale length and the inner slope of a dark halo, the stellar velocity anisotropy, and the inclination angle. Then these parameters are determined by fitting the observed line-of-sight velocity distribution.

In this work, we re-estimate ${{\rm{\Sigma }}}_{{{\rm{V}}}_{\max }}$ and ${V}_{\max }$ for the seven MW (Carina, Fornax, Sculptor, Sextans, Draco, Leo I, and Leo II) and the five M31 (And I, And II, And III, And V, and And VII) dSphs from those estimated by HC15a and add the results of the Ursa Minor dSph. This is because we have access to the latest stellar kinematic data for Draco (taken from Walker et al. 2015) and Ursa Minor, kindly provided by M. G. Walker (private communication), and because the procedure of fitting axisymmetric mass models to their kinematic data has changed slightly from previous axisymmetric works. Namely, in this work, we adopt an unbinned analysis for the comparison between data and models; this is unlike previous studies because an unbinned analysis is a more robust method for constraining dark halo parameters rather than a binned analysis (the detailed descriptions are presented in Hayashi et al. 2016). From this unbinned fitting procedure, we find that seven dSphs (Carina, Draco, Leo I, Leo II, Ursa Minor, Andromeda V, and VII) have an NFW or steeper inner density slope, while the other ones show a cored or shallower cusped dark halo. In addition, most of the dSphs with ${V}_{\max }\geqslant 25$ km s⁻¹ are settled in extended dark halos with scale lengths larger than 1.5 kpc, especially Draco and Andromeda I which have the largest scale length (~6.0 kpc). Estimating ${{\rm{\Sigma }}}_{{V}_{\max }}$ and ${V}_{\max }$ for these dSphs, we calculate an enclosed mass within ${r}_{\max }$ along the major axis of the mass distributions from estimated dark halo parameters described above. In order to determine ${r}_{\max }$, we calculate a circular velocity in gravitational potentials stemmed from axisymmetric mass distributions (Binney & Tremaine 2008)

$$\begin{eqnarray}&&{V}_{\mathrm{circ}}^{2}(R)=R| -{\rm{\nabla }}{\rm{\Phi }}| ,\end{eqnarray} \tag{ 5 }$$

where ${\rm{\nabla }}{\rm{\Phi }}$ indicate gravitational force. We estimate circular velocity along the major axis, R, using

$$\begin{eqnarray}{g}_{R}(R,z) & = & -\displaystyle \frac{\partial {\rm{\Phi }}}{\partial R}\\ & = & -2\pi {{GQa}}_{0}^{3}R{\displaystyle \int }_{0}^{\infty }d\tau \displaystyle \frac{\rho ({\tilde{m}}^{2})}{{(\tau +{a}_{0}^{2})}^{2}\sqrt{\tau +{Q}^{2}{a}_{0}^{2}}},\end{eqnarray} \tag{ 6 }$$

where τ is a new variable of integration $\tau \equiv {a}_{0}^{2}(1-{Q}^{2})[{\sinh }^{2}{u}_{m}-({(1-{Q}^{2})}^{-0.5}-1)]$ $({a}_{0}=\mathrm{const}.)$ in the spheroidal coordinate $({u}_{m},{v}_{m})$, and ${\tilde{m}}^{2}$ is defined by

$$\begin{eqnarray}&&\displaystyle \frac{{\tilde{m}}^{2}}{{a}_{0}^{2}}=\displaystyle \frac{{R}^{2}}{\tau +{a}_{0}^{2}}+\displaystyle \frac{{z}^{2}}{\tau +{Q}^{2}{a}_{0}^{2}}.\end{eqnarray} \tag{ 7 }$$

The estimates of ${{\rm{\Sigma }}}_{{V}_{\max }}$ and ${V}_{\max }$ for the 13 dSphs are listed in Table 1 and plotted in Figure 1.

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Table 1.  Estimates for ${V}_{\max }$ and ${{\rm{\Sigma }}}_{{V}_{\max }}$ of the Eight MW and the Five M31 dSphs

Object	${V}_{\max }$	${{\rm{\Sigma }}}_{{V}_{\max }}$
	(km s−1)	(M⊙ pc−2)
Milky Way
Carina	${27.9}_{-5.7}^{+10.3}$	${10.9}_{-3.7}^{+5.9}$
Fornax	${23.3}_{-1.6}^{+3.8}$	${21.0}_{-2.4}^{+4.6}$
Sculptor	${24.6}_{-2.1}^{+3.5}$	${28.1}_{-4.4}^{+9.0}$
Sextans	${25.7}_{-6.9}^{+18.6}$	${9.8}_{-3.2}^{+6.3}$
Draco	${76.4}_{-19.6}^{+25.5}$	${38.6}_{-12.1}^{+25.4}$
Leo I	${22.7}_{-3.3}^{+6.6}$	${13.4}_{-6.8}^{+19.1}$
Leo II	${27.9}_{-6.6}^{+19.9}$	${16.8}_{-7.9}^{+13.9}$
Ursa Minor	${19.7}_{-1.9}^{+3.9}$	${18.4}_{-11.1}^{+22.6}$
Andromeda
Andromeda I	${61.3}_{-17.2}^{+25.8}$	${27.7}_{-9.2}^{+21.9}$
Andromeda II	${44.3}_{-7.5}^{+9.0}$	${11.9}_{-2.7}^{+5.3}$
Andromeda III	${47.7}_{-15.6}^{+30.8}$	${21.3}_{-9.2}^{+20.9}$
Andromeda V	${27.3}_{-3.5}^{+8.8}$	${31.1}_{-16.8}^{+32.6}$
Andromeda VII	${29.4}_{-3.5}^{+8.2}$	${54.1}_{-33.3}^{+122.3}$

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Here, we note that the observational results from Sextans and Andromeda II and VII dSphs are significantly affected by the sizes of data samples and their peculiar stellar kinematics. Since Sextans might have a very large tidal radius, the currently available spectroscopic data for this galaxy are quite incomplete in the outer region. Moreover, Hayashi & Chiba (2015b) demonstrated that the lack of kinematic data samples in the outer region of a stellar system has a large impact on determining dark halo parameters, especially the axial ratio of a dark halo and its velocity anisotropy (see Figure 12 in their paper). For Andromeda II, Ho et al. (2012) first inspected the kinematical properties of this galaxy and suggested that this is a prolate rotating system, that is, their stars rotate with respect to their stellar minor axis. This is a peculiar system, to which our models cannot be properly applied. For Andromeda VII, although there is no observational evidence due to lack of a data sample, Hayashi & Chiba (2015b) suggested from the fitting results that the three-dimensional stellar density distribution of this galaxy would be preferable to the prolate system. Thus, this galaxy could also have a peculiar system like Andromeda II. Therefore, there is the possibility that the obtained parameters for these dark halos may contain large systematics.

3.1. Cosmological Simulation

In this work, we utilize the two cosmological simulations. One is the high-resolution simulation performed by Ishiyama et al. (2016), the other is the Cosmogrid simulation performed by Ishiyama et al. (2013). The former provides the higher resolution, and the latter enables us to collect more dark halo samples as it has a larger simulation box.

The run parameters in these simulations are listed in Table 2, and the best-fit cosmological parameters are consistent with the cosmic microwave background obtained by the Planck satellite (Planck Collaboration et al. 2014), namely $({{\rm{\Omega }}}_{0},{{\rm{\Omega }}}_{{\rm{\Lambda }}},h,{n}_{s},{\sigma }_{8})\,=(0.31,0.69,0.68,0.96,0.83)$ for the high-resolution simulation, and $({{\rm{\Omega }}}_{0},{{\rm{\Omega }}}_{{\rm{\Lambda }}},h,{n}_{s},{\sigma }_{8})=(0.30,0.70,0.70,1.0,0.8)$ for the Cosmogrid simulation, respectively.

Table 2.  Number of Particles, Box Length, Mass Resolution, Softening Length, and Initial Redshift of the High-resolution and Cosmogrid Simulations

	N	L	m	${\varepsilon }_{\mathrm{DM}}$	${z}_{\mathrm{ini}}$
		(Mpc)	(M⊙)	(pc)
High-resolution	20483	11.8	7.54 × 103	176.5	127
Cosmogrid	20483	30.0	1.28 × 105	175.0	65

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To identify halos and subhalos and their merger trees, we used the Rockstar (Robust Overdensity Calculation using K-Space Topologically Adaptive Refinement) halo finder⁸ (Behroozi et al. 2013) and Consistent Trees (Behroozi et al. 2013). In this paper we  use the physical values of each dark halo computed by Rockstar and Consistent Trees analysis, namely virial mass (${M}_{\mathrm{vir}}$), virial radius (${r}_{\mathrm{vir}}$), maximum circular velocity (${V}_{\max }$), and radius (${r}_{\max }$).

In the high-resolution simulation, the four MW-sized dark halos are identified, and the total mass of the halos ranges from ~0.8 to $\sim 3.0\times {10}^{12}{M}_{\odot }$ as estimated from the dynamical analysis of blue horizontal branch stars or/and dwarf satellites (e.g., Sakamoto et al. 2003; Deason et al. 2012; Kafle et al. 2014). On the other hand, in order to get a number of dark halo samples, we identify the 56 MW-sized halos with a wider mass range from ~1.0 to $\sim 6.0\times {10}^{12}{M}_{\odot }$ and select the 18 MW-sized halos that possess over 10 subhalos that satisfied some criteria (as described in the next section) among the 56 identified dark halos (as seen in Figure 2).

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In Table 3, we summarize the virial mass, virial radius, concentration indicator (as described below) and number of subhalos that satisfied the criteria (as described below) for the four MW-sized dark halos in each simulation. For the Cosmogrid simulation, we demonstrate that, out of 18 MW-like dark halos, the four host halos show relatively rapid growth at a high redshift; that is, the subhalos associated with these parent halos did fall in at some earlier time and thus might have passed through the pericenter of the host several times. Using these subhalos, we will discuss in detail their orbital evolutionary histories in Section 5.

Table 3.  Virial Mass, Virial Radius, Concentration Indicator, and Number of Subhalos within Four MW-sized Dark Halos in Each Simulation

Name	${M}_{\mathrm{vir}}$	${r}_{\mathrm{vir}}$	${\rho }_{{V}_{\max }}$	${N}_{\mathrm{subhalos}}$
	$(\times {10}^{12}{M}_{\odot })$	(kpc)	$(\times {10}^{4})$
High-resolution
H1	2.88	374	0.36	187
H2	2.40	352	0.35	146
H3	2.31	348	1.49	118
H4	1.11	272	0.61	48
Cosmogrid
CG1	4.28	420	0.13	41
CG2	2.87	368	0.44	19
CG3	3.36	388	0.77	20
CG4	3.08	377	0.75	18

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In what follows, we investigate the properties of the dark halo surface densities of subhalos within the selected host halos in each simulation.

3.2. Subhalo Criteria

Using the ${V}_{\max }$ and ${r}_{\max }$ of subhalos, which fulfill the above three criteria, we calculate their spherical-averaged ${{\rm{\Sigma }}}_{{V}_{\max }}$ derived from Equation (1). First of all, as a indicator representing the compactness of a dark halo, we utilize the mean physical density within the radius of the maximum circular velocity in units of the critical density (${\rho }_{\mathrm{crit},0}$) as defined by Diemand et al. (2007, hereafter D07),

$$\begin{eqnarray}&&{\rho }_{{V}_{\max }}=\displaystyle \frac{\overline{\rho }(\lt {r}_{\max })}{{\rho }_{\mathrm{crit},0}}=2{\left(\displaystyle \frac{{V}_{\max }}{{H}_{0}{r}_{\max }}\right)}^{2},\end{eqnarray} \tag{ 8 }$$

where H₀ denotes the Hubble constant at z = 0. This corresponds to the physical concentration indicator of a dark halo. The concentration indicator, which is commonly used in N-body simulations, is defined by the ratio between a virial and a scale radii of a halo, assuming an NFW profile. However, because density profiles of less-massive subhalos, especially those affected by tidal forces from a massive host, should deviate from their initial mass profiles (Aguilar & White 1986; Hayashi et al. 2003; Peñarrubia et al. 2008), the usually defined concentration indicator is not necessarily appropriate. Thus, we adopt an alternative ${\rho }_{{V}_{\max }}$ as the concentration indicator of subhalos in this work.

4.1. Comparison between the Observations and the Simulations

Figures 1 and 2 show the results derived from both the simulations and the observations on the ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ plane, focusing on mass scales of dSph-sized dark halos. From these figures, we demonstrate that pure dark matter simulations reproduce the dark halo surface densities derived from observations for dSphs. This means that ${{\rm{\Sigma }}}_{{V}_{\max }}$ can be unaffected by net effects of baryon physics and be dependent only on the intrinsic properties of dark halos. To confirm this accordance, we also calculate ${{\rm{\Sigma }}}_{{V}_{\max }}$ using the available catalog in the Illustris Project,¹⁰ a series of N-body and hydrodynamics simulations on cosmological volume. We use the results from the highest mass-resolution run, Illustris-1,¹¹ and select the MW-sized halos and their subhalos though the same criteria as above (i.e., ${r}_{{\rm{s}}}\gt 2{\varepsilon }_{\mathrm{DM}}$ and ${M}_{\mathrm{vir}}\gtrsim {10}^{9.8}{M}_{\odot }$). Figure 3 shows the results of the comparison with the Illustris simulation. From this figure, the effect of mass resolution emerges at low mass scales (${V}_{\max }\leqslant 30$ km s⁻¹), and it seems that these ${{\rm{\Sigma }}}_{{V}_{\max }}$ are slightly declined by the baryonic effects at higher mass scales (${V}_{\max }\geqslant 30$ km s⁻¹). However, the dark halo surface densities calculated by the Illustris simulation do not significantly differ from those observations, as do the results of the pure N-body simulations.

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This is perhaps no surprise: indeed, the radius of the maximum circular velocity of a dark halo should be much larger than the star-forming regions, which settle in the center part of a dark halo, and thus this radius would be influenced only by gravitational effects such as mass assembly history and tidal force from their host. Therefore, even if inner dark matter profiles at dwarf-galaxy scales can be transformed from cusped to cored due to energy feedback from gas outflows driven by the star formation activity of galaxies, the expelling dark matter particles from the center of a dark halo cannot escape beyond the radius of the maximum circular velocity. In other words, the cusp-to-core transformation, which is a possible solution to the core-cusp and the too-big-to-fail problem, might not affect the ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ relation. These resultants are shown by some N-body and hydrodynamical simulations of dwarf galaxies (e.g., Madau et al. 2014; Ogiya et al. 2014; Ogiya & Burkert 2015; Sawala et al. 2016). However, it is notable that using even observational results, we present and confirm the maximum circular velocity, and the radius of a dark halo is not sensitive to baryon feedback.

4.2. The Universal ${{\rm{\Sigma }}}_{{V}_{\max }}\mbox{--}{V}_{\max }$ Relation

To understand the origin of the universality of the ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ relation found for dark halos on the scales of dwarf galaxies, we investigate the evolutionary histories of subhalos in detail, especially the link between orbital evolution and ${r}_{\max }$ (and ${V}_{\max }$) evolution in the Cosmogrid simulations. The time resolutions of the high-resolution simulations are not enough to investigate the orbital evolutions of their subhalos, therefore we focus only on the subhalos in the Cosmogrid simulations. Using the merger tree data of subhalos computed by Consistent Trees, we trace the evolution of the mass distribution in satellite halos undergoing tidal stripping from a host halo.

5.1. Evolutionary History of Subhalos

The left four panels in Figure 4 show the orbital evolutionary tracks of ten representative subhalos associated with four parent dark halos derived from the Cosmogrid simulations. Out of 18 MW-like dark halos, we choose the four host halos showing relatively rapid growth at a high redshift, because subhalos associated with these parent halos did fall in at some earlier time and thus might have passed through the pericenter of the host a number of times. The basic properties of the four Cosmogrid host halos are listed in Table 3. In the left panels of Figure 4, we preferentially select subhalos that have undergone at least one pericenter passage with respect to a host halo. Eventually, however, the majority of subhalos have little experiences of pericenter passage. This implies that the subhalos, which have already passed through pericenters many times and thus have been disturbed strongly by tidal effects, have possibly been disrupted or lost their masses significantly by the present day. Such subhalos do not fulfill our subhalo criteria. Therefore, most of selected subhalos in each panel show the recent infall into their hosts.

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The right four panels in Figure 4 display the evolution in the ${r}_{\max }\,\mbox{--}{V}_{\max }$ plane of subhalos and correspond to those in the left panels. On this plane, the halo tracks start at the lower-left corner because dark halos are still small at a high redshift. After the subhalos move toward the upper right during their mass-growth phase, both their ${r}_{\max }$ and ${V}_{\max }$ start decreasing because of their infall into a host halo and associated tidal stripping. This means that tidal stripping is likely to stop the inside-out dark matter assembly by truncating its outer mass. It is worth noting from these figures that evolutionary tracks of most subhalos on ${r}_{\max }-{V}_{\max }$ follow virtually the same path, in spite of the differences in host halos and trajectories of subhalos. We interpret this interesting evolutionary path to mean that the internal structures of dark halos would be robust against several merger events (e.g., Dehnen 2005; Kazantzidis et al. 2006) and that the tidal effects on a dark halo might not deviate significantly from a ${r}_{\max }\,\mbox{--}\,{V}_{\max }$ sequence even though those effects not only truncate its dark matter at the outer region but also slightly affect its inner structures. Thus, dark halos track back their ${r}_{\max }$ and ${V}_{\max }$ to near where they were before.

Some of the subhalos, which experience frequent pericenter passage and/or have small pericenter distances, would deviate somewhat from the universal ${r}_{\max }\mbox{--}{V}_{\max }$ sequence upward or downward, because of strong tidal mass loss and the tidal disturbance of dark halo structures. As examples, we focus on the four subhalos in Figure 4: (1) the halo that experiences pericenter passage four times (red line in CG1), (2) the halo with two pericenter passages (black line in CG1), (3) the halo that undergoes infall suddenly and deeply into its host halo (blue line in CG3), and (4) the halo that undergoes the same as halo (3); (orange line in CG4). Because these subhalos have been truncated tidally, where the outer region becomes less bound with every pericenter passages, the remnants of such a strong tidal interaction have eventually low densities but are more centrally concentrated. This is why their ${r}_{\max }$ values are relatively lower than those in the ${r}_{\max }\mbox{--}{V}_{\max }$ sequence and are greatly changed.

5.2. A Possible Solution to the Origin of the Universal Relation

The evolutionary sequence on the ${r}_{\max }\,\mbox{--}{V}_{\max }$ plane described above and the deviation from it are basically consistent with the results shown in D07 (see their Figure 13). Therefore, it is suggested that subhalos within their own host halos have a common dynamical evolution, regardless of the properties of host halos. Moreover, as mentioned in D07, this sequence in all host halos can be expressed simply as the linear relation, ${r}_{\max }\propto {V}_{\max }$, and we confirm their suggested relation (dashed lines in the four right panels in Figure 4). Thus, this relation plays a key role in untangling the origin of the universal ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ relation. Based on ${r}_{\max }\propto {V}_{\max }$, the mean physical dark halo density defined by Equation (8) becomes ${\rho }_{{V}_{\max }}\propto {({V}_{\max }/{r}_{\max })}^{2}\simeq \mathrm{const}$. On the other hand, ${{\rm{\Sigma }}}_{{V}_{\max }}$ can be rewritten by ${\rho }_{{V}_{\max }}$ as ${{\rm{\Sigma }}}_{{V}_{\max }}\propto {\rho }_{{V}_{\max }}{r}_{\max }$. Consequently, we arrive at ${{\rm{\Sigma }}}_{{V}_{\max }}\propto {V}_{\max }$ using the ${r}_{\max }\,-\,{V}_{\max }$ relation.

Furthermore, subhalos evolve generally along ${r}_{\max }\propto {V}_{\max }$, although the relation holds some dispersion because the dark halo structures in these subhalos are affected by a change in mass accretion rate. Thus, the ${{\rm{\Sigma }}}_{{V}_{\max }}$ versus ${V}_{\max }$ relation that we have derived here can be nearly constant with time. Following this hypothesis, in Figure 5 we plot the redshift evolution of all of the subhalos associated with the different host halos on the ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ plane. Since the virial radius of a subhalo decreases naturally with redshift, the lower limit of ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ goes up and gets close to ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ relation at z = 0. Although the amplitude of the ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ relation at z = 5 is thus shifted upward, its slope does not change significantly. Therefore, we find that irrespective of some scatters, the relation has indeed been kept roughly invariant since its emergence.

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We have investigated a dark halo surface density inside a radius at the maximum circular velocity, ${{\rm{\Sigma }}}_{{V}_{\max }}$, first introduced by HC15a, based on the comparison between the results from the observations and those from cosmological N-body simulations. While the ${{\rm{\Sigma }}}_{{V}_{\max }}$ versus ${V}_{\max }$ relation is semi-analytically inferred from the assumption of an NFW profile combined with an empirical mass concentration relation (HC15a), this can be affected by some environmental effects, such as tidal stripping and heating, as well as halo-to-halo scatters. Therefore, in this work, we have scrutinized dark matter halos, especially subhalos associated with a MW-like dark halo, taken from cosmological pure dark matter simulations performed by Ishiyama et al. (2013, 2016) in order to understand the properties of the surface density in more detail. We have found several important results for the properties of the satellite dark halos, as follows.

• 1.  
  Dark halo surface densities derived from our simulated subhalos associated with MW-sized hosts are in remarkable agreement with those from the observations of the MW and M31 dSph satellites. This implies that the surface density is only weakly modified by baryonic feedback. Therefore, the surface density provides us with a clue for understanding the fundamental dark halo properties of the Local Group.
• 2.  
  Even if host dark halos have different masses, compactness, assembly history, and properties than their subhalos, all subhalos are found to obey the universal ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ relation. Furthermore, this universality appears to be sustained even at high redshifts.
• 3.  
  In order to understand the origin of this universal dark halo relation, we have investigated the orbital and dynamical evolutions of subhalos. It is found that most of the subhalos have several or little experiences of pericenter passage with respect to a host halo. Analyzing such a subhalo evolution, we have confirmed that most of the subhalos evolve along the ${r}_{\max }\propto {V}_{\max }$ sequence, whereas more disturbed subhalos, which have undergone pericenter passage a number of times and are very close to the center of a host halo, show somewhat of a deviation from this sequence. This suggests that the tidal force from host halos mainly removes only the outer parts of subhalos, which are accreted from the outside, so that this ${r}_{\max }\propto {V}_{\max }$ sequence remains preserved. Since the ${{\rm{\Sigma }}}_{{V}_{\max }}\,\mbox{--}{V}_{\max }$ is derived from this sequence, ${r}_{\max }\propto {V}_{\max }$, it is found to be a basic property for understanding the common dark halo surface density scales.

Following the results obtained here, there is the possibility that this universal relation for dark halos exists not only for the MW and M31, but also for larger gravitational systems such as galaxy groups and clusters. For instance, recently discovered ultra-diffuse galaxies in clusters (e.g., Koda et al. 2015; van Dokkum et al. 2015a, 2015b; Yagi et al. 2016) might be affected by strong tidal disturbances, and if so, these galaxies may show some characteristic trends in the ${{\rm{\Sigma }}}_{{V}_{\max }}$ versus ${V}_{\max }$ relation. Also, it remains yet unclear whether the ultra-faint galaxies in the MW indeed obey this universal relation, because of the paucity of observable bright stars in such faint systems. We thus require high-quality photometric and spectroscopic data of these galaxies. Current and future facilities such as the Hyper Suprime Cam (Miyazaki et al. 2012) and the Prime Focus Spectrograph (Tamura et al. 2016), to be mounted on the Subaru Telescope, and high-precision spectroscopy the Thirty Meter Telescope (Simard et al. 2016) will have the capability of obtaining more severe constraints on dark halo structures of these galaxies and enable us to understand the basic properties of dark matter in the universe.

We would like to give special thanks to Matthew G. Walker for giving us the kinematic data of Ursa Minor dwarf galaxies and for useful discussions. This work acknowledges support from MEXT Grant-in-Aid for Scientific Research on Innovative Areas, No. 16H01090 (for K.H.), No. 15H01030 (for T.I.), and No. 15H05889 and No. 16H01086 (for M.C.). This research was supported by MEXT as "Priority Issue on Post-K computer" (Elucidation of the Fundamental Laws and Evolution of the Universe) and JICFuS. G.O. acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 679145, project "COSMO-SIMS"). S.I. acknowledges the funding from Grant-in-Aid for Young Scientists (B), No. 17K17677, and from CREST, JST. Finally, Kavli IPMU is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.