Self-interacting Dark Matter Interpretation of Crater II

The satellite galaxy Crater II of the Milky Way is extremely cold and exceptionally diffuse. These unusual properties are challenging to understand in the standard model of cold dark matter. We use controlled N-body simulations to investigate the formation of Crater II in self-interacting dark matter (SIDM), where dark matter particles can scatter and thermalize. Taking the orbit motivated by the measurements from Gaia Early Data Release 3, we show a strong self-interacting cross section per particle mass of 60 cm2 g−1 is favored for Crater II. The simulated SIDM halo, with a 1 kpc core, leads to both a low stellar velocity dispersion and a large half-light radius for Crater II. These characteristics remain robust regardless of the initial stellar distribution.


INTRODUCTION
Over the past decade, astronomical surveys have discovered more than 30 satellite dwarf galaxies associated with the Milky Way, see, e.g., Simon (2019).Among them, Crater II, located 117 kpc away from the Sun, is particularly interesting, as it has the coldest velocity dispersion of σ los ≈ 2.7 km/s, while being exceptionally diffuse, with a projected half-light radius of R 1/2 ≈ 1.07 kpc (Torrealba et al. 2016;Caldwell et al. 2017;Fu et al. 2019;Vivas et al. 2020;Ji et al. 2021).Crater II has almost the lowest surface brightness among the satellites ever discovered.These unusual properties of Crater II make it an intriguing case for testing the standard model of cold dark matter (CDM).
Environmental effects can be important for the formation of Crater II.The tides from its host Milky Way zhang-xy19@mails.tsinghua.edu.cnhaiboyu@ucr.edudanengy@ucr.eduanhp@mail.tsinghua.edu.cncan strip away the halo mass and lower the velocity dispersion.Frings et al. (2017); Applebaum et al. (2021) conducted cosmological hydrodynamical CDM simulations and argued that Crater II-like galaxies could form through strong tidal interactions.However, more tailored studies found that it is difficult to simultaneously realize both low velocity dispersion and large size of Crater II in CDM (Sanders et al. 2018;Borukhovetskaya et al. 2022).The central cusp of a CDM halo is resilient to tidal stripping, and the tides must be sufficiently strong to explain the low velocity dispersion.Nevertheless, with such strong tides, the galaxy size would be truncated rapidly and it becomes much smaller than the observed size of Crater II; see Borukhovetskaya et al. (2022) for detailed discussion.It remains to be seen if baryonic feedback can modify these CDM predictions in accord with the observations of Crater II.
In this work, we investigate the formation of Crater II within the scenario of self-interacting dark matter (SIDM).Dark matter self-interactions transport heat and thermalize the inner halo; see Tulin & Yu (2018); Adhikari et al. (2022) for reviews.For an SIDM halo in the core-expansion phase, a shallow density core forms and the stellar distribution correlates with the core size.
A cored halo also boosts tidal mass loss and mild tides are allowed for lowering the halo mass.We conduct controlled N-body simulations to model the evolution of Crater II in the tidal field of the Milky Way in SIDM, as well as in CDM for comparison.We will show that in SIDM the simulated galaxies can well reproduce the unusual properties of Crater II.Taking the orbital parameters from Gaia EDR3 (Gaia Collaboration 2021; McConnachie & Venn 2020;Ji et al. 2021;Pace et al. 2022), we will show that the required self-interacting cross section is ∼ 60 cm 2 /g for Crater II.
The rest of the paper is organized as follows: In Section 2, we introduce our simulation setup.In Section 3, we discuss the properties of our simulated galaxies and compare them to the observations.We further discuss implications of our simulation results and conclude in Section 4. In Appendix A, we discuss the choice of the final snapshot.In Appendix B, we show the degeneracy effect between tidal orbit and cross section.In Appendix C, we present additional N-body simulations with live stellar particles and further confirm the results discussed in the main text.

SIMULATION SETUP
In this section, we introduce initial conditions for modeling Crater II in our N-body simulations, including initial halo and stellar density profiles and orbital parameters of our simulated satellite galaxies, the mass model of the Milky Way, and SIDM cross sections.

The dark matter halo of Crater II
We consider three cases for the self-interacting cross section per particle mass: σ/m = 10 cm 2 /g (SIDM10), 30 cm 2 /g (SIDM30), and 60 cm 2 /g (SIDM60); as well as the CDM limit for comparison.For all cases, we model the initial halo of Crater II with a Navarro-Frenk-White (NFW) profile (Navarro et al. 1997) where ρ s and r s are the scale density and radius, respectively.For CDM, SIDM10, and SIDM30, we take ρ s = 1.42 × 10 7 M ⊙ /kpc 3 and r s = 2.06 kpc.The maximum circular velocity of the halo is V max = 26.57km/s and its associated radius is r max = 4.45 kpc.For SIDM60, ρ s = 1.14 × 10 7 M ⊙ /kpc 3 and r s = 2.23 kpc; V max = 25.77km/s and r max = 4.82 kpc.The halo parameters are overall consistent with those for progenitors of simulated satellite galaxies that have similar stellar masses of Crater II (Fattahi et al. 2018).As we will show later, with the tidal orbit from the measurements of Gaia EDR3 (McConnachie & Venn 2020;Ji et al. 2021;Pace et al. 2022), only SIDM60 can reproduce the observations of Crater II.Thus we will focus on SIDM60 and CDM for the rest of the paper, and present SIDM10 and SIDM30 in Appendix B to show the degeneracy effect between orbit and cross section.
There are stringent constraints on the cross section on the scales of galaxy clusters σ/m ≲ 0.1 cm 2 /g for V max ∼ 1500 km/s (Kaplinghat et al. 2016;Harvey et al. 2015;Sagunski et al. 2021;Andrade et al. 2022;Rocha et al. 2013;Peter et al. 2013) and elliptical galaxies (Kong et al. 2024).Thus, a viable SIDM model must have a velocity-dependent cross section, which is strong in dwarf galaxies, while diminishing towards the clusters; see Tulin & Yu (2018) and references therein for the discussion about particle physics realizations of SIDM.Nevertheless, we can use an effective constant cross section for a given halo to accurately characterize its gravothermal evolution (Yang & Yu 2022;Outmezguine et al. 2023;Yang et al. 2023c,b;Fischer et al. 2023).The constant cross sections we take should be regarded as effective cross sections for the Crater II halo.
We use the public code GADGET-2 (Springel et al. 2001;Springel 2005), which is implemented with an SIDM module developed and tested in Yang & Yu (2022), and the code SpherIC (Garrison-Kimmel et al. 2013) to generate the initial condition.For all cases, the initial halo mass is 3.37 × 10 9 M ⊙ and there are 10 7 simulation particles.The particle mass is 337 M ⊙ and the Plummer softening length is 7 pc.The resolution is high enough for the purpose of this work.

The stellar component of Crater II
Since Crater II is dark matter-dominated, we can assume that stars are massless tracers of the halo potential and use the technique introduced in Bullock & Johnston (2005) to model the stellar component, i.e., the tagging method.In this work, we have further tested the method and validated its application in SIDM using N-body simulations with live stellar particles; see Appendix C.
For each simulation particle, we can attach an appropriate probability to represent a star.The equilibrium distribution function of stars is (Errani & Peñarrubia 2020) where dΩ = d 3 rd 3 v is the differential volume of phase space, Φ the potential of the satellite halo, and ν ⋆ the stellar number density.It satisfies the normalization condition dr 4πr 2 ν ⋆ (r) = N ⋆ with N ⋆ being the total number of stars.We use an Einasto profile (Einasto 1965) to model the stellar distribution of Crater II where ρ E (0) is the central density, r E is the scale radius, and we set the numerical factor α = 1.For the Einasto profile, we have For each simulation particle located at (r, v) in the phase space, the probability to represent a star is given by , where f DM (E) is the distribution function of dark matter particles.It is calculated in a similar way as in Equation 2with ν ⋆ being replaced by the number density profile of dark matter.We compute the probability at infall and reconstruct the stellar spatial distribution and kinematics from dark matter at later times by applying P (E) as a weighting factor.Torrealba et al. (2016) originally used a Plummer profile to describe the stellar distribution and reported the 2D half-light radius as R 1/2 ≈ 1.066 kpc.Borukhovetskaya et al. (2022) found that the Einasto profile in Equation 3 (α = 1) fits the Plummer projected density profile well.For the purpose of comparison, we take the same approach and choose the initial Einasto scale radius to be r E = 0.40 kpc, 0.73 kpc, and 1.37 kpc as in Borukhovetskaya et al. (2022).Since the 2D halflight radius is R 1/2 = 2.03r E , we have the initial values R 1/2 ≈ 0.8 kpc, 1.5 kpc, and 2.8 kpc.These values bracket the measured value of R 1/2 ≈ 1 kpc, and they are overall consistent with those found for progenitors of satellite galaxies that have similar stellar masses of Crater II in hydrodynamical CDM simulations (Fattahi et al. 2018).
For later snapshots, we fit the stellar distribution using the Einasto profile as well, extract the corresponding r E value, and calculate the half-light radius using the relation R 1/2 = 2.03r E .We have checked that using a Plummer profile in the tagging method gives rise to a similar evolution trajectory of the stellar distribution as with the Einasto profile.Furthermore, we have also confirmed that with the initial stellar distributions assumed in this work, the dark matter mass completely dominates over the stellar component in the entire evolution history of the simulated halo and the tagging method is well justified.

The Milky Way
We model the Milky Way with a static gravitational potential, which contains three components: a dark matter main halo, a stellar bulge, and stellar disks.The relevant parameters are given as follows.
The corresponding halo mass is M 200 ≈ 1.15 × 10 12 M ⊙ and the maximum halo circular velocity is V max ≈ 192 km/s.
• Two axisymmetric disks of a Miyamoto-Nagai profile (Miyamoto & Nagai 1975)  2011) with a circular velocity of 239 km/s at the solar radius R ⊙ = 8.3 kpc.We have checked that the pericenter of our simulated satellite halo would shift a few percent if taking the parameter values from the more recent model in McMillan (2016).The main halo is treated as a static potential, and we neglect the evaporation effect between the satellite halo and the main halo (Nadler et al. 2020;Slone et al. 2023).The approximation is justified if the cross section in the main halo is below O(1) cm 2 /g.This condition can be satisfied naturally for velocity-dependent SIDM models that evade the cluster constraints, see, e.g., Yang et al. (2023a); Nadler et al. (2023) for examples.

Orbital parameters
In Table 1, we summarize the measured orbital parameters of Crater II.Its proper motions are derived from data of the Gaia EDR3 (McConnachie & Venn 2020; Ji et al. 2021;Pace et al. 2022).In our simulation we have tested a series of obits with a wide range of pericenter r p ≈ 2.5-37.7 kpc.Our main results are based on two orbits, denoted as O2 and O3, as listed in Table 2, where we have used the same naming scheme

RESULTS
In this section, we present the properties of our simulated galaxies and compare them with the observations of Crater II.

Tidal evolution of the halo mass
In Figure 1 (left panel), we show the tidal evolution of the circular velocity at the half-light radius , a proxy for the halo mass within r = R 1/2 = 1.066 kpc, for the three cases: SIDM60 O3 (red), CDM O3 (blue), and CDM O2 (black).The horizontal line and shaded region denotes the measured value and its uncertainties V (R 1/2 ) = 4.3 ± 0.5 km/s (Caldwell et al. 2017).We choose the timescale marked by the triangle symbol as the final snapshot for analyzing other properties of the simulated galaxies, at which they are well relaxed.In Appendix A, we will show that our main results are robust to this choice.
For SIDM60 O3, V(R 1/2 ) reaches the measured value after four pericenter passages.Thus, with the orbit motivated by the Gaia EDR3 measurements, an SIDM model with σ/m ≈ 60 cm 2 /g is needed as the pericenter is large r p = 37.7 kpc.For CDM O3, V(R 1/2 ) ≈ 15 km/s, a factor of 3 larger than the measured value.If we choose the orbit O2, the CDM halo can reach V (R 1/2 ) ≈ 5 km/s after five pericenter passages.However, its pericenter r p ≈ 13.8 kpc is a factor of 2.7 smaller than that from the EDR3 measurements.
Figure 1 (right panel) shows the corresponding circular velocity profiles at the final snapshot for the three cases, as well as the initial condition.The measurements at r = R 1/2 and r = 4 3 R 1/2 are V (R 1/2 ) = 4.3 +0.5 −0.5 km/s and V ( 4 3 R 1/2 ) = 4.8 +0.6 −0.5 km/s (Caldwell et al. 2017), respectively.Tidal stripping can significantly reduce the overall halo mass, lowering the circular velocity in accord with the measurements at r ∼ R 1/2 for SIDM60 O3 and CDM O2.However, their V (r) profiles are different, which can be understood as follows.For CDM O2, r max reduces to 0.4 kpc at the final snapshot and the inner halo remains cusp.In the case of SIDM, the selfinteractions push dark matter particles outwards.Thus, compared to CDM O2, SIDM60 O3 has a lower inner density while larger r max ≈ 1.5 kpc at the final snapshot.We also see that the circular velocity of CDM O3 is too high to be consistent with the measurements.
For CDM O2, the size converges to R 1/2 ≈ 0.3-0.4kpc from R 1/2 ≈ 0.8-2.8kpc after tidal evolution; see also Borukhovetskaya et al. (2022).We have checked that our final R 1/2 values agree with the estimates based on the tidal track of CDM satellite halos in Errani et al. (2022).The same trend applies to CDM O3, but the convergence is slower as it has weaker tides.It is clear that both CDM O2 and O3 cannot explain the observations of Crater II.The former's stellar size is a factor 3 smaller than Crater II's R 1/2 ≈ 1 kpc, while the latter's velocity dispersion is a factor of 3 higher than the measured one σ los (< R 1/2 ) ≈ 2.7 km/s.Remarkably, SIDM60 O3 can simultaneously explain both stellar size and velocity dispersion of Crater II.
The upper limit of the galaxy size is largely set by r max .For the region beyond r max , stellar particles are stripped away significantly.From Figure 1  panel), we see that r max ∼ 0.4 kpc for CDM O2, while r max ∼ 1.5 kpc for SIDM60 O3 at the final snapshot.Thus the latter can retain a larger galaxy size.More specifically, in SIDM there are two competing effects in affecting the size of the satellite galaxy: tidal truncation and SIDM core formation.The former tends to reduce the size, while the latter causes it to expand (Vogelsberger et al. 2014;Carleton et al. 2019;Yang et al. 2020), see also Appendix C. If the initial distribution of stars is diffuse (R 1/2 ≈ 2.8 kpc), tidal truncation dominates and the final size is reduced.If it is compact (R 1/2 ≈ 0.8 kpc), the halo core expansion drives the increase of the stellar size.For σ/m ≈ 60 cm 2 /g, SIDM produces a halo core size of 1 kpc, which is comparable to R 1/2 of Crater II.We have also checked that the tidal radius of the SIDM60 halo is 1 kpc at the final snapshot.
In Figure 2 (right panel), we show the line-of-sight velocity dispersion profile of stellar particles at the initial (gray) and final snapshot for SIDM60 O3 (red), CDM O3 (blue), and CDM O2 (black).For each of the three cases, despite the difference among three initial σ los profiles corresponding to E1 (solid), E2 (dashed) and E3 (dotted), the final σ los profiles largely converge.For SIDM60, the profile is consistent with the measurements (Caldwell et al. 2017).It is almost flat for R ≲ 1 kpc because the halo has a shallow density core, while σ los increases in the region for R ≳ 2 kpc, where more stars are escaping at large radii.For CDM O2, σ los increases towards the center as expected from a central density cusp.For CDM O3, σ los is too high to be consistent with the measurements.We have further checked the σ los profile using data from Ji et al. (2021) and found it overall agrees with that from Caldwell et al. (2017).

The mass and density profiles
Figure 3 shows mass (left panel) and density (right panel) profiles of the dark matter (solid) and stellar (dashed) components for the three cases at the final snapshot.For comparison, we also display the favored ranges of Crater II from Caldwell et al. (2017) (cyan: halo; pink: stars; 68% credibility intervals), based on a fit using a generalized NFW profile for the halo, allowing a density core.We see that SIDM60 reproduces both halo and stellar profiles inferred from Crater II.This is not the case for the CDM halos.

DISCUSSION AND CONCLUSIONS
As we have shown, the exceptionally large size of Crater II implies that its halo has a 1 kpc density core.In CDM, a core could form due to baryonic feedback (Read & Gilmore 2005;Mashchenko et al. 2008;Pontzen & Governato 2012;Di Cintio et al. 2014 . Mass (left panel) and density (right panel) profiles of the dark matter (solid) and stellar (dashed) components for SIDM60 O3 (red), CDM O3 (blue), and CDM O2 (black) at the final snapshot.The stellar density of the simulated galaxies is normalized to match the observed central stellar density of Crater II.The cyan and pink bands denote the 68% credibility intervals for dark matter and stars, respectively, based on a fit to the kinematics of Crater II (Caldwell et al. 2017).The data points in the left panel denote the enclosed masses evaluated within R 1/2 and 4 3 R 1/2 (Caldwell et al. 2017).
et al. 2015).However, for a progenitor halo with mass 3 × 10 9 M ⊙ , similar to Crater II, the FIRE2 simulation shows the core size is O(10) pc (Lazar et al. 2020), too small to play a role.More work is needed to test if baryonic feedback can help reconcile CDM with the observations of Crater II.
Our SIDM interpretation of Crater II can be extended to Antlia II, another satellite with a large size R 1/2 ∼ 2.9 kpc and σ los ≈ 5.7 km/s (Torrealba et al. 2019), challenging CDM as well (Errani et al. 2022).Ji et al. (2021) found that there is a velocity gradient in Antlia II and a tentative one in Crater II.This may provide another test on the SIDM interpretation.Furthermore, there are over 60 confirmed or candidate satellite galaxies of the Milky Way, see, e.g., McConnachie (2012); Drlica-Wagner et al. (2020).They exhibit a great diversity in both size and enclosed mass, with R 1/2 ranging from ∼ 10 pc to a few kpc and V (R 1/2 ) ∼ 4 to more than 10 km/s.Thus it is intriguing to test if SIDM can explain the full range of the diversity in the V (R 1/2 )-R 1/2 plane of the satellite galaxies.
Cosmological N-body simulations show that strong dark matter self-interactions can diversify inner dark matter densities of satellite halos (Zavala et al. 2019;Turner et al. 2021;Correa et al. 2022;Yang et al. 2023a;Nadler et al. 2023), due to core formation and collapse (Balberg et al. 2002;Koda & Shapiro 2011).For σ/m ≳ 30-50 cm 2 /g in ultra-faint dwarf halos, the collapse could occur within the Hubble time, yielding a high central density.The collapse timescale is extremely sensitive to the concentration ∝ (σ/m) −1 c −7/2 200 (Essig et al. 2019;Kaplinghat et al. 2019;Nadler et al. 2023;Zeng et al. 2022Zeng et al. , 2023)).The concentration of the Crater II halo is close to the median and hence it is in the core-expansion phase.Nevertheless, the dense satellites of the Milky Way, such as Draco, could be in the collapse phase (Nishikawa et al. 2020;Sameie et al. 2020;Kahlhoefer et al. 2019;Correa 2021).Interestingly, σ/m ∼ 60 cm 2 /g on the Crater II mass scale is aligned with SIDM models proposed to explain the extreme diversity of dark matter distributions in other galactic systems, see, e.g., Yang et al. (2023a); Nadler et al. (2023).For the velocity-dependent SIDM model in Yang et al. (2023a), the effective cross section on average is ∼ 60 cm 2 /g at V max ∼ 5-25 km/s, the relevant range for the Crater II halo.
In summary, we have performed controlled N-body simulations and shown that the unusual properties of Crater II can be explained in SIDM.Dark matter selfinteractions lead to core formation and boost tidal mass loss for the satellite halo, resulting in a low velocity dispersion of stars even if the halo has a relatively large pericenter.At the same time, the stellar distribution expands as the halo core forms and the stellar distribution correlates with the core size.In the future, we will expand our study to other satellite galaxies of the Milky Way and explore SIDM models that can explain the full range of the diversity of the satellites, as well as galaxies in the field (Gentile et al. 2004;Kuzio de Naray et al. 2010;Oman et al. 2015;Ren et al. 2019;Piña Mancera et al. 2022;Kong et al. 2022;Montes et al. 2024;Mancera Piña et al. 2024).−0.187 (−0.148), and r p ≈ 19.9 (27.9) kpc.We see that SIDM10 O5 and SIDM30 O7 could reproduce the stellar size and velocity dispersion of Crater II as SIDM60 O3.Thus there is a degeneracy effect between cross section and orbit.Note that orbits O5 and O7 are not directly motivated by the Gaia EDR3 measurements and their associated halos would be tidally disrupted after ∼ 8 Gyr.

C. N-BODY SIMULATIONS WITH LIVE STELLAR PARTICLES
We further confirm the main results and the tagging method itself by performing additional N-body simulations with live stellar particles for SIDM60 O3.We assume the initial stellar distribution follows a Plummer profile as where M ⋆ is the total stellar mass and a is the scale radius.We take M ⋆ = 3.37 × 10 6 M ⊙ and a = 0.71 kpc.For the Plummer profile, the 2D half-light radius is R 1/2 = a, hence the initial value of R 1/2 is 0.71 kpc, which is comparable with the E1 initial condition for the Einasto profile, i.e., R 1/2 ≈ 2.03r E = 0.81 kpc.We take the Plummer profile for the stars as it is relatively straightforward to implement in the code SpherIC (Garrison-Kimmel et al. 2013).For the dark matter halo, we take ρ s = 1.02 × 10 7 M ⊙ /kpc 3 and r s = 2.30 kpc.The concentration of this halo is slightly lower than the one without stars to offset the baryonic influence of live stellar particles.This is a subtle and novel effect of SIDM and we will elaborate it further in a future publication.Figure 5 (left panel) shows the initial (gray) and final (red) surface density profiles for SIDM60 O3 from N-body simulations with live stellar particles.We construct the stellar surface density profile at the final snapshot from stellar particles (dot), fit it with a Plummer profile (dotted), and determine the half-light radius at the final snapshot R 1/2 = a ≈ 1.35 kpc.For comparison, we also use the tagging method to reconstruct the surface density (square), fit it with a Plummer profile (dashed), and find R 1/2 ≈ 1.2 kpc.We see that R 1/2 at the final snapshot is increased almost by a factor of 2 compared to the initial condition, due to SIDM core formation.Furthermore, R 1/2 inferred from the tagging method agrees with that from live stars within 11%, although the former underestimates the overall normalization of the surface density by a factor of 3. We use the tagging method for the purpose of tracking the evolution of the galaxy size, and we can add a normalization factor to match the observed surface density.Furthermore, Figure 5. Left panel: Surface density profiles for SIDM60 O3 with live stellar particles, including the initial condition (solid), the one constructed from live stellar particles at the final snapshot (dot) and its fit to a Plummer profile (dotted), as well as the one constructed from dark matter particles using the tagging method (square) and its fit to a Plummer profile (dashed).Right panel: Final mass profiles for dark matter (solid) and stars constructed from live stellar particles (dotted) and dark matter particles using the tagging method (dashed) for SIDM60 O3.The cyan and pink bands denote the 68% credibility intervals for dark matter and stars, respectively (Caldwell et al. 2017).
we have checked that for SIDM60 O3, the size evolution is almost the same whether using Einasto or Plummer profiles in the tagging method.We also note that the surface density decreases in the central regions more significantly than in the outer regions.This is because SIDM core formation leads to a shallower density profile and weaker gravitational potential.Thus, the stellar particles more likely migrate from the center towards the outer regions and compensate for the loss there, compared to the CDM limit.
Figure 5 (right panel) shows the final mass profiles for dark matter (solid) and stars constructed from live stars (dotted) and dark matter particles using the tagging method (dashed).We see that the SIDM60 simulation with live stellar particles well reproduces the inferred halo and stellar mass profiles of Crater II (Caldwell et al. 2017).
) For the thin disk, we take M d = 5.9 × 10 10 M ⊙ , a d = 3.9 kpc, and b d = 0.3 kpc.For the thick disk M d = 2 × 10 10 M ⊙ , a d = 4.4 kpc, and b d = 0.92 kpc.These values are the same as those in Borukhovetskaya et al. (2022), which are motivated by the Milky Way mass model in McMillan (

Figure 1 .
Figure2(left panel) shows the evolution of the halflight radius R 1/2 and the line-of-sight velocity dispersion σ los of the stellar particles within R 1/2 for SIDM60 O3 (red), CDM O3 (blue), and CDM O2 (black).The initial conditions R 1/2 = 2.03r E ≈ 0.8 kpc, 1.5 kpc, and 2.8 kpc are denoted as E1 (solid; dot), E2 (dashed; square), and E3 (dotted; diamond), respectively; for each case the symbols denote the evaluation at snapshots after every pericenter passage.For CDM O2, the size converges to R 1/2 ≈ 0.3-0.4kpc from R 1/2 ≈ 0.8-2.8kpc after tidal evolution; see alsoBorukhovetskaya et al. (2022).We have checked that our final R 1/2 values agree with the estimates based on the tidal track of CDM satellite halos inErrani et al. (2022).The same trend applies to CDM O3, but the convergence is slower as it has weaker tides.It is clear that both CDM O2 and O3 cannot explain the observations of Crater II.The former's stellar size is a factor 3 smaller than Crater II's R 1/2 ≈ 1 kpc, while the latter's velocity dispersion is a factor of 3 higher than the measured one σ los (< R 1/2 ) ≈ 2.7 km/s.Remarkably, SIDM60 O3 can simultaneously explain both stellar size and velocity dispersion of Crater II.The upper limit of the galaxy size is largely set by r max .For the region beyond r max , stellar particles are stripped away significantly.From Figure1(right

Figure 2 .
Figure2.Left panel: the evolution of the stellar half-light radius R 1/2 and the line-of-sight velocity dispersion of stars within R 1/2 for SIDM60 O3 (red), CDM O3 (blue), and CDM O2 (black).The circle, square, and diamond symbols denote their corresponding snapshots after every pericenter passage for evaluation.Right panel: The line-of-sight velocity dispersion profile of stellar particles at the initial (gray) and final (colored) snapshots for SIDM60 O3 (red), CDM O3 (blue), and CDM O2 (black).For both panels, the stellar distribution follows an Einasto profile with three initial R 1/2 values 0.8 kpc (E1; solid), 1.5 kpc (E2; dashed), and 2.8 kpc (E3; dotted); the data point with error bars are from the measurements of Crater II(Caldwell et al. 2017).The orbit O3 is consistent with the measurements of Gaia EDR3(McConnachie & Venn 2020;Ji et al. 2021;Pace et al. 2022), while the orbit O2 is designed such that the CDM halo can lose sufficient mass.

Figure 4 .
Figure 4. Left panel: Similar to the left panel of Figure 2, but with three sequential final snapshots for SIDM60 O3 denoted as t1, t2, and t3.The insert panel shows their corresponding locations in the V (R 1/2 )-t plane.Right panel: Similar to the left panel of Figure 2, but for two additional SIDM simulations SIDM10 O5 (orange) and SIDM30 O7 (purple).

Table 1 .
(McConnachie & Venn 2020;Ji et al. 2021;Pace et al. 2022rom the Sun D⊙ [kpc], line-of-sight velocity in the solar rest frame v los [km/s], right ascension αJ2000 and declination δJ2000 [deg], proper motions µα⋆ and µ δ [mas/yr].asinBorukhovetskayaetal.(2022).The orbit O2 has a pericenter of r p ≈ 13.8 kpc and it is designed such that the CDM halo can lose sufficient mass, while the orbit O3 has r p ≈ 37.7 kpc and it is well consistent with the measurements of Gaia EDR3(McConnachie & Venn 2020;Ji et al. 2021;Pace et al. 2022).

Table 2 .
Orbital parameters considered for our CDM and SIDM simulations.Columns from left to right: Orbit name label, proper motions µα⋆ and µ δ [mas/yr], pericenter distance [kpc], and dark matter model.The orbit O3 is consistent with the measurements of Gaia EDR3 (McConnachie & Venn 2020; Ji et al. 2021; Pace et al. 2022), while O2 is designed for the CDM halo to lose sufficient mass.