Multiple Beads on a String: Dark-matter-deficient Galaxy Formation in a Mini-Bullet Satellite–Satellite Galaxy Collision

In the idealized galaxy–galaxy collision simulations, we use the publicly available adaptive mesh refinement (AMR) code enzo (Bryan et al. 2014; Brummel-Smith et al. 2019) with its hydrodynamics solver zeus (Stone & Norman 1992a, 1992b). The grackle library (Smith et al. 2017) is used to compute radiative gas cooling and heating assuming the homogeneous ultraviolet (UV) background of Haardt & Madau (2012) at z = 0 by interpolation of the lookup table generated from the cloudy code (Ferland et al. 2017). ⁷ We refine the simulation box of (2.621 Mpc)³ down to a spatial resolution of Δx = 10 pc (level ${l}_{\max }=12$), which is eight times coarser than the simulations carried out in Paper II. It is necessary here to reduce the computational cost of simulating a larger box over a longer time than before, so we must relax the resolution, aware that it cannot resolve star cluster formation, which was a primary goal of Paper II, with its spatial resolution of Δx = 1.25 pc, but is sufficient to resolve the galaxy formation properties we need here. Our refinement strategy is super-Lagrangian, meaning that when a cell contains more mass than the mass threshold (i.e., its mass density exceeds a density threshold), that cell splits into eight child cells. At refinement level l, for gas with a star formation threshold gas number density n_th, ${M}_{\mathrm{ref},\mathrm{gas}}^{l}={2}^{-0.333(l-12)}\times {M}_{\mathrm{ref},\mathrm{gas}}^{12}$, where ${M}_{\mathrm{ref},\mathrm{gas}}^{12}$ =10, 000 M_⊙ = n_thΔx³ ≃ 2.5 M_Jeans(T = 100 K, n = 400 cm⁻³), and for (dark matter and stellar) particles, ${M}_{\mathrm{ref},\mathrm{part}}^{l}\,={2}^{-0.107(l-12)}\times {M}_{\mathrm{ref},\mathrm{part}}^{12}$, where ${M}_{\mathrm{ref},\mathrm{part}}^{12}=16,000\,\,{M}_{\odot }$. We note that while refinement proceeds down to a cell size that is small enough not to contain more than a fixed physical mass of either gas or particles, this refines the force length resolution of the gravity and gas pressure forces, but does not refine the particle mass resolution. Dark matter and stellar particles (after releasing stellar feedback) have a fixed mass at all times and do not refine.

To model feedback-regulated star formation, we adopt subgrid algorithms for under-resolved small-scale physical processes, just as we did in Paper II, but with some adjustments for the coarser resolution and an assumption of higher thermal energy released by SNe associated with the massive stars. A parcel of gas is determined to form stars according to the approach in Cen & Ostriker (1992), and the outcome of that star formation is assumed to follow the star-forming molecular cloud model (SFMC; for details, see Kim et al. 2013, 2019). In brief, an SFMC particle is created when the following criteria are met: (1) the density of a gas cell exceeds n_th = 400 cm⁻³, (2) the gas flow is converging, (3) the cooling time of the cell is shorter than its dynamical time (t_dyn), and (4) the mass in that cell is enough to create an SFMC particle heavier than m_SFMC = 5 × 10³ M_⊙ (which leads to a permanent star particle mass—i.e., a total mass that follows the initial mass function of the stars that form, of m_⋆,new = 10³ M_⊙). The SFMC particle returns 80% of its original mass to the gas in a time equal to 12t_dyn according to our assumed star formation efficiency for converting the gas-to-star mass (Krumholz & Tan 2007), along with SN thermal feedback of 10⁵¹ erg per 50 M_⊙ of the permanent star mass that peaks at 1t_dyn and 2% of the metal yield (see also Kim et al. 2011).

We follow the method used in Paper I and Paper II to initialize two progenitor galaxies by using the dice code (Perret 2016). We place two identical progenitor galaxies 60 kpc apart and set their relative velocity to be 300–600 km s⁻¹. Since the observed line-of-sight velocity difference of DF2 and DF4 is 358 km s⁻¹ (van Dokkum et al. 2022a), the relative collision velocities need to be higher than that. The pericentric distance ${r}_{\min }$ is set to be 2 kpc.

While the parameters for these idealized galaxy–galaxy collisions in Step 1 are similar to those adopted in Paper I, there are several important distinctions between the simulations here in Step 1 and those in Paper I. First, the progenitors here are taken to be spherical halos with gaseous baryons, which self-consistently form stars before their collision, while in Paper I, the intention was to model the collision of present-day galaxies with well-established disks. Second, the spatial resolution here is much higher than in Paper I, which makes a difference in the ability of the collision simulations to produce the post-collision stellar systems. To model the NGC 1052 galaxy group, it is necessary to demonstrate that a single collision with realistic parameters can naturally produce a trail of ∼10 DMDGs. The coarser spatial resolution of simulations in Paper I, of 80 pc (as opposed to the 5 pc resolution here), prevented us from forming ∼10 DMDGs there, however. In those lower-resolution simulations, fewer objects of higher mass were formed in general, and it was necessary to tune the choice of parameters just so as to maximize this number. Even so, only up to 6 DMDGs were formed. In the current paper, we believe we have converged with spatial resolution, as demonstrated by the comparison of the runs with 5 and 10 pc resolution, respectively. Moreover, the formation of ∼10 DMDGs did not require such fine-tuning as before, either.

Paper II, on the other hand, had an even higher spatial resolution than the simulations here, but only by applying the AMR to a limited zoom-in region surrounding the most massive DMDG formed by an idealized galaxy–galaxy collision. As such, it did not address the questions at issue here, of producing a trail of ∼10 DMDGs.

The progenitors are initialized with only gas and dark matter. Their dark matter halos have M₂₀₀ = 1.66 × 10¹⁰ M_⊙, J₂₀₀ = 1.03 × 10¹² M_⊙ kpc km s⁻¹, and R₂₀₀ = 51.7 kpc, and follow the Navarro-Frenk-White (NFW) profiles (Navarro et al. 1997), with the concentration parameter c = 13. ⁸ For the gas density profiles, instead of initializing the progenitor galaxies with exponential disks as in Paper I and Paper II, we this time adopt the pseudo-isothermal (PIS) profile without any rotation,

$$\begin{eqnarray}&&\rho (R)={\rho }_{0}\frac{1}{1+{\left(R/{R}_{{\rm{s}}}\right)}^{2}},\end{eqnarray} \tag{ 1 }$$

with a scale radius of R_s,gas = 2 kpc, which is more commonly observed and used to simulate low-mass satellite galaxies, to initialize the gas density profile of the progenitors (e.g., Kurapati et al. 2020). The total mass is 1.89 × 10¹⁰ M_⊙, and the gas fraction is f_gas = M_gas/M_total = 12.4%, without stars at the beginning. ⁹ The gas is initially set to a temperature of 10⁴ K and metallicity of Z = 0.1 Z_⊙ = 0.002041 to match the metallicity of a low-mass galaxy at z ∼ 1–2. We use 2 × 10⁶ dark matter particles to model each progenitor, resulting in a mass resolution of m_DM = 8.29 × 10³ M_⊙.

We perform a suite of four galaxy–galaxy collision simulations, each with a different set of initial configurations. In Table 1 we summarize this set of structural parameters we tested and the stellar masses of the most massive DMDG that resulted, the velocity difference of the most and second massive DMDGs (corresponds to DF2 and DF4), and the number of DMDGs produced at t_end, after the collisions. Among the tested results, it is notable that there are collision configurations that result in a large velocity difference (∼300 km s⁻¹) between the first and second most massive DMDGs that form, which is similar to that between DF2 and DF4 (van Dokkum et al. 2022a). We note that in these idealized collision simulations, the post-collision separation velocity does not increase with time, while in the more realistic simulations we describe below, in which the progenitors are also satellites of the massive galaxy NGC 1052, their post-collision separation velocity can increase with time. As a result, it is possible that their immediate post-collision separation velocity was somewhat lower than is observed now for DF2 and DF4.

Table 1. A Suite of 10 pc Resolution Idealized Dwarf Galaxy-Dwarf Galaxy Collision Pair Simulations Listed with Their Initial Configurations

Run Name	vcol	${r}_{\min }$	Rs,gas	Mtotal	fgas	${M}_{\star ,\mathrm{DMDG},\max }$	${\rm{\Delta }}{v}_{\mathrm{DMDG},\max 2}$	NDMDG	tend
	(km s−1)	(kpc)	(kpc)	(1010 M⊙)		(108 M⊙)	(km s−1)		(Gyr)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
1	300	2	2	1.95	0.12	3.7	123	8	0.6
2	400	2	2	1.95	0.12	2.8	202	7	0.7
3	500	2	2	1.95	0.12	0.61	278	7	0.7
4	600	2	2	1.95	0.12	1.1	306	7	0.9

Note. (1) Run name, (2) relative velocity of the two progenitors at 60 kpc distance, (3) pericentric distance (i.e., distance at closest approach), (4) scale radius of the PIS gas density profile, (5) the total mass of a progenitor, (6) gas fraction f_gas = M_gas/M_total, (7) stellar mass of the most massive DMDG formed, (8) the largest relative velocity difference between the DMDGs, (9) the number of formed DMDGs, (10) time since the pericentric approach of the two progenitor galaxies when we end the simulation.

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The time history of the mini-Bullet system from pre- to post-collision is illustrated by Figures 1, 2, and 3, which show snapshots of the results of our enzo simulation for Run 3 in Table 1. At t = − 110 Myr, two progenitor galaxies are initialized with a separation of 60 kpc, approaching at 500 km s⁻¹. Figure 1 zooms out for time-slices long before, during, and long after the progenitor collision, to show how the Mini-Bullet collision dissociates collisionless components, dark matter and stars, of these progenitors from their gas (t = 90 Myr in Figure 1). In Figure 2 we zoom in, both in time and space, to show the immediate effects of the collision on the two progenitors and the distinctions between these effects on their collisionless components (dark matter and stars), which can pass through each other, versus their collisional component, their gaseous baryonic interstellar media (ISM), which, as a fluid, cannot. Instead, the ISM of each progenitor is halted from its supersonic approach to its counterpart in the other progenitor by a strong shock that heats the gas to T > 10⁷ K, in which the sound speed is comparable to the collision velocity (t = − 30 and 0 Myr in Figures 2 and 3). There are two shocks, one on each side of their collision center, to decelerate the ISM of each incoming progenitor, producing a layer of shock-heated gas between them whose mass grows as the collision proceeds to overtake more and more of the ISM of each progenitor. The shock-heated gas begins to cool radiatively as soon as it is heated, and the shock becomes a radiative shock, for which the cooling time is short compared with the time for the shock velocity to change (t = 30 Myr in Figures 2 and 3). In these shocks, the gas can cool isobarically to well below 10⁴ K, increasing its density in inverse proportion to the temperature, so its density increases as the square of the shock Mach number. Since the gas in the ISM of each progenitor is centrally concentrated, the cooling time is shortest for shocked gas that was initially closer to the centers of the progenitors, so this is where the shocks first become radiative, and the cooling gas is subject to dynamical and gravitational instabilities. For this off-axis collision, dense gas clumps are formed due to radiative cooling near the line that connects the progenitors (y = 0 line). This results in a combination of cold gas that collapses toward their centers, while surrounding shells of still-hot shocked gas are driven to expand away from the line of collision by pressure forces, as shown in Figure 3, in the view along the line of collision (x-axis). After star formation in the collision-induced dense gas clumps, we identify seven DMDGs that are roughly aligned between the progenitors, forming an S-shaped trail (t = 390 Myr in Figure 1). Gas as the fuel of star formation is almost exhausted before this time, preventing star formation in the DMDGs on the trail. The trail of DMDGs is tidally stretched over time by the gravitational field of the separating progenitors at each end, which retain their dark matter and stars, making the S-shaped trail longer as the separation between galaxies grows (t = 690 Myr in Figure 1).

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Equipped with this knowledge from our suite of idealized galaxy–galaxy collision simulations regarding which parameters are suitable for producing DMDGs like DF2 and DF4, we now move to establish a realization of colliding progenitor pairs that are satellites of a massive host galaxy. This realization should produce two DMDGs, each with M_⋆ ≳ 10⁸ M_⊙ (for DF2 and DF4), and several UDGs along a line after ∼8 Gyr to match the observations (Román et al. 2021; van Dokkum et al. 2022a). These observations will include the radial separations and positions on the sky of DF2 and DF4, their radial velocity separation, and their radial velocities relative to that of NGC 1052.

The unknown initial conditions that will lead to these final conditions for DF2 and DF4 are the pre-collision locations and velocity vectors of the collision progenitors relative to each other and to NGC 1052. To derive these, we integrate the orbits of DF2 and DF4 backward in time to locate the collision that produced them and their velocity vectors at that time. By conducting a large enough sample of such orbit integrations, we were able to identify those for which the orbits of DF2 and DF4 once crossed in the past, indicating the time and place of the collision we postulate to have formed them. From this subset of the orbit integrations that led to collisions in the past, we refined our sample further to those for which the lookback time of the collision was long enough to explain the observationally inferred ages of the GCs of DF2 and DF4, i.e., ∼8 Gyr (van Dokkum et al. 2018c; Fensch et al. 2019; Ma et al. 2020a).

We realized that for this refined set of cases, the relative velocities of DF2 and DF4 immediately after the collision that produced them were typically lower than the currently observed radial separation velocities by ≳100 km s⁻¹, reflecting the relative acceleration of their post-collision orbits over time. This means that the immediate post-collision separation velocities were consistent with the collisions in Runs 2 and 3 in Table 1, thereby confirming that these post-collision outcomes were indeed possible from suitably high-velocity galaxy–galaxy collisions.

The next step was to select three of these orbit integrations that yielded the collisions at the right lookback time and determine the pre-collision parameters of the progenitor galaxies that would produce these immediate post-collision separations and velocities for DF2 and DF4. In principle, if we had a large enough sample of cases in Table 1, we might have had a close enough match to the orbit integrations that we could use them to identify the pre-collision configuration that would lead to the post-collision configuration derived from a given orbit integration. In practice, however, it is computationally prohibitive to run so many cases of idealized collisions for this purpose. Instead, we determined the pre-collision configuration as follows.

The exact pre-collision configurations of the two progenitors are decided by setting relative collision velocities v_col to two times the relative velocities of DF2 and DF4 Δv_DF2−DF4, resulting in the two progenitors colliding with ∼500 km s⁻¹, and pericentric distances of ${r}_{\min }\sim 2\,\mathrm{kpc}$. This follows the observation from the simulations presented in Section 2.1 that the velocity difference of the first and second most massive DMDGs formed in ${\rm{\Delta }}{v}_{\mathrm{DMDG},\max 2}$ is roughly a half of v_col of the progenitors ${\rm{\Delta }}{v}_{\mathrm{DF}2-\mathrm{DF}4}={\rm{\Delta }}{v}_{\mathrm{DMDG},\max 2}\sim 0.5{v}_{\mathrm{col}}$ (with ignorance of the long-term velocity change due to the presence of the host gravity; we refer to Section 2.1 and Table 1).

Finally, with the immediate pre-collision configurations derived in this way for each of the three selected cases above, another backward time integration was required for each, this time, of the orbits of their two progenitor galaxies that are destined to meet at their collision moment, to place them at large separations at earlier times, from which we could perform full gravitohydrodynamic simulations of their collision events from start to finish. We describe these steps in more detail below.

We use the Integrator with Adaptive Step-size control, 15th order (IAS15), a gravitational dynamics integrator, implemented in the Rebound orbit integration code (Rein & Liu 2012; Rein & Spiegel 2015), to backtrace the orbits of DF2 and DF4 based on the currently observed quantities and find the collision point of the progenitor galaxies, where DF2 and DF4 should have started to form at the same time. Then, using the confined collision parameters that can produce multiple aligned DMDGs (Section 2.1), we place the progenitor galaxies to make initial conditions for the hydrodynamic simulation (Section 2.3). Finding the collision point is not trivial because, depending on the initial locations and motions, the backtraced DF2 and DF4 may not collide. Therefore, we should find the initial conditions from the parameter space that consists of the range of current locations and motions of DF2 and DF4. ¹⁰

In Figure 4 we present an example of the three initial conditions based on the orbit integration calculation results. We place a dark matter-only massive host (black circle) with a halo mass of M_DM = 6.12 × 10¹² M_⊙ to represent the host, NGC 1052, at the center (Forbes et al. 2019). The orbits of DF2 and DF4 with the stellar mass of M_⋆ = 2 × 10⁸ M_⊙ need to be time-reversed integrated to find the collision point. The radial distances of DF2 and DF4 are constrained with a certain amount of uncertainty (22.1 ± 1.2 Mpc for DF2 and 20.0 ± 1.6 Mpc for DF4; van Dokkum et al. 2018b, 2019, 2022a), ¹¹ but the distance to NGC 1052 is not well measured. Thus, in this study, we assume the distance to NGC 1052 as 19 Mpc, consistent with previous studies using the surface brightness fluctuation method (Blakeslee et al. 2001; Tonry et al. 2001) and the Virgo-infall-corrected radial velocity (Gil de Paz et al. 2007). ¹² The transverse distances of DF2 and DF4 from NGC 1052 can be simply calculated from their angular separations (Román et al. 2021): −0.081 Mpc (DF2) and +0.168 Mpc (DF4), assuming the observed distance to DF2 and DF4. For the radial distances, we set the errors to be 0.25 × (observation error), and for the transverse distances, we adopt 0.002 Mpc as errors. The three galaxies and the observer are not exactly on the same plane, but we ignore the deviation and assume that the transverse (proper) and radial (line-of-sight) distances can be used as the x-axis and the y-axis in our orbit integration as if they were on the same plane. The radial velocities of DF2 and DF4 relative to NGC 1052 are also measured with unspecified errors (van Dokkum et al. 2022a): +315 km s⁻¹ (DF2) and −43 km s⁻¹ (DF4). Since the observation errors for radial velocities are not specified, we set their errors to be 10 km s⁻¹. We let these known parameters (radial and transverse distances and radial velocities) vary within observed values ± errors. However, since the transverse motion of DF2 and DF4 cannot be observed, we vary the transverse velocities within fixed ranges: [10 km s⁻¹, 170 km s⁻¹] for the progenitor 1 (bound satellite, dashed blue line in Figure 4) and [0 km s⁻¹, 80 km s⁻¹] for the progenitor 2 (unbound satellite, dashed red line in Figure 4) with an interval of Δv = 2.5 km s⁻¹. To confine the parameter space, we uniformly sample the parameter space, testing >1, 500, 000 cases, and find ∼3000 cases where DF2 and DF4 started from places closer than 10 kpc, the place of a Mini-Bullet galaxy collision.

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After finding the collision point, we place two progenitor galaxies (blue and red circles on dashed lines) with dark matter halo mass of M_DM = 1.42 × 10¹⁰ M_⊙ and M_gas = 2.47 × 10⁹ M_⊙ (f_gas = 0.148) without stars, ¹³ which is similar to the mass we tested, using information from the idealized two-galaxy collision simulations in Section 2.1. One of the progenitors (dashed blue) is gravitationally bound to the host and has a highly eccentric orbit. The other progenitor (dashed red) is not bound to the host, but comes from outside of the NGC 1052 system with a high velocity of >400 km s⁻¹ relative to the host, which is higher than the host virial velocity V₂₀₀ = 260 km s⁻¹.

Using these orbit integration results, we set up initial conditions consisting of the massive host and two progenitors at 0.3 Gyr before their Mini-Bullet collision, to make them relax after the artificial starbursts that occur after initialization. Their gas density distributions are given by the PIS profile, while their dark matter densities follow an NFW profile. Using the dice code (Perret 2016), we sampled the density profile of NGC 1052-like host with 10⁶ dark matter particles (m_DM,host = 6.1 × 10⁶ M_⊙) and the colliding progenitors' NFW profiles were realized with 10⁷ dark matter particles (m_DM,prog = 1.42 × 10³ M_⊙) each. Since the lookback time of 8 Gyr corresponds to z ∼ 1, we now set the progenitor halo concentrations to be c = 7, about a factor of two lower than c = 13, the value that was adopted in the idealized simulations in Table 1. This is because, for a halo with a given mass, the concentration parameter is lower for a halo that formed earlier. We summarize the sets of collision configurations and structural parameters of the progenitors we test in Table 2. Henceforth, we refer to all of the initial conditions described here in Section 2.3 as tailor-made and the cases in Table 2 as tailor-made cases (e.g., TM1 refers to tailor-made case 1). Among those initial conditions, we specifically focus on TM1, TM2, and TM3 for our analysis presented in Sections 3.1 and 3.2, as representative runs that start from different pre-collision orbital parameters of the progenitor galaxies and yield results that reproduce the observations. (Note: TM5 is a variation of TM1, with different separation distance at the closest approach for the colliding progenitors, but with a similar enough outcome to that of TM1 that we do not give a more detailed description here.)

Table 2. Initial Conditions for Simulations of Satellite–Satellite Galaxy Collisions around a Massive Host with M_⋆ = 6.12 × 10¹² M_⊙

Run name	vcol	${r}_{\min }$	Rs,gas	chalo	(x0, y0)DF2	(vx0, vy0)DF2	(x0, y0)DF4	(vx0, vy0)DF4	Orbit case
TM	(km s−1)	(kpc)	(kpc)		(Mpc)	(km s−1)	(Mpc)	(km s−1)
(''Tailor-Made'')
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
TM1	531	2	3	7	(−0.079, 3.1)	(105, 325)	(0.167, 1.0)	(150, −33)	1
TM2	468	2	3	7	(−0.081, 3.1)	(95, 305)	(0.169, 1.0)	(140, −33)	2
TM3	529	2	3	7	(−0.080, 3.1)	(125, 305)	(0.167, 1.0)	(170, −43)	3
TM4	531	1.8	3	13	(−0.079, 3.1)	(105, 325)	(0.167, 1.0)	(150, −33)	1
TM5	531	1.8	3	7	(−0.079, 3.1)	(105, 325)	(0.167, 1.0)	(150, −33)	1
TM6	531	1.5	4	13	(−0.079, 3.1)	(105, 325)	(0.167, 1.0)	(150, −33)	1
TM7	531	1.8	4	13	(−0.079, 3.1)	(105, 325)	(0.167, 1.0)	(150, −33)	1
TM8	531	2	4	13	(−0.079, 3.1)	(105, 325)	(0.167, 1.0)	(150, −33)	1

Note. The initial conditions are tailor-made in attempt to produce collisions whose outcome matches observations of the NGC 1052 group (van Dokkum et al. 2022a), including backward orbit integration described in Section 2.2. (1) Run name, (2) relative velocity of the two progenitors at pericenter, (3) pericentric distance of the progenitors, (4) gas scale radius in the PIS profile, (5) concentration parameter of the progenitor dark matter halos, (6) transverse (x) and radial (y) coordinates of DF2 relative to NGC 1052 (observed radial distance from the Earth is 20.0 Mpc), (7) transverse velocity (v_x ) and radial velocity (v_y ) of DF2 relative to NGC 1052 (currently observed to recede from NGC 1052 at 315 km s⁻¹; van Dokkum et al. 2022a), (8) transverse (x) and radial (y) coordinates of DF4 relative to NGC 1052 (observed radial distance from the Earth is 22.1 Mpc), (9) transverse velocity (v_x ) and radial velocity (v_y ) of DF4 relative to NGC 1052 (currently observed to approach NGC 1052 at −43 km s⁻¹; van Dokkum et al. 2022a), and (10) orbits of DF2 and DF4 in backward orbit integration calculation described in Section 2.2. The difference in the orbits originates from the different starting positions and velocities of DF2 and DF4 in the time-reversed orbit integrations, as listed in Columns (6)–(9), accounting for observational error and uncertainties in the transverse velocities of DF2 and DF4, which cannot be observed. Runs (TM4−8) share the starting positions and velocities of DF2 and DF4 that define Orbit 1 with TM1. To be clear, all these starting values for the backward time integration are actually final values when time is expressed as going forward to the current epoch.

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With these tailor-made initial conditions, summarized in Table 2, we again use the enzo code to simulate the high-velocity collision of two satellite galaxies and the orbital evolution of the progenitors and resultant DMDGs after the collision. The hydrodynamics and star formation physics are the same as in the idealized simulations in Section 2.1. The box size is increased to (5.243 Mpc)³ to fully capture the orbits of the product DMDGs, but the nested refinement region is set only to resolve the progenitors and product DMDGs, not the massive host galaxy. A static refinement up to level 3 (Δx = 10.24 kpc) is applied outside the nested refinement regions. We change the refinement scheme due to the refinement level change (Δx = 5 pc at level ${l}_{\max }=14$) and refine more for the particles to resolve star-dominated structures better: at a refinement level l, for gas, ${M}_{\mathrm{ref},\mathrm{gas}}^{l}={2}^{-0.264(l-14)}\times {M}_{\mathrm{ref},\mathrm{gas}}^{14}$, where ${M}_{\mathrm{ref},\mathrm{gas}}^{14}=4000\,\,{M}_{\odot }\simeq 2.5\,{M}_{\mathrm{Jeans}}^{100\,{\rm{K}}}$, and for particles, ${M}_{\mathrm{ref},\mathrm{part}}^{l}={2}^{-0.273(l-14)}\times {M}_{\mathrm{ref},\mathrm{part}}^{14}$, where ${M}_{\mathrm{ref},\mathrm{part}}^{14}=4000\,\,{M}_{\odot }$. Following the change in the cell spatial resolution and refinement criteria, we change the star formation density threshold to n_th = 1.6 × 10³ cm⁻³ and the SFMC particle mass threshold to m_SFMC = 2.5 × 10³ M_⊙ (permanent star particle mass of m_⋆,new = 5 × 10² M_⊙). The simulations are run for 2.3 Gyr, of which 0.3 Gyr are before the collision and 2 Gyr are after, at which time we analyze the properties of the product DMDGs. Due to the significant computational time needed to run the full ∼8 Gyr of simulation, we further supplement the simulations with the orbit integration code Rebound for the remaining ∼6 Gyr, to track the orbital evolution of the DMDGs and colliding progenitors during later stages.

Figures 5 and 6 depicts a time sequence of the Mini-Bullet collision of two satellite galaxies orbiting a massive host that corresponds to NGC 1052 (see Sections 2.2 and 2.3 for the exact host properties), and the resulting formation of spatially aligned DMDGs in the TM1 run is listed in Tables 2 and 3. The time of the two colliding satellite galaxies at the pericentric approach is defined as t = 0. At t = 2 Gyr, which is the last snapshot of the simulation, the progenitors are approximately 1 Mpc apart, and the sequence of DMDGs spans over 600 kpc.

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Table 3. Results from a Suite of Simulations of Satellite–Satellite Galaxy Collisions around a Massive Host

Run Name	${M}_{\star ,\mathrm{DMDG},\max 2}$	${\rm{\Delta }}{v}_{\mathrm{DMDG},\max 2}$	${\rm{\Delta }}{v}_{\mathrm{DMDG},\max 2,\mathrm{now}}$	NDMDG	${t}_{\mathrm{end},\mathrm{sim}}$	tend,orbit	Matches
(TM(''Tailor-Made''))	(108 M⊙)	(km s−1)	(km s−1)		(Gyr)	(Gyr)	NGC 1052 Group?
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
TM1	2.6, 1.5	216	324	11	2	8.375	Yes
TM2	3.4, 1.6	252	554	11	2	6.65	Yes
TM3	3.4, 1.8	215	343	9	2	8.503	Yes
TM4	1.8, -	N/A	N/A	1	2	N/A	No
TM5	2.7, 1.3	239	433	9	2	6.875	Yes
TM6	4.4, 2.6	75	N/A	2	0.7	N/A	No
TM7	2.8, 2.1	106	N/A	2	0.8	N/A	No
TM8	3.1, 2.4	52	N/A	2	0.55	N/A	No

Note. See Section 3.1 for a description and discussion of the properties of the product DMDGs. (1) Run name, (2) stellar mass of the most and second massive DMDGs ("-"=no DMDG formed), (3) line-of-sight velocity difference of the most and second massive DMDGs at t = 2 Gyr ("-"=no DMDG formed), (4) line-of-sight velocity difference of the most and second most massive DMDGs obtained from orbit integration until the distance between the two most massive DMDGs that correspond to DF2 and DF4 reaches 2.1 Mpc apart (N/A corresponds to the runs that do not involve orbit integration due to their lack of the number of DMDGs), (5) the number of formed DMDGs with M_⋆ > 10⁷ M_⊙, (6) time since the pericentric approach of the two progenitor disks when the enzo simulation ends, (7) time since the pericentric approach of the two progenitor disks when the orbit integration ends, i.e., when the distance between the two most massive DMDGs is 2.1 Mpc, (8) whether the results from TM (tailor-made) runs produce enough DMDGs to match the number of UDGs observed in the NGC 1052 group.

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The DMDGs are identified by the HOP halo finder (Eisenstein & Hut 1998). We employ the boundary of each DMDG to be the tidal radius, or the Jacobi radius, the distance at which the gravitational force of the massive host becomes equivalent to the DMDG's self-gravitation, defined by Binney & Tremaine (2008),

$$\begin{eqnarray}&&{R}_{\mathrm{tidal}}={r}_{\mathrm{host}}\displaystyle \frac{{M}_{\mathrm{DMDG}}}{3{M}_{\mathrm{host}}^{1/3}},\end{eqnarray} \tag{ 2 }$$

where r_host is the distance of the DMDG from the massive host, M_DMDG is the mass of the DMDG, and M_host = 6.12 × 10¹² M_⊙ is the mass of the massive host.

The middle row of Figure 5 illustrates the gas distribution during and after the galaxy collision. Consistent with our findings in previous studies (Paper I; Paper II), the baryonic component of the progenitor galaxies is stripped from their dark matter halos and undergoes severe shock compression during the collision process (as shown in the second, t = 0 Gyr, panel of the upper right panels). This results in a burst of star formation inside the compressed dense gas clouds (see Figures 1 and 3 in Paper II for more details). Meanwhile, the stripped gas is tidally elongated, and multiple self-gravitating gas clumps form along the line connecting the two progenitors, generating stars inside these clumps. As a result, after approximately 1 Gyr, a number of similar stellar structures are identifiable (as shown in the bottom panel of the third column in Figure 5, t = 1 Gyr). At this point, the fuel for star formation is depleted due to both the star formation burst (major) and gravitational infall to the host (minor), causing the stellar mass and population of the DMDGs to remain almost constant. Consequently, between t = 1 Gyr and t = 2 Gyr, the gravitational field of the host and the progenitors is the primary driver of the dynamical evolution of the DMDGs and the progenitors.

As summarized in Table 3, the results from the simulations successfully satisfy two of the goals listed in Section 2: the stellar mass of DF2 and DF4 and the number of product DMDGs. We find ∼10 (20; 30) self-gravitating stellar structures with M_⋆ > 10⁷ M_⊙ (M_⋆ > 10⁶ M_⊙; M_⋆ > 10⁴ M_⊙) in the simulations we analyze in detail: TM1 (tailor-made1), TM2, and TM3 (we refer to Table 3 for the exact setups). In our analysis, we restrict our focus to self-gravitating star clumps with M_⋆ > 10⁷ M_⊙ and designate them as DMDGs. Table 3 also shows the velocity differences between the two most massive DMDGs, which can be compared to the observed line-of-sight velocity difference of DF2 and DF4 at the end time of the simulations (t = 2 Gyr for TM1, TM2, and TM3 run; corresponding to ∼6 Gyr ago from the present), in the TM1 run and other simulations we test. The velocity differences are smaller, but the orbital motions for ∼6 Gyr need to be taken into account.

Notably, the formation of DMDGs with M_⋆ > 10⁸ M_⊙ and multiple DMDGs is highly sensitive to the structural parameters of the progenitors. Too small ${r}_{\min }$ and large R_s,gas result in the gathering of gas near the collision point, yielding too massive DMDGs and a small velocity difference between the most and second massive DMDGs. Eventually, the DMDGs gravitationally pull each other, and no trail of DMDGs is produced. Furthermore, the concentration of progenitor dark matter halo also affects the DMDG formation. More DMDGs are formed in the runs with lower concentration parameters, meaning that the strong tidal field during the Mini-Bullet collision can suppress gas and newly born stars from gathering and forming self-gravitating structures.

In order to compare our simulation results with observations of the NGC 1052 group, which, by design, are expected to exhibit good alignment, we further examine the orbital evolution of the DMDGs and their progenitor satellite galaxies by integrating their orbits with the Rebound code. Figure 7 displays the orbits of the product DMDGs and the progenitors, tracked in the hydrodynamic simulations from t = − 0.3 Gyr to t = 2 Gyr (solid lines) and in the orbit integration from t = 2 Gyr to t ∼ 8 Gyr (dashed lines). At the latter time, the line-of-sight distance (y-coordinates in Figures 4, 5, and 6 and x-coordinates in Figure 1 of van Dokkum et al. 2022a) between the two most massive DMDGs, which correspond to DF2 and DF4, is 2.1 Mpc apart, is set to the observed amount in van Dokkum et al. (2022a). As summarized in Table 3, in the TM1 (TM2; TM3) run, this occurs at t = 8.375 (6.65; 8.502) Gyr. The line-of-sight velocity difference of the two most massive DMDGs after the orbit integration ${\rm{\Delta }}{v}_{\mathrm{DMDG},\max 2,\mathrm{now}}=324$ in the TM1 run (554, 343 km s⁻¹ in TM2, TM3 run) is roughly in line with the observed velocity difference of DF2 and DF4, 358 km s⁻¹ (van Dokkum et al. 2022a), complying with the remaining goals of matching the velocity difference and positions of DF2 and DF4 listed in Section 2.

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Conserving the alignment of the DMDGs at t = 2 Gyr depicted in Figures 5 and 6, our orbit integration analysis reveals that the DMDGs remain spatially aligned after orbit integration, consistent with the observed peculiarity of the UDGs in the NGC 1052 group (Román et al. 2021; van Dokkum et al. 2022a). In several runs with different initial orbital and structural parameters of the colliding satellites, we consistently confirm that with appropriate initial conditions, multiple (∼10) aligned DMDGs with a considerable number of stars are formed from a single Mini-Bullet collision. However, note that the alignment of the DMDGs is not exactly on a line. There are deviations in DMDG distribution from the line, at most ∼100 kpc at t = 2 Gyr and ∼300 kpc at t ∼ 8 Gyr. Estimating these deviations in simulation could provide valuable insight for future observations in identifying which UDG is on the sequence of DMDGs and shares a common origin with DF2 and DF4, potentially providing a way to disprove or substantiate the Mini-Bullet scenario.

We present the spatial displacements of the DMDGs from the trail and the deviations in their line-of-sight velocities in Table 4, along with their positions at the end of the orbit integration (Figure 7). The displacement from the sequence of DMDGs, denoted as Δd, is defined as the distance from the DMDG to the line y = Ax + y₀ that best fits all the DMDG positions. The line-of-sight velocities (v_y ) of DMDGs other than simulated DF2 and DF4 candidates are estimated using either projected positions (x) or line-of-sight positions (y) through a simple linear relation v_y,x−pred = Ax + v_y,0 (v_y,y−pred = Ay + v_y,0) derived from the positions and velocities of the DF2 and DF4 candidates. Estimating velocities based on the linear relation derived from x and y results in significant differences between the estimated line-of-sight velocities v_y,x−pred and v_y,y−pred, with v_y,y−pred being more accurate. This is expected since the sequence of DMDGs is elongated along the y-axis and relative deviations in x are more significant than in y.

In Figure 8 we summarize the correlation between v_y and x, and v_y and y of the product DMDGs, including the simulated DF2 and DF4 candidates and the progenitors, along with linear regression fits. We perform least-square regression linear fits to the correlations using the data of the product DMDGs shown in Table 4 (not including the data of the progenitors), overplotting the fits with dashed lines. While both correlations follow linear relations fairly well, the correlation between v_y and y is tighter than that of v_y and x. The scatter in the correlation between v_y and x is crucial in interpreting the trail of UDGs in the NGC 1052 group. For instance, even if a UDG is located at the end of the trail when projected on the sky (i.e., with the greatest projected distance from NGC 1052), a different UDG, located at a smaller projected distance, can have the most extreme line-of-sight velocity.

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The implications of these findings for future observations are that determining the membership of a UDG in the NGC 1052 group within the sequence of DMDGs is a challenging task. The ideal approach involves precise measurements of line-of-sight velocities, distances, and projected distances of the aligned UDGs. However, achieving high precision in line-of-sight distance measurements is extremely difficult even with deep imaging with a significant amount of telescope time. Thus, the first step would involve establishing a correlation between the projected distances and line-of-sight velocities.

In a study by Gannon et al. (2023), the line-of-sight velocity of DF9 was measured and found to be inconsistent with the expected linear relation between the projected distances and line-of-sight velocities if it were a member of the trail along with DF2 and DF4. This caused the authors to question whether DF9 was correctly identified as a member of the trail or if it shared a common origin with the others in the trail, whether 3D geometry in projection might alter the interpretation relative to the simple linear relation if it were a member, or, finally, whether the idea of a common origin for all the galaxies in the trail was even correct. As the simulations here of the Mini-Bullet scenario predict, however, individual DMDGs are expected to exhibit deviations in their positions and velocities from that simple linear relation, so these observations of DF9 are so far not inconsistent with this. By gathering data from future deep observations of multiple UDGs on the trail, it will be possible to observe more of the details of the galaxies in the apparent trail that can then better be compared with the Mini-Bullet formation scenario of the aligned UDGs in the NGC 1052 group.

Another interesting prediction we make is the correlation between the predicted deviation of line-of-sight velocity and the transverse displacement of a DMDG from the best-fitting line of the common trail of DMDGs. This is shown in Figure 9, in which we define the deviations as the difference between predicted v_y using the projected positions (x) and the actual v_y (v_y − v_y,x−pred).

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In the TM1 run, 2 Gyr after the collision, the stellar mass of the most (second) massive DMDG is 2.6 × 10⁸ M_⊙ (1.5 × 10⁸ M_⊙), showing a good agreement with the observed stellar mass, ∼2.0 × 10⁸ M_⊙ for DF2 and ∼1.8 × 10⁸ for DF4 (van Dokkum et al. 2018b, 2019). Moreover, their line-of-sight velocity difference at the end of the orbit integration (t = 8.375 Gyr for the TM1 run) is 324 km s⁻¹, which is similar to the observed value of 358 km s⁻¹ (v_DF2 = 315 km s⁻¹ and v_DF4 = −43 km s⁻¹ relative to the host, NGC 1052; van Dokkum et al. 2022a). These results, along with those from the TM2 and TM3 runs, are summarized in Table 4. The stellar masses of the simulated DF2 and DF4 are similar to each other in all cases, with little variation from case to case. However,the line-of-sight velocities are of opposite sign relative to that of the host galaxy, with differences that vary significantly from case to case, depending on the initial orbital parameters of the progenitors and the subsequent orbits of the product DMDGs. This is because, as the orbit of the second massive DMDG (corresponding to DF4) passes closer to the host, it experiences stronger gravity, resulting in a more negative line-of-sight velocity.

van Dokkum et al. (2022a) postulated that NGC 1052-DF7 (hereafter DF7) and RCP32, located at the farthest end of the UDG trail, represent the remnants of the progenitor galaxies. Notably, a follow-up study showed that the identified GCs near RCP32 exhibit stellar populations that are more similar to those of the host galaxy NGC 1052, in contrast to the GCs associated with DF2 and DF4 (Buzzo et al. 2023). This finding provides support for the notion that RCP32 may be the preserved remnant of a post-collision satellite galaxy. In Figure 7 we illustrate the trajectories of the progenitors in our simulations and orbit integration. One of the progenitors is located farthest from the host, roughly on the trail of DMDGs, suggesting that the progenitor might correspond to RCP32. On the other hand, the other progenitor passes through and orbits the host halo rather than remaining in the sequence of the DMDGs and the other progenitor. This suggests the possibility that one of the progenitor galaxies may have experienced strong tidal forces exerted by the host, potentially rendering it challenging to be detected in the present day.

Now that we have confirmed that the spatial distribution, line-of-sight velocities, and stellar masses of the product DMDGs in our simulation are consistent with those observed by van Dokkum et al. (2022a), we turn our attention to their detailed physical properties. Figure 10 presents the stellar masses, average metallicities, and average formation time of the stars of the formed DMDGs at t = 2 Gyr, when the hydrodynamic simulation ends. The two most massive DMDGs, which correspond to DF2 and DF4, are marked with red and magenta, respectively, to distinguish them from other less massive DMDGs. In the left panel, we show the stellar masses and the average metallicities, with the error bars indicating the standard deviations of the metallicities. Overall, the average metallicities of the DMDGs exhibit a remarkable similarity with a deviation of [M/H] ∼ 0.2, regardless of their stellar mass. The right panel shows the metallicities and formation time of the stars, or the age of the stars at the end of the orbit integration on the x-axis above, which corresponds to the observed age at the current epoch. It is notable that massive DMDGs form stars for a longer period than other less massive DMDGs, along with the accretion of nearby gas.

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Even though the initial metallicity of the gas starts with Z = 0.1 Z_⊙, the metallicities range from a few Z_⊙ to ∼10 Z_⊙, which is far higher than the observed metallicities of DF2 ([M/H] = −1.07 ± 0.12; Fensch et al. 2019). We argue that this discrepancy originates from two effects: (1) pre-enrichment of the gas in the progenitor galaxies before the collision, and (2) the stellar feedback model we employ in our simulation. As described in Section 2.1, the stellar feedback is implemented with thermal feedback of SNe. However, simplifying various modes of stellar feedback, such as stellar winds and radiation feedback that affects the nearby interstellar medium over greater distances, can result in an overly efficient self-enrichment of the stars that form subsequently. This does not mean that the metal formation is overestimated, but rather that the dispersal of metals is inefficient. Especially in the case of bursty star formation resulting from shock compression of gas due to high-velocity galaxy collision, the metal dispersal should be carefully modeled (e.g., see Han et al. 2022 for the case of GC formation in cloud-cloud collisions). Including more realistic stellar feedback modes will be the subject of future studies, and our results should be interpreted as controlled simulations showing that the formed DMDGs have almost the same metallicities and ages, which can be tested in future observations.

In Figure 11 we present the temporal evolutions of the 3D stellar half-mass radii (R_1/2) of the DMDGs from the TM1, TM2, and TM3 runs. The tracking of the DMDG is based on the star particle IDs at t = 2 Gyr, and the center of the DMDG is defined as the center of mass of the identified stars at the time at which we measure R_1/2. It is worth noting that under the assumption of spherical symmetry and density profile, R_1/2 can be converted into the effective radius, or 2D projected half-light radius, R_eff, by R_eff = R_1/2/A, where A ∼ 1.3–1.35 (Wolf et al. 2010). The color-coded lines in Figure 11 indicate the 3D half-mass radii, with the distances from the host galaxy at t = 2 Gyr, r_{host,t=2 Gyr}, determining the color scheme.

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The two most massive DMDGs, denoted by larger markers (representing the simulated DF2 and DF4 candidates) exhibit up to a factor of ∼10 times more compact sizes compared to the observed R_eff of DF2 (2.2 kpc; van Dokkum et al. 2018b) and DF4 (1.6 kpc; van Dokkum et al. 2019), except for the simulated DF4 candidate in the TM1 run, which has R_1/2 ∼ 3 kpc. The fact that this DMDG has the size of a UDG is a novel finding and enhances the plausibility of the Mini-Bullet scenario. Prior to this work, in our previous works (Paper I; Paper II), it has not been demonstrated that the product can be diffuse. The size difference in the simulation runs is not negligible; the physical origin of the difference and the factors that affect the sizes of DMDGs are discussed in detail in Section 4.1.1.

Despite the suite of hydrodynamic simulations, TM1, TM2, and TM3, the start from nearly identical initial conditions, the sizes of the product DMDGs are notably different. This means that the DMDG sizes are highly sensitive to various complicated physical processes involved, especially the turbulent behavior of gas driven by stellar feedback processes during the DMDG formation. Since the simulations we perform are limited by the spatial resolution of 5 pc and the simple stellar feedback physics adopted, the goal of realistically modeling the turbulent gas emerging from cloud scale and the influence of the tidal field on the distribution of gas and stars at the same time is extremely hard to achieve. Therefore, drawing a definitive conclusion on the sizes of DMDGs in the Mini-Bullet scenario from the simulations presented in this work is challenging.

Instead, we focus on the effect of simple physical processes including the tidal field of the host and the merging of stellar structures long after the initial burst of star formation at t ≲ 0.4 Gyr. In general, the larger and more distant the DMDG from the host, the smaller its size, and vice versa. These results suggest that the primary factor influencing the sizes of the formed DMDGs is their susceptibility to the tidal forces exerted by the host galaxy, which is determined by a combination of the distance from the host and the mass of the DMDG. The post-formation evolution of the size is influenced by processes such as tidal interaction with the host and merging of star clumps. Removal of stars on the outskirts of a DMDG by tidal interaction results in a gradual decrease in size, while merging events are evident in Figure 11 as sudden changes in size. ¹⁴

We have demonstrated so far that simulations starting from initial conditions designed to lead to a satellite galaxy–galaxy collision whose products match the spatial and kinematic properties and stellar masses of the NGC 1052 group UDGs can indeed produce the observed trail of UDGs and their alignment, along with the dark matter deficiency of DF2 and DF4. However, upon more detailed inspection, the simulated UDGs do not exactly reproduce all the observed characteristics of these objects. Indeed, this is not surprising, since some details of the simulation outcome are dependent on numerical limitations and choices, such as spatial and mass resolution and the subgrid prescription for star formation and feedback physics. In this section, we focus on how the simulated outcomes differ from observational results, especially the sizes and metallicities of the stellar components and GCs hosted by the produced galaxies. We also discuss what needs to be taken into account for the comparison beyond our simulations by discussing other previous work on this subject.

4.1.1. Size of DMDGs

4.1.2. Metallicity of DMDGs

4.1.3. Survivability of Globular Clusters in the Host Tidal Field

Ogiya et al. (2022b) argued that it is challenging for the Mini-Bullet scenario to explain both the number and the spatial extent of massive GCs in DF2. Considering dynamical friction, the GC distribution should have been more extended just after their formation than it is now. Thus, the GCs are more susceptible to the strong tidal force from NGC 1052, demanding a larger number of GCs at the time of the formation of DF2, meaning that most of the stellar mass in DF2 should have been in GCs, which seems unlikely.

This argument relies on the assumption that the location of the Mini-Bullet collision was inside the virial radius of NGC 1052, however, in order to have sufficient strength of the tidal force. In fact, in the constrained initial conditions designed here to lead to a collision whose end-result matches the current positions and velocities of DF2 and DF4 after their orbital evolution for ∼8 Gyr, this collision should have taken place much farther out, at ∼2 × the virial radius of NGC 1052 (see Section 2.2 and Figure 4). Otherwise, if the collision point is too close to NGC 1052, the alignment of DMDGs does not occur, because product DMDGs close to the host will not be part of the sequence of DMDGs: they will be accelerated by gravity and end up on the other side of the host. Combining these, we claim that in the case of the NGC 1052 group system, the Mini-Bullet collision should have occurred far from the host galaxy, enabling both the formation of a trail of DMDGs and extended GC distribution in the most massive DMDGs.

We support this claim with a further quantitative analysis. Figure 12 displays the tidal radii R_tidal, stellar masses M_⋆, and the distances of the product DMDGs from the host at t = 2 Gyr, r_{host,t=2 Gyr}, for each simulation run TM1, TM2, and TM3. As a result of being farther from the host, the tidal radii R_tidal of the simulated DF2 and DF4 candidates are larger than 20 kpc, which is >2 times the current spatial extent of the DF2 GCs and large enough to retain extended GCs.

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Quantifying the probability of the Mini-Bullet event is important, as such an event is expected to be rare, considering the high v_col and the small ${r}_{\min }$ of the collision. Moreover, as discussed in Section 4.1.3, the collision should occur outside of the virial radius of the host halo, possibly reducing the likelihood of it making an occurrence even further. To address this, we now use a large-volume cosmological simulation of galaxy and large-scale structure formation in a ΛCDM universe to study the frequency of such Mini-Bullet collisions in what follows. ¹⁵

As discussed in the previous sections, in the Mini-Bullet scenario, two low-mass satellite galaxies with M₂₀₀ ∼ 10^9–10 M_⊙ collide with v_col ∼ 500 km s⁻¹ and ${r}_{\min }\lesssim {R}_{{\rm{s}},\mathrm{gas}}$. This condition is expected to be rare in the Universe, considering that the progenitor satellite galaxies collide outside the host galaxy virial radius (Figure 4) with a relative velocity well in excess of that host's virial velocity, but with such a small impact parameter that their centers approach within a small fraction of their own virial radii. Moreover, as described in Section 2.1, when one of the colliding satellites is unbound to the system and the other satellite is bound to the system, the likelihood of the collision is lower. Therefore, quantifying the statistical likelihood of the Mini-Bullet scenario is essential to study how plausible the scenario is.

Toward this end, we investigate the frequency of an occurrence of these satellite–satellite galaxy collision events in a large-volume cosmological N-body/hydrodynamics simulation of galaxy formation in ΛCDM, in a box 100 cMpc on a side, TNG100-1, from the simulation suite known collectively as IllustrisTNG (Pillepich et al. 2018a, 2018b; Marinacci et al. 2018; Naiman et al. 2018; Nelson et al. 2018; Springel et al. 2018; Nelson et al. 2019). IllustrisTNG is the recent set of cosmological simulations by the Illustris project, based on the moving-mesh code arepo (Springel 2010). The cosmological parameters adopted in the IllustrisTNG are from the Planck 2015 results: Ω_Λ,0 = 0.6911, Ω_m,0 = 0.3089, Ω_b,0 = 0.0486, σ₈ = 0.8159, n_s = 0.9667, and h = 0.6774 (Planck Collaboration et al. 2016).

TNG100-1, which has the highest resolution of the TNG100 simulations, is composed of the 1820³ baryon particles with 1.4 × 10⁶ M_⊙ and 1820³ dark matter particles with 7.5 × 10⁶ M_⊙. The particle-level data are stored in 100 snapshots from z = 20.05 to z = 0. In the halo catalogs, the halos identified with the friends-of-friends (FoF) algorithm (Davis et al. 1985) and the subhalos identified with the subfind algorithm (Springel et al. 2001; Dolag et al. 2009) are stored. The IllustrisTNG team also provides merger tree data generated by the sublink algorithm (Rodriguez-Gomez et al. 2015), which traces the descendants and progenitors of the subhalos.

In Paper I, we searched for collision-induced DMDGs directly in TNG100-1 but noted that the numerical resolution of TNG100-1 was insufficient to form DMDGs. To see DMDGs form, the simulation would have required higher mass and length resolution in both the dark matter and baryonic gas, in order to follow the shock compression of gas leading to star formation and then track the motion of these stars after they formed. Thus, in this section, we take a different approach with the aim to find not the collision-induced DMDGs themselves, but rather the number of pre-collision galaxy pairs in TNG100-1 that meet the conditions established here for being possible progenitors of the Mini-Bullet satellite–satellite galaxy collisions that form DMDGs. A similar analysis was also performed in Paper I, with criteria applied to the two-galaxy pairs found in TNG100-1. By contrast with our previous study (Paper I), however, we focus here on three-body host-satellite–satellite systems instead, to show that they can produce the aligned galaxies of the NGC 1052 system (van Dokkum et al. 2022a). This required us to examine the galaxy collisions in more detail by studying the location of the collision point and the gravitational boundedness of the colliding galaxies.

The exact search criteria are as follows. Conditions (i)–(iv) given below are applied to find the collision events that could have formed DMDGs and are based on the conditions we found to be capable of producing collision-induced DMDGs in Section 2.1 and our previous study (Paper I).

• (i)  
  The colliding galaxies collide with relative velocity v_col > 300 km s⁻¹,
• (ii)  
  the initial gas fraction of the colliding galaxies, f_gas = M_gas/M₂₀₀, is greater than 0.05,
• (iii)  
  the initial mass ratio of the colliding galaxies satisfies 1/3 < M₁/M₂ < 3, and
• (iv)  
  the initial gas mass ratio satisfies 1/3 < M_1,gas/M_2,gas < 3.

The following conditions (v) and (vi) are added to find collision events similar to the collision pairs that are studied in Section 3.1.

• (v)  
  The total mass of each colliding progenitor galaxy, M₁ and M₂, satisfies 10⁹ M_⊙ < M₁, M₂ < 10¹¹ M_⊙, and
• (vi)  
  the collision is associated with a host halo with a total mass M_200,host > 10¹¹ M_⊙.

The following conditions are newly applied in this paper.

• (vii)  
  The distance of the collision from the host halo is R > R₂₀₀ or R > R₅₀₀. The Mini-Bullet satellite–satellite galaxy collision likely to produce the NGC 1052 system should occur far from the host galaxy.
• (viii)  
  ${r}_{\min }\lt ({R}_{\mathrm{eff},1}+{R}_{\mathrm{eff},2})/2$, where R_eff,1 and R_eff,2 are the half-mass radii of the colliding galaxies. To complement the temporal resolution of the TNG100-1 simulation, ${r}_{\min }$ are calculated using the Rebound orbit integration code. Starting from the last snapshot before the collision, we take halos that are within 500 kpc from the colliding pair and have a mass of M₂₀₀ > 10⁹ M_⊙, calculating the halo orbits with the assumption that they are point masses.
• (ix)  
  We select pairs in which at least one of the colliding galaxies is gravitationally unbound to the host halo, meaning the mechanical energy E_mec = E_K + E_P > 0, where E_K is the kinetic energy and E_P the gravitational potential energy. E_P is calculated assuming that the host M_200,host is at the center of mass of the FoF halo.

Figure 13 displays the distribution of the collision pairs that satisfy the search criteria (i)–(viii) in the parameter space of v_col and ${r}_{\min }$ occurring outside of R₂₀₀ or R₅₀₀ of the host halo. With all the search criteria (i)–(ix) applied to 95 halo catalogs from z = 3 (10) to 0.01, we find 11 (36) collision pairs outside of R₂₀₀ of the host halo that are unbound to the center of mass of the host halo. ¹⁶ z = 3 corresponds to the lookback time of 11.6 Gyr, which roughly corresponds to the observed stellar age of DF2 and DF4 +2 × (measurement error). On the other hand, z = 10 is employed as a starting point of the search range z = 10–0.01 to exclude too frequent galaxy collisions at the early epoch of galaxy formation z > 10. The other numbers for less strict conditions are summarized in Table 5. In conclusion, at z < 3, in the simulated ∼(100 Mpc)³-sized universe of TNG100-1, we attest that there are ∼10 Mini-Bullet satellite–satellite galaxy collisions that are expected to produce a sequence of DMDGs even with the most conservative criteria. Since these numbers are subject to the accuracy of the orbit integration, we examine and discuss this in the Appendix.

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Table 5. The Number of Colliding Satellite–Satellite Galaxy Pairs from z = 3 to 0.01 Found in TNG100-1 that Satisfy the Search Criteria

Search Criteria	Outside of R200	Outside of R500
(1)	(2)	(3)
(i)–(vii), ${r}_{\min }\lt 15\,\mathrm{kpc}$	115 (1166)	205 (1804)
(i)–(viii)	35 (172)	53 (257)
(i)–(ix)	11 (36)	14 (44)

Note. The numbers inside parentheses are the number of pairs from z = 10 to 0.01. See Section 4.2 for details. (1) Search criteria described in Section 4.2, (2) the number of collision pairs that collide outside of R₂₀₀ of the host halo, (3) the number of collision pairs that collide outside of R₅₀₀ of the host halo.

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