Gravitational Softening Research Articles

How gravitational softening affects galaxy stability – I. Linear mode analysis of disc galaxies

Linear perturbation is used to investigate the effect of gravitational softening on the retrieved two-armed spiral eigenmodes of razor-thin stellar discs. We explore four softening kernels with different degrees of gravity bias, and with/without compact support (compact in the sense that they yield exactly Newtonian forces outside the softening kernel). These kernels are applied to two disc galaxy models with well-known unsoftened unstable modes. We illustrate quantitatively the importance of a vanishing linear gravity bias to yield accurate frequency estimates of the unstable modes. As such, Plummer softening, while very popular amongst simulators, performs poorly in our tests. The best results, with excellent agreement between the softened and unsoftened mode properties, are obtained with softening kernels that have a reduced gravity bias, obtained by compensating for the sub-Newtonian forces at small interparticle distances with slightly super-Newtonian forces at radii near the softening length. We present examples of such kernels that, moreover, are analytically simple and computationally cheap. Finally, these results light the way to the construction of softening methods with even smaller gravity bias, although at the price of increasingly complex kernels.

FIRE-2 simulations: physics versus numerics in galaxy formation

The Feedback In Realistic Environments (FIRE) project explores feedback in cosmological galaxy formation simulations. Previous FIRE simulations used an identical source code (FIRE-1) for consistency. Motivated by the development of more accurate numerics - including hydrodynamic solvers, gravitational softening, and supernova coupling algorithms - and exploration of new physics (e.g. magnetic fields), we introduce FIRE-2, an updated numerical implementation of FIRE physics for the GIZMO code. We run a suite of simulations and compare against FIRE-1: overall, FIRE-2 improvements do not qualitatively change galaxy-scale properties. We pursue an extensive study of numerics versus physics. Details of the star-formation algorithm, cooling physics, and chemistry have weak effects, provided that we include metal-line cooling and star formation occurs at higher-than-mean densities. We present new resolution criteria for high-resolution galaxy simulations. Most galaxy-scale properties are robust to numerics we test, provided: (1) Toomre masses are resolved; (2) feedback coupling ensures conservation, and (3) individual supernovae are time-resolved. Stellar masses and profiles are most robust to resolution, followed by metal abundances and morphologies, followed by properties of winds and circum-galactic media (CGM). Central (~kpc) mass concentrations in massive (L*) galaxies are sensitive to numerics (via trapping/recycling of winds in hot halos). Multiple feedback mechanisms play key roles: supernovae regulate stellar masses/winds; stellar mass-loss fuels late star formation; radiative feedback suppresses accretion onto dwarfs and instantaneous star formation in disks. We provide all initial conditions and numerical algorithms used.

The Cluster-EAGLE project: velocity bias and the velocity dispersion–mass relation of cluster galaxies

We use the Cluster-EAGLE simulations to explore the velocity bias introduced when using galaxies, rather than dark matter particles, to estimate the velocity dispersion of a galaxy cluster, a property known to be tightly correlated with cluster mass. The simulations consist of 30 clusters spanning a mass range $14.0 \le \log_{10}(M_{\rm 200c}/\mathrm{M_\odot}) \le 15.4$, with their sophisticated sub-grid physics modelling and high numerical resolution (sub-kpc gravitational softening) making them ideal for this purpose. We find that selecting galaxies by their total mass results in a velocity dispersion that is 5-10 per cent higher than the dark matter particles. However, selecting galaxies by their stellar mass results in an almost unbiased ($<5$ per cent) estimator of the velocity dispersion. This result holds out to $z=1.5$ and is relatively insensitive to the choice of cluster aperture, varying by less than 5 per cent between $r_{\rm 500c}$ and $r_{\rm 200m}$. We show that the velocity bias is a function of the time spent by a galaxy inside the cluster environment. Selecting galaxies by their total mass results in a larger bias because a larger fraction of objects have only recently entered the cluster and these have a velocity bias above unity. Galaxies that entered more than $4 \, \mathrm{Gyr}$ ago become progressively colder with time, as expected from dynamical friction. We conclude that velocity bias should not be a major issue when estimating cluster masses from kinematic methods.

The vertical structure of gaseous galaxy discs in cold dark matter haloes

AbstractWe study the vertical structure of polytropic centrifugally supported gaseous discs embedded in cold dark matter (CDM) haloes. At fixed radius, R, the shape of the vertical density profile depends weakly on whether the disc is self-gravitating (SG) or non-self-gravitating (NSG). The disc ‘characteristic’ thickness, zH, set by the midplane sound speed and circular velocity, zNSG = (cs/Vc)R, in the NSG case, and by the sound speed and surface density, $z_{\rm SG} = c_{\rm s}^2/G\Sigma$, in SG discs, is smaller than zSG and zNSG. SG discs are typically Toomre unstable, NSG discs are stable. Exponential discs in CDM haloes with roughly flat circular velocity curves ‘flare’ outwards. Flares in mono abundance or coeval populations in galaxies like the Milky Way are thus not necessarily due to radial migration. For the polytropic equation of state of the Evolution and Assembly of GaLaxies and their Environments (EAGLE) simulations, discs that match observational constraints are NSG for Md &amp;lt; 3 × 109 M⊙ and SG at higher masses, if fully gaseous. We test these analytic results using a set of idealized smoothed particle hydrodynamic simulations and find excellent agreement. Our results clarify the role of the gravitational softening on the thickness of simulated discs, and on the onset of radial instabilities. EAGLE low-mass discs are NSG so the softening plays no role in their vertical structure. High-mass discs are expected to be SG and unstable, and may be artificially thickened and stabilized unless gravity is well resolved. Simulations with spatial resolution high enough to not compromise the vertical structure of a disc also resolve the onset of their instabilities, but the converse is not true.

Modeling circumstellar disc fragmentation and episodic protostellar accretion with smoothed particle hydrodynamics in cell

We discuss the ability of the smoothed particle hydrodynamics (SPH) method combined with a grid-based solver for the Poisson equation to model mass accretion onto protostars in gravitationally unstable protostellar discs. We scrutinize important features of coupling the SPH with grid-based solvers and numerical issues associated with (1) large number of SPH neighbours and (2) relation between gravitational softening and hydrodynamic smoothing length. We report results of our simulations of razor-thin disc prone to fragmentation and demonstrate that the algorithm being simple and homogeneous captures the target physical processes - disc gravitational fragmentation and accretion of gas onto the protostar caused by inward migration of dense clumps. In particular, we obtain two types of accretion bursts: a short-duration one caused by a quick inward migration of the clump, previously reported in the literature, and the prolonged one caused by the clump lingering at radial distances on the order of 15-25~au. The latter is culminated with a sharp accretion surge caused by the clump ultimately falling on the protostar.

Spurious haloes and discreteness-driven relaxation in cosmological simulations

There is strong evidence that cosmological N-body simulations dominated by Warm Dark Matter (WDM) contain spurious or unphysical haloes, most readily apparent as regularly spaced low-mass haloes strung along filaments. We show that spurious haloes are a feature of traditional N-body simulations of cosmological structure formation models, including WDM and Cold Dark Matter (CDM) models, in which gravitational collapse proceeds in an initially anisotropic fashion, and arises naturally as a consequence of discreteness-driven relaxation. We demonstrate this using controlled N-body simulations of plane-symmetric collapse and show that spurious haloes are seeded at shell crossing by localised velocity perturbations induced by the discrete nature of the density field, and that their characteristic separation should be approximately the mean inter-particle separation of the N-body simulation, which is fixed by the mass resolution within the volume. Using cosmological N-body simulations in which particles are split into two collisionless components of fixed mass ratio, we find that the spatial distribution of the two components show signatures of discreteness-driven relaxation in their spatial distribution on both large and small scales. Adopting a spline kernel gravitational softening that is of order the comoving mean inter-particle separation helps to suppress the effect of discreteness-driven relaxation, but cannot eliminate it completely. These results provide further motivation for recent developments of new algorithms, which include, for example, revisions of the traditional N-body approach by means of spatially adaptive anistropric gravitational softenings or explicit solutions for the evolution of dark matter in phase space.

The offsets between galaxies and their dark matter in Λ cold dark matter

Abstract We use the ‘Evolution and Assembly of GaLaxies and their Environments’ (eagle) suite of hydrodynamical cosmological simulations to measure offsets between the centres of stellar and dark matter components of galaxies. We find that the vast majority (&amp;gt;95 per cent) of the simulated galaxies display an offset smaller than the gravitational softening length of the simulations (Plummer-equivalent ϵ = 700 pc), both for field galaxies and satellites in clusters and groups. We also find no systematic trailing or leading of the dark matter along a galaxy's direction of motion. The offsets are consistent with being randomly drawn from a Maxwellian distribution with σ ≤ 196 pc. Since astrophysical effects produce no feasible analogues for the $1.62^{+0.47}_{-0.49}$ kpc offset recently observed in Abell 3827, the observational result is in tension with the collisionless cold dark matter model assumed in our simulations.

Cosmological simulations of galaxy clusters with feedback from active galactic nuclei: profiles and scaling relations

We present results from a new set of 30 cosmological simulations of galaxy clusters, including the effects of radiative cooling, star formation, supernova feedback, black hole growth and AGN feedback. We first demonstrate that our AGN model is capable of reproducing the observed cluster pressure profile at redshift, z~0, once the AGN heating temperature of the targeted particles is made to scale with the final virial temperature of the halo. This allows the ejected gas to reach larger radii in higher-mass clusters than would be possible had a fixed heating temperature been used. Such a model also successfully reduces the star formation rate in brightest cluster galaxies and broadly reproduces a number of other observational properties at low redshift, including baryon, gas and star fractions; entropy profiles outside the core; and the X-ray luminosity-mass relation. Our results are consistent with the notion that the excess entropy is generated via selective removal of the densest material through radiative cooling; supernova and AGN feedback largely serve as regulation mechanisms, moving heated gas out of galaxies and away from cluster cores. However, our simulations fail to address a number of serious issues; for example, they are incapable of reproducing the shape and diversity of the observed entropy profiles within the core region. We also show that the stellar and black hole masses are sensitive to numerical resolution, particularly the gravitational softening length; a smaller value leads to more efficient black hole growth at early times and a smaller central galaxy.

Dwarf galaxies in CDM and SIDM with baryons: observational probes of the nature of dark matter

We present the first cosmological simulations of dwarf galaxies, which include dark matter self-interactions and baryons. We study two dwarf galaxies within cold dark matter, and four different elastic self-interacting scenarios with constant and velocity-dependent cross-sections, motivated by a new force in the hidden dark matter sector. Our highest resolution simulation has a baryonic mass resolution of 1.8 × 102 M⊙ and a gravitational softening length of 34 pc at z = 0. In this first study we focus on the regime of mostly isolated dwarf galaxies with halo masses ∼ 1010 M⊙ where dark matter dynamically dominates even at sub-kpc scales. We find that while the global properties of galaxies of this scale are minimally affected by allowed self-interactions, their internal structures change significantly if the cross-section is large enough within the inner sub-kpc region. In these dark-matter-dominated systems, self-scattering ties the shape of the stellar distribution to that of the dark matter distribution. In particular, we find that the stellar core radius is closely related to the dark matter core radius generated by self-interactions. Dark matter collisions lead to dwarf galaxies with larger stellar cores and smaller stellar central densities compared to the cold dark matter case. The central metallicity within 1 kpc is also larger by up to ∼15 per cent in the former case. We conclude that the mass distribution and characteristics of the central stars in dwarf galaxies can potentially be used to probe the self-interacting nature of dark matter.

Exploring the non-linear density field in the Millennium Simulations with tessellations – I. The probability distribution function

We use the Delaunay Tessellation Field Estimator (DTFE) to study the one-point density distribution functions of the Millennium (MS) and Millennium-II (MS-II) simulations. The DTFE technique is based directly on the particle positions, without requiring any type of smoothing or analysis grid, thereby providing high sensitivity to all non-linear structures resolved by the simulations. In order to identify the detailed origin of the shape of the one-point density probability distribution function (PDF), we decompose the simulation particles according to the mass of their host FoF halos, and examine the contributions of different halo mass ranges to the global density PDF. We model the one-point distribution of the FoF halos in each halo mass bin with a set of Monte Carlo realizations of idealized NFW dark matter halos, finding that this reproduces the measurements from the N-body simulations reasonably well, except for a small excess present in simulation results. This excess increases with increasing halo mass. We show that its origin lies in substructure, which becomes progressively more abundant and better resolved in more massive dark matter halos. We demonstrate that the high density tail of the one-point distribution function in less massive halos is severely affected by the gravitational softening length and the mass resolution. In particular, we find these two parameters to be more important for an accurate measurement of the density PDF than the simulated volume. Combining our results from individual halo mass bins we find that the part of the one-point density PDF originating from collapsed halos can nevertheless be quite well described by a simple superposition of a set of NFW halos with the expected cosmological abundance over the resolved mass range. The transition region to the low-density unbound material is however not well captured by such an analytic halo model.

Dynamics of core accretion

(shortened) We perform 3D hydrodynamic simulations of gas flowing around a planetary core of mass \mplan=10\me embedded in a near Keplerian background flow, using a modified shearing box approximation. We employ a nested grid hydrodynamic code with as many as six nested grids, providing spatial resolution on the finest grid comparable to the present day diameters of Neptune and Uranus. We find that a strongly dynamically active flow develops such that no static envelope can form. The activity is not sensitive to plausible variations in the rotation curve of the underlying disk. It is sensitive to the thermodynamic treatment of the gas, as modeled by prescribed equations of state (either `locally isothermal' or `locally isentropic') and the temperature of the background disk material. The activity is also sensitive to the shape and depth of the core's gravitational potential, through its mass and gravitational softening coefficient. The varying flow pattern gives rise to large, irregular eruptions of matter from the region around the core which return matter to the background flow: mass in the envelope at one time may not be found in the envelope at any later time. The angular momentum of material in the envelope, relative to the core, varies both in magnitude and in sign on time scales of days to months near the core and on time scales a few years at distances comparable to the Hill radius. We show that material entering the dynamically active environment may suffer intense heating and cooling events the durations of which are as short as a few hours to a few days. Peak temperatures in these events range from $T \sim 1000$ K to as high as $T \sim 3-4000$ K, with densities $\rho\sim 10^{-9}-10^{-8}$ g/cm$^3$. These time scales, densities and temperatures span a range consistent with those required for chondrule formation in the nebular shock model.

Gravitational softening as a smoothing operation

In self-consistent N-body simulations of collisionless systems, gravitational interactions are modified on small scales to remove singularities and simplify the task of numerically integrating the equations of motion. This `gravitational softening' is sometimes presented as an ad-hoc departure from Newtonian gravity. However, softening can also be described as a smoothing operation applied to the mass distribution; the gravitational potential and the smoothed density obey Poisson's equation precisely. While `softening' and `smoothing' are mathematically equivalent descriptions, the latter has some advantages. For example, the smoothing description suggests a way to set up N-body initial conditions in almost perfect dynamical equilibrium.

Planet-disc interaction in highly inclined systems

We study the interaction of a protoplanetary disc with a planet on a highly inclined orbit in the linear regime. The evolution of the planet is dominated by dynamical friction for planet masses above several Earth masses. Smaller planets are dominated by aerodynamic drag, especially for very high inclinations and retrograde orbits. The time-scales associated with migration and inclination damping are calculated. For certain values of the inclination, the inclination damping time-scale is longer than the migration time-scale and the disc lifetime. This result shows that highly inclined planets cannot (re-)align with the protoplanetary disc. We discuss the dependence of numerical simulations on the gravitational softening parameter. We find only a logarithmic dependence, making global three-dimensional simulations of this process computationally feasible. A large fraction of hot Jupiters is on highly inclined orbits with respect to the rotation axis of the star. On the other hand, small mass planetary systems discovered by the Kepler mission have low mutual inclinations. This shows that there are two distinct formation mechanisms at work. The process that creates inclined hot Jupiters does not operate on small mass planets because the damping time-scales are so long that these systems would still be inclined today.

Adaptive gravitational softening in gadget

Cosmological simulations of structure formation follow the collisionless evolution of dark matter starting from a nearly homogeneous field at early times down to the highly clustered configuration at redshift zero. The density field is sampled by a number of particles in number infinitely smaller than those believed to be its actual components and this limits the mass and spatial scales over which we can trust the results of a simulation. Softening of the gravitational force is introduced in collisionless simulations to limit the importance of close encounters between these particles. The scale of softening is generally fixed and chosen as a compromise between the need for high spatial resolution and the need to limit the particle noise. In the scenario of cosmological simulations, where the density field evolves to a highly inhomogeneous state, this compromise results in an appropriate choice only for a certain class of objects, the others being subject to either a biased or a noisy dynamical description. We have implemented adaptive gravitational softening lengths in the cosmological simulation code GADGET; the formalism allows the softening scale to vary in space and time according to the density of the environment, at the price of modifying the equation of motion for the particles in order to be consistent with the new dependencies introduced in the system's Lagrangian. We have applied the technique to a number of test cases and to a set of cosmological simulations of structure formation. We conclude that the use of adaptive softening enhances the clustering of particles at small scales, a result visible in the amplitude of the correlation function and in the inner profile of massive objects, thereby anticipating the results expected from much higher resolution simulations.

A natural symmetrization for the plummer potential

We propose a symmetrized form of the softened gravitational potential which is a natural extension of the Plummer potential. The gravitational potential at the position of particle i( x i , y i , z i ), induced by particle j at ( x j , y j , z j ), is given by: ϕ ij = - Gm j | r ij 2 + ϵ i 2 + ϵ j 2 | 1 / 2 , where G is the gravitational constant, m j is the mass of particle j, r ij = ∣( x i − x j ) 2 + ( y i − y j ) 2 + ( z i − z j ) 2∣ 1/2 and ϵ i and ϵ j are the gravitational softening lengths of particles i and j, respectively. This form is formally an extension of the Newtonian potential to five dimensions. The derivative of this equation in the x, y, and z directions correspond to the gravitational accelerations in these directions and they are always symmetric between two particles. When one applies this potential to a group of particles with different softening lengths, as in the case with a tree code, an averaged gravitational softening length for the group can be used. We find that the most suitable averaged softening length for a group of particles is 〈 ϵ j 2 〉 = ∑ j N m j ϵ j 2 / M , where M = ∑ j N m j and N are the mass and number of all particles in the group, respectively. The leading error related to the softening length is O ∑ j δ r j δ ϵ j 2 / r ij 3 , where δ r j is the distance between particle j and the center of mass of the group and δ ϵ j 2 = ϵ j 2 - 〈 ϵ j 2 〉 . Using this averaged gravitational softening length with the tree method, one can use a single tree to evaluate the gravitational forces for a system of particles with a wide variety of gravitational softening lengths. Consequently, this will reduce the calculation cost of the gravitational force for such a system with different softenings without the need for complicated forms of softening. We present the result of simple numerical tests. We found that our modification of the Plummer potential works well.

Global mass segregation in hydrodynamical simulations of star formation

Recent analyses of mass segregation diagnostics in star forming regions invite a comparison with the output of hydrodynamic simulations of star formation. In this work we investigate the state of mass segregation of 'stars' (i.e. sink particles in the simulations) in the case of hydrodynamical simulations which omit feedback. We first discuss methods to quantify mass segregation in substructured regions, either based on the minimum spanning tree (Allison's Lambda), or through analysis of correlations between stellar mass and local stellar surface number densities. We find that the presence of even a single 'outlier' (i.e. a massive object far from other stars) can cause the Allison Lambda method to describe the system as inversely mass segregated, even where in reality the most massive sink particles are overwhelmingly in the centres of the subclusters. We demonstrate that a variant of the Lambda method is less susceptible to this tendency but also argue for an alternative representation of the data in the plane of stellar mass versus local surface number density. The hydrodynamical simulations show global mass segregation from very early times which continues throughout the simulation, being only mildly influenced during sub-cluster merging. We find that up to approx. 2-3% of the "massive" sink particles (m > 2.5 Msun) are in relative isolation because they have formed there, although other sink particles can form later in their vicinity. Ejections of massive sinks from subclusters do not contribute to the number of isolated massive sink particles, as the gravitational softening in the calculation suppresses this process.

Edge modes in self-gravitating disc-planet interactions

We study the stability of gaps opened by a giant planet in a self-gravitating protoplanetary disc. We find a linear instability associated with both the self-gravity of the disc and local vortensity maxima which coincide with gap edges. For our models, these edge modes develop and extend to twice the orbital radius of a Saturn mass planet in discs with disc-to-star mass ratio >0.06, corresponding to a Toomre Q < 1.5 at the outer disc boundary. Unlike the local vortex-forming instabilities associated with gap edges in weakly or non-self-gravitating low viscosity discs, the edge modes are global and exist only in sufficiently massive discs, but for the typical viscosity values adopted for protoplanetary discs. Analytic modelling and linear calculations show edge modes may be interpreted as a localised disturbance associated with a gap edge inducing activity in the extended disc, through the launching of density waves excited at Lindblad resonances. Nonlinear hydrodynamic simulations are performed to investigate the evolution of edge modes in disc-planet systems. The form and growth rates of unstable modes are consistent with linear theory. Their dependence on viscosity and gravitational softening is also explored. We also performed a first study of the effect of edge modes on planetary migration. We found that if edge modes develop, then the average disc-on-planet torque becomes more positive with increasing disc mass. In simulations where the planet was allowed to migrate, although a fast type III migration could be seen that was similar to that seen in non-self-gravitating discs, we found that it was possible for the planet to interact gravitationally with the spiral arms associated with an edge mode and that this could result in the planet being scattered outwards. Thus orbital migration is likely to be complex and non monotonic in massive discs of the type we consider.

DYNAMICAL MODELING OF THE MERGER OF THE SAGITTARIUS DWARF-MILKY WAY SYSTEM

In hierarchical models of large-scale structure formation, most galaxies grow in clusters, where merging is an essential, if not dominant process. Understanding the merger process is greatly aided by examples in the vicinity of the Milky Way due to the availability of high quality positional and kinematic data. The Sagittarius Dwarf Galaxy at a distance of 25 kpc is a prime example. Detailed observations of stars in this dwarf galaxy and its associated tidal streams also provide a unique probe of the gravitational field of the Milky Way. To fully utilize this probe requires not only excellent observations but also accurate models of the merging event. We investigate the recent history of the Sagittarius Dwarf with the smoothed particle hydrodynamics/N-body GADGET-2 code, treating both the Milky Way and the dwarf galaxy as a particle system. In addition to considerations of particle number, gravitational softening, stability criteria such as Toomre Q parameter, and angular momentum transport, we add the distinction between static and dynamic potential differences as another aspect that deserves consideration. We find that the commonly employed approximation of a static Milky Way potential induces changes in the orbit corresponding to position changes on the sky of order 0.5°-1°. The weak dependence of the results on the live versus static Milky Way model justify the use of static potentials in situations which do not require very high positional accuracy.

STAR FORMATION AND FEEDBACK IN SMOOTHED PARTICLE HYDRODYNAMIC SIMULATIONS. II. RESOLUTION EFFECTS

We examine the effect of mass and force resolution on a specific star formation (SF) recipe using a set of N-body/Smooth Particle Hydrodynamic simulations of isolated galaxies. Our simulations span halo masses from 10^9 to 10^13 solar masses, more than four orders of magnitude in mass resolution, and two orders of magnitude in the gravitational softening length, epsilon, representing the force resolution. We examine the total global star formation rate, the star formation history, and the quantity of stellar feedback and compare the disk structure of the galaxies. Based on our analysis, we recommend using at least 10^4 particles each for the dark matter and gas component and a force resolution of epsilon approximately equal to 10^-3 R_vir when studying global SF and feedback. When the spatial distribution of stars is important, the number of gas and dark matter particles must be increased to at least 10^5 of each. Low mass resolution simulations with fixed softening lengths show particularly weak stellar disks due to two-body heating. While decreasing spatial resolution in low mass resolution simulations limits two-body effects, density and potential gradients cannot be sustained. Regardless of the softening, low-mass resolution simulations contain fewer high density regions where SF may occur. Galaxies of approximately 10^10 solar masses display unique sensitivity to both mass and force resolution. This mass of galaxy has a shallow potential and is on the verge of forming a disk. The combination of these factors give this galaxy the potential for strong gas outflows driven by supernova feedback and make it particularly sensitive to any changes to the simulation parameters.

Shock-induced star cluster formation in colliding galaxies

AbstractWe studied the formation process of star clusters using high-resolutionN-body/smoothed particle hydrodynamics simulations of colliding galaxies. The total number of particles is 1.2×108for our high resolution run. The gravitational softening is 5 pc and we allow gas to cool down to ~10 K. During the first encounter of the collision, a giant filament consists of cold and dense gas found between the progenitors by shock compression. A vigorous starburst took place in the filament, resulting in the formation of star clusters. The mass of these star clusters ranges from 105−8M⊙. These star clusters formed hierarchically: at first small star clusters formed, and then they merged via gravity, resulting in larger star clusters.