Self-consistent N-body Simulations Research Articles

Self-consistent N-body simulation of planetesimal-driven migration. I. The trajectories of single planets in the uniform background

Abstract Recent exoplanet observations have revealed a diversity of exoplanetary systems, which suggests the ubiquity of radial planetary migration. One powerful known mechanism of planetary migration is planetesimal-driven migration (PDM), which can let planets undergo significant migration through gravitational scattering with planetesimals. In this series of papers, we present the results of our high-resolution, self-consistent N-body simulations of PDM, in which gravitational interactions among planetesimals, the gas drag, and Type I migration are all taken into account. In this first paper (Paper I), we investigate the migration of a single planet through PDM within the framework of the classical standard disk model (the minimum-mass solar nebula model). Paper I aims to improve our understanding of planetary migration through PDM, addressing previously unexplored aspects of both the gravitational interactions among planetesimals and the interactions with disk gas. Our results show that even small protoplanets can actively migrate through PDM. Such active migration can act as a rapid radial diffusion mechanism for protoplanets and significantly influence the early stages of planetary formation (i.e., during the runaway growth phase). Moreover, a fair fraction of planets migrate outward. This outward migration may offer a potential solution for the “planet migration problem” caused by Type I migration and gives a natural mechanism for outward migration assumed in many recent scenarios for the formation of outer planets.

Impact of dynamical friction on the tidal formation of NGC 1052-DF2

ABSTRACT The formation of dark matter-deficient galaxies (DMDGs) through tidal interactions has been a subject of growing interest, particularly with the discovery of galaxies such as NGC 1052-DF2. Previous studies suggested that strong tidal forces could strip dark matter (DM) from satellite galaxies, but the role of dynamical friction in this process has been largely overlooked. In this paper, we present self-consistent N-body simulations that incorporate the effects of dynamical friction on the tidal formation of NGC 1052-DF2, and compare them with the one without dynamical friction. We find that dynamical friction significantly accelerates the decay of the satellite galaxy’s orbit, causing it to experience more frequent tidal stripping and leading to the earlier formation of a DM-deficient state, approximately $7\!-\!8$ Gyr after infall. This is a few Gyr earlier than simulations without dynamical friction. Our results suggest that DMDGs can form in a wider range of orbital configurations, particularly on more circular orbits, than previously thought. Furthermore, we find that globular clusters in the DM-deficient phase exhibit elevated velocity dispersion, providing an observational signature of this evolutionary stage. We also examine the evolution of satellite in the phase space of total energy versus angular momentum, and show that a vertically narrow feature in this phase space is a clear signature of pericentre passage. These findings broaden the understanding of how DMDGs form and highlight the critical role of dynamical friction in shaping the evolutionary history of satellite galaxies in massive haloes.

The echo of the bar buckling: Phase-space spirals in Gaia Data Release 2

We present a high-resolution numerical study of the phase-space diversity in an isolated Milky Way-type galaxy. Using a single N-body simulation (N ≈ 0.14 × 109) we explore the formation, evolution, and spatial variation of the phase-space spirals similar to those recently discovered by Antoja et al. in the Milky Way disk with Gaia Data Release 2 (DR2). For the first time in the literature we use a self-consistent N-body simulation of an isolated Milky Way-type galaxy to show that the phase-space spirals develop naturally from vertical oscillations driven by the buckling of the stellar bar. Thus, we claim that the physical mechanism standing behind the observed incomplete phase-space mixing process can be internal and not necessarily due to the perturbation induced by a massive satellite. In our model, the bending oscillations propagate outward and produce axisymmetric variations of the mean vertical coordinate and vertical velocity component of about 100 − 200 pc and 1 − 2 km s−1, respectively. As a consequence, the phase-space wrapping results in the formation of patterns with various morphologies across the disk, depending on the bar orientation, distance to the galactic center, and time elapsed since the bar buckling. Once bending waves appear, they are supported for a long time via disk self-gravity. Such vertical oscillations trigger the formation of various time-dependent phase-space spirals in the entire disk. The underlying physical mechanism implies the link between in-plane and vertical motion that leads directly to phase-space structures whose amplitude and shape are in remarkable agreement with those of the phase-space spirals observed in the Milky Way disk. In our isolated galaxy simulation, phase-space spirals are still distinguishable at the solar neighborhood 3 Gyr after the buckling phase. The long-lived character of the phase-space spirals generated by the bar buckling instability cast doubts on the timing argument used so far to get back to the time of the onset of the perturbation: phase-space spirals may have been caused by perturbations originated several gigayearrs ago, and not as recent as suggested so far.

Mergers, tidal interactions, and mass exchange in a population of disc globular clusters

We present the results of a self-consistentN-body simulation following the evolution of a primordial population of thick-disc globular clusters (GCs). We study how the internal properties of such clusters evolve under the action of mutual interactions, while they orbit a Milky Way-like galaxy. For the first time, through analytical and numerical considerations, we find that physical encounters between disc GCs are a crucial factor that contributed to the shape of the current properties of the Galactic GC system. Close passages or motion on similar orbits may indeed have a significant impact on the internal structure of clusters, producing multiple gravitationally bound sub-populations through the exchange of mass and even mergers. Our model produces two major mergers and a few small mass exchanges between pairs of GCs. Two of our GCs accrete stars from two companions, ending up with three internal sub-populations. We propose these early interactions and mergers between thick disc GCs with slightly different initial chemical compositions as a possible explanation for the spreads in metallicity observed in some of the massive Milky Ways GCs.

Formation of Tidally Induced Bars in Galactic Flybys: Prograde versus Retrograde Encounters

Abstract Bars in disk galaxies can be formed by interactions with other systems, including those of comparable mass. It has long been established that the effect of such interactions on galaxy morphology depends strongly on the orbital configuration, in particular the orientation of the intrinsic spin of the galactic disk with respect to its orbital angular momentum. Prograde encounters modify the morphology strongly, including the formation of tidally induced bars, while retrograde flybys should have little effect on morphology. Recent works on the subject reached conflicting conclusions, one using the impulse approximation and claiming no dependence on this angle in the properties of tidal bars. To resolve the controversy, we performed self-consistent N-body simulations of hyperbolic encounters between two identical Milky Way-like galaxies assuming different velocities and impact parameters, with one of the galaxies on a prograde and the other on a retrograde orbit. The galaxies were initially composed of an exponential stellar disk and an NFW dark halo, and they were stable against bar formation in isolation for 3 Gyr. We find that strong tidally induced bars form only in galaxies on prograde orbits. For smaller impact parameters and lower relative velocities, the bars are stronger and have lower pattern speeds. Stronger bars undergo extended periods of buckling instability that thicken their vertical structure. The encounters also lead to the formation of two-armed spirals with strength inversely proportional to the strength of the bars. We conclude that proper modeling of prograde and retrograde encounters cannot rely on the simplest impulse approximation.

Shape of LOSVDs in Barred Disks: Implications for Future IFU Surveys

Abstract The shape of line-of-sight velocity distributions (LOSVDs) carries important information about the internal dynamics of galaxies. The skewness of LOSVDs represents their asymmetric deviation from a Gaussian profile. Correlations between the skewness parameter (h 3) and the mean velocity ( ) of a Gauss–Hermite series reflect the underlying stellar orbital configurations of different morphological components. Using two self-consistent N-body simulations of disk galaxies with different bar strengths, we investigate correlations at different inclination angles. Similar to previous studies, we find anticorrelations in the disk area, and positive correlations in the bar area when viewed edge-on. However, at intermediate inclinations, the outer parts of bars exhibit anticorrelations, while the core areas dominated by the boxy/peanut-shaped (B/PS) bulges still maintain weak positive correlations. When viewed edge-on, particles in the foreground/background disk (the wing region) in the bar area constitute the main velocity peak, whereas the particles in the bar contribute to the high-velocity tail, generating the correlation. If we remove the wing particles, the LOSVDs of the particles in the outer part of the bar only exhibit a low-velocity tail, resulting in a negative correlation, whereas the core areas in the central region still show weakly positive correlations. We discuss implications for IFU observations on bars, and show that the variation of the correlation in the disk galaxy may be used as a kinematic indicator of the bar and the B/PS bulge.

Orthogonal Vertical Velocity Dispersion Distributions Produced by Bars

Abstract In barred galaxies, the contours of stellar velocity dispersions (σ) are generally expected to be oval and aligned with the orientation of bars. However, many double-barred (S2B) galaxies exhibit distinct σ peaks on the minor axis of the inner bar, which we termed “σ-humps,” while two local σ minima are present close to the ends of inner bars, i.e., “σ-hollows.” Analysis of numerical simulations shows that -humps or hollows should play an important role in generating the observed σ-humps+hollows in low-inclination galaxies. In order to systematically investigate the properties of in barred galaxies, we apply the vertical Jeans equation to a group of well-designed three-dimensional bar+disk(+bulge) models. A vertically thin bar can lower along the bar and enhance it perpendicular to the bar, thus generating -humps+hollows. Such a result suggests that -humps+hollows can be generated by the purely dynamical response of stars in the presence of a sufficiently massive, vertically thin bar, even without an outer bar. Using self-consistent N-body simulations, we verify the existence of vertically thin bars in the nuclear-barred and S2B models that generate prominent σ-humps+hollows. Thus, the ubiquitous presence of σ-humps+hollows in S2Bs implies that inner bars are vertically thin. The addition of a bulge makes the -humps more ambiguous and thus tends to somewhat hide the -humps+hollows. We show that may be used as a kinematic diagnostic of stellar components that have different thicknesses, providing a direct perspective on the morphology and thickness of nearly face-on bars and bulges with integral field unit spectroscopy.

Spin-up of massive classical bulges during secular evolution

Classical bulges in spiral galaxies are known to rotate but the origin of this observed rotational motion is not well understood. It has been shown recently that a low-mass classical bulge (ClB) in a barred galaxy can acquire rotation from absorbing a significant fraction of the angular momentum emitted by the bar. Our aim here is to investigate whether bars can spin up also more massive ClBs during the secular evolution of the bar, and to study the kinematics and dynamics of these ClBs. We use a set of self-consistent N-body simulations to study the interaction of ClBs with a bar that forms self-consistently in the disk. We use orbital spectral analysis to investigate the angular momentum gain by the classical bulge stars. We show that the ClBs gain, on average, about 2 - 6% of the disk's initial angular momentum within the bar region. Most of this angular momentum gain occurs via low-order resonances, particularly 5:2 resonant orbits. A density wake forms in the ClB which corotates and aligns with the bar at the end of the evolution. The spin-up process creates a characteristic linear rotation profile and mild tangential anisotropy in the ClB. The induced rotation is small in the centre but significant beyond $\sim2$ bulge half mass radii, where it leads to mass-weighted $V/\sigma \sim 0.2$, and reaches a local $V_{max}/\sigma_{in} \sim 0.5$ at around the scale of the bar. The resulting $V/\sigma$ is tightly correlated with the ratio of the bulge size to the bar size. In all models, a box/peanut bulge forms suggesting that composite bulges may be common. Bar-bulge resonant interaction in barred galaxies can provide some spin up of massive ClBs, but the process appears to be less efficient than for low-mass ClBs. Further angular momentum transfer due to nuclear bars or gas inflow would be required to explain the observed rotation if it is not primordial.

Chaotic motion and the evolution of morphological components in a time-dependent model of a barred galaxy within a dark matter halo

Studies of dynamical stability (chaotic versus regular motion) in galactic dynamics often rely on static analytical models of the total gravitational potential. Potentials based upon self-consistent N-body simulations offer more realistic models, fully incorporating the time-dependent nature of the systems. Here we aim at analysing the fractions of chaotic motion within different morphological components of the galaxy. We wish to investigate how the presence of chaotic orbits evolves with time, and how their spatial distribution is associated with morphological features of the galaxy. We employ a time-dependent analytical potential model that was derived from an N-body simulation of a strongly barred galaxy. With this analytical potential we may follow the dynamical evolution of ensembles of orbits. Using the Generalized Alignment Index (GALI) chaos detection method, we study the fraction of chaotic orbits, sampling the dynamics of both the stellar disc and of the dark matter halo. Within the stellar disc, the global trend is for chaotic motion to decrease in time, specially in the region of the bar. We scrutinized the different changes of regime during the evolution (orbits that are permanently chaotic, permanently regular, those that begin regular and end chaotic, and those that begin chaotic and end regular), tracing the types of orbits back to their common origins. Within the dark matter halo, chaotic motion also decreases globally in time. The inner halo (r < 5 kpc) is where most chaotic orbits are found and it is the only region where chaotic orbits outnumber regular orbits, in the early evolution.

Cosmological structure formation in Decaying Dark Matter models

The standard cold dark matter (CDM) model predicts too many and too dense small structures. We consider an alternative model that the dark matter undergoes two-body decays with cosmological lifetime τ into only one type of massive daughters with non-relativistic recoil velocity Vk. This decaying dark matter model (DDM) can suppress the structure formation below its free-streaming scale at time scale comparable to τ. Comparing with warm dark matter (WDM), DDM can better reduce the small structures while being consistent with high redshfit observations. We study the cosmological structure formation in DDM by performing self-consistent N-body simulations and point out that cosmological simulations are necessary to understand the DDM structures especially on non-linear scales. We propose empirical fitting functions for the DDM suppression of the mass function and the concentration-mass relation, which depend on the decay parameters lifetime τ, recoil velocity Vk and redshift. The fitting functions lead to accurate reconstruction of the the non-linear power transfer function of DDM to CDM in the framework of halo model. Using these results, we set constraints on the DDM parameter space by demanding that DDM does not induce larger suppression than the Lyman-α constrained WDM models. We further generalize and constrain the DDM models to initial conditions with non-trivial mother fractions and show that the halo model predictions are still valid after considering a global decayed fraction. Finally, we point out that the DDM is unlikely to resolve the disagreement on cluster numbers between the Planck primary CMB prediction and the Sunyaev-Zeldovich (SZ) effect number count for τ ∼ H0−1.

Formation and evolution of nuclear star clusters

AbstractNuclear stellar clusters (NSCs) are dense stellar systems known to exist at the center of most of the galaxies. Some of them host a central massive black hole (MBH). They are though to form through in-situ star formation following the infall of gas to the galactic center and/or because of the infall and merger of several stellar clusters. Here we explore the latter scenario by means of detailed self-consistent N-body simulations, proving that a NSC built by the infall and following merger of stellar clusters shows many of the observed features of the Milky Way NSC. We also explore the possibility that the infalling clusters host intermediate mass black holes (IMBHs). Once decayed to the center, the IMBHs act as massive-perturbers accelerating the relaxation of the NSC, filling the loss-cone and boosting the tidal disruption rate of stars up to a value larger than the observational estimates, therefore providing a cumulative constraint on the existence of IMBHs in NSCs. Studying how the properties of the infalling clusters map to the properties of the resulting NSC, we find that, in the IMBHs-free case, the infall mechanism is able to produce many different observational signatures in the form of age segregation.

Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models

Self-consistent N-body simulations are efficient tools to study galactic dynamics. However, using them to study individual trajectories (or ensembles) in detail can be challenging. Such orbital studies are important to shed light on global phase space properties, which are the underlying cause of observed structures. The potentials needed to describe self-consistent models are time-dependent. Here, we aim to investigate dynamical properties (regular/chaotic motion) of a non-autonomous galactic system, whose time-dependent potential adequately mimics certain realistic trends arising from N-body barred galaxy simulations. We construct a fully time-dependent analytical potential, modeling the gravitational potentials of disc, bar and dark matter halo, whose time-dependent parameters are derived from a simulation. We study the dynamical stability of its reduced time-independent 2-degrees of freedom model, charting the different islands of stability associated with certain orbital morphologies and detecting the chaotic and regular regions. In the full 3-degrees of freedom time-dependent case, we show representative trajectories experiencing typical dynamical behaviours, i.e., interplay between regular and chaotic motion for different epochs. Finally, we study its underlying global dynamical transitions, estimating fractions of (un)stable motion of an ensemble of initial conditions taken from the simulation. For such an ensemble, the fraction of regular motion increases with time.

Simulating the quartic Galileon gravity model on adaptively refined meshes

We develop a numerical algorithm to solve the high-order nonlinear derivative-coupling equation associated with the quartic Galileon model, and implement it in a modified version of the ramses N-body code to study the effect of the Galileon field on the large-scale matter clustering. The algorithm is tested for several matter field configurations with different symmetries, and works very well. This enables us to perform the first simulations for a quartic Galileon model which provides a good fit to the cosmic microwave background (CMB) anisotropy, supernovae and baryonic acoustic oscillations (BAO) data. Our result shows that the Vainshtein mechanism in this model is very efficient in suppressing the spatial variations of the scalar field. However, the time variation of the effective Newtonian constant caused by the curvature coupling of the Galileon field cannot be suppressed by the Vainshtein mechanism. This leads to a significant weakening of the strength of gravity in high-density regions at late times, and therefore a weaker matter clustering on small scales. We also find that without the Vainshtein mechanism the model would have behaved in a completely different way, which shows the crucial role played by nonlinearities in modified gravity theories and the importance of performing self-consistent N-body simulations for these theories.

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.

Dark matter direct search rates in simulations of the Milky Way and Sagittarius stream

We analyze self-consistent N-body simulations of the Milky Way disk and the ongoing disruption of the Sagittarius dwarf satellite to study the effect of Sagittarius tidal debris on dark matter detection experiments. In agreement with significant previous work, we reiterate that the standard halo model is insufficient to describe the non-Maxwellian velocity distribution of the Milky Way halo in our equilibrium halo-only and halo/galaxy models, and offer suggestions for correcting for this discrepancy. More importantly, we emphasize that the dark matter component of the leading tidal arm of the Sagittarius dwarf is significantly more extended than the stellar component of the arm, and also that the dark matter and stellar streams are not necessarily coaxial and may be offset by several kpc at the point at which they impact the Galactic disk. This suggests that the dark matter component of the Sagittarius debris is likely to have a non-negligible influence on dark matter detection experiments even when the stellar debris is centered several kpc from the solar neighborhood. Relative to models without an infalling Sagittarius dwarf, the Sagittarius dark matter debris in our models induces an energy-dependent enhancement of direct search event rates of as much as ∼ 20–45%, an energy-dependent reduction in the amplitude of the annual modulation of the event rate by as much as a factor of two, a shift in the phase of the annual modulation by as much as ∼ 20 days, and a shift in the recoil energy at which the modulation reverses phase. These influences of Sagittarius are of general interest in the interpretation of dark matter searches, but may be particularly important in the case of relatively light (mχ≲20 GeV/c2) dark matter because the Sagittarius stream impacts the solar system at high speed compared to the primary halo dark matter.

A secularly evolved model for the Milky Way bar and bulge

AbstractBars are strong drivers of secular evolution in disk galaxies. Bars themselves can evolve secularly through angular momentum transport, producing different boxy/peanut and X-shaped bulges. Our Milky Way is an example of a barred galaxy with a boxy bulge. We present a self-consistent N-body simulation of a barred galaxy which matches remarkably well the structure of the inner Milky Way deduced from star counts. In particular, features taken as signatures of a second “long bar“ can be explained by the interaction between the bar and the spiral arms of the galaxy (Martinez-Valpuesta &amp; Gerhard 2011). Furthermore the structural change in the bulge inside l = 4° measured recently from VVV data can be explained by the high-density near-axisymmetric part of the inner boxy bulge (Gerhard &amp; Martinez-Valpuesta 2012). We also compare this model with kinematic data from recent spectroscopic surveys. We use a modified version of the NMAGIC code (de Lorenzi et al. 2007) to study the properties of the Milky Way bar, obtaining an upper limit for the pattern speed of ~ 42 km/sec/kpc. See Fig. 1 for a comparison of one of our best models with BRAVA data (Kunder et al. 2012).

Dynamical evolution of a bulge in an N-body model of the Milky Way

The detailed dynamical structure of the bulge in the Milky Way is currently under debate. Although kinematics of the bulge stars can be well reproduced by a boxy-bulge, the possible existence of a small embedded classical bulge can not be ruled out. We study the dynamical evolution of a small classical bulge in a model of the Milky Way using a self-consistent high resolution N-body simulation. Detailed kinematics and dynamical properties of such a bulge are presented.

RETENTION OF A PRIMORDIAL COLD CLASSICAL KUIPER BELT IN AN INSTABILITY-DRIVEN MODEL OF SOLAR SYSTEM FORMATION

The cold classical population of the Kuiper belt exhibits a wide variety of unique physical characteristics, which collectively suggest that its dynamical coherence has been maintained through out the solar system's lifetime. Simultaneously, the retention of the cold population's relatively unexcited orbital state has remained a mystery, especially in the context of a solar system formation model, that is driven by a transient period of instability, where Neptune is temporarily eccentric. Here, we show that the cold belt can survive the instability, and its dynamical structure can be reproduced. We develop a simple analytical model for secular excitation of cold KBOs and show that comparatively fast apsidal precession and nodal recession of Neptune, during the eccentric phase, are essential for preservation of an unexcited state in the cold classical region. Subsequently, we confirm our results with self-consistent N-body simulations. We further show that contamination of the hot classical and scattered populations by objects of similar nature to that of cold classicals has been instrumental in shaping the vast physical diversity inherent to the Kuiper belt.

ECCENTRIC EVOLUTION OF SUPERMASSIVE BLACK HOLE BINARIES

In recent numerical simulations \citep{matsubayashi07,lockmann08}, it has been found that the eccentricity of supermassive black hole(SMBH) - intermediate black hole(IMBH) binaries grows toward unity through interactions with stellar background. This increase of eccentricity reduces the merging timescale of the binary through the gravitational radiation to the value well below the Hubble Time. It also gives the theoretical explanation of the existence of eccentric binary such as that in OJ287 \citep{lehto96, valtonen08}. In self-consistent N-body simulations, this increase of eccentricity is always observed. On the other hand, the result of scattering experiment between SMBH binaries and field stars \citep{quinlan96} indicated no increase of eccentricity. This discrepancy leaves the high eccentricity of the SMBH binaries in $N$-body simulations unexplained. Here we present a stellar-dynamical mechanism that drives the increase of the eccentricity of an SMBH binary with large mass ratio. There are two key processes involved. The first one is the Kozai mechanism under non-axisymmetric potential, which effectively randomizes the angular momenta of surrounding stars. The other is the selective ejection of stars with prograde orbits. Through these two mechanisms, field stars extract the orbital angular momentum of the SMBH binary. Our proposed mechanism causes the increase in the eccentricity of most of SMBH binaries, resulting in the rapid merger through gravitational wave radiation. Our result has given a definite solution to the "last-parsec problem".

On the identification of substructure in phase space using orbital frequencies

We study the evolution of satellite debris to establish the most suitable space to identify past merger events. We confirm that the space of orbital frequencies is very promising in this respect. In frequency space individual streams can be easily identified, and their separation provides a direct measurement of the time of accretion. We are able to show for a few idealized gravitational potentials that these features are preserved also in systems that have evolved strongly in time. Furthermore, this time evolution is imprinted in the distribution of streams in frequency space. We have also tested the power of the orbital frequencies in a fully self-consistent (live) N-body simulation of the merger between a disc galaxy and a massive satellite. Even in this case, streams can be easily identified and the time of accretion of the satellite can be accurately estimated.