N-body Dynamics Research Articles

No Signature of the Birth Environment of Exoplanets from Their Host Stars’ Mahalanobis Phase Space

The architectures of extrasolar planetary systems often deviate considerably from the “standard” model for planet formation, which is largely based on our own solar system. In particular, gas giants on close orbits are not predicted by planet formation theory and so some processes are thought to move the planets closer to their host stars. Recent research has suggested that hot-Jupiter host stars display a different phase space compared to stars that do not host hot Jupiters. This has been attributed to these stars forming in star-forming regions of high stellar density, where dynamical interactions with passing stars have perturbed the planets. We test this hypothesis by quantifying the phase space of planet-hosting stars in dynamical N-body simulations of star-forming regions. We find that stars that retain their planets have a higher phase space than nonhosts, regardless of their initial physical density. This is because an imprint of the kinematic substructure from the regions birth is retained, as these stars have experienced fewer and less disruptive encounters than stars whose planets have been liberated and become free-floating. However, host stars whose planets remain bound but have had their orbits significantly altered by dynamical encounters are also primarily found in high phase space regimes. We therefore corroborate other research in this area that has suggested the high phase space of hot-Jupiter host stars is not caused by dynamical encounters or stellar clustering, but rather reflects an age bias in that these stars are (kinematically) younger than other exoplanet host stars.

Dynamics of Star Cluster Formation: Mergers in Gas-rich Environments

We perform high-resolution simulations of forming star clusters as they merge inside giant molecular clouds (GMCs) using hydrodynamics coupled to N-body dynamics to simultaneously model both the gas and stars. We zoom in to previously run GMC simulations and resolve clusters into their stellar and gas components while including the surrounding GMC environment. We find that GMC gas is important in facilitating the growth of clusters in their embedded phase by promoting cluster mergers. Mergers induce asymmetric expansion of the stellar component of the clusters in our simulations. As well, mergers induce angular momentum in the clusters’ stellar and gas components. We find that mergers can lead to an increase in the amount of dense gas present in clusters if a background gas distribution is present. We predict that this can lead to new star formation that can change the overall distribution of cluster stars in velocity space. Our results suggest that subcluster mergers in the presence of background gas can imprint dynamical signatures that can be used to constrain cluster formation histories.

Milky Way globular clusters on cosmological timescales

Context. The present epoch of the Gaia success gives us a possibility to predict the dynamical evolution of our Solar System in the global Galactic framework with high precision. Aims. We statistically investigated the total interaction of globular clusters with the Solar System during six billion years of look-back time. We estimated the gravitational influence of globular clusters’ flyby onto the Oort cloud system. Methods. To perform the realistic orbital dynamical evolution for each individual cluster, we used our own high-order parallel dynamical N-body φ-GPU code that we developed. To reconstruct the orbital trajectories of clusters, we used five external dynamical time variable galactic potentials selected from the IllustrisTNG-100 cosmological database and one static potential. To detect a cluster’s close passages near the Solar System, we adopted a simple distance criteria of below 200 pc. To take into account a cluster’s measurement errors (based on Gaia DR3), we generated 1000 initial positions and velocity randomisations for each cluster in each potential. Results. We found 35 globular clusters that have had close passages near the Sun in all the six potentials during the whole lifetime of the Solar System. We can conclude that at a relative distance of 50 pc between a GC and the SolS, we obtain on average ∼15% of the close passage probability over all six billion years, and at dR = 100 pc, we get on average ∼35% of the close passage probability over all six billion years. The globular clusters BH 140, UKS 1, and Djorg 1 have a mean minimum relative distance to the Sun of 9, 19, and 17 pc, respectively. We analysed the gravitational energetic influence on the whole Oort cloud system from the closest selected globular cluster flyby. We generally found that a globular cluster with a typical mass above a few times 105 M⊙ and with deep close passages in a 1–2 pc immediately results in the ejection more than ∼30% of particles from the Oort cloud system. Conclusions. We can assume that a globular cluster with close passages near the Sun is not a frequent occurrence but also not an exceptional event in the Solar System’s lifetime.

Towards accurate field-level inference of massive cosmic structures

ABSTRACT We investigate the accuracy requirements for field-level inference of cluster and void masses using data from galaxy surveys. We introduce a two-step framework that takes advantage of the fact that cluster masses are determined by flows on larger scales than the clusters themselves. First, we determine the integration accuracy required to perform field-level inference of cosmic initial conditions on these large scales by fitting to late-time galaxy counts using the Bayesian Origin Reconstruction from Galaxies (BORG) algorithm. A 20-step COLA integrator is able to accurately describe the density field surrounding the most massive clusters in the local super-volume ($\lt 135\, {h^{-1}\mathrm{\, Mpc}}$), but does not by itself lead to converged virial mass estimates. Therefore, we carry out ‘posterior resimulations’, using full N-body dynamics while sampling from the inferred initial conditions, and thereby obtain estimates of masses for nearby massive clusters. We show that these are in broad agreement with existing estimates, and find that mass functions in the local super-volume are compatible with ΛCDM.

The HH 24 Complex: Jets, Multiple Star Formation, and Orphaned Protostars

The HH 24 complex harbors five collimated jets emanating from a small protostellar multiple system. We have carried out a multiwavelength study of the jets, their driving sources, and the cloud core hosting the embedded stellar system, based on data from the Hubble Space Telescope, Gemini, Subaru, Apache Point Observatory 3.5 m, Karl G. Jansky Very Large Array, and Atacama Large Millimeter/submillimeter Array (ALMA) telescopes. The data show that the multiple system, SSV 63, contains at least 7 sources, ranging in mass from the hydrogen-burning limit to proto-Herbig Ae stars. The stars are in an unstable nonhierarchical configuration, and one member, a borderline brown dwarf, is moving away from the protostellar system with 25 km s−1, after being ejected ∼5800 yr ago as an orphaned protostar. Five of the embedded sources are surrounded by small, possibly truncated, disks resolved at 1.3 mm with ALMA. Proper motions and radial velocities imply jet speeds of 200–300 km s−1. The two main HH 24 jets, E and C, form a bipolar jet system that traces the innermost portions of parsec-scale chains of Herbig–Haro and H2 shocks with a total extent of at least 3 pc. H2CO and C18O observations show that the core has been churned and continuously fed by an infalling streamer. 13CO and 12CO trace compact, low-velocity, cavity walls carved by the jets and an ultracompact molecular outflow from the most embedded object. Chaotic N-body dynamics likely will eject several more of these objects. The ejection of stars from their feeding zones sets their masses. Dynamical decay of nonhierarchical systems can thus be a major contributor to establishing the initial mass function.

Modelling star cluster formation: Gas accretion

ABSTRACTThe formation of star clusters involves the growth of smaller, gas-rich subclusters through accretion of gas from the giant molecular cloud (GMC) within which the subclusters are embedded. The two main accretion mechanisms responsible for this are accretion of gas from dense filaments and from the ambient background of the cloud. We perform simulations of both of these accretion processes onto gas-rich star clusters using coupled smoothed particle hydrodynamics to model the gas and N-body dynamics to model the stars. We find that, for both accretion processes, the accreting star cluster loses some of its original mass while gaining mass from either the ambient background or the dense filament. The amount of mass lost from both of these processes is small compared to the total mass of the cluster. However, in the case of accretion from a background medium, the net effect can be a decrease in the total mass of the cluster if it is travelling fast enough through the ambient medium (>4 km s−1). We find that the amount of mass lost from the cluster through filamentary accretion is independent of the density, width, or number of filaments funnelling gas into the cluster and is always such that the mass of the cluster is constantly increasing with time. We compare our results to idealized prescriptions used to model star cluster formation in larger scale GMC simulations and find that such prescriptions act as an upper limit when describing the mass of the star cluster they represent.

Early-forming Massive Stars Suppress Star Formation and Hierarchical Cluster Assembly

Feedback from massive stars plays an important role in the formation of star clusters. Whether a very massive star is born early or late in the cluster formation timeline has profound implications for the star cluster formation and assembly processes. We carry out a controlled experiment to characterize the effects of early-forming massive stars on star cluster formation. We use the star formation software suite Torch, combining self-gravitating magnetohydrodynamics, ray-tracing radiative transfer, N-body dynamics, and stellar feedback, to model four initially identical 104 M ⊙ giant molecular clouds with a Gaussian density profile peaking at 521.5 cm−3. Using the Torch software suite through the AMUSE framework, we modify three of the models, to ensure that the first star that forms is very massive (50, 70, and 100 M ⊙). Early-forming massive stars disrupt the natal gas structure, resulting in fast evacuation of the gas from the star-forming region. The star formation rate is suppressed, reducing the total mass of the stars formed. Our fiducial control model, without an early massive star, has a larger star formation rate and total efficiency by up to a factor of 3, and a higher average star formation efficiency per freefall time by up to a factor of 7. Early-forming massive stars promote the buildup of spatially separate and gravitationally unbound subclusters, while the control model forms a single massive cluster.

Exploring multiple generations of planetary embryos

Context. Global models of planet formation tend to begin with an initial set of planetary embryos for the sake of simplicity. While this approach gives valuable insights into the evolution of the initial embryos, the initial distribution itself is staked on a bold assumption. Limiting the study to an initial distribution may neglect essential physics that either precedes or follows such an initial distribution. Aims. We wish to investigate the effect of dynamic planetary embryo formation on the formation of planetary systems. Methods. The presented framework begins with an initial disk of gas, dust, and pebbles. The disk evolution, the formation of plan-etesimals and the formation of planetary embryos is modeled consistently. Embryos then grow by pebble accretion, followed by planetesimal and, eventually, gas accretion. Planet-disk interactions and N-body dynamics, along with a consideration of other simultaneously growing embryos, are included in the framework. Results. We show that the formation of planets can occur in multiple consecutive phases. Earlier generations grow massive by pebble accretion but are subject to fast type I migration and, thus, by accretion to the star. The later generations of embryos that form grow too much smaller masses by planetesimal accretion, as the amount of pebbles in the disk has vanished. Conclusions. The formation history of planetary systems may be far more complex than an initial distribution of embryos could reflect. The dynamic formation of planetary embryos needs to be considered in global models of planet formation to allow for a complete picture of the system’s evolution.

Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space

We consider the derivation of the defocusing cubic nonlinear Schrödinger equation (NLS) on $${\mathbb {R}}^{3}$$ from quantum N-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under $$H^{1}$$ regularity. The $$H^{1}$$ convergence rate estimate we obtain is almost optimal for $$H^{1}$$ datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.

Modelling star cluster formation: Mergers

Abstract Star cluster formation in giant molecular clouds involves the local collapse of the cloud into small gas-rich subclusters, which can then subsequently collide and merge to build up the final star cluster(s). In this paper, we simulate collisions between these subclusters, using coupled smooth particle hydrodynamics for the gas and N-body dynamics for the stars. We are guided by previous radiation hydrodynamics simulations of molecular cloud collapse which provide the global properties of the colliding clusters, such as their stellar and gas masses, and their initial positions and velocities. The subclusters in the original simulation were treated as sink particles which immediately merged into a single entity after the collision. We show that the more detailed treatment provides a more complex picture. At collisional velocities above ≈ 10 km/s, the stellar components of the cluster do not form a monolithic cluster within 3 Myr, although the gas may do so. At lower velocities, the clusters do eventually merge but over timescales that may be longer than the time for a subsequent collision. The structure of the resultant cluster is not well-fit by any standard density distribution, and the clusters are not in equilibrium but continue to expand over our simulation time. We conclude that the simple sink particle treatment of subcluster mergers in large-scale giant molecular cloud simulations provides an upper limit on the final cluster properties.

How Important Is Secular Evolution for Black Hole and Neutron Star Mergers in 2+2 and 3+1 Quadruple-star Systems?

Abstract Mergers of black holes (BHs) and neutron stars (NSs) result in the emission of gravitational waves that can be detected by LIGO. In this paper, we look at 2+2 and 3+1 quadruple-star systems, which are common among massive stars, the progenitors of BHs and NSs. We carry out a detailed population synthesis of quadruple systems using the Multiple Stellar Evolution code, which seamlessly takes into consideration stellar evolution, binary and tertiary interactions, N-body dynamics, and secular evolution. We find that, although secular evolution plays a role in compact object (BH and NS) mergers, (70–85)% (depending on the model assumptions) of the mergers are solely due to common envelope evolution. Significant eccentricities in the LIGO band (higher than 0.01) are only obtained with zero supernova (SN) kicks and are directly linked to the role of secular evolution. A similar outlier effect is seen in the χ eff distribution, with negative values obtained only with zero SN kicks. When kicks are taken into account, there are no systems that evolve into a quadruple consisting of four compact objects. For our fiducial model, we estimate the merger rates (in units of Gpc−3 yr−1) in 2+2 quadruples (3+1 quadruples) to be 10.8 ± 0.9 (2.9 ± 0.5), 5.7 ± 0.6 (1.4 ± 0.4), and 0.6 ± 0.2 (0.7 ± 0.3) for BH–BH, BH–NS, and NS–NS mergers, respectively. The BH–BH merger rates represent a significant fraction of the current LIGO rates, whereas the other merger rates fall short of LIGO estimates.

Beyond Bogoliubov dynamics

We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick's theorem. We determine the n-point correlation functions of the excitations around the condensate, as well as the reduced densities of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point functions.

Classical n-body scattering with long-range potentials

We consider the scattering of n classical particles interacting via pair potentials which are assumed to be ‘long-range’, i.e. of order as r tends to infinity, for some α > 0. We define and focus on the ‘free region’, the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an explicit, global surface of section for the free region. This surface of section allows us to prove the smoothness of the map sending a point to its final state and to establish a forward conjugacy between the n-body dynamics and free dynamics.

Cluster dynamics studied with the phase-space minimum spanning tree approach

The origin of weakly bound objects like clusters and hypernuclei, observed in heavy-ion collisions, is of theoretical and experimental interest. It is in the focus of the experiments at RHIC and LHC since it is not evident how such weakly bound objects can survive in an environment whose hadronic decay products point to a temperature of the order of 150 MeV. It is as well one of the key research topics in the future facilities of FAIR and NICA which are under construction in Darmstadt (Germany) and Dubna (Russia), respectively. We present here first results on the cluster dynamics within the model-independent cluster recognition library "phase-space Minimum Spanning Tree" (psMST) applied to different transport approaches: PHQMD, PHSD, SMASH and UrQMD. The psMST is based on the "Minimum Spanning Tree" (MST) algorithm for the cluster recognition which exploits correlations in coordinate space, and it is extended to correlations of baryons in the clusters in momentum space. We show the sensitivity of the cluster formation on the microscopic realization of the n-body dynamics and on the potential interactions in heavy-ion collisions.

Optimal Softening for Gravitational Force Calculations in N-body Dynamics

Abstract The choice of the optimal value of the softening length (ϵ i ) of each particle dealing with N-body simulations has a profound impact on error values in the gravitational force calculation. A slight deviation from its exact optimal value may result in a large error in the calculation. In this paper we augment the accuracy of the existing gravitational force calculation methods by providing a new technique to calculate the individual optimal values of ϵ i for various configurations of the Plummer density model. We have proposed an expression that relates each particle by considering the average characteristic length (λ i ) and density (ρ i ), unlike previous studies that considered ϵ i as an exclusive function of ρ i . We have performed gravitational force calculations for each and every particle from the Plummer density model using compact as well as noncompact gravitational force methods based on smoothed particle hydrodynamics. We have tested our new equation for the entire range of numerical simulations performed during the study. It is found that the errors in our force calculations are not only lower than those estimated from previous studies but also remain flat for various considerations of nearest neighboring particles (N neigh). The adjusted expression of ϵ λ,i in our study has less dependence on N neigh in the Plummer sphere. Finally, based on the results obtained using the method proposed in this study, we find that it remarkably improves both the accuracy as well as the stability of the gravitational force calculation.

SILCC VI – Multiphase ISM structure, stellar clustering, and outflows with supernovae, stellar winds, ionizing radiation, and cosmic rays

ABSTRACT We present simulations of the multiphase interstellar medium (ISM) at solar neighbourhood conditions including thermal and non-thermal ISM processes, star cluster formation, and feedback from massive stars: stellar winds, hydrogen ionizing radiation computed with the novel treeray radiative transfer method, supernovae (SN), and the injection of cosmic rays (CR). N-body dynamics is computed with a 4th-order Hermite integrator. We systematically investigate the impact of stellar feedback on the self-gravitating ISM with magnetic fields, CR advection and diffusion, and non-equilibrium chemical evolution. SN-only feedback results in strongly clustered star formation with very high star cluster masses, a bi-modal distribution of the ambient SN densities, and low volume-filling factors (VFF) of warm gas, typically inconsistent with local conditions. Early radiative feedback prevents an initial starburst, reduces star cluster masses and outflow rates. Furthermore, star formation rate surface densities of $\Sigma _{\dot{M}_\star } = 1.4-5.9 \times 10^{-3}$$\mathrm{M}_\odot \, \mathrm{yr}^{-1}\, \mathrm{kpc}^{-2}$, VFFwarm = 60–80 per cent as well as thermal, kinetic, magnetic, and cosmic ray energy densities of the model including all feedback mechanisms agree well with observational constraints. On the short, 100 Myr, time-scales investigated here, CRs only have a moderate impact on star formation and the multiphase gas structure and result in cooler outflows, if present. Our models indicate that at low gas surface densities SN-only feedback only captures some characteristics of the star-forming ISM and outflows/inflows relevant for regulating star formation. Instead, star formation is regulated on star cluster scales by radiation and winds from massive stars in clusters, whose peak masses agree with solar neighbourhood estimates.

Precise radial velocities of giant stars

Context. Using the Hamilton Échelle Spectrograph at Lick Observatory, we have obtained precise radial velocities (RVs) of a sample of 373 G- and K-giant stars over more than 12 yr, leading to the discovery of several single and multiple planetary systems. The RVs of the long-period (~53 yr) spectroscopic binary ε Cyg (HIP 102488) are found to exhibit additional regular variations with a much shorter period (~291 days). Aims. We intend to improve the orbital solution of the ε Cyg system and attempt to identify the cause of the nearly periodic shorter period variations, which might be due to an additional substellar companion. Methods. We used precise RV measurements of the K-giant star ε Cyg from Lick Observatory, in combination with a large set of RVs collected more recently with the SONG telescope, as well as archival data sets. We fit Keplerian and fully dynamical N-body models to the RVs in order to explore the properties of a previously known spectroscopic stellar companion and to investigate whether there is an additional planetary companion in the system. To search for long-term stable regions in the parameter space around the orbit of this putative planet, we ran a stability analysis using an N-body code. Furthermore, we explored the possibility of co-orbital bodies to the planet with a demodulation technique. We tested the hypothesis of ε Cyg being a hierarchical stellar triple by using a modified version of the N-body code. Alternative causes for the observed RV variations, such as stellar spots and oscillations, were examined by analyzing photometric data of the system and by comparing its properties to known variable stars with long secondary periods and heartbeat stars from the literature. Results. Our Keplerian model characterizes the orbit of the spectroscopic binary to higher precision than achieved previously, resulting in a semi-major axis of a = 15.8 AU, an eccentricity of e = 0.93, and a minimum mass of the secondary of msini = 0.265 M⊙. Additional short-period RV variations closely resemble the signal of a Jupiter-mass planet orbiting the evolved primary component with a period of 291 d, but the period and amplitude of the putative orbit change strongly over time. Furthermore, in our stability analysis of the system, no stable orbits could be found in a large region around the best fit. Both of these findings deem a planetary cause of the RV variations unlikely. Most of the investigated alternative scenarios also fail to explain the observed variability convincingly. Due to its very eccentric binary orbit, it seems possible, however, that ε Cyg could be an extreme example of a heartbeat system.

Semiclassical evolution with low regularity

We prove semiclassical estimates for the Schrödinger-von Neumann evolution with C1,1 potentials and density matrices whose square root have either Wigner functions with low regularity independent of the dimension, or matrix elements between Hermite functions having long range decay. The estimates are settled in different weak topologies and apply to initial density operators whose square root have Wigner functions 7 times differentiable, independently of the dimension. They also apply to the N-body quantum dynamics uniformly in N and to concentrating pure and mixed states without any regularity assumption. In a appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderón-Vaillancourt Theorem.

Modeling of the Effects of Stellar Feedback during Star Cluster Formation Using a Hybrid Gas and N-Body Method

Abstract Understanding the formation of stellar clusters requires following the interplay between gas and newly formed stars accurately. We therefore couple the magnetohydrodynamics code FLASH to the N-body code ph4 and the stellar evolution code SeBa using the Astrophysical Multipurpose Software Environment (AMUSE) to model stellar dynamics, evolution, and collisional N-body dynamics and the formation of binary and higher-order multiple systems, while implementing stellar feedback in the form of radiation, stellar winds, and supernovae in FLASH. We here describe the algorithms used for each of these processes. We denote this integrated package Torch. We then use this novel numerical method to simulate the formation and early evolution of several examples of open clusters of ∼1000 stars formed from clouds with a mass range of 103 M ⊙ to 105 M ⊙. Analyzing the effects of stellar feedback on the gas and stars of the natal clusters, we find that in these examples, the stellar clusters are resilient to disruption, even in the presence of intense feedback. This can even slightly increase the amount of dense, Jeans unstable gas by sweeping up shells; thus, a stellar wind strong enough to trap its own H ii region shows modest triggering of star formation. Our clusters are born moderately mass segregated, an effect enhanced by feedback, and retained after the ejection of their natal gas, in agreement with observations.

From Newton’s second law to Euler’s equations of perfect fluids

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty.