Disk Mass Research Articles

Dust mass in protoplanetary disks with porous dust opacities

Atacama Large Millimeter/submillimeter Array surveys have suggested that protoplanetary disks are not massive enough to form the known exoplanet population, based on the assumption that the millimeter continuum emission is optically thin. In this work, we investigate how the mass determination is influenced when the porosity of dust grains is considered in radiative transfer models. The results show that disks with porous dust opacities yield similar dust temperatures, but systematically lower millimeter fluxes, as compared to disks that incorporate compact dust grains. Moreover, we have recalibrated the relation between dust temperature and stellar luminosity for a wide range of stellar parameters. We also calculated the dust masses of a large sample of disks using the traditionally analytic approach. The median dust mass from our calculation is about six times higher than the literature result, and this is mostly driven by the different opacities of porous and compact grains. A comparison of the cumulative distribution function between disk dust masses and exoplanet masses shows that the median exoplanet mass is about two times lower than the median dust mass when grains are assumed to be porous and there are no exoplanetary systems with masses higher than the most massive disks. Our analysis suggests that adopting porous dust opacities may alleviate the mass budget problem for planet formation. As an example illustrating the combined effects of optical depth and porous dust opacities on the mass estimation, we conducted new IRAM/NIKA-2 observations toward the IRAS\,04370+2559 disk and performed a detailed radiative transfer modeling of the spectral energy distribution (SED). The best-fit dust mass is roughly 100 times higher than the value given by a traditionally analytic calculation. Future spatially resolved observations at various wavelengths are required to better constrain the dust mass.

Effect of Time-varying X-Ray Emission from Stellar Flares on the Ionization of Protoplanetary Disks

X-rays have significant impacts on cold, weakly ionized protoplanetary disks by increasing the ionization rate and driving chemical reactions. Stellar flares are explosions that emit intense X-rays and are the unique source of hard X-rays with an energy of ≳10 keV in the protoplanetary disk systems. Hard X-rays should be carefully taken into account in models as they can reach the disk midplane as a result of scattering in the disk atmospheres. However, previous models are insufficient to predict the hard X-ray spectra because of simplifications in flare models. We develop a model of X-ray spectra of stellar flares based on observations and flare theories. The flare temperature and nonthermal electron emissions are modeled as functions of flare energy, which allows us to better predict the hard X-ray photon flux than before. Using our X-ray model, we conduct radiative transfer calculations to investigate the impact of flare hard X-rays on disk ionization, with a particular focus on the protoplanetary disk around a T Tauri star. We demonstrate that for a flare with an energy of 1035 erg, X-ray photons with ≳5 keV increase the ionization rates more than galactic cosmic rays down to z ≈ 0.1R. The contribution of flare X-rays to the ionization at the midplane depends on the disk parameters such as disk mass and dust settling. We also find that the 10 yr averaged X-rays from multiple flares could certainly contribute to the ionization. These results emphasize the importance of stellar flares on the disk evolution.

Dust Drift Timescales in Protoplanetary Disks at the Cusp of Gravitational Instability

Millimeter emitting dust grains have sizes that make them susceptible to drift in protoplanetary disks due to the difference between their orbital speed and that of the gas. The characteristic drift timescale depends on the surface density of the gas. By comparing disk radius measurements from Atacama Large Millimeter/submillimeter Array CO and continuum observations at millimeter wavelengths, the gas surface density profile and dust drift time can be self-consistently determined. We find that profiles which match the measured dust mass have very short drift timescales, an order of magnitude or more shorter than the stellar age, whereas profiles for disks that are on the cusp of gravitational instability, defined via the minimum value of the Toomre parameter, Qmin∼1−2 , have drift timescales comparable to the stellar lifetime. This holds for disks with masses of dust ≳5 M ⊕ across a range of absolute ages from less than 1 Myr to over 10 Myr. The inferred disk masses scale with stellar mass as Mdisk≈M*/5Qmin . This interpretation of the gas and dust disk sizes simultaneously solves two long standing issues regarding the dust lifetime and exoplanet mass budget, and suggests that we consider millimeter wavelength observations as a window into an underlying population of particles with a wide size distribution in secular evolution with a massive planetesimal disk.

Resonant Friction on Disks in Galactic Nuclei

We argue that resonant friction has a dramatic effect on a disk whose rotation direction is misaligned with that of its host nuclear star cluster. The disk’s gravity causes gravitational perturbation of the cluster that in turn exerts a strong torque back onto the disk. We argue that this torque may be responsible for the observed disruption of the clockwise disk of young stars in the Galactic center, and show in numerical experiments that it produces the observed features in the distribution of the stars’ angular momenta. More generally, we speculate that the rotation of nuclear star clusters has a stabilizing effect on the orientation of transient massive accretion disks around the supermassive black holes residing in their centers, and thus on the directions and magnitudes of the black hole spins.

Impact of H I cooling and study of accretion disks in asymptotic giant branch wind-companion smoothed particle hydrodynamic simulations

Context. High-resolution observations reveal that the outflows of evolved low- and intermediate-mass stars harbour complex morphological structures that are linked to the presence of one or multiple companions. Hydrodynamical simulations provide a way to study the impact of a companion on the shaping of the asymptotic giant branch (AGB) star out-flow. Aims. Using smoothed particle hydrodynamic (SPH) simulations of an AGB star undergoing mass loss, which also has a binary companion, we study the impact of including H I atomic line cooling on the flow morphology. We also study how this affects the properties of the accretion disks that form around the companion. Methods. We used the PHANTOM code to perform high-resolution 3D SPH simulations of the interaction of a solar-mass companion with the outflow of an AGB star, using different wind velocities and eccentricities. We compared the model properties, computed with and without the inclusion of H I cooling. Results. The inclusion of H I cooling produces a sizeable decrease in the temperature, up to one order of magnitude, in the region closely surrounding the companion star. As a consequence, the morphological irregularities and relatively energetic (bipolar) outflows that were obtained without H I cooling no longer appear. In the case of an eccentric orbit and a low wind velocity, these morphologies are still highly asymmetric, but the same structures recur at every orbital period, making the morphology more regular. Flared accretion disks, with a (sub-)Keplerian velocity profile, are found to form around the companion in all our models with H I cooling, provided the accretion radius is small enough. The disks have radial sizes ranging from about 0.4 to 0.9 au and masses around 10−7−10−8 M⊙. For the considered wind velocities, mass accretion onto the companion is up to a factor of 2 higher than predicted by the standard Bondi Hoyle Littleton rate, ranging between ~4 to 21% of the AGB wind mass loss rate. The lower the wind velocity at the location of the companion, the larger and the more massive the disk and the higher the mass accretion efficiency. In eccentric systems, the disk size, disk mass, and mass accretion efficiency vary, depending on the orbital phase. Conclusions. H I cooling is an essential ingredient to properly model the medium around the companion where density-enhanced wind structures form and it favours the formation of an accretion disk.

Running with the bulls

Context. Stars and planets form in regions of enhanced stellar density, subjecting protoplanetary discs to gravitational perturbations from neighbouring stars. Observations in the Taurus star-forming region have uncovered evidence of at least three recent, star-disc encounters that have truncated discs (HV/DO Tau, RW Aurigae, and UX Tau), raising questions about the frequency of such events. Aims. We aim to assess the probability of observing truncating star-disc encounters in Taurus. Methods. We generated a physically motivated dynamical model including binaries and a spatial-kinematic substructure to follow the historical dynamical evolution of the Taurus star-forming region. We used this model to track star-disc encounters and the resulting outer disc truncation over the lifetime of Taurus. Results. A quarter of discs are truncated below 30 au by dynamical encounters, but this truncation mostly occurs in binaries over the course of a few orbital periods, on a timescale ≲0.1 Myr. Nonetheless, some truncating encounters still occur up to the present age of Taurus. Strongly truncating encounters (ejecting ≳10 percent of the disc mass) occur at a rate ∼10 Myr−1, sufficient to explain the encounter between HV and DO Tau ∼0.1 Myr ago. If encounters that eject only ∼1 percent of the disc mass are responsible for RW Aurigae and UX Tau, then they are also expected with encounter rate Γenc ∼ 100–200 Myr−1. However, the observed sample of recent encounters is probably incomplete, since these examples occurred in systems that are not consistent with a random drawing from the mass function. One more observed example would statistically imply additional physics, such as replenishment of the outer disc material. Conclusions. The marginal consistency of the frequency of observed recent star-disc encounters with theoretical expectations underlines the value of future large surveys searching for external structures associated with recent encounters. The outcome of such a survey offers a highly constraining, novel probe of protoplanetary disc physics.

General relativistic self-gravitating equilibrium discs around rotating neutron stars

ABSTRACT In modelling a relativistic disc around a compact object, the self-gravity of the disc is often neglected while it needs to be incorporated for more accurate descriptions in several circumstances. Extending the Komatsu–Eriguchi–Hachisu self-consistent field method, we present numerical models of a rapidly rotating neutron star with a self-gravitating disc in stationary equilibrium. In particular, our approach allows us to obtain numerical solutions involving a massive disc with the rest mass $\mathcal {O}(10^{-1})-\mathcal {O}(10^0)\, \mathrm{ M}_\odot$ closely attached to a rotating neutron star, given that the disc is mainly supported by the relativistic electron degeneracy pressure. We also assess the impact of self-gravity on the internal structure of the disc and the neutron star. These axisymmetric, stationary solutions can be employed for simulations involving the neutron star–disc system in the context of high-energy transients and gravitational-wave emissions.

The properties of magnetised cold filaments in a cool-core galaxy cluster

Filaments of cold gas ($T $ K) are found in the inner regions of many cool-core clusters. These structures are thought to play a major role in the regulation of feedback from active galactic nuclei (AGNs). We study the morphology of the filaments, their formation, and their impact on the propagation of the outflowing AGN jets. We present a set of GPU-accelerated 3D magnetohydrodynamic simulations of an idealised Perseus-like cluster using the performance portable code AthenaPK . We include radiative cooling and a self-regulated AGN feedback model that redistributes accreted material through kinetic, thermal, and magnetic feedback. We confirm that magnetic fields play an important role in both the formation and evolution of the cold material. These suppress the formation of massive cold discs and favour magnetically supported filaments over clumpy structures. Achieving resolutions of $25-50$ pc, we find that filaments are not monolithic as they contain numerous and complex magnetically supported sub-structures. We find that the mass distribution of these clumps follows a power law of slope of $ for all investigated filaments. Studying the evolution of individual filaments, we find that their formation pathways can be diverse. We find examples of filaments forming through a combination of gas uplifting and condensation, as well as systems of purely infalling clumps condensing out of the intracluster medium. The density contrast between the cold gas and the outflowing hot material leads to recurring deflections of the jets, favouring inflation of bubbles. Filaments in cool-core clusters are clumpy and contain numerous sub-structures, resulting from a complex interplay between magnetic fields, thermal instability, and jet-cloud interaction. Frequent deflections of the AGN outflows suppress jet collimation and favour the formation of large X-ray bubbles, and smaller off-axis cavities.

Disk mass after a binary neutron star merger as a constraining parameter for short gamma-ray bursts

Context. The coincident detection of GW170817 and gamma-ray burst GRB170817A marked a milestone for the connection between binary neutron star (BNS) mergers and short gamma-ray bursts (sGRBs). These mergers can lead to the formation of a black hole that is surrounded by a disk and to the generation of a powerful jet. It spends energy to break free from the merger ejecta, and then a portion of it is dissipated to produce observable emissions. Aims. Our primary goal is to enhance our comprehension of BNS mergers by constraining the disk mass for a selection of sGRBs. To do this, we used the isotropic gamma-ray luminosity and corresponding emission times as key indicators. Methods. We leveraged data from GW170817 to estimate the disk mass surrounding the BNS merger remnant, and we subsequently inferred the efficiency of the accretion onto the jet. We then statistically examined other sGRB observations to estimate whether they might have been induced by BNS mergers Results. Our findings suggest that when similar physical parameters are employed as in the only observed BNS-powered GRB event, GRB170817A, a substantial fraction of sGRBs would need an unrealistically massive disk remnant. Conclusions. This observation raises the possibility that either a different mechanism powered those events or that the post-collapse disk efficiency varies significantly in different BNS merger scenarios.

Gas dynamics around a Jupiter-mass planet

Context. Giant planets grow and acquire their gas envelope during the disk phase. At the time of the discovery of giant planets in their host disk, it is important to understand the interplay between the host disk and the envelope and circum-planetary disk properties of the planet. Aims. Our aim is to investigate the dynamical and physical structure of the gas in the vicinity of a Jupiter-mass planet and study how protoplanetary disk properties, such as disk mass and viscosity, determine the planetary system as well as the accretion rate inside the planet’s Hill sphere. Methods. We ran global 3D simulations with the grid-based code fargOCA, using a fully radiative equation of state and a dust-to-gas ratio of 0.01. We built a consistent disk structure starting from vertical thermal equilibrium obtained by including stellar irradiation. We then let a gap open with a sequence of phases, whereby we deepened the potential and increased the resolution in the planet’s neighbourhood. We explored three models. The nominal one features a disk with surface density, ∑, corresponding to the minimum mass solar nebula at the planet’s location (5.2 au), characterised by an α viscosity value of 4 10−3 at the planet’s location. The second model has a surface density that is ten times smaller than the nominal one and the same viscosity. In the third model, we also reduced the viscosity value by a factor of 10. Results. During gap formation, giant planets accrete gas inside the Hill sphere from the local reservoir. Gas is heated by compression and cools according to opacity, density, and temperature values. This process determine the thermal energy budget inside the Hill sphere. In the analysis of our disks, we find that the gas flowing into the Hill sphere is approximately scaled as the product ∑ν, as expected from viscous transport. The accretion rate of the planetary system (envelope plus circum-planetary disk) is instead scaled as √Σv, with its efficiency depending on the thermal energy budget inside the Hill sphere. Conclusions. Previous studies have shown that pressure-supported or rotationally supported structures are formed around giant planets, depending on the equation of state (EoS) or on the opacity; namely, on the dust content within the Hill sphere. In the case of a fully radiative EoS and a constant dust to gas ratio of 0.01, we find that low-mass and low-viscosity circum-stellar disks favour the formation of a rotationally supported circum-planetary disk. Gas accretion leading to the doubling time of the planetary system of > 105 years has only been found in the case of a low-viscosity disk.

A Break in the Size–Stellar Mass Relation: Evidence for Quenching and Feedback in Dwarf Galaxies

Mapping stars and gas in nearby galaxies is fundamental for understanding their growth and the impact of their environment. This issue is addressed by comparing the stellar “edges” of galaxies D stellar, defined as the outermost diameter where in situ star formation significantly drops, with the gaseous distribution parameterized by the neutral atomic hydrogen diameter measured at 1 M ⊙ pc−2, D HI. By sampling a broad H i mass range 105 M ⊙ < M HI < 1011 M ⊙, we find several dwarf galaxies with M HI < 109 M ⊙ from the field and Fornax Cluster that are distinguished by D stellar ≫ D HI. For the cluster dwarfs, the average H i surface density near D stellar is ∼0.3 M ⊙ pc−2, reflecting the impact of quenching and outside-in gas removal from ram pressure and tidal interactions. In comparison, D stellar/D HI ranges between 0.5 and 2 in dwarf field galaxies, consistent with the expectations from stellar feedback. Only more massive disk galaxies in the field can thus be characterized by the common assumption that D stellar ≲ D HI. We discover a break in the D stellar–M ⋆ relation at m break ∼ 4 × 108 M ⊙ that potentially differentiates the low-mass regime, where the influence of stellar feedback and environmental processes more prominently regulates the sizes of nearby galaxies. Our results highlight the importance of combining deep optical and H i imaging for understanding galaxy evolution.

Precession of a rotor with a different number of discrete radial elastic damping supports

A design model of a rotary system in the form of a cantilevered flexible shaft with a massive unbalanced disk at the free end is considered. In the plane of rotation of the disk, point elastically damped supports are discretely located with a clearance, stabilizing the vibrations of the rotor in the supercritical zone. With the help of the introduced precession indicator, the type and direction of the precession of the rotor during interaction with the supports is analyzed. The influence of a different number of supports on the evolution of the rotor precession in the supercritical zone after the Poincare-Andronov-Hopf bifurcation is investigated. The effect of the clearance and eccentricity of the disk mass at the contact of the disk with the supports on the precession of the rotor is analyzed.

Signatures of low-mass black hole–neutron star mergers

ABSTRACT The recent observation of the GW230529 event indicates that black hole–neutron star binaries can contain low-mass black holes. Since lower mass systems are more favourable for tidal disruption, such events are promising candidates for multimessenger observations. In this study, we employ five finite-temperature, composition-dependent matter equations of state and present results from ten 3D general relativistic hydrodynamic simulations for the mass ratios $q = 2.6$ and 5. Two of these simulations target the chirp mass and effective spin parameter of the GW230529 event, while the remaining eight contain slightly higher mass black holes, including both spinning ($a_{\mathrm{ BH}} = 0.7$) and non-spinning ($a_{\mathrm{ BH}} = 0$) models. We discuss the impact of the equation of state, spin, and mass ratio on black hole–neutron star mergers by examining both gravitational-wave and ejected matter properties. For the low-mass ratio model we do not see fast-moving ejecta for the softest equation of state model, but the stiffer model produces on the order of $10^{-6}\,\mathrm{ M}_\odot$ of fast-moving ejecta, expected to contribute to an electromagnetic counterpart. Notably, the high-mass ratio model produces nearly the same amount of total dynamical ejecta, but yields 52 times more fast-moving ejecta than the low-mass ratio system. In addition, we observe that the black-hole spin tends to decrease the amount of fast-moving ejecta while increasing significantly the total ejected mass. Finally, we note that the disc mass tends to increase as the neutron star compactness decreases.

Long-term evolutionary links between the isolated neutron star populations

ABSTRACT We have investigated the evolutionary connections of the isolated neutron star (NS) populations including radio pulsars (RPs), anomalous X-ray pulsars (AXPs), soft gamma repeaters (SGRs), dim isolated NSs (XDINs), ‘high-magnetic field’ RPs (‘HBRPs’), central compact objects (CCOs), rotating radio transients (RRATs), and long-period pulsars (LPPs) in the fallback disc model. The model can reproduce these NS families as a natural outcome of different initial conditions (initial period, disc mass, and dipole moment, μ) with a continuous μ distribution in the $\sim 10^{27} - 5 \times 10^{30}$ G cm$^3$ range. Results of our simulations can be summarized as follows: (1) A fraction of ‘HBRPs’ with relatively high μ evolve into the persistent AXP/SGR properties, and subsequently become LPPs. (2) Persistent AXP/SGRs do not have evolutionary links with CCOs, XDINs, and RRATs. (3) For a wide range of μ, most RRATs evolve passing through RP or ‘HBRP’ properties during their early evolutionary phases. (4) A fraction of RRATs which have the highest estimated birth rate seem to be the progenitors of XDINs. (5) LPPs, whose existence was predicted by the fallback disc model, are the sources evolving in the late stage of evolution before the discs become inactive. These results provide concrete support to the ideas proposing evolutionary connections between the NS families to account for the ‘birth rate problem’, the discrepancy between the cumulative birth rate estimated for these systems and the core-collapse supernova rate.

Dynamic modeling and vibration analysis for motorized spindle with uncertainties

Abstract Uncertainties inevitably exist in motorized spindle and it is difficult to accurately describe its dynamics based on a deterministic model. In this paper, a dynamic model of motorized spindle with uncertainty is proposed. The deterministic dynamic model of the motorized spindle rotor system is built by the finite element method, and the uncertainties are constructed by the polynomial chaotic expansion (PCE) method. The impact of each uncertainty parameter on the critical speed of the system is investigated by adopting the Sobol global sensitivity analysis method. The effects of uncertainties including the bearing stiffness, the material Young’s modulus, the disc mass and the unbalanced excitation on the vibration response of motorized spindle are analyzed. The modal experiments are carried out to verify the effectiveness of the model. The results show that the uncertainties have a great influence on the vibration characteristics of the motorized spindle system, in which the bearing stiffness mainly affects the high-order frequency response of the system, while the Young’s modulus affects all the order frequency response of the system. The influence of hybrid uncertainties on the frequency response of the system is more complex. This study provides a guidance to the design of motorized spindle.

Black hole-neutron star mergers with massive neutron stars in numerical relativity

We study the merger of black hole-neutron star (BH-NS) binaries in numerical relativity, focusing on the properties of the remnant disk and the ejecta, varying the mass of compactness of the NS and the mass and spin of the BH. We find that within the precision of our numerical simulations, the remnant disk mass and ejecta mass normalized by the NS baryon mass (M^rem and M^eje, respectively), and the cutoff frequency fcut normalized by the initial total gravitational mass of the system at infinite separation approximately agree among the models with the same NS compactness CNS=MNS/RNS, mass ratio Q=MBH/MNS, and dimensionless BH spin χBH irrespective of the NS mass MNS in the range of 1.092−1.691M⊙. This result shows that the merger outcome depends sensitively on Q, χBH, and CNS but only weakly on MNS. This justifies the approach of studying the dependence of NS tidal disruptions on the NS compactness by fixing the NS mass but changing the EOS. We further perform simulations with massive NSs of MNS=1.8M⊙, and compare our results of M^rem and M^eje with those given by existing fitting formulas to test their robustness for more compact NSs. We find that the fitting formulas obtained in the previous studies are accurate within the numerical errors assumed, while our results also suggest that further improvement is possible by systematically performing more precise numerical simulations. Published by the American Physical Society 2024

Protostellar Disk Formation Regimes: Angular Momentum Conservation versus Magnetic Braking

Protostellar disks around young protostars exhibit diverse properties, with their radii ranging from less than ten to several hundred astronomical units. To investigate the mechanisms shaping this disk radius distribution, we compiled a sample of 27 Class 0 and I single protostars with resolved disks and dynamically determined protostellar masses from the literature. Additionally, we derived the radial profile of the rotational-to-gravitational-energy ratio in dense cores from the observed specific angular momentum profiles in the literature. Using these observed protostellar masses and rotational energy profile, we computed theoretical disk radii from the hydrodynamic and nonideal magnetohydrodynamic (MHD) models in Y.-N. Lee et al. and generated synthetic samples to compare with the observations. In our theoretical model, the disk radii are determined by hydrodynamics when the central protostar+disk mass is low. After the protostars and disks grow and exceed certain masses, the disk radii become regulated by magnetic braking and nonideal MHD effects. The synthetic disk radius distribution from this model matches well with the observations. This result suggests that hydrodynamics and nonideal MHD can be dominant in different mass regimes (or evolutionary stages), depending on the rotational energy and protostar+disk mass. This model naturally explains the rarity of large (>100 au) disks and the presence of very small (<10 au) disks. It also predicts that the majority of protostellar disks have radii of a few tens of astronomical units, as observed.

Research on the critical speed of a double span rotor system based on finite element method

Abstract Double-span rotor systems are commonly used in aerospace engines and large turbines, where critical criticality is critical in their dynamic characteristics. This paper calculates the critical critical value of the rotor through the intervention matrix method and verifies it through the finite element method. The changes in the support and wheel disc were discussed and the results show that the increase in support stiffness, the reduction in wheel disc mass, and the increase in coupling stiffness will increase the critical stress of the double-span rotor. In addition, the increase in the position of the support and the installation position of the coupling will affect the critical point.

Revising Properties of Planet–Host Binary Systems. IV. The Radius Distribution of Small Planets in Binary Star Systems Is Dependent on Stellar Separation* *Based on observations obtained with the Hobby–Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen.

Small planets (R p ≤ 4 R ⊕) are divided into rocky super-Earths and gaseous sub-Neptunes separated by a radius gap, but the mechanisms that produce these distinct planet populations remain unclear. Binary stars are the only main-sequence systems with an observable record of the protoplanetary disk lifetime and mass reservoir, and the demographics of planets in binaries may provide insights into planet formation and evolution. To investigate the radius distribution of planets in binary star systems, we observed 207 binary systems hosting 283 confirmed and candidate transiting planets detected by the Kepler mission, then recharacterized the planets while accounting for the observational biases introduced by the secondary star. We found that the population of planets in close binaries (ρ ≤ 100 au) is significantly different from the planet population in wider binaries (ρ > 300 au) or single stars. In contrast to planets around single stars, planets in close binaries appear to have a unimodal radius distribution with a peak near the expected super-Earth peak of R p ∼ 1.3 R ⊕ and a suppressed population of sub-Neptunes. We conclude that we are observing the direct impact of a reduced disk lifetime, smaller mass reservoir, and possible altered distribution of solids reducing the sub-Neptune formation efficiency. Our results demonstrate the power of binary stars as a laboratory for exploring planet formation and as a controlled experiment of the impact of varied initial conditions on mature planet populations.

Gravitational interactions of binary systems in the massive planetesimal disc

Gravitational interactions of binary systems in the massive planetesimal disc