Gravitational Fragmentation Research Articles

MHD modeling of the molecular filament evolution

We perform numerical magnetohydrodynamic (MHD) simulations of the gravitational collapse and fragmentation of a cylindrical molecular cloud with the help of the FLASH code. The cloud collapses rapidly along it’s radius without any signs of fragmentation in the simulations without magnetic field. The radial collapse of the cloud is stopped by the magnetic pressure gradient in the simulations with parallel magnetic field. Cores with high density form at the cloud’s edges during further evolution. The core densities are n ≈ 1.7×108 and 2×10-7 cm–3 in the cases with initial magnetic field strengths B = 1.9×10-4 and 6×10-4 G, respectively. The cores move toward the cloud’s centre with supersonic speeds |vz| = 3.6 and 5.3 km/s. The sizes of the cores along the cloud’s radius and cloud’s main axis are dr = 0.0075 pc and dz = 0.025 pc, dr = 0.03 pc and dz = 0.025 pc, respectively. The masses of the cores increase during the filament evolution and lie in range of ≈(10-20)Me. According to our results, the cores observed at the edges of molecular filaments can be a result of the filament evolution with parallel magnetic field.

The Formation of Filaments and Dense Cores in the Cocoon Nebula (IC 5146)

We present 850 μm linear polarization and C18O (3 − 2) and 13CO (3 − 2) molecular line observations toward the filaments (F13 and F13S) in the Cocoon Nebula (IC 5146) using the JCMT POL-2 and Heterodyne Array Receiver Program instruments. F13 and F13S are found to be thermally supercritical with identified dense cores along their crests. Our findings include that the polarization fraction decreases in denser regions, indicating reduced dust grain alignment efficiency. The magnetic field vectors at core scales tend to be parallel to the filaments, but disturbed at the high density regions. Magnetic field strengths measured using the Davis–Chandrasekhar–Fermi method are 58 ± 31 and 40 ± 9 μG for F13 and F13S, respectively, and it reveals subcritical and sub-Alfvénic filaments, emphasizing the importance of magnetic fields in the Cocoon region. Sinusoidal C18O (3 − 2) velocity and density distributions are observed along the filaments’ skeletons, and their variations are mostly displaced by ∼1/4 × the wavelength of the sinusoid, indicating core formation occurred through the fragmentation of a gravitationally unstable filament, but with shorter core spacings than predicted. Large-scale velocity fields of F13 and F13S, studied using 13CO (3 − 2) data, present a V-shape transverse velocity structure. We propose a scenario for the formation and evolution of F13 and F13S, along with the dense cores within them. A radiation shock front generated by a B-type star collided with a sheet-like cloud about 1.4 Myr ago. The filaments became thermally critical due to mass infall through self-gravity ∼1 Myr ago, and subsequently, dense cores formed through gravitational fragmentation, accompanied by the disturbance of the magnetic field.

M17 MIR: A Massive Star Is Forming via Episodic Mass Accretion

We analyzed the Atacama Large Millimeter/submillimeter Array (ALMA) band 6 data for the outbursting massive protostar M17 MIR. The ALMA CO J = 2–1 data reveal a collimated and bipolar north–south outflow from M17 MIR. The blueshifted outflow exhibits four CO knots (N1 to N4) along the outflow axis, while the redshifted outflow appears as a single knot (S1). The extremely high velocity (EHV) emissions of N1 and S1 are jetlike and contain subknots along the outflow axis. Assuming the nearest EHV subknots trace the ejecta from the accretion outbursts in the past decades, a tangential ejection velocity of ∼421 km s−1 is derived for M17 MIR. Assuming the same velocity, the dynamical times of the multiple ejecta, traced by the four blueshifted CO knots, range from 20 to 364 yr. The four blueshifted CO knots imply four clustered accretion outbursts with a duration of tens of years in the past few hundred years. The intervals between the four clustered accretion outbursts are also about tens of years. These properties of the four clustered accretion outbursts are in line with the disk gravitational instability and fragmentation model. The episodic accretion history of M17 MIR traced by episodic outflow suggests that a massive star can form from a lower-mass protostar via frequent episodic accretion events triggered by disk gravitational instability and fragmentation. The first detection of the knotty outflow from an outbursting massive protostar suggests that mass ejections accompanied with accretion events could serve as an effective diagnostic tool for the episodic accretion histories of massive protostars.

The backreaction of stellar wobbling on accretion discs of massive protostars

black In recent years, it has been demonstrated that massive stars see their infant circumstellar medium shaped into a large irradiated, gravitationally unstable accretion disc during their early formation phase. Such discs constitute the gas reservoir from which nascent high-mass stars gain a substantial fraction of their mass by episodic accretion of dense gaseous circumstellar clumps, simultaneously undergoing accretion-driven bursts and producing close-orbit spectroscopic companions of the young high-mass stellar object. We aim to evaluate the effects of stellar motion caused by the disc non-axisymmetric gravitational field on the disc evolution and its spatial morphology. In particular, we analyse the disc's propensity to gravitational instability and fragmentation and the disc's appearance in synthetic millimetre band images pertinent to the ALMA facility. We employed three-dimensional radiation-hydrodynamical simulations of the surroundings of a young massive star in the non-inertial spherical coordinate system, adopting the highest spatial resolution to date and including the indirect star-disc gravitational potential caused by the asymmetries in the circumstellar disc. The resulting disc configurations were post-processed with the radiation transfer tool RADMC-3D and CASA software to obtain synthetic images of the disc. We confirm that the early evolution of the accretion disc is notably different when stellar wobbling is taken into account. The redistribution of angular momentum in the system makes the disc smaller and rounder, reduces the number of circumstellar gaseous clumps formed via disc gravitational fragmentation, and prevents the ejection of gaseous clumps from the disc. The synthetic predictive images at millimetre wavelengths of the accretion disc that includes stellar wobbling are in better agreement with the observations of the surroundings of massive young stellar objects, namely black AFGL 4176 mm1, G17.64+0.16, and G353.273 than our simulations of numerical hydrodynamics that omit this physical mechanism. Our work confirms that stellar wobbling is an essential ingredient to account for in numerical simulations of accretion discs of massive protostars.

Transport-driven super-Jeans fragmentation in dynamical star-forming regions

ABSTRACT The Jeans criterion is one cornerstone in our understanding of gravitational fragmentation. A critical limitation of the Jeans criterion is that the background density is assumed to be a constant, which is often not true in dynamic conditions such as star-forming regions. For example, during the formation phase of the high-density gas filaments in a molecular cloud, a density increase rate $\dot{\rho }$ implies a mass accumulation time of $t_{\rm acc}= \rho / \dot{\rho }= - \rho (\nabla \cdot (\rho \vec{v}))^{-1}$. The system is non-stationary when the mass accumulation time becomes comparable to the free-fall time $t_{\rm ff} = 1 / \sqrt{G \rho }$. We study fragmentation in non-stationary settings, and find that accretion can significantly increase in the characteristic mass of gravitational fragmentation (λJeans, aac = λJeans(1 + tff/tacc)1/3, $m_{\rm Jeans,\, acc} = m_{\rm Jeans} (1 + t_{\rm ff} / t_{\rm acc})$). In massive star-forming regions, this mechanism of transport-driven super-Jeans fragmentation can contribute to the formation of massive stars by causing order-of-magnitude increases in the mass of the fragments.

Formation of a wide-orbit giant planet in a gravitationally unstable subsolar-metallicity protoplanetary disc

ABSTRACT Direct imaging observations of planets revealed that wide-orbit (>10 au) giant planets exist even around subsolar-metallicity host stars and do not require metal-rich environments for their formation. A possible formation mechanism of wide-orbit giant planets in subsolar-metallicity environments is the gravitational fragmentation of massive protoplanetary discs. Here, we follow the long-term evolution of the disc for 1 Myr after its formation, which is comparable to disc lifetime, by way of a two-dimensional thin-disc hydrodynamic simulation with the metallicity of 0.1 $\rm {Z}_{\odot }$. We find a giant protoplanet that survives until the end of the simulation. The protoplanet is formed by the merger of two gaseous clumps at ∼0.5 Myr after disc formation, and then it orbits ∼200 au from the host star for ∼0.5 Myr. The protoplanet’s mass is ∼10 MJ at birth and gradually decreases to 1 MJ due to the tidal effect from the host star. The result provides the minimum mass of 1 MJ for the protoplanet in our simulation. We anticipate that the mass of a protoplanet experiencing reduced mass loss thanks to the protoplanetary contraction in higher resolution simulations can increase to ∼10 MJ. We argue that the disc gravitational fragmentation would be a promising pathway to form wide-orbit giant planets with masses of ≥1 MJ in subsolar-metallicity environments.

Kinematics and stability of high-mass protostellar disk candidates at sub-arcsecond resolution

Context. The fragmentation mode of high-mass molecular clumps and the accretion processes that form the most massive stars (M ≳ 8 M⊙) are still not well understood. A growing number of case studies have found massive young stellar objects (MYSOs) to harbour disk-like structures, painting a picture that the formation of high-mass stars may proceed through disk accretion, similar to that of lower-mass stars. However, the properties of such structures have yet to be uniformly and systematically characterised. Aims. The aim of this work is to uniformly study the kinematic properties of a large sample of MYSOs and characterise the stability of possible circumstellar disks against gravitational fragmentation. Methods. We have undertaken a large observational programme (CORE) making use of interferometric observations from the Northern Extended Millimetre Array (NOEMA) for a sample of 20 luminous (L > 104 L⊙) protostellar objects in the 1.37 mm wavelength regime in both continuum and spectral line emission, reaching 0.4″ resolution (800 au at 2 kpc). Results. We present the gas kinematics of the full sample and detect dense gas emission surrounding 15 regions within the CORE sample. Using the dense gas tracer CH3CN, we find velocity gradients across 13 cores perpendicular to the directions of bipolar molecular outflows, making them excellent disk candidates. The extent of the CH3CN emission tracing the disk candidates varies from 1800 to 8500 au. Analysing the free-fall to rotational timescales, we find that the sources are rotationally supported. The rotation profiles of some disk candidates are well described by differential rotation while for others the profiles are poorly resolved. Fitting the velocity profiles with a Keplerian model, we find protostellar masses in the range of ~ 10–25 M⊙. Modelling the level population of CH3CN (12K–11K) K = 0–6 lines, we present temperature maps and find median temperature in the range 70–210 K with a diversity in distributions. Radial profiles of the specific angular momentum (j) for the best disk candidates span a range of 1–2 orders of magnitude, on average ~10−3 km s−1 pc, and they follow j ∝ r1.7, which is consistent with a poorly resolved rotating and infalling envelope-disk model. Studying the Toomre stability of the disk candidates, we find almost all (11 out of 13) disk candidates to be prone to fragmentation due to gravitational instabilities at the scales probed by our observations, as a result of their high disk to stellar mass ratio. In particular, disks with masses greater than ~ 10–20% of the mass of their host (proto)stars are Toomre unstable, and more luminous YSOs tend to have disks that are more massive compared to their host star and hence more prone to fragmentation. Conclusions. In this work, we show that most disk structures around high-mass YSOs are prone to disk fragmentation early in their formation due to their high disk to stellar mass ratio. This impacts the accretion evolution of high-mass protostars which will have significant implications for the formation of the most massive stars.

On the origin of planetary-mass objects in NGC 1333

ABSTRACT The dominant formation mechanism of brown dwarfs and planetary-mass objects (PMOs) in star-forming regions is presently uncertain. Do they form like stars, via the collapse and fragmentation of cores in giant molecular clouds, or do they form like planets in the discs around stars and are ejected via dynamical interactions? In this paper, we quantify the spatial distribution of substellar objects in NGC 1333, in particular focusing on PMOs that have been the target of recent deep imaging observations. We find that these objects have a spatial distribution that is indistinguishable from the stars, and more massive brown dwarfs. We also analyse N-body simulations and find that a population of ejected planets would have a significantly different spatial and kinematic distribution from stars, and brown dwarfs that also formed through gravitational collapse and fragmentation. We therefore conclude that the low-mass substellar objects in NGC 1333 formed more like stars than planets, although we predict that a population of hitherto undetected ejected PMOs may be lurking in this and other star-forming regions.

The kinetic and magnetic energy budget of hub-filament systems during the gravitational fragmentation of molecular clouds

ABSTRACT We present a numerical study of the balance between the gravitational (Eg), kinetic (Ek), and magnetic (Em) energies of structures within a hub-filament system in a simulation of the formation and global hierarchical collapse (GHC) of a giant molecular cloud. For structures defined by various density thresholds, and at different evolutionary stages, we investigate the scaling of the virial parameter, α, with mass M, and of the Larson ratio, ${\cal {L}}_{\rm v}\equiv \sigma _{\rm v}/R^{1/2}$, with column density Σ, where σv is the 1D velocity dispersion, and R is an effective radius. We also investigate these scalings for the corresponding magnetic parameters αm and ${\cal {L}}_{\rm {m}}$. Finally, we compare our numerical results with an observational sample of massive clumps. We find that: 1) αm and ${\cal {L}}_{\rm {m}}$ follow similar α–M and ${\cal {L}}$–Σ scalings as their kinetic counterparts, although the ratio Em/Ek decreases as |Eg| increases. 2) The largest objects, defined by the lowest thresholds, tend to appear gravitationally bound (and magnetically supercritical), while their internal substructures tend to appear unbound (and subcritical). This suggests that the latter are being compressed by the infall of their parent structures, and supports earlier suggestions that the measured mass-to-magnetic flux ratio μ decreases inwards in a centrally-peaked cloud under ideal MHD. 3) The scatter in the α–M and ${\cal {L}}$–Σ plots is reduced when Ek and Em are plotted directly against Eg, suggesting that the scatter is due to an ambiguity between mass and size. 4) The clumps in our GHC simulation follow the same trends as the observational sample of massive clumps in the ${\cal {L}}$–Σ and α–M diagrams. We conclude that the main controlling parameter of the energy budget in the structures is Eg, with the kinetic and magnetic energies being derived from it.

3D Radiation Hydrodynamic Simulations of Gravitational Instability in AGN Accretion Disks: Effects of Radiation Pressure

We perform 3D radiation hydrodynamic local shearing-box simulations to study the outcome of gravitational instability (GI) in optically thick active galactic nuclei (AGNs) accretion disks. GI develops when the Toomre parameter Q T ≲ 1, and may lead to turbulent heating that balances radiative cooling. However, when radiative cooling is too efficient, the disk may undergo runaway gravitational fragmentation. In the fully gas-pressure-dominated case, we confirm the classical result that such a thermal balance holds when the Shakura–Sunyaev viscosity parameter (α) due to the gravitationally driven turbulence is ≲0.2, corresponding to dimensionless cooling times Ωt cool ≳ 5. As the fraction of support by radiation pressure increases, the disk becomes more prone to fragmentation, with a reduced (increased) critical value of α (Ωt cool). The effect is already significant when the radiation pressure exceeds 10% of the gas pressure, while fully radiation-pressure-dominated disks fragment at t cool ≲ 50 Ω−1. The latter translates to a maximum turbulence level α ≲ 0.02, comparable to that generated by magnetorotational instability. Our results suggest that gravitationally unstable (Q T ∼ 1) outer regions of AGN disks with significant radiation pressure (likely for high/near-Eddington accretion rates) should always fragment into stars, and perhaps black holes.

Gravito-turbulence in local disc simulations with an adaptive moving mesh

ABSTRACTSelf-gravity plays an important role in the evolution of rotationally supported systems such as protoplanetary discs, accretion discs around black holes, or galactic discs, as it can both feed turbulence and lead to gravitational fragmentation. While such systems can be studied in the shearing box approximation with high local resolution, the large density contrasts that are possible in the case of fragmentation still limit the utility of Eulerian codes with constant spatial resolution. In this paper, we present a novel self-gravity solver for the shearing box based on the TreePM method of the moving-mesh code arepo. The spatial gravitational resolution is adaptive, which is important to make full use of the quasi-Lagrangian hydrodynamical resolution of the code. We apply our new implementation to two- and three-dimensional, self-gravitating discs combined with a simple β-cooling prescription. For weak cooling we find a steady, gravito-turbulent state, while for strong cooling the formation of fragments is inevitable. To reach convergence for the critical cooling efficiency above which fragmentation occurs, we require a smoothing of the gravitational force in the two-dimensional case that mimics the stratification of the three-dimensional simulations. The critical cooling efficiency we find, β ≈ 3, as well as the box-averaged quantities characterizing the gravito-turbulent state, agrees well with various previous results in the literature. Interestingly, we observe stochastic fragmentation for β > 3, which slightly decreases the cooling efficiency required to observe fragmentation over the lifetime of a protoplanetary disc. The numerical method outlined here appears well suited to study the problem of galactic discs as well as the magnetized, self-gravitating discs.

The Role of Filamentary Structures in the Formation of Two Dense Cores, L1544 and L694-2

We present mapping results of two prestellar cores, L1544 and L694-2, embedded in filamentary clouds in C18O (3–2), 13CO (3–2), 12CO (3–2), HCO+ (4–3), and H13CO+ (4–3) lines with the James Clerk Maxwell Telescope to examine the role of the filamentary structures in the formation of dense cores in the clouds, with new distance estimates for L1544 ( pc) and L694-2 ( pc). From these observations, we found that the nonthermal velocity dispersion of two prestellar cores and their surrounding clouds is smaller than or comparable to the sound speed. This may indicate that the turbulence has already been dissipated for both filaments and cores during their formation time. We also found a λ/4 shift between the periodic oscillations in the velocity and the column density distributions, implying the possible presence of gravitational core-forming flow motion along the axis of the filament. The mass accretion rates due to these flow motions are estimated to be 2–3 M ⊙ Myr−1, being comparable to that for Serpens cloud but much smaller than those for the Hub filaments, cluster, or high-mass forming filaments by 1 or 2 orders of magnitude. From this study, we suggest that the filaments in our targets might be formed from the shock compression of colliding clouds, and then the cores are formed by gravitational fragmentation of the filaments to evolve to the prestellar stage. We conclude that the filamentary structures in the clouds play an important role in the entire process of formation of dense cores and their evolution.

Protostellar-disc fragmentation across all metallicities

ABSTRACT Cosmic metallicity evolution possibly creates the diversity of star formation modes at different epochs. Gravitational fragmentation of circumstellar discs provides an important formation channel of multiple star systems, including close binaries. We here study the nature of disc fragmentation, systematically performing a suite of 2D radiation-hydrodynamic simulations, in a broad range of metallicities, from the primordial to the solar values. In particular, we follow relatively long-term disc evolution over 15 kyr after the disc formation, incorporating the effect of heating by the protostellar irradiation. Our results show that the disc fragmentation occurs at all metallicities 1–$0 \, \rm {Z}_{\odot }$, yielding self-gravitating clumps. Physical properties of the clumps, such as their number and mass distributions, change with the metallicity due to different gas thermal evolution. For instance, the number of clumps is the largest for the intermediate metallicity range of 10−2–$10^{-5} \, \rm {Z}_{\odot }$, where the dust cooling is effective exclusively in a dense part of the disc and causes the fragmentation of spiral arms, although the disc might fragment at a similar rate, also at lower metallicities 10−6–$0 \, \rm {Z}_{\odot }$ with higher spatial resolution. The disc fragmentation is more modest for 1–$0.1 \, \rm {Z}_{\odot }$, thanks to the disc stabilization by the stellar irradiation. Such metallicity dependence agrees with the observed trend that the close binary fraction increases with decreasing metallicity in the range of 1–$10^{-3} \, \rm {Z}_{\odot }$.

Chaotic and Clumpy Galaxy Formation in an Extremely Massive Reionization-era Halo

Abstract The SPT 0311–58 system at z = 6.900 is an extremely massive structure within the reionization epoch and offers a chance to understand the formation of galaxies at an extreme peak in the primordial density field. We present 70 mas Atacama Large Millimeter/submillimeter Array observations of the dust continuum and [C ii] 158 μm emission in the central pair of galaxies and reach physical resolutions of ∼100–350 pc, among the most detailed views of any reionization-era system to date. The observations resolve the source into at least a dozen kiloparsec-size clumps. The global kinematics and high turbulent velocity dispersion within the galaxies present a striking contrast to recent claims of dynamically cold thin-disk kinematics in some dusty galaxies just 800 Myr later at z ∼ 4. We speculate that both gravitational interactions and fragmentation from massive parent disks have likely played a role in the overall dynamics and formation of clumps in the system. Each clump individually is comparable in mass to other 6 < z < 8 galaxies identified in rest-UV/optical deep field surveys, but with star formation rates elevated by a factor of ~3-5. Internally, the clumps themselves bear close resemblance to greatly scaled-up versions of virialized cloud-scale structures identified in low-redshift galaxies. Our observations are qualitatively similar to the chaotic and clumpy assembly within massive halos seen in simulations of high-redshift galaxies.

Evolution of the Angular Momentum during Gravitational Fragmentation of Molecular Clouds*

Abstract We investigate the origin of the observed scaling j ∼ R 3/2 between the specific angular momentum j and the radius R of molecular clouds (MCs) and their their substructures, and of the observed near independence of β, the ratio of rotational to gravitational energy, from R. To this end, we measure the angular momentum (AM) of “Lagrangian” particle sets in a smoothed particle hydrodynamics (SPH) simulation of the formation, collapse, and fragmentation of giant MCs. The Lagrangian sets are initially defined as connected particle sets above a certain density threshold at a certain time t def, and then the same set of SPH particles is followed either forward or backward in time. We find the following. (i) The Lagrangian particle sets evolve along the observed j–R relation when the volume containing them also contains a large number of other “intruder” particles. Otherwise, they evolve with j ∼ cst. (ii) Tracking Lagrangian sets to the future, we find that a subset of the SPH particles participates in the collapse, while the rest disperses away. (iii) These results suggest that the Lagrangian sets of fluid particles exchange their AM with other neighboring fluid particles via turbulent viscosity. (iv) We conclude that the j–R relation arises from a global tendency toward gravitational contraction, mediated by AM loss via turbulent viscosity, which induces fragmentation into dense, low-AM clumps, and diffuse, high-AM envelopes, which disperse away, limiting the mass efficiency of the fragmentation.

Disk fragmentation in high-mass star formation

Context. Increasing evidence suggests that, similar to their low-mass counterparts, high-mass stars form through a disk-mediated accretion process. At the same time, formation of high-mass stars still necessitates high accretion rates, and hence, high gas densities, which in turn can cause disks to become unstable against gravitational fragmentation. Aims. We study the kinematics and fragmentation of the disk around the high-mass star forming region AFGL 2591-VLA 3 which was hypothesized to be fragmenting based on the observations that show multiple outflow directions. Methods. We use a new set of high-resolution (0′′.19) IRAM/NOEMA observations at 843 μm towards VLA 3 which allow us to resolve its disk, characterize the fragmentation, and study its kinematics. In addition to the 843 μm continuum emission, our spectral setup targets warm dense gas and outflow tracers such as HCN, HC3N and SO2, as well as vibrationally excited HCN lines. Results. The high resolution continuum and line emission maps reveal multiple fragments with subsolar masses within the inner ~1000 AU of VLA 3. Furthermore, the velocity field of the inner disk observed at 843 μm shows a similar behavior to that of the larger scale velocity field studied in the CORE project at 1.37 mm. Conclusions. We present the first observational evidence for disk fragmentation towards AFGL 2591-VLA 3, a source that was thought to be a single high-mass core. While the fragments themselves are low-mass, the rotation of the disk is dominated by the protostar with a mass of 10.3 ± 1.8 M⊙. These data also show that NOEMA Band 4 can obtain the highest currently achievable spatial resolution at (sub-)mm wavelengths in observations of strong northern sources.

GRB variabilities and following gravitational waves induced by gravitational instability in NDAFs

ABSTRACT The present work proposes a new formalism for the inner regions of a neutrino-dominated accretion flows (NDAFs) by considering the self-gravity, where the neutrino opacity is high enough to make neutrinos trapped becoming a dominant factor in the transportation of energy and angular momentum over the magneto rotational instability. We investigate the possibility of gravitational instability and fragmentation to model the highly variable structure of the prompt emission in gamma-ray bursts (GRBs). The results lead us to introduce the gravitational instability, in these inner regions, as a source of a new viscosity that is of the same functional form as that of the β-prescription of viscosity. Such a consideration brings about fragmentation in the unstable inner disc. In addition, we find the consequent clumpy structure of this area capable to account for the temporal variability of GRB’s light curve, especially for the lower choices of the parameter β, ∼10−5. Finally, we predict the formation of gravitational waves through the migration of fragments before being tidally disrupted. These waves appear to be detectable via a range of current and future detectors from LIGO to Cosmic Explorer.

Gravitational fragmentation of extremely metal-poor circumstellar discs

ABSTRACT We study the gravitational fragmentation of circumstellar discs accreting extremely metal-poor ($Z \le 10^{-3}\, \mathrm{Z}_{\odot }$) gas, performing a suite of 3D hydrodynamic simulations using the adaptive mesh refinement code enzo. We systematically follow the long-term evolution for 2 × 103 yr after the first protostar’s birth, for the cases of Z = 0, 10−5, 10−4, and $10^{-3}\, \mathrm{Z}_{\odot }$. We show that evolution of number of self-gravitating clumps qualitatively changes with Z. Vigorous fragmentation induced by dust cooling occurs in the metal-poor cases, temporarily providing ∼10 self-gravitating clumps at Z = 10−5 and $10^{-4}\, \mathrm{Z}_{\odot }$. However, we also show that the fragmentation is a very sporadic process; after an early episode of the fragmentation, the number of clumps continuously decreases as they merge away in these cases. The vigorous fragmentation tends to occur later with the higher Z, reflecting that the dust-induced fragmentation is most efficient at the lower density. At $Z = 10^{-3}\, \mathrm{Z}_{\odot }$, as a result, the clump number stays smallest until the disc fragmentation starts in a late stage. We also show that the clump mass distribution depends on the metallicity. A single or binary clump substantially more massive than the others appear only at $Z = 10^{-3}\, \mathrm{Z}_{\odot }$, whereas they are more evenly distributed in mass at the lower metallicities. We suggest that the disc fragmentation should provide the stellar multiple systems, but their properties drastically change with a tiny amount of metals.

Features of the Dynamical Evolution of a Massive Disk of Trans-Neptunian Objects

—The dynamical features of a massive disk of distant trans-Neptunian objects are considered in the model of the formation of small bodies in the Hill region of a giant gas-dust clump that arose as a result of gravitational instability and fragmentation of the protoplanetary disk. The dynamical evolution of the orbits of small bodies under the action of gravitational perturbations from the outer planets and self-gravity of the disk has been studied for a time interval of the order of a billion years. It is shown that the secular effects of the gravitational influence of a massive disk of small bodies lead to an increase in the eccentricities of the orbits of individual objects. The result of this dynamical behavior is the creation of a flux of small bodies coming close to the orbit of Neptune. The change in the number of objects surviving in the observable region of distant trans-Neptunian objects (the region of orbits with perihelion distances of 40 < q < 80 AU and semimajor axes 150 < a < 1000 AU), over time depends on the initial mass of the disk. For disks with masses exceeding several Earth masses, there is a tendency to a decrease in the number of distant trans-Neptunian objects surviving in the observable region after evolution for a time interval of the order of the age of the Solar System, with an increase in the initial mass. On the other hand, for most objects, orbital eccentricities decrease under the influence of the self-gravity of the disk. Therefore, the main part of the disk is preserved in the region of heliocentric distances exceeding 100 AU.

Testing the effect of resolution on gravitational fragmentation with Lagrangian hydrodynamic schemes

ABSTRACT To study the resolution required for simulating gravitational fragmentation with newly developed Lagrangian hydrodynamic schemes, meshless finite-volume method (MFV) and meshless finite-mass method, we have performed a number of simulations of the Jeans test and compared the results with both the expected analytical solution and results from the more standard Lagrangian approach: smoothed particle hydrodynamics (SPH). We find that the different schemes converge to the analytical solution when the diameter of a fluid element is smaller than a quarter of the Jeans wavelength, λJ. Among the three schemes, SPH/MFV shows the fastest/slowest convergence to the analytical solution. Unlike the well-known behaviour of Eulerian schemes, none of the Lagrangian schemes investigated displays artificial fragmentation when the perturbation wavelength, λ, is shorter than λJ, even at low numerical resolution. For larger wavelengths (λ &amp;gt; λJ), the growth of the perturbation is delayed when it is not well resolved. Furthermore, with poor resolution, the fragmentation seen with the MFV scheme proceeds very differently compared to the converged solution. All these results suggest that, when unresolved, the ratio of the magnitude of hydrodynamic force to that of self-gravity at the sub-resolution scale is the largest/smallest in MFV/SPH, the reasons for which we have discussed in detail. These tests are repeated to investigate the effect of kernels of higher order than the fiducial cubic spline. Our results indicate that the standard deviation of the kernel is a more appropriate definition of the ‘size’ of a fluid element than its compact support radius.