Final Eccentricity Research Articles

White dwarf eccentricity fluctuation and dissipation by AGB convection

ABSTRACT Millisecond pulsars with white dwarf companions have typical eccentricities $e\sim 10^{-6}{\!-\!}10^{-3}$. The eccentricities of helium white dwarfs are explained well by applying the fluctuation–dissipation theorem to convective eddies in their red giant progenitors. We extend this theory to more massive carbon–oxygen (CO) white dwarfs with asymptotic giant branch (AGB) progenitors. Due to the radiation pressure in AGB stars, the dominant factor in determining the remnant white dwarf’s eccentricity is the critical residual hydrogen envelope mass $m_{\rm env}$ required to inflate the star to giant proportions. Using a suite of mesa stellar evolution simulations with $\Delta m_{\rm c}=10^{-3}\, {\rm M}_{\odot }$ core-mass intervals, we resolved the AGB thermal pulses and found that the critical $m_{\rm env}\propto m_{\rm c}^{-6}$. The resulting eccentricity $e\sim 3\times 10^{-3}$ is almost independent of the remnant CO white dwarf’s mass $m_{\rm c}$. Nearly all of the measured eccentricities lie below this robust theoretical limit, indicating that the eccentricity is damped during the common-envelope inspiral that follows the unstable Roche lobe overflow of the AGB star. Specifically, we focused on white dwarfs with median masses $m_{\rm c}\gt 0.6\, {\rm M}_{\odot }$. These massive white dwarfs begin their inspiral with practically identical orbital periods and eccentricities, eliminating any dependence on the initial conditions. For this sub-sample, we find an empirical relation $e\propto P^{3/2}$ between the final period and eccentricity that is much tighter than previous studies – motivating theoretical work on the eccentricity evolution during the common envelope phase. The eccentricities of lower mass CO white dwarfs may be explained by alternative formation channels.

Comparison of interventional treatment options for coronary calcified nodules: A sub-analysis of the ROTA.shock trial

BackgroundThe optimal treatment for coronary calcified nodules (CNs) is still unclear. The aim of this study was to compare the modification of these lesions by coronary intravascular lithotripsy (IVL) and rotational atherectomy (RA) using optical coherence tomography (OCT). MethodsROTA.shock was a 1:1 randomized, prospective, double-arm multi-center non-inferiority trial that compared the use of IVL and RA with percutaneous coronary intervention (PCI) in severely calcified lesions. In 19 of the patients out of this study CNs were detected by OCT in the target lesion and were treated by either IVL or RA. ResultsThe mean angle of CNs was significantly larger in final OCT scans than before RA (92 ± 17° vs. 68 ± 7°; p = 0.01) and IVL (89 ± 18° vs. 60 ± 10°; p = 0.03). The CNs were thinner upon final scans than in initial native scans (RA: 17.8 ± 7.8 mm vs. 38.6 ± 13.1 mm; p = 0.02; IVL: 16.5 ± 9.0 mm vs. 37.2 ± 14.3 mm; p = 0.02). Nodule volume did not differ significantly between native and final OCT scans (RA: 0.66 ± 0.12 mm3 vs. 0.61 ± 0.33 mm3; p = 0.68; IVL: 0.64 ± 0.19 mm3 vs. 0.68 ± 0.22 mm3; p = 0.74). Final stent eccentricity was high with 0.62 ± 0.10 after RA and 0.61 ± 0.09 after IVL. ConclusionRA or IVL are unable to reduce the volume of the calcified plaque. CN modulation seems to be mainly induced by the stent implantation and not by RA or IVL.

Design and research of a heavy load fatigue test system based on hydraulic control for full-size marine low-speed diesel engine bearings

As the power density design indicators of marine low-speed diesel engines gradually increased, resulting the loads of marine low-speed diesel engine bearings become increasingly complex. It's easy to cause crankshaft fatigue damage, leading to a very significant economic loss. Due to the maximum thrust loads of full-size-bearing is larger than 2000kN, the bearings performance tests are very expensive on a complete diesel engine. So, newly developed bearings must be verified on a specialized test rig. The research goal is to design a novel fatigue life test system (Bearing diameter: 300mm–520mm, Speed: 5r/min-200r/min, Thrust: 0kN–2800kN) that can simulate the working conditions of full-size low-speed marine diesel engine bearings. Adopting the advanced hydraulic-based control passive pressurization technology. Multi-platform joint simulation technology is used to make sure that the maximum thrust of the test rig is proportional to the eccentricity of the eccentric shaft. The final eccentricity of the shaft is determined by 3mm for test rig. The results based on 475mm×150mm bearing is published for the first time. The test load can be well enveloped in the load of the target diesel engine, with maximum load fluctuation less than 0.2%. The test rig can meet the performance testing requirements of full-size bearings.

Adhesion dynamics under time-varying deposition: A study on robotic assisted extrusion

Recent advances in robotic assisted-additive manufacturing (RA-AM) have enabled rapid material extrusion-based processing with comprehensive data collection. The following study investigates the adhesion dynamics of the initial printed layer across parameters such as surface energies, stand-off heights, and extrusion speeds of up to 100 mm/s, using an applied in-situ thermal analysis technique. Observations indicate that the characteristic length parameter, Lc < 0.05 mm, is adequate in anchoring the thermal melt, which adheres to the substrate when the nozzle proximity to the surface increases. Up to 100% molten area is contacting the surface prior to translation, and a final eccentricity over 0.85 has been observed. Through an analysis of variance, operational parameters of lower nozzle heights, printing speeds, and higher surface energy were statistically significant. The resultant in-situ characterization-driven data, was used to train a convolutional neural network (CNN). The model tested at an accuracy of 90.9%, and was able to distinguish between failed prints and initially adhered structures.

Revisiting the common envelope evolution in binary stars: A new semianalytic model for N -body and population synthesis codes

We present a novel way of modeling common envelope evolution in binary and few-body systems. We consider the common envelope inspiral as driven by a drag force with a power-law dependence in relative distance and velocity. The orbital motion is resolved either by direct N-body integration or by solving the set of differential equations for the orbital elements as derived using perturbation theory. Our formalism can model the eccentricity during the common envelope inspiral, and it gives results consistent with smoothed particles hydrodynamical simulations. We apply our formalism to common envelope events from binary population synthesis models and find that the final eccentricity distribution resembles the observed distribution of post-common-envelope binaries. Our model can be used for time-resolved common-envelope evolution in population synthesis calculations or as part of binary interactions in direct N-body simulations of star clusters.

Investigation of stiffness and damping coefficients in fluid film bearing with bio-oils and mineral-based oil

Rotating machinery utilizes fluid film bearings because of their good load carrying capacity and inherent damping properties. Fluid film bearings offer stiffness and damping parameters to ensure dynamic stability. The damping and stiffness characteristics of oil film significantly change the critical speed and unbalance response of the rotor. The damping and stiffness coefficients (direct, cross-coupled) may lead to system instability, also known as self-excited vibrations. Bio-oils have good tribological and physiochemical properties. They are also environment-friendly and bio-degradable. In this work, bio-oils (rapeseed, palm olein, and soybean) are evaluated in terms of their dynamic characteristics to that of mineral-based SAE40 lubricant for journal-bearing applications. The analytical extraction of damping and stiffness coefficients is performed based on the experimental dynamic viscosity data at different temperatures. In comparison to SAE40, the bio-oils have shown improved responsiveness. Bio-oils showed lower cross-coupled stiffness coefficients than SAE40, the primary source of rotor instability. The main contribution of this study is that it provides damping and stiffness coefficients for various bio-oils at different eccentricity ratios and viscosities for different types of oils. This may be used to characterize the dynamic properties of the fluid film bearing under various operating circumstances. It is concluded that the difference in cross-coupled stiffness favors bio-oils. Whereas, the difference in cross-coupled damping favors SAE40 by around 6%. This discrepancy is decreased to less than 2% for all oil’s final eccentricity ratio values (bio-oils and SAE40).

Modelling and simulation of hot direct extrusion process for optimal product characteristics: Single and multi-response optimization approach

The study of eccentricity minimization in cylindrical products helps to reduce the mechanical vibrations and wear of related mechanical parts such as bearings, columns and gears which positively affects in maintenance costs savings and increasing production quality reliability. The main purpose of this paper is to investigate the effect of the eccentricity between the billet material and the die parts on the quality of the final product in the direct extrusion process. The input parameters to produce a cylindrical product shape are optimized in MINITAB based on Taguchi method and ANOVA. The selected material of the billet is the aluminium alloy AA2024, and the die material is Steel H13. The inputs parameters are the temperature, the die angle, the ram speed, and the presumed eccentricity. The finite element model of the process is simulated in DFORM-3D for providing the extrusion information such as the pressure, the effective stress and strain, the final product eccentricity, and the roundness error. The study is carried out on two cases of the presumed eccentricity in addition to a case of zero eccentricity. The single and multi-response optimizations are executed to obtain the optimum parameters for the minimum product eccentricity and roundness error.

Tidal Evolution of the Eccentric Moon around Dwarf Planet (225088) Gonggong

Abstract Recent astronomical observations revealed that (225088) Gonggong, a 1000 km sized trans-Neptunian dwarf planet, hosts an eccentric satellite, Xiangliu, with an eccentricity of approximately 0.3. As the majority of known satellite systems around trans-Neptunian dwarf planets have circular orbits, the observed eccentricity of the Gonggong–Xiangliu system may reflect the singular properties of the system. In this study, we assumed that the Gonggong–Xiangliu system formed via a giant impact and we investigated the subsequent secular tidal evolution of this system under the simplifying assumptions of homogeneous bodies and of zero orbital inclination. We conducted simulations of coupled thermal–orbital evolution using the Andrade viscoelastic model and included higher-order eccentricity functions. The distribution of the final eccentricity from a large number of simulations with different initial conditions revealed that the radius of Xiangliu is not larger than 100 km. We also derived the analytical solution of the semilatus rectum evolution, a function of the radius of Xiangliu. From the point of view of the final semilatus rectum, the radius of Xiangliu was estimated to be close to 100 km. Together with the results of the Hubble Space Telescope observations, our findings suggest Gonggong and Xiangliu have similar albedos.

Modelling punching shear failure at edge slab-column connections by means of nonlinear joint elements

This paper is concerned with the modelling of punching shear failure at edge columns of flat slabs without shear reinforcement. Punching failure at edge columns is much less researched than at interior columns despite typical buildings having more edge than interior columns. The paper uses nonlinear finite element analysis (NLFEA) to study the influence on punching resistance of parameters including column aspect ratio and loading eccentricity. The NLFEA is carried out using both 3D solid elements and a Joint Shell Punching Model (JSPM) which combines nonlinear shell elements with nonlinear joint elements that incorporate the failure criterion of the Critical Shear Crack Theory (CSCT). Both constant and varying loading eccentricities are investigated since near failure eccentricity at edge column connections typically reduces below that calculated with elastic analysis due to moment being redistributed from the support to span. This reduction in eccentricity is beneficial since punching resistance is shown to depend on the final loading eccentricity. Consequently, designing for punching on the basis of elastic edge column moments is overly conservative.

The eccentricity distribution of giant planets and their relation to super-Earths in the pebble accretion scenario

Observations of the population of cold Jupiter planets (r &gt;1 AU) show that nearly all of these planets orbit their host star on eccentric orbits. For planets up to a few Jupiter masses, eccentric orbits are thought to be the outcome of planet–planet scattering events taking place after gas dispersal. We simulated the growth of planets via pebble and gas accretion as well as the migration of multiple planetary embryos in their gas disc. We then followed the long-term dynamical evolution of our formed planetary system up to 100 Myr after gas disc dispersal. We investigated the importance of the initial number of protoplanetary embryos and different damping rates of eccentricity and inclination during the gas phase for the final configuration of our planetary systems. We constrained our model by comparing the final dynamical structure of our simulated planetary systems to that of observed exoplanet systems. Our results show that the initial number of planetary embryos has only a minor impact on the final orbital eccentricity distribution of the giant planets, as long as the damping of eccentricity and inclination is efficient. If the damping is inefficient (slow), systems with a larger initial number of embryos harbour larger average eccentricities. In addition, for slow damping rates, we observe that scattering events are already common during the gas disc phase and that the giant planets that formed in these simulations match the observed giant planet eccentricity distribution best. These simulations also show that massive giant planets (above Jupiter mass) on eccentric orbits are less likely to host inner super-Earths as they get lost during the scattering phase, while systems with less massive giant planets on nearly circular orbits should harbour systems of inner super-Earths. Finally, our simulations predict that giant planets are not single, on average, but they live in multi-planet systems.

Companion-driven evolution of massive stellar binaries

ABSTRACT At least $70\, {\rm per\, cent}$ of massive OBA-type stars reside in binary or higher order systems. The dynamical evolution of these systems can lend insight into the origins of extreme phenomena such as X-ray binaries and gravitational wave sources. In one such dynamical process, the Eccentric Kozai–Lidov (EKL) mechanism, a third companion star alters the secular evolution of a binary system. For dynamical stability, these triple systems must have a hierarchical configuration. We explore the effects of a distant third companion’s gravitational perturbations on a massive binary’s orbital configuration before significant stellar evolution has taken place (≤10 Myr). We include tidal dissipation and general relativistic precession. With large (38 000 total) Monte Carlo realizations of massive hierarchical triples, we characterize imprints of the birth conditions on the final orbital distributions. Specifically, we find that the final eccentricity distribution over the range of 0.1–0.7 is an excellent indicator of its birth distribution. Furthermore, we find that the period distributions have a similar mapping for wide orbits. Finally, we demonstrate that the observed period distribution for approximately 10-Myr-old massive stars is consistent with EKL evolution.

Comparative analysis of lunar capture braking method based on particle swarm optimization

Purpose The purpose of this paper is to verify the feasibility of lunar capture braking through three methods based on particle swarm optimization (PSO) and compare the advantages and disadvantages of the three strategies by analyzing the results of the simulation. Design/methodology/approach The paper proposes three methods to verify capture braking based on PSO. The constraints of the method are the final lunar orbit eccentricity and the height of the final orbit around the Moon. At the same time, fuel consumption is used as a performance indicator. Then, the PSO algorithm is used to optimize the track of the capture process and simulate the entire capture braking process. Findings The three proposed braking strategies under the framework of PSO algorithm are very effective for solving the problem of lunar capture braking. The simulation results show that the orbit in the opposite direction of the trajectory has the most serious attenuation at perilune, and it should consume the least amount of fuel in theoretical analysis. The methods based on the fixed thrust direction braking and thrust uniform rotation braking can better ensure the final perilune control accuracy and fuel consumption. As for practice, the fixed thrust direction braking method is better realized among the three strategies. Research limitations/implications The process of lunar capture is a complicated process, involving effective coordination between multiple subsystems. In this article, the main focus is on the correctness of the algorithm, and a simplified dynamic model is adopted. At the same time, because the capture time is short, the lunar curvature can be omitted. Furthermore, to better compare the pros and cons of different braking modes, some influence factors and perturbative forces are not considered, such as the Earth’s flatness, light pressure and system noise and errors. Practical implications This paper presents three braking strategies that can satisfy all the constraints well and optimize the fuel consumption to make the lunar capture more effective. The results of comparative analysis demonstrate that the three strategies have their own superiority, and the fixed thrust direction braking is beneficial to engineering realization and has certain engineering practicability, which can also provide reference for lunar exploration orbit design. Originality/value The proposed capture braking strategies based on PSO enable effective capture of the lunar module. During the lunar exploration, the capture braking phase determines whether the mission will be successful or not, and it is essential to control fuel consumption on the premise of accuracy. The three methods in this paper can be used to provide a study reference for the optimization of lunar capture braking.

Migration of planets in circumbinary discs

Aims. The discovery of planets in close orbits around binary stars raises questions about their formation. It is believed that these planets formed in the outer regions of the disc and then migrated through planet-disc interaction to their current location. Considering five different systems (Kepler-16, -34, -35, -38, and -413) we model planet migration through the disc, with special focus on the final orbital elements of the planets. We investigate how the final orbital parameters are influenced by the disc and planet masses. Methods. Using two-dimensional, locally isothermal, and viscous hydrodynamical simulations, we first model the disc dynamics for all five systems, followed by a study of the migration properties of embedded planets with different masses. To strengthen our results, we apply two grid-based hydrodynamical codes using different numerics (PLUTO and FARGO3D). Results. For all systems, we find that the discs become eccentric and precess slowly. We confirm the bifurcation feature in the precession period – gap-size diagram for different binary mass ratios. The Kepler-16, -35, -38, and -413 systems lie on the lower branch and Kepler-34 on the upper one. For systems with small binary eccentricity, we find a new non-monotonic, loop-like feature. In all systems, the planets migrate to the inner edge of the disc cavity. Depending on the planet-disc mass ratio, we observe one of two different regimes. Massive planets can significantly alter the disc structure by compressing and circularising the inner cavity and they remain on nearly circular orbits. Lower-mass planets are strongly influenced by the disc, their eccentricity is excited to high values, and their orbits are aligned with the inner disc in a state of apsidal corotation. In our simulations, the final locations of the planets are typically too large with respect to the observations because of the large inner gaps of the discs. The migrating planets in the most eccentric discs (around Kepler-34 and -413) show the largest final eccentricity in agreement with the observations.

Formation of solar system analogues – I. Looking for initial conditions through a population synthesis analysis

Population synthesis models of planetary systems developed during the last $\sim$15 years could reproduce several of the observables of the exoplanet population, and also allowed to constrain planetary formation models. We present our planet formation model, which calculates the evolution of a planetary system during the gaseous phase. The code incorporates relevant physical phenomena for the formation of a planetary system, like photoevaporation, planet migration, gas accretion, water delivery in embryos and planetesimals, a detailed study of the orbital evolution of the planetesimal population, and the treatment of the fusion between embryos, considering their atmospheres. The main goal of this work, unlike other works of planetary population synthesis, is to find suitable scenarios and physical parameters of the disc to form solar system analogs. We are specially interested in the final planet distributions, and in the final surface density, eccentricity and inclination profiles for the planetesimal population. These final distributions will be used as initial conditions for N-body simulations, to study the post-oligarchic formation in a second work. We then consider different formation scenarios, with different planetesimal sizes and different type I migration rates. We find that solar system analogs are favored in massive discs, with low type I migration rates, and small planetesimal sizes. Besides, those rocky planets within their habitables zones are dry when discs dissipate. At last, the final configurations of solar system analogs include information about the mass and semimajor-axis of the planets, water contents, and the properties of the planetesimal remnants.

Erratum: Cross-sections for planetary systems interacting with passing stars and binaries

Most planetary systems are formed within stellar clusters, and these environments can shape their properties. This paper considers scattering encounters between solar systems and passing cluster members, and calculates the corresponding interaction cross sections. The target solar systems are generally assumed to have four giant planets, with a variety of starting states, including circular orbits with the semimajor axes of our planets, a more compact configuration, an ultra-compact state with multiple mean motion resonances, and systems with massive planets. We then consider the effects of varying the cluster velocity dispersion, the relative importance of binaries versus single stars, different stellar host masses, and finite starting eccentricities of the planetary orbits. For each state of the initial system, we perform an ensemble of numerical scattering experiments and determine the cross sections for eccentricity increase, inclination angle increase, planet ejection, and capture. This paper reports results from over 2 million individual scattering simulations. Using supporting analytic considerations, and fitting functions to the numerical results, we find a universal formula that gives the cross sections as a function of stellar host mass, cluster velocity dispersion, starting planetary orbital radius, and final eccentricity. The resulting cross sections can be used in a wide variety of applications. As one example, we revisit constraints on the birth aggregate of our Solar System due to dynamical scattering and find $N<10^4$ (consistent with previous estimates).

ECCENTRIC JUPITERS VIA DISK–PLANET INTERACTIONS

Numerical hydrodynamics calculations are performed to determine conditions under which giant planet eccentricities can be excited by parent gas disks. Unlike in other studies, Jupiter-mass planets are found to have their eccentricities amplified --- provided their orbits start eccentric. We disentangle the web of co-rotation, co-orbital, and external resonances to show that this finite-amplitude instability is consistent with that predicted analytically. Ellipticities can grow until they reach of order the disk's aspect ratio, beyond which the external Lindblad resonances that excite eccentricity are weakened by the planet's increasingly supersonic epicyclic motion. Forcing the planet to still larger eccentricities causes catastrophic eccentricity damping as the planet collides into gap walls. For standard parameters, the range of eccentricities for instability is modest; the threshold eccentricity for growth ($\sim$$0.04$) is not much smaller than the final eccentricity to which orbits grow ($\sim$$0.07$). If this threshold eccentricity can be lowered (perhaps by non-barotropic effects), and if the eccentricity driving documented here survives in 3D, it may robustly explain the low-to-moderate eccentricities $\lesssim 0.1$ exhibited by many giant planets (including Jupiter and Saturn), especially those without planetary or stellar companions.

Testing eccentricity pumping mechanisms to model eccentric long-period sdB binaries with MESA

Hot subdwarf-B stars in long-period binaries are found to be on eccentric orbits, even though current binary-evolution theory predicts those objects to be circularised before the onset of Roche-lobe overflow (RLOF). We aim to find binary-evolution mechanisms that can explain these eccentric long-period orbits, and reproduce the currently observed period-eccentricity diagram. Three different processes are considered; tidally-enhanced wind mass-loss, phase-dependent RLOF on eccentric orbits and the interaction between a circumbinary disk and the binary. The binary module of the stellar-evolution code MESA (Modules for Experiments in Stellar Astrophysics) is extended to include the eccentricity-pumping processes. The effects of different input parameters on the final period and eccentricity of a binary-evolution model are tested with MESA. The end products of models with only tidally-enhanced wind mass-loss can indeed be eccentric, but these models need to lose too much mass, and invariably end up with a helium white dwarf that is too light to ignite helium. Within the tested parameter space, no sdBs in eccentric systems are formed. Phase-dependent RLOF can reintroduce eccentricity during RLOF, and could help to populate the short-period part of the period-eccentricity diagram. When phase-dependent RLOF is combined with eccentricity pumping via a circumbinary disk, the higher eccentricities can be reached as well. A remaining problem is that these models favour a distribution of higher eccentricities at lower periods, while the observed systems show the opposite. The models presented here are potentially capable of explaining the period-eccentricity distribution of long-period sdB binaries, but further theoretical work on the physical mechanisms is necessary.

Final source eccentricity measured by HBT interferometry with the event shape selection

Azimuthal angle dependence of the pion source radii has been measured applying the event shape engineering technique at the PHENIX experiment. When events with higher magnitude of second-order flow vector are selected, the oscillation of the source radii is enhanced as well as v2 which leads to the enhancement of the measured final source eccentricity. The event twist effect in the spatial source distribution in the final state has been also explored with AMPT model. Results indicate a possible twisted source due to the initial longitudinal fluctuations.

Precision tube production: Influencing the eccentricity and residual stresses by tilting and shifting

Tube drawing is considered to be one of the most effective and flexible methods to reduce easily tube dimensions, to improve the surface finish, and to design mechanical properties. However, during the production of pre-tubes, it is common to obtain a tube which could still remain eccentric despite being reduced during the drawing processes. This eccentricity causes an increase in weight since dimensioning has to be based on the weakest location of the tube. Moreover, high values of residual stresses can exist after the drawing process, which – in the case of tensile stresses – can promote failure of the tube subject to tension loading. In this paper, the effects of tilting the die and shifting the tube on the eccentricity and the residual stresses were experimentally and numerically determined using as-received tubes and after their subsequent processing. The relative eccentricity change was measured for each tilting angle and shifting value. It has been shown that when a tilted die is used, the final eccentricity varies depending on the tilting angle. For example, using an angle of 5° resulted in a decrease of about 40% compared to the standard condition. By increasing the tilting angle, the surface and sub-surface residual stresses can be decreased. A 3D model using ABAQUS software was developed and verified to describe the evolution of residual stresses and eccentricities.

Cross-sections for planetary systems interacting with passing stars and binaries

Most planetary systems are formed within stellar clusters, and these environments can shape their properties. This paper considers scattering encounters between solar systems and passing cluster members, and calculates the corresponding interaction cross sections. The target solar systems are generally assumed to have four giant planets, with a variety of starting states, including circular orbits with the semimajor axes of our planets, a more compact configuration, an ultra-compact state with multiple mean motion resonances, and systems with massive planets. We then consider the effects of varying the cluster velocity dispersion, the relative importance of binaries versus single stars, different stellar host masses, and finite starting eccentricities of the planetary orbits. For each state of the initial system, we perform an ensemble of numerical scattering experiments and determine the cross sections for eccentricity increase, inclination angle increase, planet ejection, and capture. This paper reports results from over 2 million individual scattering simulations. Using supporting analytic considerations, and fitting functions to the numerical results, we find a universal formula that gives the cross sections as a function of stellar host mass, cluster velocity dispersion, starting planetary orbital radius, and final eccentricity. The resulting cross sections can be used in a wide variety of applications. As one example, we revisit constraints on the birth aggregate of our Solar System due to dynamical scattering and find $N<10^4$ (consistent with previous estimates).