Angular Deficit Research Articles

Interactions among Noninteracting Particles in Planet Formation Simulations

Abstract Over the course of recent decades, N-body simulations have become a standard tool for quantifying the gravitational perturbations that ensue in planet-forming disks. Within the context of such simulations, massive non-central bodies are routinely classified into “big” and “small” particles, where big objects interact with all other objects self-consistently, while small bodies interact with big bodies but not with each other. Importantly, this grouping translates to an approximation scheme where the orbital evolution of small bodies is dictated entirely by the dynamics of the big bodies, yielding considerable computational advantages with little added cost in terms of astrophysical accuracy. Here we point out, however, that this scheme can also yield spurious dynamical behavior where, even in the absence of big bodies within a simulation, indirect coupling among small bodies can lead to excitation of the constituent “non-interacting” orbits. We demonstrate this self-stirring by carrying out a sequence of numerical experiments, and confirm that this effect is largely independent of the time-step or the integration algorithm employed. Furthermore, adopting the growth of angular momentum deficit as a proxy for dynamical excitation, we explore its dependence on time, the cumulative mass of the system, as well as the total number of particles present in the simulation. Finally, we examine the degree of such indirect excitation within the context of conventional terrestrial planet formation calculations, and conclude that although some level of caution may be warranted, this effect plays a negligible role in driving the simulated dynamical evolution.

Predicting the long-term stability of compact multiplanet systems

We combine analytical understanding of resonant dynamics in two-planet systems with machine-learning techniques to train a model capable of robustly classifying stability in compact multiplanet systems over long timescales of [Formula: see text] orbits. Our Stability of Planetary Orbital Configurations Klassifier (SPOCK) predicts stability using physically motivated summary statistics measured in integrations of the first [Formula: see text] orbits, thus achieving speed-ups of up to [Formula: see text] over full simulations. This computationally opens up the stability-constrained characterization of multiplanet systems. Our model, trained on ∼100,000 three-planet systems sampled at discrete resonances, generalizes both to a sample spanning a continuous period-ratio range, as well as to a large five-planet sample with qualitatively different configurations to our training dataset. Our approach significantly outperforms previous methods based on systems' angular momentum deficit, chaos indicators, and parametrized fits to numerical integrations. We use SPOCK to constrain the free eccentricities between the inner and outer pairs of planets in the Kepler-431 system of three approximately Earth-sized planets to both be below 0.05. Our stability analysis provides significantly stronger eccentricity constraints than currently achievable through either radial velocity or transit-duration measurements for small planets and within a factor of a few of systems that exhibit transit-timing variations (TTVs). Given that current exoplanet-detection strategies now rarely allow for strong TTV constraints [S. Hadden, T. Barclay, M. J. Payne, M. J. Holman, Astrophys. J. 158, 146 (2019)], SPOCK enables a powerful complementary method for precisely characterizing compact multiplanet systems. We publicly release SPOCK for community use.

The study of the generalized boson oscillator in a chiral conical space–time

Our work mainly study the relativistic generalized boson oscillator namely generalized Duffin–Kemmer–Petiau (DKP) oscillator with the function [Formula: see text] considered as the Cornell potential under the chiral conical space–time background. We obtain the wave function and energy spectrum of radial equation by using commonly used the Nikiforov–Uvarov method. It is shows that the energy spectrum of the generalized DKP oscillator depend explicitly on the angular deficit [Formula: see text], related rotation parameter [Formula: see text] and torsion parameter [Formula: see text], which characterize the global structure of the metric in the chiral conical space–time. In addition, the Cornell potential parameters [Formula: see text] have non-negligible influence on the energy spectrum of the studied systems.

Flyby encounters between two planetary systems II: exploring the interactions of diverse planetary system architectures

ABSTRACT Planetary systems formed in clusters may be subject to stellar encounter flybys. Here, we create a diverse range of representative planetary systems with different orbital scales and planets’ masses and examine encounters between them in a typical open cluster. We first explore the close-in multisuper Earth systems ≲0.1 au. They are resistant to flybys in that only ones inside a few au can destabilize a planet or break the resonance between such planets. But these systems may capture giant planets on to wide orbits from the intruding star during distant flybys. If so, the original close-in small planets’ orbits may be tilted together through Kozai–Lidov mechanism, forming a ‘cold’ system that is significantly inclined against the equator of the central host. Moving to the intermediately placed planets around solar-like stars, we find that the planets’ mass gradient governs the systems’ long-term evolution post-encounter: more massive planets have better chances to survive. Also, a system’s angular momentum deficit, a quantity describing how eccentric/inclined the orbits are, measured immediately after the encounter, closely relates to the longevity of the systems – whether or not and when the systems turn unstable in the ensuing evolution millions of years post-encounter. We compare the orbits of the surviving planets in the unstable systems through (1) the immediate consequence of the stellar fly or (2) internal interplanetary scattering long post-encounter and find that those for the former are systematically colder. Finally, we show that massive wide-orbit multiplanet systems like that of HR 8799 can be easily disrupted and encounters at a few hundreds of au suffice.

Study of the Dirac oscillator in the presence of vector and scalar potentials in the cosmic string space-time

In this paper, we use the functional Bethe ansatz method to solve the radial problem of the Dirac oscillator in cosmic string space-time, and its general solution under the Killingbeck potential plus isotonic oscillator potential in the limit of the spin and the pseudo-spin symmetries are further presented. Corresponding to the expressions of energies and wave function of bound state and first excited state are given. Furthermore, some particular cases including the Cornell potential, the Kratzer potential, the Killingbeck potential and the isotonic oscillator potentials are also addressed. It shows that the energy levels of the systems depend explicitly on the potential parameters [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and the angular deficit parameter [Formula: see text] which characterize topological defect.

Self-force from a conical singularity, without renormalization

We develop an approach to calculate the self-force on a charged particle held in place in a curved spacetime, in which the particle is attached to a massless string and the force is measured by the string's tension. The calculation is based on the Weyl class of static and axially symmetric spacetimes, and the presence of the string is manifested by a conical singularity; the tension is proportional to the angular deficit. A remarkable and appealing aspect of this approach is that the calculation of the self-force requires no renormalization of the particle's field. This is in contract with traditional methods, which incorporate a careful and elaborate subtraction of the singular part of the field. We implement the approach in a number of different situations. First, we examine the case of an electric charge in Schwarzschild spacetime, and recover the classic Smith-Will force in addition to a purely gravitational contribution to the self-force. Second, we turn to the case of electric and magnetic dipoles in Schwarzschild spacetime, and correct expressions for the self-force previously obtained in the literature. Third, we replace the electric charge by a scalar charge, and recover Wiseman's no-force result, which we generalize to a scalar dipole. And fourth, we calculate the force exerted on extended bodies such as Schwarzschild black holes and Janis-Newman-Winicour objects, which describe scalarized naked singularities.

Stability analysis of three exoplanet systems

ABSTRACT The orbital solutions of published multiplanet systems are not necessarily dynamically stable on time-scales comparable to the lifetime of the system as a whole. For this reason, dynamical tests of the architectures of proposed exoplanetary systems are a critical tool to probe the stability and feasibility of the candidate planetary systems, with the potential to point the way towards refined orbital parameters of those planets. Such studies can even help in the identification of additional companions in such systems. Here, we examine the dynamical stability of three planetary systems, orbiting HD 67087, HD 110014, and HD 133131A. We use the published radial velocity measurements of the target stars to determine the best-fitting orbital solutions for these planetary systems using the systemic console. We then employ the N-body integrator mercury to test the stability of a range of orbital solutions lying within 3σ of the nominal best fit for a duration of 100 Myr. From the results of the N-body integrations, we infer the best-fitting orbital parameters using the Bayesian package astroemperor. We find that both HD 110014 and HD 133131A have long-term stable architectures that lie within the 1σ uncertainties of the nominal best fit to their previously determined orbital solutions. However, the HD 67087 system exhibits a strong tendency towards instability on short time-scales. We compare these results to the predictions made from consideration of the angular momentum deficit criterion, and find that its predictions are consistent with our findings.

Exact solution of the (1+2)-dimensional generalized Kemmer oscillator in the cosmic string background with the magnetic field**Project supported by the National Natural Science Foundation of China (Grant Nos. 11465006 and 11565009).

We study a two-dimensional generalized Kemmer oscillator in the cosmic string spacetime with the magnetic field to better understand the contribution from gravitational field caused by topology defects, and present the exact solutions to the generalized Kemmer equation in the cosmic string with the Morse potential and Coulomb-liked potential through using the Nikiforov–Uvarov (NU) method and biconfluent Heun equation method, respectively. Our results give the topological defect’s correction for the wave function, energy spectrum and motion equation, and show that the energy levels of the generalized Kemmer oscillator rely on the angular deficit α connected with the linear mass density m of the cosmic string and characterized the metric’s structure in the cosmic string spacetime.

Normalized angular momentum deficit: a tool for comparing the violence of the dynamical histories of planetary systems

Context. Population studies of the orbital characteristics of exoplanets in multi-planet systems have highlighted the existence of an anticorrelation between the average orbital eccentricity of planets and the number of planets of their host system, that is, its multiplicity. This effect was proposed to reflect the varying levels of violence in the dynamical evolution of planetary systems. Aims. Previous work suggested that the relative violence of the dynamical evolution of planetary systems with similar orbital architectures can be compared through the computation of their angular momentum deficit (AMD). We investigated the possibility of using a more general metric to perform analogous comparisons between planetary systems with different orbital architectures. Methods. We considered a modified version of the AMD, the normalized angular momentum deficit (NAMD), and used it to study a sample of 99 multi-planet systems containing both the currently best-characterized extrasolar systems and the solar system, that is, planetary systems with both compact and wide orbital architectures. Results. We verified that the NAMD allows us to compare the violence of the dynamical histories of multi-planet systems with different orbital architectures. We identified an anticorrelation between the NAMD and the multiplicity of the planetary systems, of which the previously observed eccentricity–multiplicity anticorrelation is a reflection. Conclusions. Our results seem to indicate that phases of dynamical instabilities and chaotic evolution are not uncommon among planetary systems. They also suggest that the efficiency of the planetary formation process in producing high-multiplicity systems is likely to be higher than that suggested by their currently known population.

Vacuum polarization in the background of conical singularity

We consider the gravity-induced effects associated with a massless scalar field living in a higher-dimensional spacetime being the tensor product of Minkowski space and spherically-symmetric space with angle deficit. These spacetimes are considered as simple models for a multidimensional global monopole or cosmic string with flat extra dimensions, where the deficit of solid angle is proportional to Newton constant and tension. Thus, we refer to them as conical backgrounds. In terms of the angular deficit value, we derive the perturbative expression for the scalar Green’s function and compute it to the leading order. With the use of this Green’s function we compute the renormalized vacuum expectation value of the scalar-field’s energy-momentum tensor. We make some general note on the linear-on-curvature part of the trace of even coefficients of Schwinger-deWitt expansion.

Characteristic radiological findings for revision surgery after balloon kyphoplasty

Balloon kyphoplasty (BKP) sometimes fails to improve patients’ outcomes, with revision surgery, using anterior or posterior reconstruction, being required. The purpose of this study was to investigate the radiological risk factors of failure after BKP in the treatment of osteoporotic vertebral fractures (OVFs). This case-control study included 105 patients treated with single BKP and 14 patients who required revision BKP. We evaluated radiological findings differentiating both groups, using plain radiography and computed tomography, before BKP. Angular flexion-extension motion was significantly greater in the revision than BKP group. While the frequency of pedicle fracture and posterior wall injury was not different between the groups, a split type fracture was more frequent in the revision group. Split type fracture had the highest adjusted odds ratio (OR) for revision (16.5, p = 0.018). Angular motion ≥14° increased the risk for revision surgery by 6-fold (p = 0.013), with endplate deficit having an OR of revision of 5.0 (p = 0.032). The revision rate after BKP was 3.8%, with split type fracture, greater angular motion and large endplate deficit being risk factors for revision. Treatment strategies for patients with these risk factors should be carefully evaluated, considering the inherent difficulties in performing revision surgery after BKP.

Secular dynamics in extrasolar systems with two planets in mutually inclined orbits

AbstractWe revisit the problem of the secular dynamics in two-planet systems in which the planetary orbits exhibit a high value of the mutual inclination. We propose a ‘basic hamiltonian model’ for secular dynamics, parameterized in terms of the system’s Angular Momentum Deficit (AMD). The secular Hamiltonian can be obtained in closed form, using multipole expansions in powers of the distance ratio between the planets, or in the usual Laplace-Lagrange form. The main features of the phase space (number and stability of periodic orbits, bifurcations from the main apsidal corotation resonances, Kozai resonance etc.) can all be recovered by choosing the corresponding terms in the ‘basic Hamiltonian’. Applications include the semi-analytical determination of the actual orbital state of the system using Hamiltonian normalization techniques. An example is discussed referring to the system of two outermost planets of the ν-Andromedae system.

Low-eccentricity migration of ultra-short-period planets in multiplanet systems

ABSTRACT Recent studies suggest that ultra-short-period planets (USPs), Earth-sized planets with sub-day periods, constitute a statistically distinct sub-sample of Kepler planets: USPs have smaller radii (1–1.4R⊕) and larger mutual inclinations with neighbouring planets than nominal Kepler planets, and their period distribution is steeper than longer period planets. We study a ‘low-eccentricity’ migration scenario for the formation of USPs, in which a low-mass planet with initial period of a few days maintains a small but finite eccentricity due to secular forcings from exterior companion planets, and experiences orbital decay due to tidal dissipation. USP formation in this scenario requires that the initial multiplanet system have modest eccentricities (≳0.1) or angular momentum deficit. During the orbital decay of the innermost planet, the system can encounter several apsidal and nodal precession resonances that significantly enhance eccentricity excitation and increase the mutual inclination between the inner planets. We develop an approximate method based on eccentricity and inclination eigenmodes to efficiently evolve a large number of multiplanet systems over Gyr time-scales in the presence of rapid (as short as ∼100 yr) secular planet–planet interactions and other short-range forces. Through a population synthesis calculation, we demonstrate that the ‘low-e migration’ mechanism can naturally produce USPs from the large population of Kepler multis under a variety of conditions, with little fine-tuning of parameters. This mechanism favours smaller inner planets with more massive and eccentric companion planets, and the resulting USPs have properties that are consistent with observations.

Energy Dissipation in Large Collisions—No Change in Planet Formation Outcomes

Abstract It is often asserted that more accurate treatment of large collisions in planet formation simulations will lead to vastly different results—in particular a lower final angular momentum deficit (AMD—commonly used to measure orbital excitement). As nearly all simulations to date consider perfect merging (100% energy dissipation) during embryo–embryo collisions, and typically end up with an overexcited final terrestrial planetary system, it has been suggested that a better treatment of energy dissipation during large collisions could decrease the final dynamical excitation (or AMD). Although some work related to energy dissipation has been done (mostly during the runaway growth phase when planetesimals grow into protoplanets), this had never been fully tested in the post-runaway phase, where protoplanets (embryos) grow chaotically into planets via large collisions among themselves. In this work, we test varying amounts of energy dissipation within embryo–embryo collisions, by assuming a given coefficient of restitution for collisions. Our results show that varying the level of energy dissipated within embryo–embryo collisions do not play any important role in the final terrestrial planetary system. We have found a strong linear correlation in our results related to the final number of planets formed and the final AMD. Additionally, reproducing the current radial mass concentration of the terrestrial planets, even when starting from an annulus of material, is challenging when modeling growth from planetesimals to planets.

Secular spin-axis dynamics of exoplanets

Context. Seasonal variations and climate stability of a planet are very sensitive to the planet obliquity and its evolution. This is of particular interest for the emergence and sustainability of land-based life, but orbital and rotational parameters of exoplanets are still poorly constrained. Numerical explorations usually realised in this situation are therefore in heavy contrast with the uncertain nature of the available data. Aims. We aim to provide an analytical formulation of the long-term spin-axis dynamics of exoplanets, linking it directly to physical and dynamical parameters, but still giving precise quantitative results if the parameters are well known. Together with bounds for the poorly constrained parameters of exoplanets, this analysis is designed to enable a quick and straightforward exploration of the spin-axis dynamics. Methods. The long-term orbital solution is decomposed into quasi-periodic series and the spin-axis Hamiltonian is expanded in powers of eccentricity and inclination. Chaotic zones are measured by the resonance overlap criterion. Bounds for the poorly known parameters of exoplanets are obtained from physical grounds (rotational breakup) and dynamical considerations (equipartition of the angular momentum deficit). Results. This method gives accurate results when the orbital evolution is well known. The detailed structure of the chaotic zones for the solar system planets can be retrieved from simple analytical formulas. For less-constrained planetary systems, the maximal extent of the chaotic regions can be computed, requiring only the mass, the semi-major axis, and the eccentricity of the planets present in the system. Additionally, some estimated bounds of the precession constant allow to classify which observed exoplanets are necessarily out of major spin-orbit secular resonances (unless the precession rate is affected by the presence of massive satellites).

Secular Transport during Disk Dispersal: The Case of Kepler-419

Abstract Due to fortuitous circumstances, the two giant planets around Kepler-419 have well characterized three-dimensional orbits. They are nearly coplanar to each other; the inner one has a large eccentricity (≃0.82); and the apses of the two orbits librate around anti-alignment. Such a state defies available proposals for large eccentricities. We argue that it is instead uniquely produced by a decaying protoplanetary disk. When the disk was massive, its precessional effect on the planets forced the two apses to center around an anti-aligned state. And as the disk is gradually eroded, the pair of planets are adiabatically transported to a new state where most of the eccentricity (or rather, the angular momentum deficit) is transferred to the inner planet, and the two apses are largely anti-aligned. During this transport, any initial mutual inclination may be reduced or enhanced; either may be compatible with the current constraints. So a primordial disk can drive up planet eccentricities both in resonant planet pairs (as has been shown for GJ 876) and in secularly-interacting, non-resonant pairs. The mechanism discussed here may be relevant for forming hot Jupiters and for explaining the observed eccentricities of warm and cold giant planets.

The study of a half-spin relativistic particle in the rotating cosmic string space–time

In this paper, a charged half-spin particle depicted by the Dirac equation in the presence of a uniform magnetic field and a mixed potential are analyzed in the rotating cosmic string space–time. In order to facilitate this study, we assume that the symmetrical center of the potential is on the string and the magnetic field is parallel to the string. Based on the functional analysis method, we obtain the energy eigenvalues for different physical situations. It shows that the energy levels of the system depend explicitly on the angular deficit [Formula: see text] and the rotational parameter [Formula: see text] which characterize the global structure of the metric in the space–time of the rotating cosmic string.

Hill stability in the AMD framework

In a two-planet system, a topological boundary that is created by Sundman (1912, Acta Math., 36, 105) inequality can forbid close encounters between the two planets for an infinite time. A system is said to be Hill stable if it verifies this topological condition. Hill stability is widely used in the study of extrasolar planet dynamics. However, the coplanar and circular orbit approximation is often used. In this paper, we explain how the Hill stability can be understood in the framework of angular momentum deficit (AMD). In the secular approximation, AMD allows us to discriminate between a priori stable systems and systems for which a more in-depth dynamical analysis is required. We show that the general Hill stability criterion can be expressed as a function of only semimajor axes, masses, and total AMD of the system. The proposed criterion is only expanded in the planets-to-star mass ratio ε and not in the semimajor axis ratio, eccentricities, nor the mutual inclination. Moreover, the expansion in ε remains excellent up to values of about 10−3 even for two planets with very different mass values. We performed numerical simulations in order to highlight the sharp change of behavior between Hill stable and Hill unstable systems. We show that Hill stable systems tend to be very regular, whereas Hill unstable systems often lead to rapid planet collisions. We also note that Hill stability does not provide protection from the ejection of the outer planet.

A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems

The inclinations of exoplanets detected via radial velocity method are essentially unknown. We aim to provide estimations of the ranges of mutual inclinations that are compatible with the long-term stability of the system. Focusing on the skeleton of an extrasolar system, i.e., considering only the two most massive planets, we study the Hamiltonian of the three-body problem after the reduction of the angular momentum. Such a Hamiltonian is expanded both in Poincar\'e canonical variables and in the small parameter $D_2$, which represents the normalised Angular Momentum Deficit. The value of the mutual inclination is deduced from $D_2$ and, thanks to the use of interval arithmetic, we are able to consider open sets of initial conditions instead of single values. Looking at the convergence radius of the Kolmogorov normal form, we develop a reverse KAM approach in order to estimate the ranges of mutual inclinations that are compatible with the long-term stability in a KAM sense. Our method is successfully applied to the extrasolar systems HD 141399, HD 143761 and HD 40307.

Non-Abelian cosmic string in the Starobinsky model of gravity

In this paper, we analyze numerically the behavior of the solutions corresponding to a non-Abelian cosmic string in the framework of the Starobinsky model, i.e. where [Formula: see text]. We perform the calculations for both an asymptotically flat and asymptotically (anti)-de Sitter spacetimes. We found that the angular deficit generated by the string decreases as the parameter [Formula: see text] increases, in the case of a null cosmological constant. For a positive cosmological constant, we found that the cosmic horizon is affected in a nontrivial way by the parameter [Formula: see text].