Eccentricity Evolution Research Articles

Long-term evolution of the inclined geosynchronous orbit in Beidou Navigation Satellite System

China’s Beidou Navigation Satellite System (BDS), unlike other navigation satellite systems, uses several inclined geosynchronous orbits (IGSO) to enhance the accuracy of regional or global navigation. In order to maintain a safe space environment in the vicinity of its operational orbit, it is necessary that the decommissioned satellites be well disposed of. To understand the underlying dynamics that affect the BDS IGSO, we study this problem from two aspects. In this paper, we first theoretically analyze the problem using the simplified models with 1 and 2 degrees of freedoms (1-/2-dof). Then we extensively investigate the numerically propagated orbits for 200 and 1000years, applying the results from these simplified models and seeking proper explanations for the underlying dynamics. We especially focus on the eccentricity evolution, which is a major concern regarding the collision hazard. We expect to understand the underlying dynamics governing the long-term evolution of BDS IGSO and gain helpful insight into future disposal strategies.

Retrograde binaries of massive black holes in circumbinary accretion discs

We explore the hardening of a massive black hole binary embedded in a circum-binary gas disc when the binary and the gas are coplanar and the gas is counter-rotating. The secondary black hole, revolving in the direction opposite to the gas, experiences a drag from gas-dynamical friction and from direct accretion of part of it. Using two-dimensional (2D) hydrodynamical grid simulations we investigate the effect of changing the accretion prescriptions on the dynamics of the secondary black hole which in turn affect the binary hardening and eccentricity evolution. We find that realistic accretion prescriptions lead to results that differ from those inferred assuming accretion of all the gas within the Roche Lobe of the secondary black hole. Different accretion prescriptions result in different disc's surface densities which alter the black hole's dynamics back. Full 3D SPH realizations of a number of representative cases, run over a shorter interval of time, validate the general trends observed in the less computationally demanding 2D simulations. Initially circular black hole binaries increase only slightly their eccentricity which then oscillates around small values (<0.1) while they harden. By contrast, initially eccentric binaries become more and more eccentric. A semi-analytical model describing the black hole's dynamics under accretion only explores the late evolution stages of the binary in an otherwise unperturbed retrograde disc to illustrate how eccentricity evolves with time in relation to the shape of the underlying surface density distribution.

DIFFICULTY IN THE FORMATION OF COUNTER-ORBITING HOT JUPITERS FROM NEAR-COPLANAR HIERARCHICAL TRIPLE SYSTEMS: A SUB-STELLAR PERTURBER

ABSTRACT Among 100 transiting planets with a measured projected spin–orbit angle λ, several systems are suggested to be counter-orbiting. While these cases may be due to the projection effect, the mechanism that produces a counter-orbiting planet has not been established. A promising scenario for counter-orbiting planets is the extreme eccentricity evolution in near-coplanar hierarchical triple systems with eccentric inner and outer orbits. We examine this scenario in detail by performing a series of systematic numerical simulations, and consider the possibility of forming hot Jupiters (HJs), especially a counter-orbiting one under this mechanism with a distant sub-stellar perturber. We incorporate quadrupole and octupole secular gravitational interaction between the two orbits, and also short-range forces (correction for general relativity, star and inner planetary tide, and rotational distortion) simultaneously. We find that most systems are tidally disrupted and that a small fraction of the surviving planets turn out to be prograde. The formation of counter-orbiting HJs in this scenario is possible only in a very restricted parameter region, and thus is very unlikely in practice.

Frequency and time-domain inspiral templates for comparable mass compact binaries in eccentric orbits

Inspiraling compact binaries with non-negligible orbital eccentricities are plausible gravitational wave (GW) sources for the upcoming network of GW observatories. In this paper, we present two prescriptions to compute post-Newtonian (PN) accurate inspiral templates for such binaries. First, we adapt and extend the post-circular scheme of Yunes {\it et al.} [Phys. Rev. D 80, 084001 (2009)] to obtain a Fourier-domain inspiral approximant that incorporates the effects of PN-accurate orbital eccentricity evolution. This results in a fully analytic frequency-domain inspiral waveform with Newtonian amplitude and 2PN order Fourier phase while incorporating eccentricity effects up to sixth order at each PN order. The importance of incorporating eccentricity evolution contributions to the Fourier phase in a PN consistent manner is also demonstrated. Second, we present an accurate and efficient prescription to incorporate orbital eccentricity into the quasi-circular time-domain {\texttt{TaylorT4}} approximant at 2PN order. New features include the use of rational functions in orbital eccentricity to implement the 1.5PN order tail contributions to the far-zone fluxes. This leads to closed form PN-accurate differential equations for evolving eccentric orbits and the resulting time-domain approximant is accurate and efficient to handle initial orbital eccentricities $\leq 0.9$. Preliminary GW data analysis implications are probed using match estimates.

Scattering experiments meetN-body – I. A practical recipe for the evolution of massive black hole binaries in stellar environments

The N-independence observed in the evolution of massive black hole binaries (MBHBs) in recent simulation of merging stellar bulges suggests a simple interpretation beyond complex time-dependent relaxation processes. We conjecture that the MBHB hardening rate is equivalent to that of a binary immersed in a field of unbound stars with density $\rho$ and typical velocity $\sigma$, provided that $\rho$ and $\sigma$ are the stellar density and the velocity dispersion at the influence radius of the MBHB. By comparing direct N-body simulations to an hybrid model based on 3-body scattering experiments, we verify this hypothesis: when normalized to the stellar density and velocity dispersion at the binary influence radius, the N-body MBHB hardening rate approximately matches that predicted by 3-body scatterings in the investigated cases. The eccentricity evolution obtained with the two techniques is also in reasonable agreement. This result is particularly practical because it allows to estimate the lifetime of MBHBs forming in dry mergers based solely on the stellar density profile of the host galaxy. We briefly discuss some implications of our finding for the gravitational wave signal observable by pulsar timing arrays and for the expected population of MBHBs lurking in massive ellipticals.

ECCENTRICITY EVOLUTION THROUGH ACCRETION OF PROTOPLANETS

Most of super-Earths detected by the radial velocity (RV) method have significantly smaller eccentricities than the eccentricities corresponding to velocity dispersion equal to their surface escape velocity ("escape eccentricities"). If orbital instability followed by giant impacts among protoplanets that have migrated from outer region is considered, it is usually considered that eccentricities of the merged bodies become comparable to those of orbital crossing bodies, which are excited up to their escape eccentricities by close scattering. However, the eccentricity evolution in the {\it in situ} accretion model has not been studied in detail. Here, we investigate the eccentricity evolution through {\it N}-body simulations. We have found that the merged planets tend to have much smaller eccentricities than the escape eccentricities due to very efficient collision damping. If the protoplanet orbits are initially well separated and their eccentricities are securely increased, an inner protoplanet collides at its apocenter with an outer protoplanet at its pericenter. The eccentricity of the merged body is the smallest for such configuration. Orbital inclinations are also damped by this mechanism and planets tend to share a same orbital plane, which is consistent with {\it Kepler} data. Such efficient collision damping is not found when we start calculations from densely packed orbits of the protoplanets. If the protoplanets are initially in the mean-motion resonances, which corresponds to well separated orbits, the {\it in situ} accretion model well reproduces the features of eccentricities and inclinations of multiple super-Earths/Earth systems discovered by RV and {\it Kepler} surveys.

Dynamical evolution of massive black holes in galactic-scaleN-body simulations – introducing the regularized tree code ‘rvine’

We present a hybrid code combining the OpenMP-parallel tree code VINE with an algorithmic chain regularization scheme. The new code, called "rVINE", aims to significantly improve the accuracy of close encounters of massive bodies with supermassive black holes in galaxy-scale numerical simulations. We demonstrate the capabilities of the code by studying two test problems, the sinking of a single massive black hole to the centre of a gas-free galaxy due to dynamical friction and the hardening of a supermassive black hole binary due to close stellar encounters. We show that results obtained with rVINE compare well with NBODY7 for problems with particle numbers that can be simulated with NBODY7. In particular, in both NBODY7 and rVINE we find a clear N-dependence of the binary hardening rate, a low binary eccentricity and moderate eccentricity evolution, as well as the conversion of the galaxy's inner density profile from a cusp to a a core via the ejection of stars at high velocity. The much larger number of particles that can be handled by rVINE will open up exciting opportunities to model stellar dynamics close to SMBHs much more accurately in a realistic galactic context. This will help to remedy the inherent limitations of commonly used tree solvers to follow the correct dynamical evolution of black holes in galaxy scale simulations.

Tidal synchronization of close-in satellites and exoplanets: II. Spin dynamics and extension to Mercury and exoplanet host stars

This paper deals with the application of the creep tide theory (Ferraz-Mello, CeMDA 116, 109, 2013) to the rotation of close-in satellites, Mercury, close-in exoplanets and their host stars. The solutions show two extreme cases: close-in giant gaseous planets, with fast relaxation (low viscosity) and satellites and Earth-like planets, with slow relaxation (high viscosity). The rotation of close-in gaseous planets follows the classical Darwinian pattern: it is tidally driven towards a stationary solution which is synchronized, but, if the orbit is elliptical, with a frequency larger than the orbital mean-motion. The rotation of rocky bodies, however, may be driven to several attractors whose frequencies are 1/2,1,3/2,2,5/2 ... times the mean-motion. The number of attractors increases with the viscosity of the body and with the orbital eccentricity. The classical example is Mercury, whose rotational period is 2/3 of the orbital period (3/2 attractor). The planet behaves as a molten body with a relaxation that allowed it to cross the 2/1 attractor without being trapped, but not to escape being trapped in the 3/2 one. In that case, the relaxation is estimated to lie in the interval 4.6 -- 27 x 10^{-9} s^{-1} (equivalent to a quality factor roughly constrained to the interval 5<Q<50). The stars have relaxation similar to the hot Jupiters and their rotation is also driven to the only stationary solution existing in these cases. However, solar-type stars may lose angular momentum due to stellar wind, braking the rotation and displacing the attractor towards larger periods. Old active host stars with big close-in companions generally have rotational periods larger than the orbital periods of the companions. The paper also includes the study of the energy dissipation and the evolution of the orbital eccentricity.

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.

PLANET FORMATION IN BINARIES: DYNAMICS OF PLANETESIMALS PERTURBED BY THE ECCENTRIC PROTOPLANETARY DISK AND THE SECONDARY

Detections of planets in eccentric, close (separations of ~20 AU) binary systems such as \alpha Cen or \gamma Cep provide an important test of planet formation theories. Gravitational perturbations from the companion are expected to excite high planetesimal eccentricities resulting in destruction, rather than growth, of objects with sizes of up to several hundred km in collisions of similar-size bodies. It was recently suggested that gravity of a massive axisymmetric gaseous disk in which planetesimals are embedded drives rapid precession of their orbits, suppressing eccentricity excitation. However, disks in binaries are themselves expected to be eccentric, leading to additional planetesimal excitation. Here we develop secular theory of eccentricity evolution for planetesimals perturbed by the gravity of an elliptical protoplanetary disk (neglecting gas drag) and the companion. For the first time we derive an expression for the disturbing function due to an eccentric disk, which can be used for a variety of other astrophysical problems. We obtain explicit analytical solutions for planetesimal eccentricity evolution and delineate four different regimes of dynamical excitation. We show that in systems with massive (>10^{-2}M_Sun) disks, planetesimal eccentricity is usually determined by the gravity of the eccentric disk alone, and is comparable to the disk eccentricity. As a result, the latter imposes a lower limit on collisional velocities of solids, making their growth problematic. This fragmentation barrier can be removed if the gaseous disk rapidly precesses or if its own self-gravity is efficient at lowering disk eccentricity.

DYNAMIC STABILITY OF THE SOLAR SYSTEM: STATISTICALLY INCONCLUSIVE RESULTS FROM ENSEMBLE INTEGRATIONS

Due to the chaotic nature of the Solar System, the question of its long-term stability can only be answered in a statistical sense, for instance, based on numerical ensemble integrations of nearby orbits. Destabilization of the inner planets, leading to close encounters and/or collisions can be initiated through a large increase in Mercury's eccentricity, with a currently assumed likelihood of ~1%. However, little is known at present about the robustness of this number. Here I report ensemble integrations of the full equations of motion of the eight planets and Pluto over 5 Gyr, including contributions from general relativity. The results show that different numerical algorithms lead to statistically different results for the evolution of Mercury's eccentricity (eM). For instance, starting at present initial conditions (eM ~= 0.21), Mercury's maximum eccentricity achieved over 5 Gyr is on average significantly higher in symplectic ensemble integrations using heliocentricthan Jacobi coordinates and stricter error control. In contrast, starting at a possible future configuration (eM ~= 0.53), Mercury's maximum eccentricity achieved over the subsequent 500 Myr is on average significantly lower using heliocentric than Jacobi coordinates. For example, the probability for eM to increase beyond 0.53 over 500 Myr is >90% (Jacobi) vs. only 40-55% (heliocentric). This poses a dilemma as the physical evolution of the real system - and its probabilistic behavior - cannot depend on the coordinate system or numerical algorithm chosen to describe it. Some tests of the numerical algorithms suggest that symplectic integrators using heliocentric coordinates underestimate the odds for destabilization of Mercury's orbit at high initial eM.

Evolution of eccentricity and orbital inclination of migrating planets in 2:1 mean motion resonance

We determine, analytically and numerically, the conditions needed for a system of two migrating planets trapped in a 2:1 mean motion resonance to enter an inclination-type resonance. We provide an expression for the asymptotic equilibrium value that the eccentricity $e_{\rm i}$ of the inner planet reaches under the combined effects of migration and eccentricity damping. We also show that, for a ratio $q$ of inner to outer masses below unity, $e_{\rm i}$ has to pass through a value $e_{\rm i,res}$ of order 0.3 for the system to enter an inclination-type resonance. Numerically, we confirm that such a resonance may also be excited at another, larger, value $e_{\rm i, res} \simeq 0.6$, as found by previous authors. A necessary condition for onset of an inclination-type resonance is that the asymptotic equilibrium value of $e_{\rm i}$ is larger than $e_{\rm i,res}$. We find that, for $q \le 1$, the system cannot enter an inclination-type resonance if the ratio of eccentricity to semimajor axis damping timescales $t_e/t_a$ is smaller than 0.2. This result still holds if only the eccentricity of the outer planet is damped and $q \lesssim 1$. As the disc/planet interaction is characterized by $t_e/t_a \sim 10^{-2}$, we conclude that excitation of inclination through the type of resonance described here is very unlikely to happen in a system of two planets migrating in a disc.

On the relativistic Lagrange-Laplace secular dynamics for extrasolar systems

AbstractWe study the secular dynamics of extrasolar planetary systems by extending the Lagrange-Laplace theory to high order and by including the relativistic effects. We investigate the long-term evolution of the planetary eccentricities via normal form and we find an excellent agreement with direct numerical integrations. Finally we set up a simple analytic criterion that allows to evaluate the impact of the relativistic effects in the long-time evolution.

LINEAR COROTATION TORQUES IN NON-BAROTROPIC DISKS

I derive a fully analytic expression for the linear corotation torque to first order in eccentricity for planets in non-barotropic protoplanetary disks, taking into account the effect of disk entropy gradients. This torque formula is applicable to both the co-orbital corotation torque and the non co-orbital corotation torques -- for planets in orbits with non-zero eccentricity -- in disks where the thermal diffusivity and viscosity are sufficient to maintain linearity of these interactions. While the co-orbital corotation torque is important for migration of planets in Type I migration, the non co-orbital corotation torque plays an important role in the eccentricity evolution of giant planets that have opened gaps in the disk. The presence of an entropy gradient in the disk can significantly modify the corotation torque in both these cases.

On some long time dynamical features of the Trojan asteroids of Jupiter

The equation of motion of long periodic libration around the Lagrangian point $$L_4$$ in the restricted three-body problem is investigated. The range of validity of an approximate analytical solution in the tadpole region is determined by numerical integration. The predictions of the model of libration are tested on the Trojan asteroids of Jupiter. The long time evolution of the orbital eccentricity and the longitude of the perihelion of the Trojan asteroids, under the effect of the four giant planets, is also investigated and a slight dynamical asymmetry is shown between the two groups of Trojans at $$L_4$$ and $$L_5$$ .

Secular dynamics in hierarchical three-body systems

The secular approximation for the evolution of hierarchical triple configurations has proven to be very useful in many astrophysical contexts, from planetary to triple-star systems. In this approximation the orbits may change shape and orientation, on time-scales longer than the orbital time-scales, but the semimajor axes are constant. For example, for highly inclined triple systems, the Kozai–Lidov mechanism can produce large-amplitude oscillations of the eccentricities and inclinations. Here we revisit the secular dynamics of hierarchical triple systems. We derive the secular evolution equations to octupole order in Hamiltonian perturbation theory. Our derivation corrects an error in some previous treatments of the problem that implicitly assumed a conservation of the z-component of the angular momentum of the inner orbit (i.e. parallel to the total angular momentum of the system). Already to quadrupole order, our results show new behaviours including the possibility for a system to oscillate from prograde to retrograde orbits. At the octupole order, for an eccentric outer orbit, the inner orbit can reach extremely high eccentricities and undergo chaotic flips in its orientation. We discuss applications to a variety of astrophysical systems, from stellar triples to merging compact binaries and planetary systems. Our results agree with those of previous studies done to quadrupole order only in the limit in which one of the inner two bodies is a massless test particle and the outer orbit is circular; our results agree with previous studies at octupole order for the eccentricity evolution, but not for the inclination evolution.

Origin and Implications of high eccentricities in massive black hole binaries at sub-pc scales

We outline the eccentricity evolution of sub-parsec massive black hole binaries (MBHBs) forming in galaxy mergers. In both stellar and gaseous environments, MBHBs are expected to grow large orbital eccentricities before they enter the gravitational wave (GW) observational domain. We re-visit the predicted eccentricities detectable by space based laser interferometers (as the proposed ELISA/NGO) for both environments. Close to coalescence, many MBHBs will still maintain detectable eccentricities, spanning a broad range from < 10−5 up to 0.5. Stellar and gas driven dynamics lead to distinct distributions, with the latter favoring larger eccentricities. At larger binary separations, when emitted GWs will be observed by pulsar timing arrays (PTAs), the expected eccentricities are usually quite large, in the range 0.01 – 0.7, which poses an important issue for signal modelling and detection algorithms. In this window, large eccentricities also have implications on proposed electromagnetic counterparts to the GW signal, which we briefly review.

Planet-Disk Interaction and Orbital Evolution

As planets form and grow within gaseous protoplanetary disks, the mutual gravitational interaction between the disk and planet leads to the exchange of angular momentum and migration of the planet. We review current understanding of disk-planet interactions, focusing in particular on physical processes that determine the speed and direction of migration. We describe the evolution of low-mass planets embedded in protoplanetary disks and examine the influence of Lindblad and corotation torques as a function of the disk properties. The role of the disk in causing the evolution of eccentricities and inclinations is also discussed. We describe the rapid migration of intermediate-mass planets that may occur as a runaway process and examine the transition to gap formation and slower migration driven by the viscous evolution of the disk for massive planets. The roles and influence of disk self-gravity and magnetohydrodynamic turbulence are discussed in detail, as a function of the planet mass, as is the evolution of multiple planet systems. Finally, we address the question of how well global models of planetary formation that include migration are able to match observations of extrasolar planets.

PLANETESIMAL DYNAMICS IN INCLINED BINARY SYSTEMS: THE ROLE OF GAS-DISK GRAVITY

We investigate the effects of gas-disk gravity on the planetesimal dynamics in inclined binary systems, where the circumprimary disk plane is tilted by a significant angle ($i_B$) with respect to the binary disk plane. Our focus is on the Lidov-Kozai mechanism and the evolution of planetesimal eccentricity and inclination. Using both analytical and numerical methods, we find that, on one hand, the disk gravity generally narrows down the Kozai-on region, i.e., the Lidov-Kozai effect can be suppressed in certain parts of (or even the whole of) the disk, depending on various parameters. In the Kozai-off region, planetesimals would move on orbits close to the mid-plane of gas-disk, with the relative angle ($i^{'}$) following a small amplitude periodical oscillation. On the other hand, when we include the effects of disk gravity, we find that the Lidov-Kozai effect can operate even at arbitrarily low inclinations ($i_B$), although lower $i_B$ leads to a smaller Kozai-on region. Furthermore, in the Kozai-on region, most planetesimals' eccentricities can be excited to extremely high values ($\sim 1$), and such extreme high eccentricities usually accompany orbital flipping, i.e., planetesimal orbit flips back and forth between anterograde and retrograde. Once a planetesimal reaches very high orbital eccentricity, gas drag damping will shrink the planetesimal orbit, forming a "hot planetesimal" on a near circular orbit very close to the primary star. Such a mechanism, if replacing the planetesimals and gas drag damping with Jupiters and tidal damping respectively, may lead to frequent production of hot-Jupiters.

Massive black hole binary plane reorientation in rotating stellar systems

Abstract We study the evolution of the orientation of the orbital plane of massive black hole binaries (BHBs) in rotating stellar systems in which the total angular momentum of the stellar cusp is misaligned with respect to that of the binary. We compare results from direct summation N-body simulations with predictions from a simple theoretical model. We find that the same encounters between cusp stars and the BHBs that are responsible for the hardening and eccentricity evolution of the binary lead to a reorientation of the binary orbital plane. In particular, binaries whose angular momentum is initially misaligned with respect to that of the stellar cusp tend to realign their orbital planes with the angular momentum of the cusp on a time-scale of a few hardening times. This is due to angular momentum exchange between stars and the BHBs during close encounters, and may have important implications for the relative orientation of host galaxies and radio jets.