Hyperbolic Orbits Research Articles

Implications of the interstellar object 1I/'Oumuamua for planetary dynamics and planetesimal formation

’Oumuamua, the first bona fide interstellar planetesimal, was discovered passing through our Solar system on a hyperbolic orbit. This object was likely dynamically ejected from an extrasolar planetary system after a series of close encounters with gas giant planets. To account for ’Oumuamua’s detection, simple arguments suggest that ∼1 M⊕ of planetesimals are ejected per solar mass of Galactic stars. However, that value assumes mono-sized planetesimals. If the planetesimal mass distribution is instead top-heavy, the inferred mass in interstellar planetesimals increases to an implausibly high value. The tension between theoretical expectations for the planetesimal mass function and the observation of ’Oumuamua can be relieved if a small fraction (⁠|${\sim } 0.1{\rm -}1 \, \rm {per\,cent}$|⁠) of planetesimals are tidally disrupted on the pathway to ejection into ’Oumuamua-sized fragments. Using a large suite of simulations of giant planet dynamics including planetesimals, we confirm that 0.1–1 per cent of planetesimals pass within the tidal disruption radius of a gas giant on their pathway to ejection. ’Oumuamua may thus represent a surviving fragment of a disrupted planetesimal. Finally, we argue that an asteroidal composition is dynamically disfavoured for ’Oumuamua, as asteroidal planetesimals are both less abundant and ejected at a lower efficiency than cometary planetesimals.

Periodic and Degenerate Orbits Around the Equilibrium Points in the Relativistic Restricted Three-Body Problem

In the present work, the periodic orbits around the relativistic restricted three-body problem (in brief relativistic RTBP) is treated. The equations of motion of the relativistic restricted three-body problem are presented. The locations of the equilibrium points are computed. The stability of equilibrium points is outlined. Elliptic and hyperbolic orbits in the neighborhood of the collinear points are obtained and then represented graphically for the whole domain of mass ratio as well as for different initial conditions. Degenerate hyperbolic orbits around the collinear points are obtained. Elliptic orbits in the neighborhood of the triangular point L4 are deduced and then represented graphically for the whole domain of mass ratio, for a selected range of frequency, for selected range of time as well as for different initial conditions. All obtained figures are analyzed dynamically.

Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds

We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of Sarig [29] for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a fixed constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.

Morse-Smale systems and horseshoes for three dimensional singular flows

We prove that for every three-dimensional vector field, either it can be accumulated by Morse-Smale ones, or it can be accumulated by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit in the $C^1$ topology.

Exponential convergence to time-periodic viscosity solutions in time-periodic Hamilton-Jacobi equations

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold, where the Hamiltonian satisfies the condition: The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit. It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a 1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.

Dynamical behaviour of a compound elastic pendulum

The aim of the paper is a comprehensive study of the compound elastic pendulum (CEP) with two degrees of freedom to point out the main complex (chaotic) dynamics that it can exhibit. The simplest way to find complex behavior in a nonintegrable Hamiltonian system is firstly to look for homoclinic (heteroclinic) orbit(s). Here, under suitable assumptions, we detect the existence of a homoclinic orbit of CEP and present the equation for it. Moreover, we show that for any value of the small parameter the system has a hyperbolic periodic orbit, whose invariant manifolds intersect themselves transversally.

Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities

This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under natural geometrical assumptions on the underlying space $M$, we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to $D$. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich stadia.

Analysis of Position and Velocity Variations for Hyperbolic Orbits and Application to Flyby Anomaly

Analysis of Position and Velocity Variations for Hyperbolic Orbits and Application to Flyby Anomaly

Col-OSSOS: Colors of the Interstellar Planetesimal 1I/‘Oumuamua

Abstract The recent discovery by Pan-STARRS1 of 1I/2017 U1 (‘Oumuamua), on an unbound and hyperbolic orbit, offers a rare opportunity to explore the planetary formation processes of other stars and the effect of the interstellar environment on a planetesimal surface. 1I/‘Oumuamua’s close encounter with the inner solar system in 2017 October was a unique chance to make observations matching those used to characterize the small-body populations of our own solar system. We present near-simultaneous g′, r′, and J photometry and colors of 1I/‘Oumuamua from the 8.1 m Frederick C. Gillett Gemini-North Telescope and gri photometry from the 4.2 m William Herschel Telescope. Our g′r′J observations are directly comparable to those from the high-precision Colours of the Outer Solar System Origins Survey (Col-OSSOS), which offer unique diagnostic information for distinguishing between outer solar system surfaces. The J-band data also provide the highest signal-to-noise measurements made of 1I/‘Oumuamua in the near-infrared. Substantial, correlated near-infrared and optical variability is present, with the same trend in both near-infrared and optical. Our observations are consistent with 1I/‘Oumuamua rotating with a double-peaked period of 8.10 ± 0.42 hr and being a highly elongated body with an axial ratio of at least 5.3:1, implying that it has significant internal cohesion. The color of the first interstellar planetesimal is at the neutral end of the range of solar system g − r and r − J solar-reflectance colors: it is like that of some dynamically excited objects in the Kuiper Belt and the less-red Jupiter Trojans.

1I/2017 U1 (‘Oumuamua) is Hot: Imaging, Spectroscopy, and Search of Meteor Activity

Abstract 1I/2017 U1 (‘Oumuamua), a recently discovered asteroid in a hyperbolic orbit, is likely the first macroscopic object of extrasolar origin identified in the solar system. Here, we present imaging and spectroscopic observations of ‘Oumuamua using the Palomar Hale Telescope as well as a search of meteor activity potentially linked to this object using the Canadian Meteor Orbit Radar. We find that ‘Oumuamua exhibits a moderate spectral gradient of , a value significantly lower than that of outer solar system bodies, indicative of a formation and/or previous residence in a warmer environment. Imaging observation and spectral line analysis show no evidence that ‘Oumuamua is presently active. Negative meteor observation is as expected, since ejection driven by sublimation of commonly known cometary species such as CO requires an extreme ejection speed of ∼40 m s−1 at ∼100 au in order to reach the Earth. No obvious candidate stars are proposed as the point of origin for ‘Oumuamua. Given a mean free path of ∼109 ly in the solar neighborhood, ‘Oumuamua has likely spent a very long time in interstellar space before encountering the solar system.

Implications for Planetary System Formation from Interstellar Object 1I/2017 U1 (‘Oumuamua)

Abstract The recently discovered minor body 1I/2017 U1 (‘Oumuamua) is the first known object in our solar system that is not bound by the Sun’s gravity. Its hyperbolic orbit (eccentricity greater than unity) strongly suggests that it originated outside our solar system; its red color is consistent with substantial space weathering experienced over a long interstellar journey. We carry out a simple calculation of the probability of detecting such an object. We find that the observed detection rate of 1I-like objects can be satisfied if the average mass of ejected material from nearby stars during the process of planetary formation is ∼20 Earth masses, similar to the expected value for our solar system. The current detection rate of such interstellar interlopers is estimated to be 0.2 yr−1, and the expected number of detections over the past few years is almost exactly one. When the Large Synoptic Survey Telescope begins its wide, fast, deep all-sky survey, the detection rate will increase to 1 yr−1. Those expected detections will provide further constraints on nearby planetary system formation through a better estimate of the number and properties of interstellar objects.

Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions

Small noise can induce rare transitions between metastable states, which can be characterized by Maximum Likelihood Paths (MLPs). Nongradient systems contrast gradient systems in that MLP does not have to cross the separatrix at a saddle point, but instead possibly at a point on a hyperbolic periodic orbit. A numerical approach for identifying such unstable periodic orbits is proposed based on String method. In a special class of nongradient systems (‘orthogonal-type’), there are provably local MLPs that cross such saddle point or hyperbolic periodic orbit, and the separatrix crossing location determines the associated local maximum of transition rate. In general cases, however, the separatrix crossing may not determine a unique local maximum of the rate, as we numerically observed a counter-example in a sheared 2D-space Allen–Cahn SPDE. It is a reasonable conjecture that there are always local MLPs associated with each attractor on the separatrix, such as saddle point or hyperbolic periodic orbit; our numerical experiments did not disprove so.

A new method of patched-conic for interplanetary orbit

In this paper, a new method of patched-conic for designing the interplanetary orbit is proposed. It is based on iterative computation and the equations are derived. The new method avoids the errors caused by assuming the velocity vector at the boundary of planetary sphere of influence parallel to the asymptote of the hyperbolic orbit. Based on the orbit constraint, the spacecraft’s velocity vector and the position vector at the boundary of planetary sphere of influence are calculated during each iteration to meet the continuous condition. This method is suitable for implementation in computer program and has a fast convergence speed.

Gravitational radiation driven capture in unequal mass black hole encounters

The gravitational radiation driven capture (GR capture) between unequal mass black holes without spins has been investigated with numerical relativistic simulations. We adopt the parabolic approximation which assumes that the gravitational wave radiation from a weakly hyperbolic orbit is the same as that from the parabolic orbit having the same pericenter distance. Using the radiated energies from the parabolic orbit simulations, we have obtained the critical impact parameter ($b_{\rm crit}$) for the GR capture for weakly hyperbolic orbit as a function of initial energy. The most energetic encounters occur around the boundary between the direct merging and the fly-by orbits, and can emit several percent of initial total ADM energy at the peak. When the total mass is fixed, energy and angular momentum radiated in the case of unequal mass black holes are smaller than those of equal mass black holes having the same initial orbital angular momentum for the fly-by orbits. We have compared our results with two different Post-Newtonian (PN) approximations, the exact parabolic orbit (EPO) and PN corrected orbit (PNCO). We find that the agreement between the EPO and the numerical relativity breaks down for very close encounters ($\it{e.g.}$, $b_{\rm crit} \lesssim 100$ M), and it becomes worse for higher mass ratios. For instance, the critical impact parameters can differ by more than $50\%$ from those obtained in EPO if the relative velocity at infinity $v_{\infty}$ is larger than 0.1 for the mass ratio of $m_{1}/m_{2}=16$. The PNCO gives more consistent results than EPO, but it also underestimates the critical impact parameter for the GR capture at $b_{\rm crit} \lesssim40$ M.

Parameterization of Invariant Manifolds for Periodic Orbits (II): A Posteriori Analysis and Computer Assisted Error Bounds

In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier–Taylor parameterizations of local stable/unstable manifolds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical methods developed in Castelli et al. (SIAM J Appl Dyn Syst 14(1):132–167, 2015) in order to obtain a high order Fourier–Taylor series expansion of the parameterization. The main result of the present work is an a-posteriori theorem which provides mathematically rigorous error bounds. The hypotheses of the theorem are checked with computer assistance. The argument relies on a sequence of preliminary computer assisted proofs where we validate the numerical approximation of the periodic orbit, its stable/unstable normal bundles, and the jets of the manifold to some desired order M. We illustrate our method by implementing validated computations for two dimensional manifolds in the Lorenz equations in $$\mathbb {R}^3$$ and a three dimensional manifold of a suspension bridge equation in $$\mathbb {R}^4$$ .

New State Transition Matrix for Relative Motion on an Arbitrary Keplerian Orbit

A new formation of the relative state transition matrix has been developed for the spacecraft relative motion without using the Tschauner–Hempel solutions. The relative state transition matrix is obtained by expressing the inertial two-body state transition matrix in the rotating frame through a series of coordinate transformations. Two different approaches are used for exploiting the inertial two-body state transition matrix. One is to use the parametric linearization technique in terms of the orbital elements. The other one is to adopt the universal variables and Lagrange coefficients to achieve the same goal. This new state transition matrix for spacecraft relative motion is time explicit and universally applicable to elliptic, parabolic, and hyperbolic orbits. The numerical results demonstrate that the new state transition matrix can achieve high-precision state propagation for spacecraft relative motion.

Collisions with meteoroid streams as one possible mechanism for the formation of hyperbolic cometary orbits

This paper presents the results of a statistical analysis of the dynamic parameters of 300 comets that have osculating hyperbolic orbits. It is shown that such comets differ from other comets by their large perihelion distances and by a predominance of retrograde motion. It is shown that the values of i, the inclination of the hyperbolic comets, are in comparative excess over the interval 90–120°. The dominance by q, the perihelion distance, renders it difficult to suggest that the excess hyperbolic velocity of these comets can be the result of physical processes that take place in their nuclei. Aspects of the following working hypothesis, that the hyperbolic excess of parameter e might be formed after comets pass through meteoroid streams, are also studied. To evaluate this hypothesis, the distribution of the orbits of hyperbolic comets relative to the plane of motion of 112 established meteoroid streams are analyzed. The number (N) of orbit nodes for hyperbolic comets with respect to the plane of each stream at various distances is calculated. To determine the degree of redundancy of N, a special computing algorithm was applied that provided the expected value nav as well as the standard deviation σ for the number of cometary nodes at the plane of each stream. A comparative analysis of the N and nav values that take σ into account suggests an excess in 40 stream cases. This implies that the passage of comets through meteoroid streams can lead to an acceleration of the comets’ heliocentric velocity.

Improvement of the position of planet X based on the motion of nearly parabolic comets

Based on the motion of nearly parabolic comets, we have improved the position of planet X in its orbit obtained by Batygin and Brown (2016). By assuming that some of the comets discovered to date could have close encounters with this planet, we have determined the comets with a small minimum orbit intersection distance with the planet. Five comets having hyperbolic orbits before their entry into the inner Solar system have been separated out from the general list. By assuming that at least one of them had a close encounter with the planet, we have determined the planet’s possible position. The planet’s probable ephemeris positions at the present epoch have been obtained by assuming the planet to have prograde and retrograde motions. In the case of a prograde motion, the planet is currently at a distance Δ whose value belongs to the interval Δ ∈ (1110, 1120) AU and has a right ascension α and declination δ within the intervals α ∈ (83◦, 90◦) and δ ∈ (8◦, 10◦); the true anomaly υ belongs to the interval υ ∈ (176◦, 184◦). In the case of a retrograde motion: α ∈ (48◦, 58◦), δ ∈ (−12◦, −6◦), Δ ∈ (790, 910) AU, and υ ∈ (212◦, 223◦). It should be noted that in the case of a retrograde motion of the planet, its ephemeris position obtained from the motion of comets agrees with the planet’s position obtained byHolman and Payne (2016) from highly accurate Cassini observations and is consistent with the results of Fienga et al. (2016).

Direct transfer trajectory design options for interplanetary orbiter missions using an iterative patched conic method

In a direct interplanetary transfer, the spacecraft moves from a parking orbit of the departure planet to a parking orbit of the arrival planet. The transfer trajectory must be designed such that the specified arrival parking orbit conditions are achieved. For a fixed departure epoch and flight duration, there are four distinct transfer trajectory design options in a direct transfer. The conventional patched conic method, the most widely used analytical trajectory design method, does not identify these design options. An iterative patched conic method that identifies these distinct design options is developed and presented in this paper. This method involves two iterative processes: (i) iteration on the hyperbolic orbit characteristics using an analytical tuning strategy to achieve the hyperbolic excess velocity vector at the patch point, (ii) iteration on the patch points at the sphere of influence. The performance of the proposed method is compared with the conventional and V-infinity tuned patched conic methods. A design analysis tool, based on the proposed method, is developed and tested in various orbiter mission scenarios.

Strong scarring of logarithmic quasimodes

We consider a semiclassical (pseudo)differential operator on a compact surface (M,g), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit γ at some energy E0. For any ϵ>0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ϵh|logh| in the semiclassical limit, but which still exhibit a strong scar on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.