Kepler Energy Research Articles

A new correction method for quasi-Keplerian orbits

A pure two-body problem has seven integrals including the Kepler energy, the Laplace vector and the angular momentum vector. However, only five of them are independent. When the five independent integrals are preserved, the two other dependent integrals are naturally preserved from a theoretical viewpoint; but they may not necessarily be from a numerical computational viewpoint. Because of this, we use seven scale factors to adjust the integrated positions and velocities so that the adjusted solutions strictly satisfy the seven constraints. Noticing the existence of the two dependent integrals, we adopt the Newton iterative method combined with singular value decomposition to calculate these factors. This correction scheme can be applied to perturbed two-body and N-body problems in the solar system. In this case, the seven quantities associated with each planet slowly vary with time. More accurate values can be given to the seven slowly-varying quantities by integrating the integral invariant relations of these quantities and the equations of motion. They should be satisfied with the adjusted solutions. Numerical tests show that the new method can significantly reduce the rapid growth of numerical errors for all orbital elements.

The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

ABSTRACT A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 108 yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.

Initial Parameter Analysis about Resonant Orbits in Earth-Moon System

The initial parameters about resonant orbits in the Earth-Moon system were investigated in this study. Resonant orbits with different ratios are obtained in the two-body problem and planar circular restricted three-body problem (i.e., PCRTBP). It is found that the eccentricity and initial phase are two important initial parameters of resonant orbits that affect the closest distance between the spacecraft and the Moon. Potential resonant transition or resonant flyby may occur depending on the possibility of the spacecraft approaching the Moon. Based on an analysis of ballistic capture and flyby, the Kepler energy and the planet’s perturbed gravitational sphere are used as criteria to establish connections between the initial parameters and the possible “steady” resonant orbits. The initial parameter intervals that can cause instability of the resonant orbits in the CRTBP are obtained. Examples of resonant orbits in 1:2 and 2:1 resonances are provided to verify the proposed criteria.

K2 Ultracool Dwarfs Survey. III. White Light Flares Are Ubiquitous in M6-L0 Dwarfs

Abstract We report the white light flare rates for 10 ultracool dwarfs using Kepler K2 short-cadence data. Among our sample stars, two have spectral type M6, three are M7, three are M8, and two are L0. Most of our targets are old low-mass stars. We identify a total of 283 flares in all of the stars in our sample, with Kepler energies in the range log E Kp ∼ (29–33.5) erg. Using the maximum-likelihood method of line fitting, we find that the flare frequency distribution (FFD) for each star in our sample follows a power law with slope −α in the range −(1.3–2.0). We find that cooler objects tend to have shallower slopes. For some of our targets, the FFD follows either a broken power law, or a power law with an exponential cutoff. For the L0 dwarf 2MASS J12321827-0951502, we find a very shallow slope (−α = −1.3) in the Kepler energy range (0.82–130) × 1030 erg: this L0 dwarf has flare rates which are comparable to those of high-energy flares in stars of earlier spectral types. In addition, we report photometry of two superflares: one on the L0 dwarf 2MASS J12321827-0951502 and another on the M7 dwarf 2MASS J08352366+1029318. In the case of 2MASS J12321827-0951502, we report a flare brightening by a factor of ∼144 relative to the quiescent photospheric level. Likewise, for 2MASS J08352366+1029318, we report a flare brightening by a factor of ∼60 relative to the quiescent photospheric level. These two superflares have bolometric (ultraviolet/optical/infrared) energies 3.6 × 1033 erg and 8.9 × 1033 erg respectively, while the full width half maximum timescales are very short, ∼2 min. We find that the M8 star TRAPPIST-1 is more active than the M8.5 dwarf 2M03264453+1919309, but less active than another M8 dwarf (2M12215066-0843197).

Trajectory exploration within asynchronous binary asteroid systems using refined Lagrangian coherent structures

Ground observations have found that asynchronous systems constitute most of the population of the near-Earth binary asteroids. This paper concerns the trajectory of a particle in the asynchronous system which is systematically described using periodic ellipsoidal and spherical body models. Due to the non-autonomous characteristics of the asynchronous system, Lagrangian coherent structures (LCS) are employed to identify the various dynamical behaviors. To enhance the accuracy of LCS, a robust LCS finding algorithm is developed incorporating hierarchical grid refinement, one-dimensional search and variational theory verification. In this way, the intricate dynamical transport boundaries are detected efficiently. These boundaries indicate that a total of 15 types of trajectories exist near asynchronous binary asteroids. According to their Kepler energy variations, these trajectories can be grouped into four basic categories, i.e., transitory, escape, impact and flyby trajectories. Furthermore, the influence of the ellipsoid’s spin period on the dynamical behavior is discussed in the context of the change of dynamical regions. We found that the transitory and impact motions occur easily in the synchronous-like binary systems, in which the rotation period of the ellipsoid is nearly equal to that of the mutual orbit. Meanwhile, the results confirm a positive correlation between the spinning rate of the ellipsoid and the probability of the escape and flyby trajectories. The LCS also reveal a marked increase in trajectory diversity after a larger initial energy is selected.

Tidal power: A different perspective

Kepler Energy is taking a different approach to tidal power, applying a design that can be used in benign sea conditions in a location closer to end-demand and which could generate utility-scale quantities of power, as Peter Dixon explains.

VELOCITY CORRECTIONS TO KEPLER ENERGY AND LAPLACE INTEGRAL

For each celestial body of multi-planet systems, there are two slowly varying quantities or quasi-integrals, Kepler energy and Laplace integral, which are closely associated with the orbital semimajor axis and eccentricity, respectively. To correct numerical errors of the quantities, we give an extension of Nacozy's approach and develop a new manifold correction method, where corresponding reference values of the quantities at every integration step are obtained from integral invariant relations, and only velocity corrections are used to approximately satisfy the two quasi-integrals. As a result, the scheme does enhance the quality of the integration by significantly raising the accuracy of the two elements. Especially, it is superior to the existing dual scaling method in the improvement of eccentricity in general when the adopted integrator provides a sufficient precision to the eccentricity.

Velocity scaling method to correct individual Kepler energies

As variants of the velocity–position single scaling method of Fukushima, which extends Nacozy’s manifold correction scheme to monitor the integrated position and velocity by using the integral invariant relation and the same spatial scale transformation, a new velocity scaling method and a new position scaling method for correcting the varying Kepler energy of each body in an n-body problem of planetary dynamics are presented. Compared with Fukushima’s idea, the new schemes are simple to operate. Like other existing methods including the method of Fukushima and of Wu et al., the two new methods not only are almost the same effectiveness in significantly improving the orbital semi-major axis or mean anomaly at the epoch, but also can raise the accuracy of numerical integration by several orders. In particular, the new velocity scaling method as well as the method of Wu et al. is the most convenient in application.

Numerical Comparison of Two-Body Regularizations

We numerically compare four schemes to regularize a three-dimensional two-body problem under perturbations: the Sperling-Burdet (S-B), Kustaanheimo-Stiefel (K-S), and Burdet-Ferrandiz (B-F) regularizations, and a three-dimensional extension of the Levi-Civita (L-C) regularization we developed recently. As for the integration time of the equation of motion, the least time is needed for the unregularized treatment, followed by the K-S, the extended L-C, the B-F, and the S-B regularizations. However, these differences become significantly smaller when the time to evaluate perturbations becomes dominant. As for the integration error after one close encounter, the K-S and the extended L-C regularizations are tied for the least error, followed by the S-B, the B-F, and finally the unregularized scheme for unperturbed orbits with eccentricity less than 2. This order is not changed significantly by various kinds of perturbations. As for the integration error of elliptical orbits after multiple orbital periods, the situation remains the same except for the rank of the S-B scheme, which varies from the best to the second worst depending on the length of integration and/or on the nature of perturbations. Also, we confirm that Kepler energy scaling enhances the performance of the unregularized, K-S, and extended L-C schemes. As a result, the K-S and the extended L-C regularizations with Kepler energy scaling provide the best cost performance in integrating almost all the perturbed two-body problems.

Efficient Integration of Highly Eccentric Orbits by Scaling Methods Applied to Kustaanheimo-Stiefel Regularization

We apply our single scaling method to the numerical integration of perturbed two-body problems regularized by the Kustaanheimo-Stiefel (K-S) transformation. The scaling is done by multiplying a single scaling factor with the four-dimensional position and velocity vectors of an associated harmonic oscillator in order to maintain the Kepler energy relation in terms of the K-S variables. As with the so-called energy rectification of Aarseth, the extra cost for the scaling is negligible, since the integration of the Kepler energy itself is already incorporated in the original K-S formulation. On the other hand, the single scaling method can be applied at every integration step without facing numerical instabilities. For unperturbed cases, the single scaling applied at every step gives a better result than either the original K-S formulation, the energy rectification applied at every apocenter, or the single scaling method applied at every apocenter. For the perturbed cases, however, the single scaling method applied at every apocenter provides the best performance for all perturbation types, whether the main source of error is truncation or round-off.

Efficient Orbit Integration by Linear Transformation for Consistency of Kepler Energy, Full Laplace Integral, and Angular Momentum Vector

By adopting a general linear transformation as the method of manifold correction, we modify our dual scaling method to integrate quasi-Keplerian orbits numerically. The new method adjusts the integrated position and velocity at each integration step in order to exactly satisfy the relations for the Kepler energy, angular momentum vector, and the full Laplace vector. In the case of no perturbation, the integration errors in all the orbital elements except the mean longitude at the epoch, which grows linearly with time, are reduced to the level of the machine epsilon throughout the integration. For perturbed orbits, the integration errors in position are smaller than with the previous methods of manifold correction. Since its wide applicability is unchanged and the cost of additional computation is similarly negligible, we recommend the new method as the best of our methods of manifold correction.

Efficient Orbit Integration by Scaling and Rotation for Consistency of Kepler Energy, Laplace Integral, and Angular Momentum Direction

By adding the orbital angular momentum vector as another auxiliary quantity to be integrated, we extend our scaling methods to integrate quasi-Keplerian orbits numerically in order to suppress the growth of integration errors in the inclination and the longitude of the ascending node. This time, the method follows the time evolution of the angular momentum vector, as well as the time development of the Kepler energy and/or the Laplace integral, in addition to integrating the usual equation of motion. By using a rotation that is independent of the application of the spatial scaling, the new method adjusts the position and velocity integrated rigorously at each integration step in order to align both perpendicular to the integrated angular momentum vector. The direction and the angle of the rotation are determined uniquely from the position, the velocity, and the angular momentum vector integrated. As with the original scaling methods, the new method is simple to implement, fast to compute, and applicable to a wide variety of integration methods, perturbation types, and complexities of problems. Although this addition provides no significant decrease in the position error, the new method is superior to the original scaling methods in the sense that it enhances the quality of the integration by significantly reducing the errors of the orbital plane at the cost of a negligibly small amount of additional computation.

Efficient Orbit Integration by Dual Scaling for Consistency of Kepler Energy and Laplace Integral

By adding the Laplace integral as the second auxiliary quantity to be integrated, we extend our scaling method to integrate quasi-Keplerian orbits numerically in order to suppress the growth of integration errors not only in the semimajor axis but also in the other orbital elements, especially in the eccentricity and in the longitude of pericenter. This time, the method simultaneously follows the time evolution of the Kepler energy and the Laplace integral in addition to integrating the usual equation of motion. By using a dual spatial scale transformation, it adjusts the position and velocity that are integrated in order to satisfy both the Kepler energy relation and another functional relation derived from the Laplace integral rigorously at each integration step. The scale factors for the position and for the velocity are set separately and are determined by solving a set of linear equations. Just as with the original scaling method, the new method is simple to implement, fast to compute, and applicable to a wide variety of integration methods, perturbation types, and problem complexities. With the exception of the J2 perturbation, the new method is superior to the original scaling method because it achieves significantly fewer integration errors for the physical properties such as the shape and the orientation of orbits at the cost of a negligibly small amount of additional computation.

Efficient Orbit Integration by Scaling for Kepler Energy Consistency

Extending Nacozy's idea of manifold correction by using the concept of the integral invariant relation, we propose a new approach to numerically integrate quasi-Keplerian orbits. The method integrates the time evolution of the Kepler energy and the usual equation of motion simultaneously. Then it directly adjusts the integrated position and velocity by a spatial scale transformation in order to satisfy the Kepler energy relation rigorously at every integration step. The scale factor is determined by solving an associated cubic equation precisely with the help of Newton's method. In treating multiple bodies, the Kepler energies are integrated for each body and the scale factors are adjusted separately. The implementation of the new method is simple, the added cost of computation is low, and its applicability is wide. Numerical experiments show that the scaling reduces the integration error drastically. In the case of pure Keplerian orbits, the truncation error grows linearly with respect to time, and the round-off error grows more slowly than that. When perturbations exist, a component that grows with the second or a higher power of time appears in the truncation error, but its magnitude is reduced significantly as compared with the case without scaling. The rate of decrease varies roughly as the 5/4 to 5/2 power of the strength of the perturbing acceleration, where the power index depends on the type of perturbation. The method seems to suppress the accumulation of round-off errors in the perturbed cases, although the details remain to be investigated. The new approach provides a fast and high-precision device with which to simulate the orbital motions of major and minor planets, natural and artificial satellites, comets, and space vehicles at a negligible increase in computational cost.

Long-Term Integration Error of Kustaanheimo-Stiefel Regularized Orbital Motion

We confirm that the positional error of a perturbed two-body problem expressed in the Kustaanheimo-Stiefel (K-S) variable is proportional to the fictitious time s, which is the independent variable in the K-S transformation. This property does not depend on the type of perturbation, on the integrator used, or on the initial conditions, including the nominal eccentricity. The error growth of the physical time evolution and the Kepler energy is proportional to s if the perturbed harmonic oscillator part of the equation of motion is integrated by a time-symmetric integration formula, such as the leapfrog or the symmetric multistep method, and is proportional to s2 when using traditional integrators, such as the Runge-Kutta, Adams, Stormer, and extrapolation methods. Also, we discovered that the K-S regularization avoids the step size resonance/instability of the symmetric multistep method that appears in the unregularized cases. Therefore, the K-S regularized equation of motion is useful to investigate the long-term behavior of perturbed two-body problems, namely, those used for studying the dynamics of comets, minor planets, the Moon, and other natural and artificial satellites.

Orbital evolution of long-period comets I. Trivariate Monte Carlo simulation

The mean orbital evolution of long-period comets for 16 representative initial orbits to short-period comets is calculated by a Monte Carlo method. First, trivariate perturbation distributions of barycentric Kepler energy, total angular momentum, and its z component in single encounters of comets with Jupiter are obtained numerically. Their characteristics are examined in detail and the distributions are found to be simple, symmetric, and easy to handle. Second, utilizing these distributions, we have done trivariate Monte Carlo simulations of the orbital evolution of long-period comets, with special emphasis on high-inclination orbits. About half of the 16 initial orbits are traced up to 5000 returns. For each of these orbits, the mean values of semimajor axis, perihelion distance, and inclination; their standard deviations, survival, and capture rates; as well as time scales of orbital evolution are calculated as functions of return number. Survival rates of the initial orbits with high inclination (∼90°) and small perihelion distance (∼1–2 AU) have been found to be only two or three times smaller than those of the main-source orbits of short-period comets established quantitatively by Everhart. The time scales of orbitsl evolution of the former, however, are nearly 10 times longer than the latter. There is a general trend that, for smaller perihelion distance, the survival efficiency becomes higher. The results of this paper should be considered a basis for a succeeding paper (Paper II) in which the physical lifetime of comets will be determined, and a comparison with the orbital data will be done.