Step-size effect in the time-transformed leapfrog integrator on elliptic and hyperbolic orbits

ABSTRACT A drift-kick-drift (DKD) type leapfrog symplectic integrator applied for a time-transformed separable Hamiltonian (or time-transformed symplectic integrator; TSI) has been known to conserve the Kepler orbit exactly. We find that for an elliptic orbit, such feature appears for an arbitrary step size. But it is not the case for a hyperbolic orbit: When the half step size is larger than the conjugate momenta of the mean anomaly, a phase transition happens and the new position jumps to the non-physical counterpart of the hyperbolic trajectory. Once it happens, the energy conservation is broken. Instead, the kinetic energy minus the potential energy becomes a new conserved quantity. We provide a mathematical explanation for such phenomenon. Our result provides a deeper understanding of the TSI method, and a useful constraint of the step size when the TSI method is used to solve the hyperbolic encounters. This is particular important when an (Bulirsch–Stoer) extrapolation integrator is used together, which requires the convergence of integration errors.

Planets in Binaries: Formation and Dynamical Evolution

Binary systems are very common among field stars, yet the vast majority of known exoplanets have been detected around single stars. While this relatively small number of planets in binaries is probably partly due to strong observational biases, there is, however, statistical evidence that planets are indeed less frequent in binaries with separations smaller than 100 au, strongly suggesting that the presence of a close-in companion star has an adverse effect on planet formation. It is indeed possible for the gravitational pull of the second star to affect all the different stages of planet formation, from proto-planetary disk formation to dust accumulation into planetesimals, to the accretion of these planetesimals into large planetary embryos and, eventually, the final growth of these embryos into planets. For the crucial planetesimal-accretion phase, the complex coupling between dynamical perturbations from the binary and friction due to gas in the proto-planetary disk suggests that planetesimal accretion might be hampered due to increased, accretion-hostile impact velocities. Likewise, the interplay between the binary’s secular perturbations and mean motion resonances lead to unstable regions, where not only planet formation is inhibited, but where a massive body would be ejected from the system on a hyperbolic orbit. The amplitude of these two main effects is different for S- and P-type planets, so that a comparison between the two populations might outline the influence of the companion star on the planet formation process. Unfortunately, at present the two populations (circumstellar or circumbinary) are not known equally well and different biases and uncertainties prevent a quantitative comparison. We also highlight the long-term dynamical evolution of both S and P-type systems and focus on how these different evolutions influence the final architecture of planetary systems in binaries.

Cloudlet capture by transitional disk and FU Orionis stars

After its formation, a young star spends some time traversing the molecular cloud complex in which it was born. It is therefore not unlikely that, well after the initial cloud collapse event which produced the star, it will encounter one or more low mass cloud fragments, which we call “cloudlets” to distinguish them from full-fledged molecular clouds. Some of this cloudlet material may accrete onto the star+disk system, while other material may fly by in a hyperbolic orbit. In contrast to the original cloud collapse event, this process will be a “cloudlet flyby” and/or “cloudlet capture” event: A Bondi–Hoyle–Lyttleton type accretion event, driven by the relative velocity between the star and the cloudlet. As we will show in this paper, if the cloudlet is small enough and has an impact parameter similar or less than GM*/v∞2 (with v∞ being the approach velocity), such a flyby and/or capture event would lead to arc-shaped or tail-shaped reflection nebulosity near the star. Those shapes of reflection nebulosity can be seen around several transitional disks and FU Orionis stars. Although the masses in the those arcs appears to be much less than the disk masses in these sources, we speculate that higher-mass cloudlet capture events may also happen occasionally. If so, they may lead to the tilting of the outer disk, because the newly infalling matter will have an angular momentum orientation entirely unrelated to that of the disk. This may be one possible explanation for the highly warped/tilted inner/outer disk geometries found in several transitional disks. We also speculate that such events, if massive enough, may lead to FU Orionis outbursts.

On Solving Hyperbolic Trajectory Using New Predictor-Corrector Quadrature Algorithms

In this Paper, we proposed two new predictor corrector methods for solving Kepler's equation in hyperbolic case using quadrature formula which plays an important and significant rule in the evaluation of the integrals. The two procedures are developed that, in two or three iterations, solve the hyperbolic orbit equation in a very efficient manner, and to an accuracy that proves to be always better than 10-15. The solution is examined with and with grid size , using the first guesses hyperbolic eccentric anomaly is and , where is the eccentricity and is the hyperbolic mean anomaly.

Tangent-Impulse Interception for a Hyperbolic Target

The two-body interception problem with an upper-bounded tangent impulse for the interceptor on an elliptic parking orbit to collide with a nonmaneuvering target on a hyperbolic orbit is studied. Firstly, four special initial true anomalies whose velocity vectors are parallel to either of the lines of asymptotes for the target hyperbolic orbit are obtained by using Newton-Raphson method. For different impulse points, the solution-existence ranges of the target true anomaly for any conic transfer are discussed in detail. Then, the time-of-flight equation is solved by the secant method for a single-variable piecewise function about the target true anomaly. Considering the sphere of influence of the Earth and the upper bound on the fuel, all feasible solutions are obtained for different impulse points. Finally, a numerical example is provided to apply the proposed technique for all feasible solutions and the global minimum-time solution with initial coasting time.

The occurence of interstellar meteoroids in the vicinity of the Earth

If interstellar meteors are present among the registered meteor orbits, the distribution of the excesses of their heliocentric velocities should correspond to the distribution of radial velocities of close stars. Hence, for the velocity vi = 20 kms−1 of an interstellar meteor (with respect to the Sun) we obtain a heliocentric velocity vH = 46.6 kms−1 of an interstellar meteor arriving at the Earth. Moreover, a concentration of radiants to the Sun's apex should be observed. An analysis of the hyperbolic meteors among the 4581 photographic orbits of the IAU Meteor Data Center showed that the identification of the vast majority of the hyperbolic orbits in these catalogues has been caused by an erroneous determination of their heliocentric velocity and/or other parameters. Any error in the determination of vH, especially near the parabolic limit, can create an artificial hyperbolic orbit that does not really exist. On the basis of photographic meteors from the IAU MDC, the proportion of possible interstellar meteors decreased significantly (greater than 1 order of magnitude) after error analysis and does not exceed the value 2.5 × 10−4. Neither any concentration of radiants to the Sun's apex, nor any distribution following the motion of interstellar material has been found.