True Orbit Research Articles

Modelling the orbital histories of satellites of Milky Way-mass galaxies: testing static host potentials against cosmological simulations

ABSTRACT Understanding the evolution of satellite galaxies of the Milky Way (MW) and M31 requires modelling their orbital histories across cosmic time. Many works that model satellite orbits incorrectly assume or approximate that the host halo gravitational potential is fixed in time and is spherically symmetric or axisymmetric. We rigorously benchmark the accuracy of such models against the FIRE-2 cosmological baryonic simulations of MW/M31-mass haloes. When a typical surviving satellite fell in ($3.4\!-\!9.7\, \rm {Gyr}$ ago), the host halo mass and radius were typically 26–86 per cent of their values today, respectively. Most of this mass growth of the host occurred at small distances, $r\lesssim 50\, \rm {kpc}$, opposite to dark matter only simulations, which experience almost no growth at small radii. We fit a near-exact axisymmetric gravitational potential to each host at z = 0 and backward integrate the orbits of satellites in this static potential, comparing against the true orbit histories in the simulations. Orbital energy and angular momentum are not well conserved throughout an orbital history, varying by 25 per cent from their current values already $1.6\!-\!4.7\, \rm {Gyr}$ ago. Most orbital properties are minimally biased, ≲10 per cent, when averaged across the satellite population as a whole. However, for a single satellite, the uncertainties are large: recent orbital properties, like the most recent pericentre distance, typically are ≈20 per cent uncertain, while earlier events, like the minimum pericentre or the infall time, are ≈40–80 per cent uncertain. Furthermore, these biases and uncertainties are lower limits, given that we use near-exact host mass profiles at z = 0.

-chain closing lemma for certain partially hyperbolic diffeomorphisms

Abstract For every $r\in \mathbb {N}_{\geq 2}\cup \{\infty \}$ , we prove a $C^r$ -orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with one-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism f, if a point y is chain attainable from x through pseudo-orbits, then for any neighborhood U of x and any neighborhood V of y, there exist true orbits from U to V by arbitrarily $C^r$ -small perturbations. As a consequence, we prove that for $C^r$ -generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity.

Standing and traveling waves in a model of periodically modulated one-dimensional waveguide arrays.

In the present work we study coherent structures in a one-dimensional discrete nonlinear Schrödinger lattice in which the coupling between waveguides is periodically modulated. Numerical experiments with single-site initial conditions show that, depending on the power, the system exhibits two fundamentally different behaviors. At low power, initial conditions with intensity concentrated in a single site give rise to transport, with the energy moving unidirectionally along the lattice, whereas high-power initial conditions yield stationary solutions. We explain these two behaviors, as well as the nature of the transition between the two regimes, by analyzing a simpler model where the couplings between waveguides are given by step functions. For the original model, we numerically construct both stationary and moving coherent structures, which are solutions reproducing themselves exactly after an integer multiple of the coupling period. For the stationary solutions, which are true periodic orbits, we use Floquet analysis to determine the parameter regime for which they are spectrally stable. Typically, the traveling solutions are characterized by having small-amplitude oscillatory tails, although we identify a set of parameters for which these tails disappear. These parameters turn out to be independent of the lattice size, and our simulations suggest that for these parameters, numerically exact traveling solutions are stable.

A torsion-based solution to the hyperbolic regime of the J_2-problem

A popular intermediary in the theory of artificial satellites is obtained after the elimination of parallactic terms from the J_2-problem Hamiltonian. The resulting quasi-Keplerian system is in turn converted into the Kepler problem by a torsion. When this reduction process is applied to unbounded orbits, the solution is made of Keplerian hyperbolae. For this last case, we show that the torsion-based solution provides an effective alternative to the Keplerian approximation customarily used in flyby computations. Also, we check that the extension of the torsion-based solution to higher orders of the oblateness coefficient yields the expected convergence of asymptotic solutions to the true orbit.

An 8-bit integer true periodic orbit PRNG based on delayed Arnold’s cat map

Designing a pseudo-random number generator (PRNG) exhibiting true periodic orbits with large periods is promising for embedded applications. Implementing chaotic behaviors on a finite precision digital platform using conventional unsigned integer or floating-point arithmetic yields short period limit-cycles. It leads the output to diverge from the true chaotic orbits. This paper presents an unsigned integer encoded PRNG based on the delayed quantized Arnold’s Cat Map (QACM) that exhibits true periodic orbits with large periods. The proposed PRNG is obtained by delaying one of the 2-D QACM outputs with a linear feedback shift register (LFSR) thereby increasing its period. By investigating its period, we show that the periodicity increases with an increase in the delay and bit-precision. We also present the proposed PRNG unit’s hardware architecture and its synthesis results targeting on a Xilinx Zynq 7000 Field Programmable Gate Array (FPGA) with 8-bit precision and delay δ=58. Further, we synthesize the design using 90 nm CMOS technology for ASIC realization. The synthesis results demonstrate the throughput as 2.208 Gbps and 5.46 Gbps at 138 MHz and 342 MHz operating frequency on FPGA and ASIC respectively. We experimentally evaluate the performance of random numbers using the NIST 800-22-1 tests along with the correlation and key sensitivity tests.

Comment on “Tidally Synchronized Solar Dynamo: A Rebuttal” by Nataf (Solar Phys. 297, 107, 2022)

Nataf (Solar Phys.297, 107, 2022) has recently asserted that the hypothesis that the solar dynamo may be synchronized by planetary tidal forces is unsupported by any evidence. He reached this conclusion by adopting a simplistic tidal model (which was discussed in his Appendix A) that relies solely on the effect of Mercury, Venus, Earth, and Jupiter whose orbits were assumed to be circular. His model was unable to produce tides with periods compatible with those of the Schwabe 11-year solar cycle. I demonstrate here that the modeling in Nataf (2022) is erroneous and that a correct modeling and interpretation of the planetary tidal function, which accounts for all planets and their true orbits, fits well with the spectral requirements of the Schwabe 11-year solar cycle. This result has been already shown and discussed in a substantial body of scholarly research on the subject, which Nataf apparently ignored. A recent and extended review of the empirical and theoretical evidences supporting the planetary synchronized solar dynamo theory was offered by Scafetta and Bianchini (Front. Astron. Space Sci.9, 937930, 2022).

Asteroids seen by JWST-MIRI: Radiometric size, distance, and orbit constraints

Infrared measurements of asteroids are crucial for the determination of physical and thermal properties of individual objects, and for understanding the small-body populations in the solar system as a whole. However, standard radiometric methods can only be applied if the orbit of an object is known, hence its position at the time of the observation. With JWST-MIRI observations the situation will change and many unknown, often very small, solar system objects will be detected. Later orbit determinations are difficult due to the faintness of the objects and the lack of dedicated follow-up concepts. We present MIRI observations of the outer-belt asteroid (10920) 1998 BC1 and an unknown object, detected in all nine MIRI bands in close apparent proximity to (10920). We developed a new method called STM-ORBIT to interpret the multi-band measurements without knowing the object’s true location. The power of the new technique is that it determines the most-likely heliocentric and observer-centric distance and phase angle ranges, allowing us to make a radiometric size estimate. The application to the MIRI fluxes of (10920) was used to validate the method. It leads to a confirmation of the known radiometric size-albedo solution, and puts constraints on the asteroid’s location and orbit in agreement with its true orbit. To back up the validation of the method, we obtained additional ground-based light curve observations of (10920), combined withGaiadata, which indicate a very elongated object (a/b ≥1.5), with a spin-pole at (λ,β)ecl= (178°, +81°), with an estimated error of about 20°, and a rotation period of 4.861191 ± 0.000015 h. A thermophysical study of all available JWST-MIRI and WISE measurements leads to a size of 14.5–16.5 km (diameter of an equal-volume sphere), a geometric albedopVbetween 0.05 and 0.10, and a thermal inertia in the range 9–35 (best value 15) J m−2s−0.5K−1. For the newly discovered MIRI object, the STM-ORBIT method revealed a size of 100–230 m. The new asteroid must be on a low-inclination orbit (0.7° <i <2.0°) and it was located in the inner main-belt region during JWST observations. A beaming parameterηlarger than 1.0 would push the size even below 100 meters, a main-belt regime that has escaped IR detections so far. This kind of MIRI observations can therefore contribute to formation and evolution studies via classical size-frequency studies, which are currently limited to objects larger than about one kilometer in size. We estimate that MIRI frames with pointings close to the ecliptic and short integration times of only a few seconds will always include a few asteroids; most of them will be unknown objects.

Uncovering the true periods of the young sub-Neptunes orbiting TOI-2076

Context. TOI-2076 is a transiting three-planet system of sub-Neptunes orbiting a bright (G = 8.9 mag), young (340 ± 80 Myr) K-type star. Although a validated planetary system, the orbits of the two outer planets were unconstrained as only two non-consecutive transits were seen in TESS photometry. This left 11 and 7 possible period aliases for each. Aims. To reveal the true orbits of these two long-period planets, precise photometry targeted on the highest-probability period aliases is required. Long-term monitoring of transits in multi-planet systems can also help constrain planetary masses through TTV measurements. Methods. We used the MonoTools package to determine which aliases to follow, and then performed space-based and ground-based photometric follow-up of TOI-2076 c and d with CHEOPS, SAINT-EX, and LCO telescopes. Results. CHEOPS observations revealed a clear detection for TOI-2076 c at $P = 21.02538_{ - 0.00074}^{ + 0.00084}$ d, and allowed us to rule out three of the most likely period aliases for TOI-2076 d. Ground-based photometry further enabled us to rule out remaining aliases and confirm the P = 35.12537 ± 0.00067 d alias. These observations also improved the radius precision of all three sub-Neptunes to 2.518 ± 0.036, 3.497 ± 0.043, and 3.232 ± 0.063 R⊕. Our observations also revealed a clear anti-correlated TTV signal between planets b and c likely caused by their proximity to the 2:1 resonance, while planets c and d appear close to a 5:3 period commensurability, although model degeneracy meant we were unable to retrieve robust TTV masses. Their inflated radii, likely due to extended H-He atmospheres, combined with low insolation makes all three planets excellent candidates for future comparative transmission spectroscopy with JWST.

A medium-scale study of using machine learning fusion to improve TLE prediction precision without external information

This paper presents a machine learning (ML) approach that can improve orbit prediction precision and reduce uncertainties with a properly designed fusion strategy. The ML models are trained based on historical TLE sets and then ML-modifications are generated for the future states and uncertainty predictions. The ML-modification is first interpreted as a pseudo-measurement of the true orbit prediction error and then fused with the conventional orbital state and uncertainty propagation result. The regularized particle filter (PF) with progressive correction and systematic sampling is adopted for uncertainty propagation while using the Simplified General Perturbations #4 (SGP4) model. Comprehensive experiments are conducted and analyzed on over 100 resident space objects (RSOs) in different orbit types. The results demonstrate that the fusion strategy enables the ML approach to work with the advanced PF prediction method. The prediction precision can be significantly improved for the majority of the cases. The prediction accuracy is observed to be improved in some cases but without a clear pattern. The results and analysis also suggest that future studies of a monitoring system facilitating the fusion process would be helpful in practical applications.

Scheduling Direct Imaging Observations Based on Radial Velocity Orbital Fits: Best Practices for Translating Orbits and Failure Modes

Abstract Future direct imaging mission concepts are planning to observe planets discovered via Doppler spectroscopy, with observation scheduling decisions based on radial velocity (RV) fit orbits to increase the mission’s odds of making a detection. However, some orbital parameters, such as inclination and planet radius, that are necessary to determine when a planet is detectable via imaging cannot be resolved with only RV data. This work studies how to best use a RV fit to determine when a planet will be detectable with direct imaging and quantify the impacts of RV-measurement precision on direct imaging observations. Beginning by defining a “true” planet, we simulate RV observations, fit the RV observations via Bayesian orbit fitting, construct many Keplerian orbits based on the RV fit and priors on the unknown orbital parameters, and propagate the constructed orbits and the true orbit for 20 yr to determine when the constructed orbits deviate significantly. We identify four methods of constructing orbits, ultimately finding that creating and sampling high-likelihood orbital parameters from a multivariate Gaussian produces the best results for high RV error and that creating and sampling from a simple normal distribution for each orbital parameter independently performs the best for low RV error. Further, we establish two modes of failure: intermittent failure, which captures when the constructed orbits incorrectly indicate that a planet is detectable; and dispersion failure, when the constructed orbits have dispersed so much that they no longer provide useful information.

Calibration of ISA accelerometer sensing axes for the BepiColombo mission

ISA is a three-axis accelerometer embarked on-board BepiColombo, a joint ESA/JAXA space mission launched on the 19th of October 2018 ​at 22.45 (local time) and devoted to put two spacecraft (MPO and MMO) around Mercury (2025), for a deeper investigation of the planet and its environment. ISA aims at measuring, directly and not by modelling, first time for a planet other than Earth, the non-gravitational perturbations the MPO will undergo during its stay around Mercury. Such data will be used within the frame of the Radio-Science Experiments to reconstruct, a posteriori and with very high accuracy, the true gravitational orbit of the spacecraft and, at last, the gravity field of Mercury. Three single accelerometers, each one characterised by its own sensing axis, i.e. the direction along which it is sensitive to accelerations, collect the non-gravitational perturbations encountered by the MPO. The arrangement of sensing axes, whose knowledge is essential to reconstruct the sensed accelerations, was measured with very high accuracy by the Experimental Gravitation group at IAPS/INAF in Rome, which manages ISA and holds the PI-ship. The group developed a calibration procedure to identify on-ground the direction of ISA sensing axes with respect to an optical reference frame. An experimental facility was designed and assembled to accomplish the measurement. In November 2015, the calibration was performed on the ISA Flight Model, currently in cruise phase towards Mercury. Results of the measurement campaign are reported and prove the determination of ISA sensing axes with an accuracy within 20–30 arcseconds per single axis, well below the 100 arcseconds required by the Radio-Science needs.

Stability and bifurcation analyses for exterior resonant families in Earth-Moon system

In this work, stability and bifurcation analyses for exterior resonant orbit families have been investigated for a test spacecraft around Earth-Moon system. The initial states are obtained for certain resonance ratio orbits in the two-body problem. The shooting method is applied to compute the true resonant orbits families based on the planar circular restricted three-body problem. By means of numerical approach, the 1:2 and 2:3 families of resonant orbits are studies, and the distribution of eigenvalues, stability curves are presented. Inparticular, some critical points, corresponding to period-doubling and tangent bifurcations, are found in the stability curves. The bifurcation and stability analyses show overall features of the resonant orbits. This study may provide alternative orbital design ideas.

A Theoretical Optimum Tilt Angle Model for Solar Collectors from Keplerian Orbit

Solar energy has been extensively used in industry and everyday life. A more suitable solar collector orientation can increase its utilization. Many studies have explored the best orientation of the solar collector installation from the perspective of data analysis and local-area cases. Investigating the optimal tilt angle of a collector from the perspective of data analysis, or guiding the angle of solar collector installation, requires an a priori theoretical tilt angle as a support. However, none of the current theoretical studies have taken the real motion of the Sun into account. Furthermore, a complete set of theoretical optimal tilt angles for solar energy is necessary for worldwide locations. Therefore, from the view of astronomical mechanics, considering the true orbit of the Sun, a mathematical model that is universal across the globe is proposed: the Kepler motion model is constructed from the solar orbit and transformed into the local Earth coordinate system. After that, the calculation of the optimal tilt angle solution is given. Finally, several examples are shown to demonstrate the variation of the optimal solar angle with month and latitude. The results show that for daily fixed solar collectors, the altitude angle of the collector should be about 6° above the noon solar altitude angle in summer and 6° lower in winter. For annual fixed collectors, the tilt angle should be slightly higher than the latitude. In summary, this study demonstrates that when a location is specified, this model can be used to calculate the theoretical optimum tilt angle of solar collectors for that position.

Utilizing true periodic orbits in chaos-based cryptography

Digital realizations of chaos-based cryptosystems suffer from lack of a reliable method for implementation. The common choice for implementation is to use fixed or floating-point arithmetic directly. However, such an approach produces unreliable and erroneous pseudo-chaotic orbits, which are not the original orbits of the chaotic map. Also, using fixed/floating-point arithmetic is not a common approach of modern cryptography, which mostly deals with the manipulation of bits with simple logic operations. For these reasons, chaos-based cryptography and modern cryptography have a gap between them. Recently, new approaches are proposed for the digital implementation of chaotic systems. The LSB (Least Significant Bit) extension method is one such approach, which allows the production of true periodic and true chaotic orbits with simple logic operations. However, true orbits have not been utilized in any study yet. In this paper, true periodic orbits are utilized for the first time in a chaos-based cryptosystem. It turns out that true periodic orbits have many advantages over pseudo-chaotic orbits, but using them directly is not secure. Therefore, new design approaches are proposed for securely utilizing true periodic orbits. These design approaches fundamentally depend on structures called M-BSCM (Modified Binary Shift Chaotic Map). Accordingly, a chaos-based cryptosystem is modified for the utilization of true periodic orbits by replacing the chaotic maps inside it with M-BSCMs. The realizations are performed on both software and FPGA (Field-Programmable Gate Array) hardware. Such an approach will help to close the gap between chaos-based cryptography and modern cryptography.

A More Reliable Orbit Initialization Method for LEO Precise Orbit Determination Using GNSS

Precise orbit determination (POD) using GNSS has been rapidly developed and is the mainstream technology for the navigation of low Earth orbit (LEO) satellites. The initialization of orbit parameters is a key prerequisite for LEO POD processing. For a LEO satellite equipped with a GNSS receiver, sufficient discrete kinematic positions can be obtained easily by processing space-borne GNSS data, and its orbit parameters can thus be estimated directly in iterative manner. This method of direct iterative estimation is called as the direct approach, which is generally considered highly reliable, but in practical applications it has risk of failure. Stability analyses demonstrate that the direct approach is sensitive to oversized errors in the starting velocity vector at the reference time, which may lead to large errors in design matrix because the reference orbit may be significantly distorted, and eventually cause the divergence of the orbit parameter estimation. In view of this, a more reliable method, termed the progressive approach, is presented in this paper. Instead of estimating the orbit parameters directly, it first fits the discrete kinematic positions to a reference ephemeris in the form of the GNSS broadcast ephemeris, which construct a reference orbit that is smooth and close to the true orbit. Based on the reference orbit, the starting orbit parameters are computed in sufficient accuracy, and then the final orbit parameters are estimated with a high accuracy by using discrete kinematic positions as measurements. The stability analyses show that the design matrix errors are reduced in the progressive approach, which would assure more robust orbit parameter estimation than the direct estimation approach. Various orbit initialization experiments are performed on the KOMPSAT-5 and FY3C satellites. The results have fully verified the high reliability of the proposed progressive approach.

Performance Analysis of the Korean Positioning System Using Observation Simulation

The Korean government has a plan to build a new regional satellite navigation system called the Korean Positioning System (KPS). The initial KPS constellation is designed to consist of seven satellites, which include three geostationary Earth orbit (GEO) satellites and four inclined geosynchronous orbit (IGSO) satellites. KPS will provide an independent positioning, navigation, and timing (PNT) service in the Asia-Oceania region and can also be compatible with GPS. In the simulation for KPS, we employ 24 GPS as designed initially and 7 KPS satellites. Compared to the true orbit that we simulated, the averaged root mean square (RMS) values of orbit-only signal-in-space ranging errors (SISRE) are approximately 4.3 and 3.9 cm for KPS GEO and IGSO. Two different positioning solutions are analyzed to demonstrate the KPS performance. KPS standard point positioning (SPP) errors in the service area are about 4.7, 3.9, and 7.1 m for east (E), north (N), and up (U) components, respectively. The combined KPS+GPS SPP accuracy can be improved by 25.0%, 31.8%, and 35.0% compared to GPS in E, N, and U components. The averaged position errors for KPS kinematic precise point positioning (KPPP) are less than 10 cm. In the fringe of the KPS service area, however, the position RMS errors can reach about 40 cm. Unlike KPS, GPS solutions show high positioning accuracy in the KPS service area. The combined KPS+GPS can be improved by 28.7%, 27.1%, and 30.5% compared to GPS in E, N, and U components, respectively. It is noted that KPS can provide better performance with GPS in the Asia-Oceania region.

Numerical study on (ω, Lδ)-Lipschitz shadowing of stochastic differential equations

This paper focuses on the feasibility and numerical implementations of (ω, Lδ)-Lipschitz shadowing of a class of stochastic differential equations via numerical analysis. A new notion of random Lipschitz shadowing and exact estimation of shadowing distance of stochastic differential equations are investigated. Moreover, the measurability of the true orbits is also studied. Numerical simulations of chaotic orbits of stochastic Lu¨ system are provided to illustrate the effectiveness of the proposed theorem.

Orbital shadowing and stability for vector fields

The orbital shadowing property means that there exists a true orbit close to the approximate orbit in the sense of Hausdroff distance. In this paper, we show that in any dimension the C1 interior of the set of vector fields having the orbital shadowing property is contained in the set of Ω-stable vector fields. We also discuss the structural stability of vector fields having the orbital shadowing property.

Uncertainty Analysis of the Short-Arc Initial Orbit Determination

The solution of the short-arc angles-only orbit determination problem has large uncertainty because the topocentric range is not observable. For a certain angular observation tracklet with measurement noise, there exist numerous potential orbits, all of which are compatible with the observations. However, the solution of a deterministic initial orbit determination algorithm is usually far different from the true orbit, especially for the semimajor axis and the eccentricity. A new sampling method is proposed to describe the probability distribution of the orbit determination solutions. Firstly, a series of orbits are sampled in the semimajor axis - eccentricity plane. A chi-square test method is proposed to select candidate orbits from the sample orbits. The weights of the candidate orbits are calculated to measure their probability being the true orbit. Finally, the kernel density estimation algorithm is used to estimate the probability density function of the true orbits. With some a priori assumptions, the candidate orbits can be further screened, and their weight can be modified. The a priori knowledge can significantly improve the accuracy of the orbit determination solution.

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

AbstractWe present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic.We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three‐body problem and the spatial circular restricted three‐body problem.Our method differs, in several crucial aspects, from earlier works. Unlike the well‐known “two‐dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower‐dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry‐Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.