Orbit Segments Research Articles

A solar rotation signature in cosmic dust: Frequency analysis of dust particle impacts on the Wind spacecraft

Dust particle impacts on the Wind spacecraft were detected with its plasma wave instrument Wind/WAVES. Frequency analysis on the resulting dust impact time series has revealed spectral peaks indicative of a solar rotation signature. We investigated whether this solar rotation signature is embedded in the interplanetary or in the interstellar dust (ISD) and whether it is caused by co-rotating interaction regions (CIRs), by the sector structure of the interplanetary magnetic field (IMF), or by external effects. We performed frequency analysis on different subsets of the data to investigate the origin of these spectral peaks, comparing segments of Wind's orbit when the spacecraft moved against or with the ISD inflow direction and comparing the time periods of the ISD focusing phase and the ISD defocusing phase of the solar magnetic cycle. A superposed epoch analysis of the number of dust impacts during CIRs was used to investigate the systematic effect of CIRs. Case studies of time periods with frequent or infrequent occurrences of CIRs were performed and compared to synthetic data of cosmic dust impacts affected by CIRs. We performed similar case studies for time periods with a stable or chaotic IMF sector structure. The superposed epoch analysis was repeated for a time series of the spacecraft floating potential. Spectral peaks were found at the solar rotation period of $ sim d $ and its harmonics at $13.5\ d $ and $9\ d $. This solar rotation signature may affect both interplanetary and interstellar dust. The appearance of this signature correlates with the occurrence of CIRs but not with the stability of the IMF sector structure. The CIRs cause, on average, a reduction in the number of dust impact detections. Periodic changes of the spacecraft's floating potential were found to partially counteract this reduction by enhancing the instrument's sensitivity to dust impacts; these changes of the floating potential are thus unlikely to be the cause of the solar rotation signature.

Disappearing Solar Wind at Mars: Changes in the Mars‐Solar Wind Interaction

AbstractOn 26 December 2022 the solar wind density dropped by over an order of magnitude and remained low for about a day. We have utilized in‐situ plasma measurements made by the Mars Atmosphere and Volatile EvolutioN mission to determine how this change affected the Mars‐solar wind interaction. During this time period, on inbound orbit segments, MAVEN sampled the terminator ionosphere, which switched from a magnetized to unmagnetized state immediately following the minimum in solar wind density. The magnetic field amplitude was typically 5–10 nT within the upper ionosphere prior to the event and consistently <1 nT after. During the event the magnetic pressure dominated immediately above the ionosphere while within the ionosphere the ionospheric plasma pressure dominated. The high altitude terminator ionosphere remained in this unmagnetized state throughout the event, suggesting that it was the new equilibrium state of the system. The terminator upper ionosphere returned to its original magnetized state once the solar wind density had recovered. The outbound orbit segments sampled the dayside subsolar region which remained magnetized throughout the event: the magnetization state of the ionosphere varied locally, dependent upon the solar zenith angle and corresponding incident solar wind dynamic pressure. Such conditions are different to the commonly reported unmagnetized ionospheric state at Venus during solar maximum conditions, where the interplanetary magnetic field is repelled from the entire dayside ionosphere. Drastic changes in the upstream solar wind are able to change the Mars‐solar wind interaction state on timescales less than one MAVEN orbit (∼3.5 hr).

Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four-body problem

We develop constructive, computer assisted arguments for proving the existence of transverse homoclinic connections for hyperbolic periodic orbits in Hamiltonian systems. These arguments are applied at a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four-body problem: a six dimensional vector field. The idea is to formulate the desired connecting orbits as solutions of certain two point boundary value problems for orbit segments which originate and terminate on the local stable/unstable manifolds attached to a periodic orbit. These boundary value problems are studied via a Newton-Kantorovich argument in an appropriate Cartesian product of Banach algebras of rapidly decaying sequences of Chebyshev coefficients. Perhaps the most delicate part of the problem is controlling the boundary conditions, which must lie on the local stable/unstable manifolds of the periodic orbit. For this portion of the problem we use a parameterization method to develop Fourier-Taylor approximations equipped with mathematically rigorous a-posteriori error bounds. This requires validated computation of a finite number of Fourier-Taylor coefficients via Newton-Kantorovich arguments in appropriate Cartesian product of rapidly decaying sequences of Fourier coefficients, followed by a fixed point argument to bound the tail terms of the Taylor expansion. After some geometric considerations in the energy manifold, transversality of the connection follows as a direct consequence of the Newton-Kantorovich argument.

On Feldman–Katok metric and entropy formulas

In this paper, we study the Feldman–Katok metric and give corresponding entropy formulas by replacing Bowen metric with Feldman–Katok metric. It turns out that the Feldman–Katok metric is the weakest one, in the sense of allowing desynchrony and jump in the process of measuring orbit segments’ distance, that makes the entropy formulas valid. Some related topics are also discussed.

Energy Transition Density of Driven Chaotic Systems: A Compound Trace Formula

Oscillations in the probability density of quantum transitions of the eigenstates of a chaotic Hamiltonian within classically narrow energy ranges have been shown to depend on closed compound orbits. These are formed by a pair of orbit segments, one in the energy shell of the original Hamiltonian and the other in the energy shell of the driven Hamiltonian, with endpoints that coincide. Viewed in the time domain, the same pair of trajectory segments arises in the semiclassical evaluation of the trace of a compound propagator: the product of the complex exponentials of the original Hamiltonian and of its driven image. It is shown here that the probability density is the double Fourier transform of this trace, and that the closed compound orbits emulate the role played by the periodic orbits in Gutzwiller’s trace formula in its semiclassical evaluation. The phase of the oscillations with the energies or the evolution parameters agree with those previously obtained, whereas the amplitude of the contribution of each closed compound orbit is more compact and independent of any feature of the Weyl–Wigner representation in which the calculation was carried out.

Existence and uniqueness of equilibrium states for systems with specification at a fixed scale: an improved Climenhaga–Thompson criterion* *This research has been supported [in part] by CAPES—Finance Code 001 and CNPq-grants. MJP was partially supported by Grant Cientista do Nosso Estado-FAPERJ.

We consider the uniqueness of equilibrium states for dynamical systems that satisfy certain weak, non-uniform versions of specification, expansivity, and the Bowen property at a fixed scale. Following Climenhaga–Thompson’s approach which was originally due to Bowen and Franco, we prove that equilibrium states are unique even when the weak specification assumption only holds on a small collection of orbit segments. This improvement will be crucial in a subsequent work, where we will prove that (open and densely) every Lorenz attractor supports a unique measure of maximal entropy.

Classification of Cassini’s Orbit Regions as Magnetosphere, Magnetosheath, and Solar Wind via Machine Learning

Several machine learning algorithms and feature subsets from a variety of particle and magnetic field instruments on-board the Cassini spacecraft were explored for their utility in classifying orbit segments as magnetosphere, magnetosheath or solar wind. Using a list of manually detected magnetopause and bow shock crossings from mission scientists, random forest (RF), support vector machine (SVM), logistic regression (LR) and recurrent neural network long short-term memory (RNN LSTM) classification algorithms were trained and tested. A detailed error analysis revealed a RNN LSTM model provided the best overall performance with a 93.1% accuracy on the unseen test set and MCC score of 0.88 when utilizing 60 min of magnetometer data (|B|, Bθ, Bϕ and BR) to predict the region at the final time step. RF models using a combination of magnetometer and particle data, spanning H+, He+, He++ and electrons at a single time step, provided a nearly equivalent performance with a test set accuracy of 91.4% and MCC score of 0.84. Derived boundary crossings from each model’s region predictions revealed that the RNN model was able to successfully detect 82.1% of labeled magnetopause crossings and 91.2% of labeled bow shock crossings, while the RF model using magnetometer and particle data detected 82.4 and 74.3%, respectively.

Detection of Snow Cover from Historical and Recent AVHHR Data—A Thematic TIMELINE Processor

Global snow cover forms the largest and most transient part of the cryosphere in terms of area. On the local and regional scale, small changes can have drastic effects such as floods and droughts, and on the global scale is the planetary albedo. Daily imagery of snow cover forms the basis of long-term observation and analysis, and only optical sensors offer the necessary spatial and temporal resolution to address decadal developments and the impact of climate change on snow availability. The MODIS sensors have been providing this daily information since 2000; before that, only the Advanced Very High-Resolution Radiometer (AVHRR) from the National Oceanographic and Atmospheric Administration (NOAA) was suitable. In the TIMELINE project of the German Aerospace Center, the historic AVHRR archive in HRPT (High Resolution Picture Transmission) format is processed for the European area and, among other processors, one output is the thematic product ‘snow cover’ that will be made available in 1 km resolution since 1981. The snow detection is based on the Normalized Difference Snow Index (NDSI), which enables a direct comparison with the MODIS snow product. In addition to the NDSI, ERA5 re-analysis data on the skin temperature and other level 2 TIMELINE products are included in the generation of the binary snow mask. The AVHRR orbit segments are projected from the swath projection into LAEA Europe, aggregated into daily coverages, and from this, the 10-day and monthly snow covers are finally calculated. In this publication, the snow cover algorithm is presented, as well as the results of the first validations and possible applications of the final product.

Consolidating ICESat-2 Ocean Wave Characteristics with CryoSat-2 during the CRYO2ICE Campaign

Using the Ice, Cloud, and land Elevation Satellite 2 (ICESat-2) global high-resolution elevation measurements, it is possible to distinguish individual surface ocean waves. With the vast majority of ocean surveying missions using radar satellites, ICESat-2 observations are an important addition to ocean surveys. ICESat-2 can also provide additional observations not possible with radar. In this paper, we consolidate the ICESat-2 ocean observations by comparing the significant wave height (SWH) with coincident CryoSat-2 radar observations during the CRYO2ICE campaign from August 2020 to August 2021. We use 136 orbit segments, constrained to the Pacific and Atlantic oceans as well as the Bering Sea, to compare observations to show the level of agreement between these systems. Three models based on ICESat-2 are used in the comparison: the standard ocean data output (ATL12), a method of modeling the individual surface waves using the geolocated photons and, functioning as a baseline, an approach using the standard deviation of the ocean surface. We find the following correlations between the SWHs from the models and the SWHs from CryoSat-2: 0.97 for ATL12, 0.95 for the observed waves model, and 0.97 for the standard deviation model. In the same comparison, we find mean differences relative to the observed SWHs for each model, as well as errors, which increase as the SWH increases. The SWH observed from ICESat-2 is found to agree with observations from CryoSat-2, with limitations due to changes in the sea state between the satellite observations. Observing the individual surface waves from ICESat-2 can therefore provide additional observed properties of the sea state that can be used alongside other global observations.

The ACTIONFINDER: An Unsupervised Deep Learning Algorithm for Calculating Actions and the Acceleration Field from a Set of Orbit Segments

Abstract We introduce the ACTIONFINDER, a deep learning algorithm designed to transform a sample of phase-space measurements along orbits in a static potential into action and angle coordinates. The algorithm finds the mapping from positions and velocities to actions and angles in an unsupervised way, by using the fact that points along the same orbit have identical actions. Here we present the workings of the method and test it on simple axisymmetric models, comparing the derived actions to those generated with the Torus Mapping technique. We show that it recovers the torus actions for halo-type orbits in a realistic model of the Milky Way to ∼0.6% accuracy with as few as 1024 input phase-space measurements. These actions are much better conserved along orbits than those estimated with the Stäckel fudge. In our case, the reciprocal mapping from actions and angles to positions and velocities can also be learned. One of the advantages of the ACTIONFINDER is that it does not require the underlying potential to be known in advance—indeed it is designed to return the acceleration field. We expect the algorithm to be useful for analyzing the properties of dynamical systems in numerical simulations. However, our ultimate goal with this effort will be to apply it to real stellar streams to recover the Galactic acceleration field in a way that is relatively agnostic about the underlying dark matter properties or the behavior of gravity.

Pristine PSP/WISPR Observations of the Circumsolar Dust Ring near Venus's Orbit

Abstract The Parker Solar Probe mission (PSP) has completed seven orbits around the Sun. The Wide-field Imager for Solar Probe (WISPR) on PSP consists of two visible light heliospheric imagers, which together image the interplanetary medium between 13.°5 and 108° elongation. The PSP/WISPR nominal science observing window occurs during the solar encounters, which take place when the spacecraft (S/C) is within 0.25 au from the Sun. During Orbit 3, an extended science campaign took place while PSP transited between 0.5 and 0.25 au (during both inbound and outbound orbit segments). PSP mission operations implemented a variety of 180° S/C rolls about the S/C-Sun pointing axis during the extended science window. The vantage of the PSP location, combined with the different S/C roll orientations, allowed us to unveil a circumsolar dust density enhancement associated with Venus’s orbit. Specifically, we observed an excess brightness band of about 1% at its center over the brightness of the background zodiacal light in all PSP/WISPR images obtained during the extended campaign. We explain this brightness band as due to an increase in the density of the circumsolar dust orbiting the Sun close to the Venusian orbit. The projected latitudinal extent of the ring is estimated at about 0.043 au ± 0.004 au, exhibiting an average density enhancement of the order of 10%. Here, we report and characterize the first comprehensive, pristine observations of the plane-of-sky projection of the dust ring in almost its full 360° longitudinal extension.

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of $\mathbb{R}^3$ with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.

GNSS Radio Occultation Simulation Using Multiple Phase Screen Orbit Sampling

Wave optics propagators (WOPs) are commonly used to describe the propagation of radio signals through earth’s atmosphere. In radio occultation (RO) context, multiple phase screen (MPS) method has been used to model the effects of the atmosphere in Global Navigation Satellite System (GNSS) signals during an occultation event. WOP implementation includes, in addition to MPS, a diffraction integral as the final step to calculate the radio signal measured in the low-earth orbit (LEO) satellite. This approach considers vacuum as the propagation medium at high altitudes, which is not always the case when the ionosphere is taken into account in simulations. An alternative approach is using MPS all the way to LEO in order to sample the GNSS signal in orbit. This approach, named MPS orbit sampling (MPS-OS), is evaluated in this letter. Different scenarios of setting occultation assuming a short segment of the LEO orbit have been simulated using MPS and MPS-OS. Results have been compared to Abel transform references. Furthermore, a long segment scenario has been evaluated as well. A comparison of bending angle (BA) and residual ionospheric error (RIE) shows the equivalence between MPS and MPS-OS results. The main application of MPS-OS should be in occultation events with long segments of orbit and including ionosphere, in which a standard WOP may not be appropriate.

Charging of Geostationary Satellite Electro-L2 in the Earth Shadow

Magnetosphere plasma electron and proton fluxes in the energy range 0.04–11.3 keV were measured using electrostatic spectrometers established onboard the geostationary Electro-L2 spacecraft (point of standing 76° EL). Measurements were done in the lit orbit segments and in the Earth shadow during the periods of vernal and autumn equinox in 2016–2017. In terms of the observation data, fluxes, and parameters of the plasma particles for different geomagnetic activity levels and potentials of the spacecraft metal base were determined. The measured values of the negative spacecraft metal base potential achieve 5–10 kV in the Earth shadow. Interpretation of the measurement results was done using the results of the spacecraft charging modeling obtained with the COULOMB-2 program package. Electric field gauges (EFGs) and electrostatic discharge (ESD) gauges were established on the spacecraft surface as well. The electrostatic spectrometers data and the COULOMB-2 results coincide with the EFGs data. ESDs were registered onboard the spacecraft during the spacecraft entrance into the Earth shadow and exit from the shadow.

Hybrid Differential Dynamic Programming in the Circular Restricted Three-Body Problem

This paper presents the application of differential dynamic programming to low-thrust trajectory optimization in the three-body problem along with supporting techniques. A penalty method is suggested for path inequality constraints and used to enforce a minimum radius constraint. Time variable formulations are presented for the flight time and a moving target. The latter is used to access all points along a desired periodic orbit as valid terminal states. Multiphase differential dynamic programming is applied to discontinuous initial guesses that combine segments of different periodic orbits. Numerical examples include multirevolution transfers between distant retrograde orbits, a heteroclinic transfer between planar Lyapunov orbits, and transfers between L1 and L2 halo orbits in the Earth–moon circular restricted three-body problem.

Origin and continuation of 3/2, 5/2, 3/1, 4/1 and 5/1 resonant periodic orbits in the circular and elliptic restricted three-body problem

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.

Motion planning for Unmanned Surface Vehicle based on Trajectory Unit

Aiming at fine motion control on a small scale area for unmanned surface vehicle (USV), an approach for motion planning based on Trajectory Unit has been proposed. By making use of USV's hydrodynamic model and combining corresponding rules, the method generates a Trajectory Unit set which contains different orbit segments. Through the location and direction that the set of tracks can reach, the route searching in research waters can be done. Finally, a practical motion route will be planned for an USV. The experimental results show that, the planning route can not only avoid obstacles, but also meet the movement characteristics of an USV. And the approach has realized the fine motion control for an USV in a small range of scenarios.

Geostationary Belt – State’s Territory or Province of Mankind?

Abstract The geostationary orbit is a special area in outer space. Because of its distinctive characteristics, it has constantly been the subject of economic and political desirability. Space powers, taking advantage of their technological superiority and rules applied by the International Telecommunication Union (ITU) retained a privileged position. Developing countries, responding to this state of affairs, have taken a number of measures to improve their positions. Some of them posed a challenge to the main regulation of space law (Bogota declaration was an attempt to exercise a national sovereignty over the segments of the geostationary orbit), some are based on the use of the legal gaps in ITU regulations. Given these circumstances, the specific case of geostationary belt contributes to the debate on the regulations governing space exploration.

Tangencies Between Global Invariant Manifolds and Slow Manifolds Near a Singular Hopf Bifurcation

Invariant manifolds of equilibria and periodic orbits are key objects that organize the behavior of a dynamical system both locally and globally. If multiple time scales are present in the dynamical system, there also exist so-called slow manifolds, that is, manifolds along which the flow is very slow compared with the rest of the dynamics. In particular, slow manifolds are known to organize the number of small oscillations of what are known as mixed-mode oscillations (MMOs). Slow manifolds are locally invariant objects that may interact with invariant manifolds, which are globally invariant objects; such interactions produce complicated dynamics about which only little is known from a few examples in the literature. We study the transition through a quadratic tangency between the unstable manifold of a saddle-focus equilibrium and a repelling slow manifold in a system where the corresponding equilibrium point undergoes a supercritical singular Hopf bifurcation. We compute the manifolds as families of orbit segments with a two-point boundary value problem setup and track their intersections, referred to as connecting canard orbits, as a parameter is varied. We describe the local and global properties of the manifolds, as well as the role of the interaction as an organizer of large-amplitude oscillations in the dynamics. We find and describe recurrent dynamics in the form of MMOs, which can be continued in parameters to Shilnikov homoclinic bifurcations. We detect and identify two such homoclinic orbits and describe their interactions with the MMOs.

Solar wind‐ and EUV‐dependent models for the shapes of the Martian plasma boundaries based on Mars Express measurements

AbstractThe long operational life (2003‐present) of Mars Express (MEX) has allowed the spacecraft to make plasma measurements in the Martian environment over a wide range of upstream conditions. We have analyzed ∼7000 MEX orbits, covering three orders of magnitude in solar wind dynamic pressure, with data from the on board Analyzer of Space Plasmas and Energetic Particles (ASPERA‐3) package, mapping the locations where MEX crosses the main plasma boundaries, induced magnetosphere boundary (IMB), ionosphere boundary (IB), and bow shock (BS). A coincidence scheme was employed, where data from the Ion Mass Analyzer (IMA) and the Electron Spectrometer (ELS) had to agree for a positive boundary identification, which resulted in crossings from 1083 orbit segments that were used to create dynamic two‐parameter (solar wind density, nsw, and velocity vsw) dependent global dynamic models for the IMB, IB, and BS. The modeled response is found to be individual to each boundary. The IMB scales mainly dependent on solar wind dynamic pressure and EUV intensity. The BS location closely follows the location of the IMB at the subsolar point, though under extremely low nsw and vsw the BS assumes a more oblique shape. The IB closely follows the IMB on the dayside and changes its nightside morphology with different trends for nsw and vsw. We also investigate the influence of extreme ultraviolet (EUV) radiation on the IMB and BS, finding that increased EUV intensity expands both boundaries.