Relative Orbit Research Articles

Absolute and relative POD of LEO satellites in formation flying: Undifferenced and uncombined approach

Absolute or relative precise orbit determination (POD) is an essential prerequisite for many low earth orbit (LEO) missions. The POD of LEO satellites typically relays on processing the onboard global navigation satellite system (GNSS) measurements. The absolute POD is usually based on an ionosphere-free (IF) combination, and currently, integer ambiguity resolution (IAR) can be achieved only when external GNSS satellite phase bias (SPB) products are used. The use of these products is not flexible in multi-frequency/multi-constellation scenarios and is difficult to achieve in real-time missions. For relative POD, the double-differenced (DD) with IAR is the most general method. However, the differencing process amplifies observation noise and loses the opportunity to impose dynamic constraints on some eliminated parameters. In this contribution, based on the use of undifferenced and uncombined (UDUC) observations, a new model for both absolute and relative POD is proposed. In this model, the ambiguities of common-view satellites are constructed into DD form, thus IAR can be achieved without any external SPB products. Working with the UDUC observations, multi-frequency scenarios can be easily applied, and residuals can be separated for each frequency. In addition, with precise GNSS satellite clock/orbit products, both the absolute and relative orbits can be derived, which supports absolute and relative LEO POD. Based on onboard GPS observations of T-A and T-B satellites in formation flying, the performance of the UDUC POD model with DD ambiguity was evaluated. With the UDUC algorithm and IAR, the proposed model presented a consistency of 2.8–3.8 cm in 3D with the reference orbits, and the orbit difference was reduced by 16.3% and 10.6% for T-A and T-B compared with the IF-based POD, respectively. In addition, the relative orbit of the two satellites derived from the proposed model showed a consistency of 1.1–1.5 mm, which proved the feasibility of the UDUC POD model with DD ambiguity for formation flying missions.

The nearby astrometric-spectroscopic binary star Hip 68682

The nearby astrometric-spectroscopic binary star Hip 68682 has an orbital period of ∼ 9.88 yr. The dynamical state (component masses and kinematic parameters) of this system is redetermined by fitting all the observational data available to us, including the radial velocity data (RVD), the relative position data (RPD), and the Hipparcos Intermediate Astrometric Data (HIAD). The precision-weighted sum of squared residuals calculated from our results is significantly smaller than the previous ones. With the simulated data based on Chinese Space Station Telescope (CSST), the relative orbit could be improved further.Based on the known apparent magnitude information and the fitted parallax 60.59 ± 0.36 mas, we derive the absolute V magnitudes of the two components as 5.21 mag and 8.91 mag, respectively. The component masses are determined to be 0.946 ± 0.024M⊙ and 0.526 ± 0.014M⊙. These improved dynamical masses are helpful to constrain the stellar empirical mass–luminosity relation (MLR) for low mass stars. However, significant improvement of mass–luminosity relation for this mass region still needs more data.

Orbit size and estimated eye size in dinosaurs and other archosaurs and their implications for the evolution of visual capabilities

ABSTRACT Vision is one of the most important senses for animals, allowing them to interact with their environment and with further implications for evolutionary histories. However, relevant soft tissues, such as the eye and associated structures, are not preserved in fossil vertebrates, limiting our knowledge of their visual capabilities. Here, we quantified absolute and relative orbit size for 400 species of dinosaurs and other extinct archosaurs using linear measurements of the preserved skeletal elements as a proxy for visual capabilities. Our results demonstrate that the orbit makes up on average 20% of skull size with a strong and consistent correlation across all sampled groups. This trend is largely independent of temporal distribution, species richness, and phylogeny. In fact, relative orbit size is narrowly constrained and did not surpass 45% of skull size, suggesting physiological and functional controls. Estimated eye size was found to be absolutely larger in herbivores, whereas carnivores tended to have smaller eyes absolutely and compared with skull size. Relatively large eyes only occurred in small-bodied species and vice versa. However, eye size alone was not sufficient to discriminate between different activity patterns or to characterize visual capabilities in detail.

Subgroups of continuous full groups and relative continuous orbit equivalences of one-sided topological Markov shifts

Let A be an irreducible non-permutation matrix with entries in {0,1}. We study a family ΓA,f of subgroups of the continuous full group ΓA of the one-sided topological Markov shift (XA,σA). The subgroup ΓA,f is indexed by an integer valued continuous function f on XA such that ΓA,0=ΓA the non-amenable continuous full group and ΓA,1=ΓAAF the amenable AF full group. We will first prove that a spatial realization theorem for the subgroups ΓA,f under certain conditions on the functions f. We will second introduce relative versions of continuous orbit equivalence to study classification of the subgroups ΓA,f related to their associated groupoids and C⁎-algebras.

Continuous Low-Thrust Maneuver Planning for Space Gravitational Wave Formation Reconfiguration Based on Improved Particle Swarm Optimization Algorithm

This study proposes a three-spacecraft formation reconfiguration strategy of minimum fuel for space gravitational wave detection missions in the high Earth orbit (105 km). For solving the limitations of measurement and communication in long baseline formations, a control strategy of a virtual formation is applied. The virtual reference spacecraft provides a desired relative state between the satellites, which is then used to control the motion of the physical spacecraft to maintain the desired formation. A linear dynamics model based on relative orbit elements' parameterization is used to describe the relative motion in the virtual formation, which facilitates the inclusion of J2, SRP, and lunisolar third-body gravity effects and provides a direct insight into the relative motion geometry. Considering the actual flight scenarios of gravitational wave formations, a formation reconfiguration strategy based on continuous low thrust is investigated to achieve the desired state at a given time while minimizing interference to the satellite platform. The reconfiguration problem is considered a constrained nonlinear programming problem, and an improved particle swarm algorithm is developed to solve this problem. Finally, the simulation results demonstrate the performance of the proposed method in improving the maneuver sequence distribution and optimizing maneuver consumption.

Uniformly distributed orbits in $$\mathbb {T}^d$$ and singular substitution dynamical systems

We find sufficient conditions for the singularity of a substitution $\mathbb{Z}$-action's spectrum, which generalize the conditions given in arXiv:2003.11287, Theorem 2.4, and we also obtain a similar statement for a collection of substitution $\mathbb{R}$-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector $\mathbf{v}\in\mathbb{Q}^d$, we find necessary and sufficient conditions for the orbit of $\omega\mathbf{v}$ to be uniformly distributed modulo $1$ for almost every $\omega\in\mathbb{R}$. We use our results to find new examples of singular substitution $\mathbb{Z}$- and $\mathbb{R}$-actions.

Extrasolar Planet Candidates Identified by Single Transit from TESS

The discovery of planets around stars other than Sun was a grand milestone to understand origin and evolution of life in Universe. 2019 Nobel Prize for Physics was awarded to the first discovery of exoplanet around Sun-like stars, Peg 51b, in 1995. With development of 25 years, the current number of confirmed exoplanets have reached 4,000. The population of exoplanets presents great diversity and complexity beyond the planets in Solar system, severely challenging our understanding on the origin and evolution of planets. Moreover, clues of extrasolar life have been partially disclosed by detection of exoplanets in habitable zone with temporal surface temperature. In recent years, several promising surveys of exoplanets have been carried out to search for more exoplanets, compiling more complete sample of exoplanets and detecting Earth-like exoplanets. Transiting Exoplanet Survey Satellite (TESS) launched in 2018, is the most import facility for this purpose. It monitors the brightness of nearby stars in all sky with high cadences, searching for exoplanetary transit events. The survey is sensitive to detect terrestrial exoplanets. In this work, we reanalyzed the high quality light curves of TESS with algorithm developed by ourselves and visual examination. Our work was designed complementary to the automatic pipeline developed by TESS science team. TESS pipeline is not optimal for long-period exoplanets, for which only one transit event occurring in the observation sectors of 27.4 days, appearing as single transit single. Those long-period planets are more likely to be habitable with liquid water on their surface in relative large orbits. We took efforts to recover exoplanets with single transit in TESS. First, we identified candidates with significant narrow dimming in light curves uniformly by our automatic program. The singles were further examined visually and fitted by transit profile, selecting transit events. More precise modeling was carried out by MCMC analysis for transit exoplanet candidates. As a preliminary attempt, we only processed the light curves in sector 1 only. Full TESS dataset in all 26 sectors will be studied in future. In summary, we detected 25 exoplanet candidates in publicly accessible light curves in TESS sector 1. Among them, 14 candidates are known exoplanets, and 6 are candidates identified by TESS pipeline in catalog of Tess Objects of Interest (TOI). 5 exoplanet candidates with single transit signal are newly identified by their profiles, and the properties of these system were inferred in this work. The orbital parameters and physical properties derived by us are consistent with those in literature for all the 14 known exoplanets. 2 among 6 of exoplanet candidates in TOI catalog are single transit objects, our results are consistent with values reported in TOI catalog. For the new exoplanet candidates, we report their inferred orbits and physical sizes. We plan to implement our analysis for all light curves in 26 sectors of TESS, Over 100 exoplanet candidates are expected to be recovered, significantly contributing to the current population of 4,000 known exoplanets.

Robust relative orbit synchronization for spacecraft cluster: A distributed learning sliding mode control approach

This paper investigates the issue of robust relative orbit synchronization control for spacecraft cluster subjects to space perturbation. First, the relative orbit dynamics of the leader-follower spacecraft cluster is established. Combined with P-type learning algorithm and sliding mode control (SMC) strategy, a novel distributed learning SMC (LSMC) strategy with the consensus algorithm is explored such that all the follower spacecrafts converge to the target relative position and their velocities tend to the leader spacecraft. The distributed P-type learning algorithm is used to update and compensate space perturbation for all the follower spacecrafts. The proposed LSMC approach has the potential to achieve higher control accuracy than the traditional SMC one while it has the lower computation burden than the distributed adaptive SMC one. The stability analysis of the proposed distributed LSMC strategy is provided in detail. Finally, a numerical example is simulated to illustrate the effectiveness and superiority of the proposed distributed LSMC-based relative orbit synchronization control for spacecraft cluster.

Monitoring of Paddy and Maize Fields Using Sentinel-1 SAR Data and NGB Images: A Case Study in Papua, Indonesia

This study addresses the question of how to evaluate the growth stage of food crops, for instance, paddy (Oryza sativa) and maize (Zea mays), from two different sensors in selected developed areas of Papua Province of Indonesia. Level-1 Ground Range Detected (L1 GRD) images from Sentinel-1 Synthetic Aperture Radar (SAR) data were used to investigate the growth of paddy and maize crops. An NGB camera was then used to obtain the Green Normalized Difference Vegetation Index (GNDVI), and the Enhanced Normalized Difference Vegetation Index (ENDVI) as in situ measurement. Afterwards, the results were analyzed based on the Radar Vegetation Index (RVI) and the Vertical-Vertical (VV) and Vertical Horizontal (VH) band backscatters at incidence angles of 30.55°–45.88°, and 30.59°–46.16° in 2021 and 2022, respectively. The findings showed that Sigma0_VV_db and sigma0_VH_db had a strong correlation (R2 above 0.900); however, polarization modification is required, specifically in the maize field. The RVI calculated and backscatter changes in this study were comparable to the in situ measurements, specifically those of paddy fields, in 2022. Even though the results of this study were not able to prove the RVI values from the two relative orbits (orbit31 and orbit155) due to the different angle incidences and the availability of the Sentinel-1 SAR data set over the study area, the division of SAR image data based on each relative orbit adequately represents the development of crops in our study areas. The significance of this study is expected to support food crop security and the implementation of development plans that contribute to the local government’s goals and settings.

Naturally bounded relative motion for formation flying near triangular libration points

This paper proposes a numerical searching and ergodic mappings method to seek the naturally bounded relative orbits near the triangular libration points in the Earth-Moon system. An orbital energy saved numerical continuation method along one axis is proposed and the Poincaré map near the triangular libration points is obtained, based on which two characteristic parameters are defined to generate the naturally bounded relative motions, i.e., the crossing interval period and the reference angle. Then, a numerical searching method is employed to generate an ergodic mapping and a neural network is utilized to realize the transformation from the ergodic mappings to orbital states. By pairing the ergodic mappings of different initial orbital states, the naturally bounded relative motion realized can remain for a long time and does not diverge.

Effects of hydrogen bonds on two-photon absorption of Green fluorescent protein chromophore analogue

Effects of hydrogen bonds on two-photon absorption (TPA) of a new donor-acceptor type green fluorescent protein chromophore analogue are investigated by employing a combined molecular dynamics and quantum chemistry method. The probable configurations of the chromophore in water are extracted from molecular dynamics simulation and the TP A properties of more than twenty hydrogen bond complexes are computed by quadratic response theory. Thereby, the structure and property relations are established. Three types of hydrogen bonds including O⋯H–O, N–H⋯O and N⋯H–O can be formed between the chromophore and water molecules. The O⋯H–O induces a little decrease of TPA cross section with a red-shifted wavelength. The N–H⋯O gives rise to a great enhancement of TP A at a longer wavelength, while the N⋯H–O decreases TP A significantly and makes the wavelength blue-shifted. The reasons for these effects are rationalized well by using a two-state model analysis. The related molecular orbitals are also plotted to visualize the charge transfer characters. In addition, the averaged TP A spectrum is obtained by calculating the probabilities of various hydrogen bond complexes. Our research could provide a good insight into the design of two-photon materials by making use of hydrogen bond networks.

Cooperative Angles-Only Relative Navigation Algorithm for Multi-Spacecraft Formation in Close-Range

As to solve the collaborative relative navigation problem for near-circular orbiting small satellites in close-range under GNSS denied environment, a novel consensus constrained relative navigation algorithm based on the lever arm effect of the sensor offset from the spacecraft center of mass is proposed. Firstly, the orbital propagation model for the relative motion of multi-spacecraft is established based on Hill-Clohessy-Wiltshire dynamics and the line-of-sight measurement under sensor offset condition is modeled in Local Vertical Local Horizontal frame. Secondly, the consensus constraint model for the relative orbit state is constructed by introducing the geometry constraint between the spacecraft, based on which the consensus unscented Kalman filter is designed. Thirdly, the observability analysis is done and the necessary conditions of the sensor offset to make the state observable are obtained. Lastly, digital simulations are conducted to verify the proposed algorithm, where the comparison to the unconstrained case is also done. The results show that the estimated error of the relative position converges very quickly, the location error is smaller than 10 m under the condition of 10−3 rad level camera and 5 m offset.

Mission Design and Orbit-Attitude Control Algorithms Development of Multistatic SAR Satellites for Very-High-Resolution Stripmap Imaging

This study designs a multistatic synthetic aperture radar (SAR) formation-flying system for very-high-resolution stripmap imaging (VHRSI) using manufacturable SAR microsatellites. Multistatic SAR formation specifications for VHRSI are derived based on the SAR image theory. For the simultaneous multi-satellite operation, the advantages of the autonomous orbit and attitude control are prominent in terms of the workload of the ground station or the efficient performance of missions. Therefore, the autonomous relative-orbit-control algorithm using relative orbital elements is developed to maintain the designed multistatic SAR formation. Additionally, an autonomous attitude-control algorithm for multistatic SAR imaging is designed by applying the optimal right-ascension of the descending node (RADN) sector concept. Finally, the resolution improvement of VHRSI is verified through multistatic SAR imaging simulations. The multistatic SAR formation is designed with three satellites separated by 7.5 km each in the along-track direction. Autonomous relative orbit control maintains the relative position error within 45 m (3σ). Additionally, the autonomous attitude control simulation verifies that the satellites perform attitude maneuvers suitable for the operation mode, and the pointing error is maintained within 0.0035° (3σ). The spatial resolution of the multistatic SAR system for VHRSI is 0.95 × 0.96 m, which satisfies the very-high-spatial-resolution requirement.

Implications of the Milky Way Travel Velocity for Dynamical Mass Estimates of the Local Group

The total mass of the Local Group (LG) is a fundamental quantity that enables interpreting the orbits of its constituent galaxies and placing the LG in a cosmological context. One of the few methods that allows inferring the total mass directly is the “Timing Argument,” which models the relative orbit of the Milky Way (MW) and M31 in equilibrium. The MW itself is not in equilibrium, a byproduct of its merger history and including the recent pericentric passage of the Large Magellanic Cloud (LMC), and recent work has found that the MW disk is moving with a lower bound “travel velocity” of ∼32 km s−1 with respect to the outer stellar halo. Previous Timing Argument measurements have attempted to account for this nonequilibrium state, but have been restricted to theoretical predictions for the impact of the LMC specifically. In this paper, we quantify the impact of a travel velocity on recovered LG mass estimates using several different compilations of recent kinematic measurements of M31. We find that incorporating the measured value of the travel velocity lowers the inferred LG mass by 10%–12% compared to a static MW halo. Measurements of the travel velocity with more distant tracers could yield even larger values, which would further decrease the inferred LG mass. Therefore, the newly measured travel velocity directly implies a lower LG mass than from a model with a static MW halo and must be considered in future dynamical studies of the Local Volume.

Symmetry-reduced dynamic mode decomposition of near-wall turbulence

Data-driven dimensionality reduction methods such as proper orthogonal decomposition and dynamic mode decomposition have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration, as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for fluid flows in rectangular channels by formulating a continuous symmetry reduction method that eliminates the translations in the streamwise and spanwise directions simultaneously. We demonstrate our method by computing the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding windows of data obtained from the transitional plane-Couette and turbulent plane-Poiseuille flow simulations. In the former setting, SRDMD captures the dynamics in the vicinity of the invariant solutions with translation symmetries, i.e. travelling waves and relative periodic orbits, whereas in the latter, our calculations reveal episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.

Design of Ganymede-Synchronous Frozen Orbit around Europa

A Ganymede-synchronous frozen orbit around Europa provides a stable spatial geometry between a Europa probe and a Ganymede lander, which facilitates the observation of Ganymede and data transmission between probes. However, the third-body gravitation perturbation of Ganymede continues to accumulate and affect the long-term evolution of the Europa probe. In this paper, the relative orbit of Ganymede with respect to Europa is considered to accurately capture the perturbation potential. The orbital evolution behaviors of the Europa probe under the non-spherical gravitation of Europa and the third-body gravitation of Jupiter and Ganymede are studied based on a double-averaging framework. Then, the initial orbital conditions of the Ganymede-synchronous frozen orbit are developed. A station-keeping maneuver was performed to maintain the orbital elements to achieve the Ganymede-synchronous and frozen behaviors. A numerical simulation showed that the consumption for orbital maintenance is acceptable.

Proximity Operations About Apophis Through Its 2029 Earth Flyby

The dynamics and control of a satellite in proximity to the asteroid Apophis across its Earth close approach in 2029 is evaluated and investigated. First, the feasibility of carrying out close proximity operations about Apophis when in its heliocentric orbit phase is evaluated and shown to be feasible. Then three different types of close proximity motion relative to Apophis are analyzed that will enable a spacecraft to take observations throughout the Earth close approach. These are maintaining a relative orbit that is somewhat distant from Apophis, hovering along the Earth–Apophis line, or maintaining orbit about Apophis through the flyby. Each of these are shown to be feasible, albeit challenging, and some basic aspects of these operations are noted and discussed.

Optimal Projected Circular Orbit for Spacecraft Formation Flying Near a Slowly Rotating Asteroid

In spacecraft formation flying missions, the projected circular (relative) orbit (PCO), which appears circular when viewed along the zenith-nadir direction of the nominal chief spacecraft, is suitable for constructing a synthetic aperture radar. In this paper, the analysis of a PCO for spacecraft formation flying near a slowly rotating asteroid is conducted. By matching the in-plane and out-of-plane relative motion frequency, the critical values of the initial phase angle that characterizes the PCO are identified, resulting in the minimum precession of relative orbit. The analysis is then utilized to develop the fuel-optimal control law for establishing a target PCO. In particular, the fuel-optimal problem is addressed with an indirect method, where a homotopy approach is adopted, and the initial (unknown) costate vector is calculated with a scaling technique, so that the sensitivity to the initial guess problem is mitigated.

Analytical Nonlinear Observability Analysis for Spacecraft Autonomous Relative Navigation

Spacecraft autonomous relative navigation is widely required in the field of on-orbit services. In previous works, it has been shown that the monocular sequential images (MSIs) is a cost-effective autonomous relative navigation method. However, given the lack of size and initial orbit of noncooperative targets, the relative navigation system (RNS) with MSIs cannot ensure to estimate the entire relative motion states during the whole observation process. It is critical to perform an online observability analysis for RNS with MSIs to ensure autonomous relative motion estimation validity and accuracy. An analytical nonlinear observability analysis method is proposed in this article to improve the computational efficiency. We study the attributes of the relative orbit dynamics model and Lie algebra operations, and find that the observability matrix is only related to the limited states. The relationship among those states, rank, and the Frobenius norm of observability matrix is demonstrated, which simplify the observability criteria conditions and the degree measurements in analytical forms. In addition, based on the proposed method, we analyze the influence of the orbital manifolds on the observability degree of RNS with MSIs. Several suggestions are proposed for orbital manifolds detecting. The conclusion is a theoretical supplement to the past numerical research results. Finally, we implement the numerical simulations, and the current study’s findings verified the effectiveness of the proposed observability analysis method.

The geologically oldest specimen of Pterodactylus: a new exquisitely preserved skeleton from the Upper Jurassic (Kimmeridgian) Plattenkalk deposits of Painten (Bavaria, Germany)

Pterodactylus from the uppermost Jurassic of southern Germany represents one of the most iconic pterosaurs, due to its status of being the first member of the Pterosauria to have been described and named. During the early phase of pterosaur research, Pterodactylus was a wastebasket taxon containing dozens of sometimes distantly related assigned species. Decades later, a comprehensive revision of the genus significantly reduced the number of species. To date, only one species remains in the genus, Pterodactylus antiquus, although the referral of several specimens to this taxon and the taxonomic relationships of them is still debated. Thus far, the genus has been only reported from the Upper Jurassic Plattenkalk deposits of Bavaria, and all of these occurrences are Tithonian in age. Here we describe the first record of Pterodactylus from the Torleite Formation near Painten (Bavaria), which represents the first occurrence of the genus from the Kimmeridgian. The specimen is a complete, articulated and exquisitely preserved skeleton of a small-sized individual. Aside from its old geological age, it is a typical representative of the genus, greatly resembling other specimens from younger strata. Certain characters, such as the overall size, skull length, relative orbit size, and phalangeal formula indicate that the specimen from Painten represents a juvenile to young subadult individual, an ontogenetic stage rarely found among Pterodactylus specimens. The find significantly expands the temporal range of the taxon and represents one of the best-preserved specimens of the genus reported so far.