Orbital Integrals Research Articles

Equilibrium dynamical models in the inner region of the Large Magellanic Cloud based on Gaia DR3 kinematics

The Large Magellanic Cloud (LMC) contains complex dynamics driven by both internal and external processes. The external forces are due to tidal interactions with the Small Magellanic Cloud and the Milky Way, while internally its dynamics mainly depend on the stellar, gas, and dark matter mass distributions. Despite this complexity, simple physical models often provide valuable insights into the primary driving factors. We used Gaia Data Release 3 (DR3) to explore how well equilibrium dynamical models based on the Jeans equations and the Schwarzschild orbit superposition method are able to describe the LMC's five-dimensional phase-space distribution and line-of-sight (LOS) velocity distribution, respectively. In the Schwarzschild model, we incorporated a triaxial bar component for the first time and derived the LMC's bar pattern speed. We fit comprehensive Jeans dynamical models to all Gaia DR3 stars with proper motion and LOS velocity measurements found in the footprint of the VISTA near-infrared survey of the Magellanic System using a discrete maximum likelihood approach. These models are very efficient at discriminating genuine LMC member stars from Milky Way foreground stars and background galaxies. They constrain the shape, orientation, and enclosed mass of the galaxy under the assumption of axisymmetry. We used the Jeans model results as a stepping stone to more complex two-component Schwarzschild models, which include an axisymmetric disc and a co-centric triaxial bar, which we fit to the LMC Gaia DR3 LOS velocity field using a $ minimisation approach. The Jeans models describe the rotation and velocity dispersion of the LMC disc well, and we find an inclination angle of $ circ circ $, line of nodes orientation of $ circ circ $, and an intrinsic thickness of the disc of $q_0^d = b a (minor to major axis ratio). However, bound to axisymmetry, these models fail to properly describe the kinematics in the central region of the galaxy dominated by the LMC bar. We used the derived disc orientation and the Gaia DR3 density image of the LMC to obtain the intrinsic shape of the bar. Using these two components as input to our Schwarzschild models, we performed orbit integration and weighting in a rotating reference frame fixed to the bar, deriving an independent measurement of the LMC bar pattern speed of $ Both the Jeans and Schwarzschild models predict the same enclosed mass distribution within a radius of $6.2$\,kpc of $ Msun.

Optimizing surgical approaches for lacrimal gland adenoid cystic carcinoma to minimize cross-organ invasion.

To evaluate the outcomes of eye-sparing surgery for lacrimal gland adenoid cystic carcinoma and the impact on tumor recurrence and orbital integrity. The study enrolled four patients with recurrent lacrimal gland adenoid cystic carcinoma. The outcome focused on the relevance of the integrity of the lateral orbital wall to the occurrence of extraorbital metastasis in the local recurrence of lacrimal gland adenoid cystic carcinoma. Three patients underwent eye-sparing surgery via lateral orbitotomy without postoperative radiotherapy, and one patient who underwent eye-sparing surgery via sub-brow approach. These four patients all demonstrated a recurrence involving the invasion of extraorbital tissues as metastatic form through surgical bone seams. Preserving intact orbital bone tissue is crucial for mitigating direct cross-organ metastasis of lacrimal gland adenoid cystic carcinoma. The findings suggest avoiding the lateral orbitotomy approach with no or limited orbital bone wall invasion.

One stream or two - exploring Andromeda’s North West stream

Abstract We present results of our dynamical stream modelling for the North West Stream in the outer halo of the Andromeda galaxy (M31). Comprising two main segments, the North West Stream was thought to be a single structured arching around M31. However, recent evidence suggests that it is two separate, unrelated, streams. To test this hypothesis we use observational data from 6 fields associated with the upper segment of the North West Stream together with 8 fields and 5 globular clusters associated with the lower segment to constrain model orbits. We fit both segments of the stream using a fixed potential model for M31 and an orbit integrator to compare orbits with the observed streams. We measure the central tracks and predict proper motions for for the upper segment (lower segment) finding ${\mu ^*_{\alpha }}$ = 0.078$^{+0.015}_{-0.012}$ (0.085$^{+0.001}_{-0.002}$) mas/yr and μδ = $-0.05^{+0.008}_{-0.009}$ ($-0.095^{+0.003}_{-0.005}$) mas/yr. Our results support the hypothesis that the dwarf spheroidal galaxy Andromeda XXVII is the progenitor of the upper segment of the North West Stream and that the upper and lower segments do not comprise a single structure. We propose that the upper segment, which appears to be on an infall trajectory with M31, be renamed the “Andromeda XXVII Stream” and the lower segment, also apparently infalling towards M31, retain the name “North West Stream”.

Photometric and Kinematic Studies of Open Clusters Ruprecht 1 and Ruprecht 171

ABSTRACTThis study outlines a detailed investigation using CCD UBV and Gaia DR3 data sets of the two open clusters Ruprecht 1 (Rup‐1) and Ruprecht 171 (Rup‐171). Fundamental astrophysical parameters such as color excesses, photometric metallicities, ages, and isochrone distances were based on UBV‐data analyses, whereas membership probability calculations, structural and astrophysical parameters, as well as the kinematic analyses were based on Gaia DR3‐data. We identified 74 and 596 stars as the most probable cluster members with membership probabilities over 50% for Rup‐1 and Rup‐171, respectively. The color excesses were obtained as and mag for Rup‐1 and Rup‐171, respectively. Photometric metallicity analyses were performed by considering F‐G type main‐sequence member stars and found to be and dex for Rup‐1 and Rup‐171, respectively. Ages and distances were based on both UBV and Gaia‐data analyses; according to isochrone‐fitting these values were estimated to be Myr, pc for Rup‐1 and Myr, pc for Rup‐171. The present‐day mass function slope of Rup‐1 was estimated as and Rup‐171 as . Galactic orbit integration analyses showed that both of the clusters might be formed outside the solar circle.

Nonplanar Nanographene: A Hydrocarbon Hole-Transporting Material That Competes with Triarylamines.

Hole-transporting materials (HTMs) are essential for optoelectronic devices, such as organic light-emitting diodes (OLEDs), dye-sensitized solar cells, and perovskite solar cells. Triarylamines have been employed as HTMs since they were introduced in 1987. However, heteroatoms or side chains embedded in the core skeleton of triarylamines can cause thermal and chemical stability problems. Herein, we report that hexabenzo[a,c,fg,j,l,op]tetracene (HBT), a small nonplanar nanographene, functions as a hydrocarbon HTM with hole transport properties that match those of triarylamine-based HTMs. X-ray structural analysis and theoretical calculations revealed effective multidirectional orbital interactions and transfer integrals for HBT. In-depth experimental and theoretical analyses revealed that the nonplanarity-inducing annulative π-extension can achieve not only a stable amorphous state in bulk films, but also a higher increase in the highest occupied molecular orbital level than conventional linear or cyclic π-extension. Furthermore, an in-house manufactured HBT-based OLED exhibited excellent performance, featuring superior curves for current density-voltage, external quantum efficiency-luminance, and lifetime compared to those of representative triarylamine-based OLEDs. A notable improvement in device lifetime was observed for the HBT-based OLED, highlighting the advantages of the hydrocarbon HTM. This study demonstrates the immense potential of small nonplanar nanographenes for optoelectronic device applications.

Nonplanar Nanographene: A Hydrocarbon Hole‐Transporting Material That Competes with Triarylamines

AbstractHole‐transporting materials (HTMs) are essential for optoelectronic devices, such as organic light‐emitting diodes (OLEDs), dye‐sensitized solar cells, and perovskite solar cells. Triarylamines have been employed as HTMs since they were introduced in 1987. However, heteroatoms or side chains embedded in the core skeleton of triarylamines can cause thermal and chemical stability problems. Herein, we report that hexabenzo[a,c,fg,j,l,op]tetracene (HBT), a small nonplanar nanographene, functions as a hydrocarbon HTM with hole transport properties that match those of triarylamine‐based HTMs. X‐ray structural analysis and theoretical calculations revealed effective multidirectional orbital interactions and transfer integrals for HBT. In‐depth experimental and theoretical analyses revealed that the nonplanarity‐inducing annulative π‐extension can achieve not only a stable amorphous state in bulk films, but also a higher increase in the highest occupied molecular orbital level than conventional linear or cyclic π‐extension. Furthermore, an in‐house manufactured HBT‐based OLED exhibited excellent performance, featuring superior curves for current density–voltage, external quantum efficiency–luminance, and lifetime compared to those of representative triarylamine‐based OLEDs. A notable improvement in device lifetime was observed for the HBT‐based OLED, highlighting the advantages of the hydrocarbon HTM. This study demonstrates the immense potential of small nonplanar nanographenes for optoelectronic device applications.

Irregular fixation – II. The orbits of irregular satellites

ABSTRACT Irregular satellites (ISs) are believed to have been captured during the Solar system’s dynamical history and provide clues for the Solar system’s formation and evolution. ISs occupy a large fraction of the Hill sphere of their host planet and their orbits are highly perturbed by the Sun. We use a novel formalism developed in Paper I to characterize their orbits in terms of an effective secular Hamiltonian (the Brown Hamiltonian) that accounts for their large orbital separations. We find that prograde satellites generally follow the Brown Hamiltonian, while retrograde satellites (which extend further) deviate more significantly. Nevertheless, the phase portrait is much better described by the Brown Hamiltonian for all satellites. We construct a semi-analytical criterion that predicts the librating orbit based on the effective energy due to the Brown Hamiltonian. We also check our results with highly accurate N-body integrations of satellite orbits, where initial conditions are loaded directly from the updated ephemeris from the NASA Horizons data base. Although the retrograde librating orbits occupy more area in the parameter space, the vast majority of librating ISs are prograde. Using our method, we find 13 librating satellites, 8 of them previously known to librate, and the rest shown to librate for the first time. Further observations of existing and new satellites could shed more light on the dynamical history of the Solar system and satellite formation and test our results.

On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of \mathrm{SL}_{2} to automorphic representations on \mathrm{PGL}_{2} . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of \mathrm{SL}_{3} to \mathrm{PGL}_{3} , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation \Theta_{\mathrm{SO}_8} for split \mathrm{SO}_{8}(\mathbb{A}) , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on \mathrm{PGL}_{3}(\mathbb{A}) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of \mathrm{SL}_{3}(\mathbb{A}) . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

A detailed chemical study of the extreme velocity stars in the galaxy

ABSTRACT Two decades on, the study of hypervelocity stars is still in its infancy. These stars can provide novel constraints on the total mass of the Galaxy and its dark matter distribution. However how these stars are accelerated to such high velocities is unclear. Various proposed production mechanisms for these stars can be distinguished using chemo-dynamic tagging. The advent of Gaia and other large surveys have provided hundreds of candidate hyper velocity objects to target for ground-based high-resolution follow-up observations. We conduct high-resolution spectroscopic follow-up observations of 16 candidate late-type hyper velocity stars using the Apache Point Observatory and the McDonald Observatory. We derive atmospheric parameters and chemical abundances for these stars. We measure up to 22 elements, including the following nucleosynthetic families: $\alpha$ (Mg, Si, Ca, and Ti), light/odd-Z (Na, Al, V, Cu, and Sc), Fe-peak (Fe, Cr, Mn, Co, Ni, and Zn), and neutron capture (Sr, Y, Zr, Ba, La, Nd, and Eu). Our kinematic analysis shows one candidate is unbound, two are marginally bound, and the remainder are bound to the Galaxy. Finally, for the three unbound or marginally bound stars, we perform orbit integration to locate possible globular cluster or dwarf galaxy progenitors. We do not find any likely candidate systems for these stars and conclude that the unbound stars are likely from the the stellar halo, in agreement with the chemical results. The remaining bound stars are all chemically consistent with the stellar halo as well.

Kinematic origin of white dwarfs in the solar neighborhood

Context. White dwarfs are considered to be efficient cosmochronometers. Thanks to the recent space-borne mission Gaia, a nearly complete sample up to about 100 parsecs from the Sun has been compiled. However, the Galaxy, as a dynamic system, implies that these objects may have very diverse origins. It is therefore of paramount importance to characterize the origins of white dwarfs from the different Galactic structure components found in our solar neighborhood. Aims. We aim to compute the Galactic orbits for white dwarfs of the thin and thick disk, as well as the halo components observed in our solar neighborhood. On the basis of these determinations, we analyze the most probable regions of the Galaxy where they could have formed, along with the distribution of their orbital parameters and the observational biases introduced when constructing the local sample. Methods. We used a detailed Galactic orbit integration package, in conjunction with a detailed population synthesis code specifically designed to replicate the different Galactic components of the white dwarf population. Synthetic stars were generated based on the current observational sample and their orbital integration allowed for the reconstruction of the population’s history. Results. Our kinematic analysis of the white dwarf population reveals the ephemeral nature of the concept of the solar neighborhood, as the majority of thin-disk, thick-disk, and halo white dwarfs will have left our 100 pc neighborhood in approximately 3.30 Myr, 1.05 Myr, and 0.6 Myr, respectively. Moreover, the spatial distribution of the integrated thin-disk orbits suggests that 68% of these stars were formed at less than 1 kpc from the Sun, while most of the thick-disk members have undergone radial disk migration. Halo members are those stars that typically belong to the “inner halo”, given that their orbits mostly planar and do not extend beyond R = 20 − 25 kpc. Despite the observational bias, which mostly affects the oldest stars in the thick disk and halo, we show that the wider distribution of orbital parameters is well represented by the sample. Conclusions. The solar neighborhood is a transitory concept, whereby its current population of white dwarfs originates from larger regions of the Galaxy. This fact must be taken into account when analyzing the overall properties of such a population, such as its age distribution, metallicities, and formation history. Even so, the kinematic properties observed by recent missions such as Gaia are representative of the total population up to a distance of approximately 500 pc.

A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.

Calculating Global Navigation Satellite System Satellite Velocities and Accelerations by Utilizing the Orbit Fitting and Orbit Integration Methods

Calculating Global Navigation Satellite System Satellite Velocities and Accelerations by Utilizing the Orbit Fitting and Orbit Integration Methods

Testing MOND using the dynamics of nearby stellar streams

The stellar halo of the Milky Way is built up at least in part from debris from past mergers. The stars from these merger events define substructures in phase space, for example in the form of streams, which are groups of stars that move on similar trajectories. The nearby Helmi streams discovered more than two decades ago are a well-known example. Using 6D phase-space information from the Gaia space mission, recent work showed that the Helmi streams are split into two clumps in angular momentum space. This substructure can be explained and sustained in time if the dark matter halo of the Milky Way takes a prolate shape in the region probed by the orbits of the stars in the streams. Here, we explore the behaviour of the two clumps identified in the Helmi streams in a modified Newtonian dynamics (MOND) framework to test this alternative model of gravity. We performed orbit integrations of Helmi streams member stars in a simplified MOND model of the Milky Way and using the more sophisticated phantom of RAMSES simulation framework. We find with both approaches that the two Helmi streams clumps do not retain their identity and dissolve after merely Myr. This extremely short timescale would render the detection of two separate clumps very unlikely in MONDian gravity. The observational constraints provided by the streams, which MOND fails to reproduce in its current formulation, could potentially also be used to test other alternative gravity models.

Arithmetic fundamental lemma for the spherical Hecke algebra

We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity conjecture on Hecke operators. Then we formulate the arithmetic fundamental lemma conjecture for the spherical Hecke algebra. We also formulate a conjecture on the abundance of spherical Hecke functions with identically vanishing first derivative of orbital integrals. We prove these conjectures for the case U(1)×U(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{U} (1)\ imes \ extrm{U} (2)$$\\end{document}.

Efficient screening and catalytic mechanism of TM@β-Te for nitrogen reduction reaction

At present, nitrogen reduction reaction (NRR) has become one critical method for NH3 production through the single-atom catalyst (SAC). A variety of TM@β-Te SACs, constructed by 28 transition metal (TM) atoms anchoring on monovacant β-Te substrate, are systematically investigated by using DFT calculations. An efficient “six-step” screening scheme is proposed to screen out potential NRR catalysts, during which the implicit and explicit solvent models are involved to evaluate competitive reactions of H2O and H+. The final selected (Ta, W)@β-Te catalysts have limiting potentials (UL) of −0.32 V and −0.37 V, respectively. Three intrinsic descriptors of φ (the ratio of number of valence electrons and electronegativity of TM atom), ΔG∗N(adsorption energy of a single N atom on the substrate) and ICOHP (Crystal orbital Hamiltonian integrals for TM-N bonds) are proposed and their correlations with UL are investigated to expound the catalytic activity and mechanism. The relationships of dN≡N ∼ φ, ΔG∗N2∼ φ, ΔG∗N∼ φ and ICOHP ∼ φ resemble “volcanoes”, where (Ta, W)@β-Te are located near the vertex as potential catalysts, which can be verified by scaling relationship of ICOHP and ΔG∗N. Moreover, the UL ∼ φ, UL ∼ ΔG∗Nand UL ∼ ICOHP can be used to determine the potential catalysts from rough catalyst screening to accurate determination of potential determining step (PDS) in catalytic reaction. The constant potential model is adopted to investigate the influences of electrode potentials on adsorption strength of NRR intermediates. This work is informative for the research of NRR process and catalyst design of monoalkenes, which offers an efficient and dependable approach for screening excellent NRR catalysts and revealing the origin of catalytic activity.

Delivery of Dust Particles from Protoplanetary Disks onto Circumplanetary Disks of Giant Planets

The principal regular satellites of gas giants are thought to be formed by the accumulation of solid materials in circumplanetary disks (CPDs). While there has been significant progress in the study of satellite formation in CPDs, details of the supply of satellite building blocks to CPDs remain unclear. We perform the orbital integration of solid particles in the protoplanetary disk (PPD) approaching a planet, considering the gas drag force by using the results of three-dimensional hydrodynamical simulations of a local region around the planet. We investigate the planetary mass dependence of the capture positions and the capture rates of dust particles accreting onto the CPD. We also examine the degree of dust retention in the accreting gas onto the CPD, which is important for determining the ratio of the dust-to-gas inflow rates, a key parameter in satellite formation. We find that the degree of dust retention increases with increasing planetary mass for a given dust scale height in the PPD. In the case of a small planet (M p = 0.2M Jup), most particles with insufficient initial altitudes in the PPD are isolated from the gas in the accreting region. On the other hand, in the case of a massive planet (M p = 1M Jup), dust particles can be coupled to the vertically accreting gas, even when the dust scale height is about 10%–30% of the gas scale height. The results of this study can be used for models of dust delivery and satellite formation in the CPDs of gas giants of various masses, including exoplanets.

Анализ регулярности/хаотичности динамики шаровых скоплений в центральной области Млечного Пути

We analyzed the regularity/chaoticity of the orbits of 45 globular clusters in the central region of the Galaxy with a radius of 3.5 kpc, which are subject to the greatest influence from an elongated rotating bar. Various analysis methods were used, namely, methods for calculating the maximum characteristic Lyapunov exponents, the method of Poincaré sections, the frequency method based on the calculation of fundamental frequencies, as well as the visual assessment method. Bimodality was discovered in the histogram of the distribution of positive Lyapunov exponents, calculated in the classical version, without renormalization of the shadow orbit, which makes it possible to implement a probabilistic method of GC classification. To construct the orbits of globular clusters, we used a gravitational potential model with a bar in the form of a triaxial ellipsoid, described in detail in the work of Bajkova et al., Izvestiya GAO in Pulkovo, 2023, 228, 1. The following bar parameters were adopted: mass 1010M⊙, semimajor axis length 5 kpc, bar viewing angle 25o, rotation speed 40 km s−1 kpc−1. To form the 6D-phase space required for orbit integration, the most accurate astrometric data to date from the Gaia satellite (EDR3) (Vasiliev, Baumgardt, 2021), as well as new refined average distances to globular clusters (Baumgardt, Vasiliev, 2021) were used. A classification is made of globular clusters with regular and chaotic dynamics. As the analysis showed, globular clusters with small pericentric distances and large eccentricities are most susceptible to the influence of the bar and demonstrate the greatest chaos. It is shown that the results of classifying globular clusters by the nature of their orbital dynamics, obtained using the various methods of analysis considered in the work, correlate well with each other.

Post-new-horizons Orbits and Masses for the Satellites of Pluto

We report on ephemerides for the satellites of Pluto based on the large set of astrometric measurements. Our orbit fit yielded the following masses with 1σ uncertainties: GM Pluto = 869.3 ± 0.4 km3 s−2, GM Charon = 106.1 ± 0.3 km3 s−2, GM Nix = 1.50 ± 0.52 × 10−3 km3 s−2, GM Hydra = 2.01 ± 0.27 × 10−3 km3 s−2, corresponding to the densities of ρ Pluto = 1.853 ± 0.004 g cm−3, ρ Charon = 1.705 ± 0.006 g cm−3, ρ Nix = 0.88 ± 0.31 g cm−3, and ρ Hydra = 1.21 ± 0.19 g cm−3. Masses of Kerberos and Styx remain unconstrained, and it is unlikely that we will be able to measure them even if we obtain abundant 1 mas precision astrometry during the next 20 yr. We summarize the results of orbit integration in terms of osculating and precessing ellipse model mean elements. All satellites reside in near-circular orbits, and Kerberos and Styx have 0.4 deg and 0.3 deg inclinations with respect to Charon’s orbit plane. The nodal regression periods for Kerberos and Hydra are ∼9 and ∼14 yr respectively. We found that Charon’s orbit pole can be approximated as: R.A. = 133.0073 + 0.0036T deg, and decl. = –6.2449 + 2.5 × 10−5 T deg, where T is Julian centuries from the epoch J2000, based on 5000 yr of orbit integration.

Jupiter Co-Orbital Comet P/2023 V6 (PANSTARRS): Orbital History and Modern Activity State

The discovery of the transient Jupiter co-orbital comet P/2019 LD2 (ATLAS) drew significant interest. Not only will LD2 transition between being a Centaur and a Jupiter-family comet (JFC) in 2063, the first time this process can be observed as it happens, it is also very active for its large heliocentric distance. We present observations and orbital integrations of the newly discovered transient Jupiter co-orbital comet P/2023 V6 (PANSTARRS), the second such object known. Despite similar modern orbits, V6 is significantly (15×) less active than LD2 and most JFCs as determined via Af ρ measurements at the same R H . We find that V6 is co-orbital between 2020 and 2044, twice the duration of LD2, but it will not become a JFC soon. We interpret these differences in activity as evolutionary, with V6 having lost a significant fraction of its near-surface ice compared to LD2 by previously being warmer. While V6's pre-encounter orbit was somewhat warmer than LD2's, future thermal modeling will be needed to understand if this can explain their differences or if a more significant difference further into the past is required. This is more evidence that LD2 is a pristine and ice-rich object, and thus it may display very strong activity when it becomes a JFC. We use the differences between V6 and LD2 to discuss the interpretation of cometary activity at large heliocentric distances as well as the small end of the cratering record of the Galilean satellites. Continuing observations of both objects are highly encouraged.

Orbit Determination of High Orbit Spacecraft based on BDS-3 using Extended Kalman Filter

The positioning of highly elliptical orbit (HEO) spacecraft is very important in deep space exploration and other missions. The Beidou Navigation Satellite System (BDS-3), which was completed in 2020, has put forward new ideas for using GNSS technology to make up for the lack of earth-based TT&C systems. At the same time, the selection of the gravity field order and the perturbation forces must be considered when using dynamical methods for HEO spacecraft positioning. In this paper, simulation experiments are carried out based on the Two-Line Orbital Element (TLE) data of BDS. Two HEO satellites with different semimajor-axis (67509.6 km, 212857.3 km) are selected as the target spacecraft orbits. The single point positioning (SPP) method and Extended Kalman Filter (EKF) method are used for HEO spacecrafts positioning respectively. The experiments show that a 20×20 order gravity field model can be chosen for the orbital integral of HEO spacecraft orbits, and the integration error caused by solar gravity and lunar gravity should be taken into account with the change of orbital altitude. Adding Geostationary Orbit (GEO) and Inclined GeoSynchronous Orbit (IGSO) navigation satellites improved the visibility of these two orbits by 19.45% and 16.64%, respectively, and improved the GDOP value by 14.1% and 3.8%, respectively, compared to observing MEO satellites only. The use of the EKF method can effectively improve the positioning accuracy of the HEO spacecraft compared to the SPP method. For the spacecraft in HEO-1, the RMS value of the positioning errors of the EKF method was 67.82 m, and the accuracy of each direction was improved by 44.28%, 38.85%, and 38.62%, respectively. For the spacecraft in HEO-2, the positioning errors in the x direction of EKF method was within 12 km and within 2 km in the y and z directions. The accuracy of these three directions was improved by 84.91%, 79.24%, and 58.64%, respectively.