A unified state model of orbital trajectory and attitude dynamics

The trajectory and attitude dynamics of an orbital spacecraft are defined by a unified state model, which enables efficient and rapid machine computation for mission analysis, orbit determination and prediction, satellite geodesy and reentry analysis. The state variables are momenta — a general form for attitude, and a parametric form for orbital motion. The orbital parameters are the velocity state characteristics of the orbital hodograph. The coordinate variables are sets of four Euler parameters, which define the rotation transformation by the quaternion algebra. The unified state model possesses many analytical properties which are invaluable for dynamical system synthesis, numerical analysis and machine solution: regularization, unified matrix algebra, state graphs and transforms. The analytic partials of position and velocity with the state and coordinate variables are presented, as well as representative perturbation functions such as air drag, gravitational potential harmonics, and propulsion thrust.     

Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $J_2$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.     

GPS-Based On-Board Orbit Determination of a Satellite Using Extended H ∞ Filtering Algorithms

In this paper the extended H filtering algorithms for the design of the GPS-based on-board autonomous navigation system for a low earth orbit (LEO) satellite are introduced. The dynamic process models for the estimation of position, velocity and acceleration from the GPS measurements are established. The nominal orbit of the small LEO satellite is determined by using the 7th–8th order Runge—Kutta algorithms. Three filtering approaches are applied to smooth the orbit solutions, respectively, based upon the simulated GPS pseudo range observables using the Satellite Navigation Tool Box. The simulation shows that the observed orbit errors obtained by using the extended H filtering algorithms can be reduced to a lower level than the observed orbit errors in the sense of RMS within 12 h of tracking time by using the H filtering algorithms and the extended Kalman filtering algorithms under the appropriately designed parameters. Based upon the position errors predicted by the three filtering algorithms after the last observation, we find that the extended H filtering algorithm provides the least position errors of the user satellite.This revised version was published online in October 2005 with corrections to the Cover Date.     

Determination of the orbit of a natural satellite with hyperbolic flybys

We show that, when a natural satellite like Titan is invisible (e.g., due to an opaque atmosphere) its planetary orbit and its mass can be determined by tracking a spacecraft in close flybys. This is an important problem in the Cassini mission to the Saturnian system, which will be greatly improved by a good astrometric model for all its main components; in particular, an accuracy of a few hundred meters for the orbit of Titan is necessary to allow a measurement of its moment of inertia. The orbit of the spacecraft is the union of elliptical arcs, joined by short hyperbolic transitions: a problem of singular perturbation theory, whose solution leads to a matching condition between the inner hyperbolic orbit and the elliptical orbital elements. Since the inner elements are given in terms of the relative position and velocity of the spacecraft, accurate Doppler measurements in both regions can provide a satisfactory determination of Titan's position and velocity, hence of its Keplerian elements. The errors in this determination are discussed on the basis of the expected Allan deviation of the Doppler method; it is found that the driving errors are those in the elliptical arcs; the fractional errors in Titan's orbital elements are expected to be 10^–7. It is also possible to measure the mass of the satellite; however, when the eccentricity e of the flybys is large, the mass and a scaling transformation are highly correlated and the fractional error in the mass is expected to be e times worse.     

Non-gravitational force modeling of Comet 81P/Wild 2: II. Rotational evolution

In this paper, we have studied both the dynamical and the rotational evolution of an 81P/Wild 2-like comet under the effects of the outgassing-induced force and torque. The main aim is to study if it is possible to reproduce the non-gravitational orbital changes observed in this comet, and to establish the likely evolution of both orbital and rotational parameters. To perform this study, a simple thermophysical model has been used to estimate the torque acting on the nucleus. Once the torque is calculated, Euler equations are solved numerically considering a nucleus mass directly estimated from the changes in the orbital elements (as determined from astrometry). According to these simulations, when the water production rate and changes in orbital parameters for 1997, as well as observational rotational parameters for 2004 are imposed as constraints, the change in the orbital period of 81P/Wild 2, , will decrease so that to , which is similar to the actual tendency observed from 1988 up to 1997. This nearly constant decreasing can be explained as due to a slight drift of the spin axis orientation towards larger ecliptic longitudes. After studying the possible spin axis orientations proposed for 1997, simulations suggest that the spin obliquity and argument (I,Φ)=(56°,167°) is the most likely. As for rotational evolution, changes per orbit smaller than 10% of the actual spin velocity are probable, while the most likely value corresponds to a change between 2 and 7% of the spin velocity. Equally, net changes in the spin axis orientation of 4°-8° per orbit are highly expected.     

The equations of relative motion in the orbital reference frame

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.     

Atmospheric trajectory and orbit of the Omolon meteorite

Abstract— The Omolon meteorite fell on 1981 May 15 at 17:10 U.T. to a point with the coordinates φ = 64°01′08″ N, λ = 161°48′30″ E. This is the fifth pallasite that was observed at the moment of its fall and the largest of the pallasites known worldwide (250 kg). The history of the observation, search, and finding of the meteorite is briefly described. From the size of the meteorite and the funnel that it produced, the velocity of its encounter with the ground is estimated by aerodynamic formulas to be 220 m/s. An attempt at estimating the meteorite's initial velocity and mass from its terminal values (which yielded the mass range of 390–490 kg that corresponds to the velocity range of 12–15 km/s) was successful for the mass but unsuccessful for the velocity and the incidence angle, because the problem was ill posed. The position of the radiant is determined from the available observations to be α = 176.4°, δ = +24.1° (Leo). The radiant was situated at an elongation of 29° from the antapex, which means that this was an overtaking meteorite and its entry velocity did not exceed 16 km/s. Three variants of the calculation of the orbital elements—for an entry velocity of 12, 14, and 16 km/s—are presented. In all the three cases, the meteoroid's orbit is close to the orbits of Apollo asteroids and to the orbits of iron meteoroids observed as fireballs with bright iron lines in their spectra. The Omolon meteorite was probably a fragment of an Apollo M-type asteroid. This study is the first attempt at calculating the orbit of a pallasite.     

Transit timing effects due to an exomoon – II

In our previous paper, we evaluated the transit duration variation (TDV) effect for a co-aligned planet-moon system at an orbital inclination of   i = 90°  . Here, we will consider the effect for the more general case of   i ≤ 90°  and an exomoon inclined from the planet-star plane by Euler rotation angles  α, β  and γ. We find that the TDV signal has two major components, one due to the velocity variation effect described in our first paper and one new component due to transit impact parameter variation. By evaluating the dominant terms, we find the two effects are additive for prograde exomoon orbits, and deductive for retrograde orbits. This asymmetry could allow for future determination of the orbital sense of motion. We re-evaluate the ratio of TDV and transit timing variation effects, η, in the more general case of an inclined planetary orbit with a circular orbiting moon and find that it is still possible to directly determine the moon's orbital separation from just the ratio of the two amplitudes, as first proposed in our previous paper.     

A special perturbation method in orbital dynamics

The special perturbation method considered in this paper combines simplicity of computer implementation, speed and precision, and can propagate the orbit of any material particle. The paper describes the evolution of some orbital elements based in Euler parameters, which are constants in the unperturbed problem, but which evolve in the time scale imposed by the perturbation. The variation of parameters technique is used to develop expressions for the derivatives of seven elements for the general case, which includes any type of perturbation. These basic differential equations are slightly modified by introducing one additional equation for the time, reaching a total order of eight. The method was developed in the Grupo de Dinámica de Tethers (GDT) of the UPM, as a tool for dynamic simulations of tethers. However, it can be used in any other field and with any kind of orbit and perturbation. It is free of singularities related to small inclination and/or eccentricity. The use of Euler parameters makes it robust. The perturbation forces are handled in a very simple way: the method requires their components in the orbital frame or in an inertial frame. A comparison with other schemes is performed in the paper to show the good performance of the method.     

Sensitivity study of high eccentricity orbits for Mars gravity recovery

By linear perturbation theory, a sensitivity study is presented to calculate the contribution of the Mars gravity field to the orbital perturbations in velocity for spacecrafts in both low eccentricity Mars orbits and high eccentricity orbits(HEOs). In order to improve the solution of some low degree/order gravity coefficients, a method of choosing an appropriate semimajor axis is often used to calculate an expected orbital resonance, which will significantly amplify the magnitude of the position and velocity perturbations produced by certain gravity coefficients. We can then assess to what degree/order gravity coefficients can be recovered from the tracking data of the spacecraft. However, this existing method can only be applied to a low eccentricity orbit, and is not valid for an HEO. A new approach to choosing an appropriate semimajor axis is proposed here to analyze an orbital resonance. This approach can be applied to both low eccentricity orbits and HEOs. This small adjustment in the semimajor axis can improve the precision of gravity field coefficients and does not affect other scientific objectives.     

The effect of an electric field induced by a time-dependent ring current on the particle drift motion

The effect of an electric field induced by a rapidly decaying ring current on the motion of charged particles in the magnetosphere has been investigated using Euler potentials. For a model consisting of the earth dipole and the symmetric ring current, the electric field satisfies the condition E . B = 0.

Under this circumstance, the E × B drift of the particle can be identified as the motion of the magnetic field lines and vice versa. The time dependent electric field induced can be evaluated in a Spherical polar coordinate system by the formula

where and β are Euler potentials.

A model calculation on the particle drift velocity v_D = E × B/B² shows that the radial component of the drift velocity is in good agreement with those deduced from whistler duct studies.     

Families of periodic orbits in Hill’s problem with solar radiation pressure: application to Hayabusa 2

This work studies periodic solutions applicable, as an extended phase, to the JAXA asteroid rendezvous mission Hayabusa 2 when it is close to target asteroid 1999 JU3. The motion of a spacecraft close to a small asteroid can be approximated with the equations of Hill’s problem modified to account for the strong solar radiation pressure. The identification of families of periodic solutions in such systems is just starting and the field is largely unexplored. We find several periodic orbits using a grid search, then apply numerical continuation and bifurcation theory to a subset of these to explore the changes in the orbit families when the orbital energy is varied. This analysis gives information on their stability and bifurcations. We then compare the various families on the basis of the restrictions and requirements of the specific mission considered, such as the pointing of the solar panels and instruments. We also use information about their resilience against parameter errors and their ground tracks to identify one particularly promising type of solution.     

FMG载荷地面试观测导行跟踪系统的设计与实现

全日面矢量磁像仪(Full-disk vector MagnetoGraph, FMG)是先进天基太阳天文台(Advanced Space-based Solar Observatory, ASO-S)卫星的3台主载荷之一,为开展FMG全系统性能测试和定标试验,已搭建用于FMG外场测试的地面试观测平台.利用该平台模拟FMG在轨跟踪状态,研制了基于全日面太阳图像的望远镜导行系统.该系统通过大面阵CCD (Charge Coupled Device)采集太阳像、多重逻辑条件判定、微调恒动跟踪速度校正偏移等策略,实现了RMS (Root Mean Square)优于1′′/30 min的跟踪精度.通过分析FMG方案阶段试观测的太阳纵向磁图,开启导行30 min后磁图特征点在赤经方向的偏移比恒动条件下减少17.5′′,提升了磁图空间分辨率.测试过程中该系统达到设计指标且工作稳定,为FMG地面试观测提供了良好的技术支撑.     

A simple procedure to extend the Gauss method of determining orbital parameters from three to N points

A simple procedure is developed to determine orbital elements of an object orbiting in a central force field which contribute more than three independent celestial positions. By manipulation of formal three point Gauss method of orbit determination, an initial set of heliocentric state vectors r _i and $\dot{\mathbf{r}}_{i}$ is calculated. Then using the fact that the object follows the path that keep the constants of motion unchanged, I derive conserved quantities by applying simple linear regression method on state vectors r _i and $\dot{\mathbf{r}}_{i}$ . The best orbital plane is fixed by applying an iterative procedure which minimize the variation in magnitude of angular momentum of the orbit. Same procedure is used to fix shape and orientation of the orbit in the plane by minimizing variation in total energy and Laplace Runge Lenz vector. The method is tested using simulated data for a hypothetical planet rotating around the sun.     

A method of astronomical autonomous orbit and attitude determinations for satellites

A method of realtime autonomous orbit determination for earth satellites using the extended Kalman filtering is proposed. The observed quantities are: the satellite-sun direction vector measured by a sun sensor, the satellite-earth and satellite-moon direction vectors measured by an ultraviolet sensor, and the geocentric distance measured by a radar altimeter. At the same time the satellite attitude to the earth is also determined. Results of our simulation of the autonomous orbit determination show that the precision of the orbit determinations is better than 200 m. The effects of the sampling period, orbital inclination, orbital eccentricity and orbital altitude on the precision of orbit determination are analyzed and compared, and certain principles helpful for improving the precision of orbit determination are suggested.     

Lunar orbit determination in the presence of unmodeled accelerations

A technique for estimating the state of an artificial satellite in the presence of unmodeled accelerations is presented. The unmodeled acceleration is approximated by a first-order Gauss-Markov sequence which can be separated into a timewise correlated component and a purely random component. Using this approximation, a sequential procedure for estimating the position, velocity, and the unmodeled acceleration is developed. The method is evaluated by reducing range-rate observations obtained by tracking the Apollo 10 and 11 spacecraft during the lunar orbit phase of the mission. Numerical results are presented which show that the observation residual pattern lies within the observation noise standard deviation. The values of the estimated components of the unmodeled acceleration are repeatable from orbit to orbit within a given mission and from mission to mission when the same ground track is covered. Finally, the variation in the radial component of the unmodeled acceleration shows a high correlation with the reported location of the lunar surface mascons.     

The evolution of co‐orbiting material in the orbit of 2201 Oljato from 1980 to 2012 as deduced from Pioneer Venus Orbiter and Venus Express magnetic records

Asteroid 2201 Oljato passed through perihelion inside the orbit of Venus near the time of its conjunction with Venus in 1980, 1983, and 1986. During those three years, many interplanetary field enhancements (IFEs) were observed by the Pioneer Venus Orbiter (PVO) in the longitude sector where the orbit of Oljato lies inside Venus' orbit. We attribute IFEs to clouds of fine‐scale, possibly highly charged dust picked up by the solar wind after an interplanetary collision between objects in the diameter range of 10–1000 m. We interpret the increase rate in IFEs at PVO in these years as due to material in Oljato's orbit colliding with material in, or near to, Venus' orbital plane and producing a dust‐anchored structure in the interplanetary magnetic field. In March 2012, almost 30 yr later, with Venus Express (VEX) now in orbit, the Oljato‐Venus geometry is similar to the one in 1980. Here, we compare IFEs detected by VEX and PVO using the same IFE identification criteria. We find an evolution with time of the IFE rate. In contrast to the results in the 1980s, the recent VEX observations reveal that at solar longitudes in which the Oljato orbit is inside that of Venus, the IFE rate is reduced to the level even below the rate seen at solar longitudes where Oljato's orbit is outside that of Venus. This observation implies that Oljato not only lost its co‐orbiting material but also disrupted the “target material,” with which the co‐orbiting material was colliding, near Venus.     

Sun–Earth L_{1} and L_{2} to Moon transfers exploiting natural dynamics

This paper examines the design of transfers from the Sun–Earth libration orbits, at the \(L_{1}\) and \(L_{2}\) points, towards the Moon using natural dynamics in order to assess the feasibility of future disposal or lifetime extension operations. With an eye to the probably small quantity of propellant left when its operational life has ended, the spacecraft leaves the libration point orbit on an unstable invariant manifold to bring itself closer to the Earth and Moon. The total trajectory is modeled in the coupled circular restricted three-body problem, and some preliminary study of the use of solar radiation pressure is also provided. The concept of survivability and event maps is introduced to obtain suitable conditions that can be targeted such that the spacecraft impacts, or is weakly captured by, the Moon. Weak capture at the Moon is studied by method of these maps. Some results for planar Lyapunov orbits at \(L_{1}\) and \(L_{2}\) are given, as well as some results for the operational orbit of SOHO.     

Effects of gravity-gradient torque on the rotational motion of A triaxial satellite in a precessing elliptic orbit

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the effects of gravity-gradient torque on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is free to precess and spin. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite to second order inn/0, wheren is the orbital mean motion of the center of mass and0 is a reference value of the magnitude of the satellite's rotational angular velocity. The differential equations derivable from the transformed Hamiltonian are integrable and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed and a comparison of theoretical results with observations is made.     

Elliptic-type motion in Fock's gravitational field

The elliptic-type motion in the gravitational field found by Fock as exact solution of Einstein's vacuum equations in the case of spherical symmetry (Solution called here Fock's gravitational field) is studied by means of a classic method based on the perturbation theory. Regarding the deviations of the orbit from a Kepleian orbit as perturbations, the first and second order variations of the Keplerian orbital elements over one nodal period as well as those of the nodal period itself are determined.