On the equinoctial orbit elements

This paper investigates the equinoctial orbit elements for the two-body problem, showing that the associated matrices are free from singularities for zero eccentricities and zero and ninety degree inclinations. The matrix of the partial derivatives of the position and velocity vectors with respect to the orbit elements is given explicitly, together with the matrix of inverse partial derivatives, in order to facilitate construction of the matrizant (state transition matrix) corresponding to these elements. The Lagrange and Poisson bracket matrices are also given. The application of the equinoctial orbit elements to general and special perturbations is discussed.This work was initiated while the second author was a Postdoctoral Scholar in the School of Engineering and Applied Science, University of California, Los Angeles.     

Satellite Relative Motion Using Differential Equinoctial Elements

For precise control, to minimize the fuel consumption, and to maximize the lifetime of satellite formations a precise analytic solution is needed for the relative motion of satellites. Based on the relationship between the relative states and the differential orbital elements, the state transition matrix for the linearized relative motion that includes the effects due to the reference orbit eccentricity and the gravitational perturbations is derived. This method is called the Geometric Method. To avoid any singularities at zero eccentricity and zero inclination, equinoctial variables are used to derive the relative motion state transition matrices for both mean and osculating elements. This approach can be extended easily to include other perturbing forces.     

Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.     

Nonlinear analytic solution for perturbed relative motion using differential equinoctial elements

This paper develops a nonlinear analytic solution for satellite relative motion in J₂-perturbed elliptic orbits by using the geometric method that can avoid directly solving the complex differential equations. The differential equinoctial elements (DEEs) are used to remove any singularities for zero-eccentricity or zero-inclination orbits. Based on the relationship between the relative states and the DEEs, state transition tensors (STTs) for transforming the osculating DEEs and propagating the mean DEEs have been derived. The formulation of these STTs has been split into a set of vector and matrix operations, which avoids directly expanding the complex second-order terms, and thus, the obtained STTs could be easy-to-understand and easy-to-code. Numerical results show that the proposed nonlinear solution is valid for zero-eccentricity and zero-inclination reference orbit and is more accurate than the previous linear or nonlinear methods for the long-term prediction of satellite relative motion.     

Long-term evolution of orbits about a precessing oblate planet: 3. A semianalytical and a purely numerical approach

Construction of an accurate theory of orbits about a precessing and nutating oblate planet, in terms of osculating elements defined in a frame associated with the equator of date, was started in Efroimsky and Goldreich (2004) and Efroimsky (2004, 2005, 2006a, b). Here we continue this line of research by combining that analytical machinery with numerical tools. Our model includes three factors: the J ₂ of the planet, its nonuniform equinoctial precession described by the Colombo formalism, and the gravitational pull of the Sun. This semianalytical and seminumerical theory, based on the Lagrange planetary equations for the Keplerian elements, is then applied to Deimos on very long time scales (up to 1 billion years). In parallel with the said semianalytical theory for the Keplerian elements defined in the co-precessing equatorial frame, we have also carried out a completely independent, purely numerical, integration in a quasi-inertial Cartesian frame. The results agree to within fractions of a percent, thus demonstrating the applicability of our semianalytical model over long timescales. Another goal of this work was to make an independent check of whether the equinoctial-precession variations predicted for a rigid Mars by the Colombo model could have been sufficient to repel its moons away from the equator. An answer to this question, in combination with our knowledge of the current position of Phobos and Deimos, will help us to understand whether the Martian obliquity could have undergone the large changes ensuing from the said model (Ward 1973; Touma and Wisdom 1993, 1994; Laskar and Robutel 1993), or whether the changes ought to have been less intensive (Bills 2006; Paige et al. 2007). It has turned out that, for low initial inclinations, the orbit inclination reckoned from the precessing equator of date is subject only to small variations. This is an extension, to non-uniform equinoctial precession given by the Colombo model, of an old result obtained by Goldreich (1965) for the case of uniform precession and a low initial inclination. However, near-polar initial inclinations may exhibit considerable variations for up to ±10 deg in magnitude. This result is accentuated when the obliquity is large. Nevertheless, the analysis confirms that an oblate planet can, indeed, afford large variations of the equinoctial precession over hundreds of millions of years, without repelling its near-equatorial satellites away from the equator of date: the satellite inclination oscillates but does not show a secular increase. Nor does it show secular decrease, a fact that is relevant to the discussion of the possibility of high-inclination capture of Phobos and Deimos. We use the term “precession” in its general meaning, which includes any change of the instantaneous spin axis. So generally defined precession embraces the entire spectrum of spin-axis variations—from the polar wander and nutations through the Chandler wobble through the equinoctial precession.     

PUVM2测轨方法的误差估计及其传播

本文根据误差理论,对ＰＵＶＭ２测轨方法的误差及其传播规律进行了初步的分析和研究,给出了卫星的轨道根数σ和空间位置ｒ↑→的内符合误差估计公式。     

Luni-solar perturbations of an Earth satellite

Luni-solar perturbations of the orbit of an artificial Earth satellite are given by modifying the analytical theory of an artificial lunar satellite derived by the author in recent papers. Expressions for the first-order changes, both secular and periodic, in the elements of the geocentric Keplerian orbit of the earth satellite are given, the moon's geocentric orbit, including solar perturbations in it, being found by using Brown's lunar theory.The effects of Sun and Moon on the satellite orbit are described to a high order of accuracy so that the theory may be used for distant earth satellites.     

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.     

Quasi-Tangency Points on the Orbits of a Small Body and a Planet at the Low-Velocity Encounter

We propose a method for selecting a low-velocity encounter of a small body with a planet from the evolution of the orbital elements. Polar orbital coordinates of the quasi-tangency point on the orbit of a small body are determined. Rectangular heliocentric coordinates of the quasi-tangency point on the orbit of a planet are determined. An algorithm to search for low-velocity encounters in the evolution of the orbital elements of small bodies is described. The low-velocity encounter of comet 39P/Oterma with Jupiter is considered as an example.     

TV Observation of the Daytime Meteor Shower; The Arietids

We have carried out multi-station TV observations since 1994 in order to determine the orbit of the Arietid daytime meteor stream. In 1999, one possible Arietid meteor was recorded by our simultaneous observations and its orbit was determined. In 2003, two Arietid meteors were observed from two stations of our observing site, those orbits were determined precisely, the orbital elements were in good agreement with each other. This is the first time that determination of the precise orbit of the Arietids has been made from optical observations. The orbit of these Arietid meteors, and comparison with the orbit obtained from radar observations are discussed.     

Night-time geomagnetic variations at low latitudes

Using hourly values of the magnetic elements H, D and Z for 1964, 1965, their variation during night-time hours is examined from both their monthly means and from a previously used harmonic analysis method. The data set used represents quiet magnetic conditions. Consistent changes during the night are often found. Seasonal changes are also examined and it seems necessary to modify the Malin—Isikara hypothesis of a moving ring current by including a seasonal modulation of ring current strength with equinoctial maximum and a local time-varying component or partial ring current. The night-time D component shows considerable asymmetry between North and South hemispheres and this might be due to field-aligned current structure. There is a small amount of evidence for a night-time westward equatorial electrojet enhancement.     

卫星轨道预报的一种分析方法

人造地球卫星的轨道预报是空间环境监测和实时跟踪测量中一个重要环节，由于监测对象众多，要求精度也不太高，通常采用分析法预报．在已有分析法得到t时刻平均根数的基础上给出一种轨道预报方法，由t时刻的平均根数给出该时刻卫星的位置和速度，在此基础上将地球非球形引力摄动的周期项直接用卫星直角坐标的位置和速度分量表示，这样可以避免在计算轨道根数变化的周期项时出现的奇点问题，从而对根数的选择无特殊要求，可适用于各种轨道，简化预报程序和相应的软件，提高预报效率。     

The orbital evolution of comet Bowell (1980b)

Planetary perturbations and orbital evolution of the elements of the comet Bowell (1980b) are calculated. The sudden change of all the elements of the orbit on February 1981 is caused by Jupiter's perturbation.     

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.     

Interannual variability of the solar constant

It can be concluded from the calculations performed of interannual variations of the distance between the Sun and the Earth in the moments of the Earth’s position in the equinoctial and solstitial points that the mean amplitude (approximately the same for all the equinoctial and solstitial points) is determined to be equal to 5700 km (at the maximum values being approximately equal to 15000 km). The values of the solar constant have been calculated on the basis of the data of varying distances, and the values of its interannual variability (for the period from 1900 up to 2050) have determined. Based on the analysis of the series, new periodic characteristics of a long-term variation of the solar constant, related to the celestial-mechanical process, namely, to the perturbed orbital motion of the Earth, are obtained. A three-year cycle is distinguished in the interannual variability of the solar constant, which alternates with a two-year cycle every eight and eleven years. The amplitude of the interannual variability in the series of equinoctial and solstitial points is on average about 0.1 W/m² (about 0.008% of the solar constant value). This is comparable to the interannual variability of the solar constant in the eleven-year cycle of the solar activity. The series obtained can be represented by alternation of eleven-year and eight-year cycles. The eleven-year cycle is composed of three three-year cycles and one two-year cycle, and the eight-year cycle is composed of two three-year cycles and one two-year cycle.     

CHAMP加速仪资料的快速校标研究

对星载加速仪进行校标是有效利用星载加速仪测量数据的基础,目前校标方法都是建立在星载GPS资料处理的基础上,对处理软件和计算设备的要求都非常高.为了满足高层大气阻力研究的需要,提出了一种快速高效的校标方法,即利用GFZ公布的CHAMP卫星快速轨道作为观测资料,采用有尺度因子和线性偏差的加速仪测量值代替非引力模型摄动加速度...     

Some investigations into the atmospheric drag problem

This paper calls into question the validity of the well-known formulae for the perturbations in the Keplerian elements, over one revolution of an orbit, for the motion of a drag-perturbed artificial satellite. These formulae are derived from Gauss's form of the planetary equations, by averaging over a single revolution of the orbit, and using the eccentric anomaly as the independent variable.It is shown that for light balloon-type satellites in near-circular orbits neither the eccentric anomaly nor the true longitude is a suitable choice of independent variable for the averaging procedure. Under these circumstances, it would seem that simple formulae for the variations in the elements cannot be derived from Gauss's equations.     

An Accuracy Analysis of the SGP4/SDP4 Model

Based on the latest release of the SGP4/SDP4 (Simplified General Perturbation Version 4/ Simplified Deep-space Perturbation Version 4) model, in this paper we have designed an orbit determination program. Through calculations for the 1120 objects with various types and orbital elements selected from the space objects database, we have obtained the accuracies of the orbit determination prediction dealt with various types of space objects by the SGP4/SDP4 model. The results show that the accuracies of the near-earth objects are in the order of magnitude of 100 meters; the averages of the orbit determination accuracies of the semi-synchronous and geosynchronous orbits are, respectively, 0.7 and 1.9 km. The orbit determination accuracies of the elliptical orbit objects are related to their eccentricities. Except for few elliptical orbit objects with e > 0.8, the orbit determination errors of the vast majority of the elliptical orbit objects are all less than 10 km. By using the SGP4/SDP4 model to make 3 days predictions for near-earth objects, 30 days for semi-synchronous orbit objects, 15 days for geosynchronous orbit objects and 1 day for elliptical orbit objects, the errors of prediction generally don’t exceed 40 km.     

Influence of fast interstellar gas flow on the dynamics of dust grains

The orbital evolution of a dust particle under the action of a fast interstellar gas flow is investigated. The secular time derivatives of Keplerian orbital elements and the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit are derived. The secular time derivatives of the semi-major axis, eccentricity, and of the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit constitute a system of equations that determines the evolution of the particle’s orbit in space with respect to the gas flow velocity vector. This system of differential equations can be easily solved analytically. From the solution of the system we found the evolution of the Keplerian orbital elements in the special case when the orbital elements are determined with respect to a plane perpendicular to the gas flow velocity vector. Transformation of the Keplerian orbital elements determined for this special case into orbital elements determined with respect to an arbitrary oriented plane is presented. The orbital elements of the dust particle change periodically with a constant oscillation period or remain constant. Planar, perpendicular and stationary solutions are discussed. The applicability of this solution in the Solar System is also investigated. We consider icy particles with radii from 1 to 10 μm. The presented solution is valid for these particles in orbits with semi-major axes from 200 to 3000 AU and eccentricities smaller than 0.8, approximately. The oscillation periods for these orbits range from 10⁵ to 2 × 10⁶ years, approximately.     

The perturbing effects on the planetary satellite

The problem triaxial satellite having a plane of dynamical symmetry in the restricted problem of three bodies has been studied. The first integrals are established and the general solution of the problem can be written in quadratures. The results show that the semi-major axis of the satellite orbit and its rotational angular momentum remain unchanged. The singular solution of this problem has been considered and the elements of satellite orbit can be determined.