Minimum altitude variation orbits. Analysis of characteristics and stability

The article discusses the regularities of satellite motion in almost circular orbits under the influence of the second zonal harmonic of the geopotential. The aim of the research is to determine the parameters of orbits with a minimum change in radius and to study the properties of these orbits. It is shown that the problem of determining the parameters of orbits with a minimum change in radius is of theoretical and practical interest. These orbits are the closest to Keplerian circular orbits. The practical interest in such orbits is determined by the possibility of using them for scientific research and Earth observation systems. Based on the analysis of the literature, it was concluded that the solution of the problem under consideration is not complete by now: the algorithm for determining the parameters of the orbits are not well founded and unnecessarily complicated; there is no analytical analysis of the stability of the orbits of the minimum change in radius. The efficiency of application of the previously developed theory of describing the motion of satellites in almost circular orbits for determining the parameters of orbits with a minimum change in radius is shown. For this purpose, the solutions of the first approximation of the motion of satellites in almost circular orbits under the influence of the second zonal harmonic of the geopotential have been improved. These solutions make it easy to determine the parameters of the orbits of the minimum change in radius. The averaged equations of the second approximation of the influence of the second zonal harmonic on the satellite motion are constructed and, on their basis, the stability of the orbits with a minimum change in radius is proved. It is shown that the second approximation in small parameters completely describes the main regularities of the long-period satellite motion under the influence of the second zonal harmonic of the geopotential. With the help of numerical studies, the instability of orbits with a minimum change in radius is shown with allowance for the effect of higher order harmonics of the geopotential. Analysis of the area of possible application of orbits with a minimum change in radius showed that such orbits can be of practical importance for very low and ultra low orbits, where the control action on the satellite movement is carried out at least once every two days.

Global searches of frozen orbits around an oblate Earth-like planet

A frozen orbit is beneficial for observation owing to its stationary apsidal line. The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J2 and limited high-order terms, which cannot meet the stringent demands of all missions. In this study, the gravitational field is expanded to J15 terms and the Hamiltonian canonical form described by the Delaunay variables is used. The zonal harmonic coefficients of the Earth are chosen as the sample. Short-periodic terms are eliminated based on the Hori-Lie transformation. An algorithm is developed to solve all equilibrium points of the Hamiltonian function. A stable frozen orbit with an argument of perigee that equals neither 90° nor 270° is first reported in this paper. The local stability and topology of the equilibrium points are obtained from their eigenvalues. The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods. The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case. The analytical results can be applied to other Earth-like planets and asteroids.

Analytic Inversion of Closed form Solutions of the Satellite’s J2 Problem

This report provides some closed form solutions -and their inversion- to a satellite’s bounded motion on the equatorial plane of a spheroidal attractor (planet) considering the J2 spherical zonal harmonic. The equatorial track of satellite motion- assuming the co-latitude φ fixed at π/2- is investigated: the relevant time laws and trajectories are evaluated as combinations of elliptic integrals of first, second, third kind and Jacobi elliptic functions. The new feature of this report is: from the inverse t = t(c) we get the period T of some functions c(t) of mechanical interest and then we construct the relevant c(t) expansion in Fourier series, in such a way performing the inversion. Such approach-which led to new formulations for time laws of a J2 problem- is benchmarked by applying it to the basic case of keplerian motion, finding again the classic results through our different analytic path.

Analytical solutions for low-thrust orbit transfers

This paper presents analytical solutions for the estimation of the $$\Delta V$$ Δ V cost of the transfer of a spacecraft subject to a low-thrust action. The equations represent an extension of solutions already available in the literature. Moreover, the paper presents novel analytical solutions for low-thrust transfers under the effect of the second-order zonal harmonics of the Earth’s gravitational potential. In particular, the paper is divided into two parts. The first part presents analytical expressions for the $$\Delta V$$ Δ V cost of transfers. All analytical equations were validated through numerical integration of the dynamics of the spacecraft. The second part of the paper introduces new analytical equations for low-thrust transfers between circular inclined orbits with different values of the right ascension of the ascending node, under the effect of the second-order zonal harmonic of the Earth’s gravitational potential. Both in the first and second parts, analytic solutions for the variation with time of the orbital elements during the transfer are presented. The proposed equations are applicable to low-thrust transfer realised through a long spiral trajectory.

The J 2 Relative Perturbation Analysis of Satellite Formation under the Requirement of Relative Position Maintenance with Millimeter-Level Accuracy

With high-precision DEM (Digital Elevation Model) and GMTI (Ground Moving Target Indicator) as the demand background, the influence of J 2 zonal harmonic term perturbation on the relative motion of the millimeter-level short-range leader-follower satellites in near-circular orbit is studied through the relative perturbation method. An equation of motion that can describe the motion of the leader-follower satellites under the influence of J 2 perturbation in near-circular orbit is derived, and the characteristics of the trajectory of in-plane periodic motion are analyzed. A study shows that under the influence of the relative perturbation of the J 2 term, the in-plane periodic motion of the leader-follower satellites in near-circular orbit is a symmetrical closed “drop-shaped” trajectory with a period of 2 π / n . By comparing with the results of numerical simulations, the correctness of the conclusions obtained in this paper is verified. According to the research results, it can be known that only using a thruster as the actuator to maintain the relative position can no longer meet the requirements of the long-term mm-level relative position maintenance. In the future, a new technical approach needs to be explored to achieve the long-term relative position maintenance with millimeter-level control accuracy.

Method for calculating the perturbed trajectoryof a two-impulse flight between the halo orbit in the vicinity of the L2 point of the Sun — Earth systemand the near-lunar orbit

The paper introduces a new method for solving the problem of calculating the perturbed trajectory of a two-impulse flight between a near-lunar orbit and a halo orbit in the vicinity of the L2 point of the Sun — Earth system. Unlike traditional numerical methods, this method has better convergence. Accelerations from the gravitational forces of the Earth, the Moon and the Sun as point masses and acceleration from the second zonal harmonic of the geopotential are taken into account at all sections of the trajectory. The calculation of the flight path is reduced to solving a two-point boundary value problem for a system of ordinary differential equations. The developed method is based on the parameter continuation method and does not require the choice of an initial approximation for solving the boundary value problem. The last section of the paper provides examples and results of the analysis based on this method.

The effect of zonal harmonics on dynamical structures in the circular restricted three-body problem near the secondary body

The circular restricted three-body model is widely used for astrodynamical studies in systems where two major bodies are present. However, this model relies on many simplifications, such as point-mass gravity and planar, circular orbits of the bodies, and limiting its accuracy. In an effort to achieve higher-fidelity results while maintaining the autonomous simplicity of the classic model, we employ zonal harmonic perturbations since they are symmetric about the z-axis, thus bearing no time-dependent terms. In this study, we focus on how these perturbations affect the dynamic environment near the secondary body in real systems. Concise, easily implementable equations for gravitational potential, particle motion, and modified Jacobi constant in the perturbed model are presented. These perturbations cause a change in the normalized mean motion, and two different formulations are addressed for assigning this new value. The shifting of collinear equilibrium points in many real systems due to $$J_2$$ J 2 of each body is reported, and we study how families of common periodic orbits—Lyapunov, vertical, and southern halo—shift and distort when $$J_2$$ J 2 , $$J_4$$ J 4 , and $$J_6$$ J 6 of the primary and $$J_2$$ J 2 of the secondary body are accounted for in the Jupiter–Europa and Saturn–Enceladus systems. It is found that these families of periodic orbits change shape, position, and energy, which can lead to dramatically different dynamical behavior in some cases. The primary focus is on moons of the outer planets, many of which have very small odd zonal harmonic terms, or no measured value at all, so while the developed equations are meant for any and all zonal harmonic terms, only even terms are considered in the simulations. Early utilization of this refined CR3BP model in mission design will result in a more smooth transition to full ephemeris model.

Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair $$h_1$$ h 1 , $$h_2$$ h 2 of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$ f = h 1 - x ¯ h 2 . We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

Balanced Low Earth Satellite Orbits

The present work aims at constructing an atlas of the balanced Earth satellite orbits with respect to the secular and long periodic effects of Earth oblateness with the harmonics of the geopotential retained up to the 4th zonal harmonic. The variations of the elements are averaged over the fast and medium angles, thus retaining only the secular and long periodic terms. The models obtained cover the values of the semi-major axis from 1.1 to 2 Earth’s radii, although this is applicable only for 1.1 to 1.3 Earth’s radii due to the radiation belts. The atlas obtained is useful for different purposes, with those having the semi-major axis in this range particularly for remote sensing and meteorology.

Analysis of the Orbital Approach Dynamics of the Space Debris Collector to the Fragment of Debris by the Method of Thrust Reversal with Interruption

The debris collector and a debris fragment move along random noncoplanar orbits in the altitude range of 400--2000 km. The thrust of the promising engine is 5000--25 000 N, the specific impulse of the promising fuel is not lower than 20 000 m/s. The remaining fuel after approach is not less than the specified. The debris collector undocks from the base station, transfers from its orbital plane to the debris fragment orbital plane, performs phasing, approaches the fragment, grabs it and returns to the base station. The paper considers only the stage of orbital approach. The duration of the entire flight mission is limited to one day. The phasing time is insufficient, therefore, at the start time of the orbital approach, the distance to the target is ~ 100 km, the relative velocity is ~ 1 km/s. On the other hand, for reliable and safe grabbing of a debris fragment, it is necessary to provide a distance of ~ 1 m and a relative velocity of ~ 1 m/s. It is shown that this can be achieved by approach using the method of thrust reversal with interruption. An effective algorithm of approach with target is proposed. An analysis of the orbital approach dynamics was performed by joint numerical integration of the orbital motion equations of the debris collector and the debris fragment by the 4th-order Runge --- Kutta method. Approach is performed in 6 cycles. In each cycle, the engine turns on three times. Two cycles are performed by sustainer engines, four cycles are performed by auxiliary engines of lower thrust. The fuel depletion and the non-sphericity of the Earth's gravitational field according to the 2nd zonal harmonic are taken into account. Calculation example is considered. Convergence estimates of the integration procedure by the resultant distance to the target and the resultant relative velocity are given. Resultant orbital approach is oscillation process with heavy damping. Damping is ensured by multiple firings of the sustainer (auxiliary) engine