Approximate analytic solution of motion for spacecraft around an oblate planet

Abstract

The oblateness of a planet has an important effect on the surrounding spacecraft and is usually regarded as a significant perturbation factor to spacecraft’s motion. In this paper, an approximate analytic solution of motion for spacecraft which considering this perturbation factor is investigated. The perturbed dynamic model which taking true anomaly as the independent variable is firstly built in non-inertial coordinate system moving with a Kepler two-body orbit, then the six-dimensional state transition matrix about the state deviation vector is introduced. After that, the expression of J2 term’s gravitational potential in this non-inertial coordinate system is obtained based on spherical trigonometry, and the analytic solution for every variable of the state deviation vector is then derived by complicated mathematic transformation. Numerical simulation results done by the presented method have been compared with that done by numerical integration as well as an existing first-order analytic solution. The comparison results show that the presented method has high accuracy, and is more accurate than the first-order analytic solution within one or few orbit revolutions, which makes it more suitable for short-term orbit prediction of the satellites, especially for those suborbital flight vehicles.

Keywords

Perturbed dynamic model planet’s oblateness spacecraft mean orbit elements approximate analytic solution orbit perturbation

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References

Melton

. Fundamentals of astrodynamics and applications. J Guid Control Dyn 1998; 21: 672–672.

Russell

. Survey of spacecraft trajectory design in strongly perturbed environments. J Guid Control Dyn 2012; 35: 705–720.

Battin RH. An introduction to the mathematics and methods of astrodynamics. American Institute of Aeronautics and Astronautics, 1987, p.83.

Zes D. Exo-atmospheric intercept with J2 correction. In: AIAA Guidance, navigation, and control conference and exhibit, Boston, MA, 2013, paper no. 98-4305.

Kozai

. The motion of a close earth satellite. Astron J 1959; 64: 367–377.

Kozai

. Secular perturbations of asteroids with high inclination and eccentricity. Astron J 1962; 67: 591–598.

Brouwer

. Solution of the problem of artificial satellite theory without drag. Astron J 1959; 64: 378–397.

Barrar

. Some remarks on the motion of a satellite of an oblate planet. Astron J 1961; 66: 11–15.

Ren

. Analytic solution of the state transition matrix for free flight perturbation equations of ballistic missile. Chin Space Sci Technol 1983; 1: 11–16.

10.

Schaub

Alfriend

. J 2 invariant relative orbits for spacecraft formations. Celest Mech Dyn Astron 2001; 79: 77–95.

11.

Schaub

Alfriend

. Impulsive feedback control to establish specific mean orbit elements of spacecraft formations. J Guid Control Dyn 2012; 24: 739–745.

12.

Schaub

Vadali

Junkins

et al.

Spacecraft formation flying control using mean orbit elements. J Astronaut Sci 2000; 48: 69–87.

13.

Zhong

Gurfil

. Mean orbital elements estimation for autonomous satellite guidance and orbit control. J Guid Control Dyn 2013; 36: 1624–1641.

14.

Gurfil

. Analysis of J2-perturbed motion using mean non-osculating orbital elements. Celest Mech Dynl Astron 2004; 90: 289–306.

15.

Rosengren

Scheeres

. On the Milankovitch orbital elements for perturbed Keplerian motion. Celest Mech Dynl Astron 2014; 118: 197–220.

16.

Wang

Gurfil

. Dynamical modeling and lifetime analysis of geostationary transfer orbits. Acta Astronaut 2016; 128: 262–276.

17.

Srivastava

Kumar

Kushvah

. The effects of oblateness and solar radiation pressure on halo orbits in the photogravitational Sun-Earth system. Acta Astronaut 2016; 129: 389–399.

18.

Gomes

Oliveira

GMC

Prado

AFBA

et al.

Close approach of a cloud of particles around an oblate planet. Comput Appl Math 2015; 35: 1–11.

19.

Liu

Zhang

et al.

Gravitational orbit–attitude coupling dynamics of a large solar power satellite. Aerosp Sci Technol 2017; 62: 46–54.

20.

Zheng

. Research on effect of geophysical disturbance factors and the compensation method for hit accuracy of long-range ballistic missile, Changsha: National University of Defense Technology, 2006.

21.

Zheng

Tang

. Flight dynamics of ballistic missile in gravity anomaly field, Beijing: National Defense Industry Press, 2009.

22.

Zhou

Zheng

Tang

. Ballistic error propagation algorithm for glide trajectory based on perturbation theory. Proc IMechE, Part G: J Aerospace Engineering. 2017; 231: 1574–1592.