Fourier Series Approximations toJ2-Bounded Equatorial Orbits

Abstract

The current paper offers a comprehensive dynamical analysis and Fourier series approximations ofJ2-bounded equatorial orbits. The initial conditions of heterogeneous families ofJ2-perturbed equatorial orbits are determined first. Then the characteristics of two types ofJ2-bounded orbits, namely, pseudo-elliptic orbit and critical circular orbit, are studied. Due to the ambiguity of the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions, showing little physical insight into the problem, a new scheme, termed Fourier series expansion, is adopted for approximation herein. Based on least-squares fitting to the coefficients, the solutions are expressed with arbitrary high-order Fourier series, since the radius and the flight time vary periodically as a function of the polar angle. As a consequence, the solutions can be written in terms of elementary functions such as cosines, rather than complex mathematical functions. Simulations enhance the proposed approximation method, showing bounded and negligible deviations. The approximation results show a promising prospect in preliminary orbits design, determination, and transfers for low-altitude spacecrafts.

A Simple Time Domain Collocation Method to Precisely Search for the Periodic Orbits of Satellite Relative Motion

“…Suppose the projected closed orbit can be expressed in a system like (15), and , , , V , V , and V are forced to have the same frequency . Then collocate points, normally spaced equally in a time domain, on the slowly drifting trajectory.…”

“…The TDC method proposed by Dai et al [11] is essentially one of the weighted residual methods and has been successfully employed to solve a variety of the nonlinear dynamical problems [12][13][14]. In the TDC method, the desired periodic solution is preassumed as a truncated Fourier series first, which has been adopted in the previous works by Wang et al [15] and Kasdin and Gurfil [16]. Then, the approximate solution of the Fourier series is substituted into the equations of nonlinear system, resulting in a residual error function.…”