Optimal Reentry Time Estimation of an Upper Stage from Geostationary Transfer Orbit

Abstract

A STABLE space-debris environment is defined as an environment permitting safe space operations in the long-term future [1]. A simple way to achieve a stable space debris environment is to limit the lifetime of man-made orbiting vehicles to 25 years [2]. A spent stage that evolves along a geostationary transfer orbit (GTO) is subject to a higher collision threat potential; that is, since its path will necessarily traverse the region between geostationary orbit and low Earth orbit. Subsequently, minimization of the spent stage lifetime is one important objective in space debris mitigation efforts. The impact of natural forces such as drag and lunisolar perturbations on the lifetime of high-eccentricity orbits (less than 0.5) is the subject of earlier studies [2–7]. The effectiveness of lunisolar perturbations in reducing the lifetime of eccentric orbits with high-apogee altitudes is also the subject of previous investigations [2]. Measures such as deboosting compromise the payload capacity of the launch vehicle. Therefore, the decay of the spent stages must be carefully monitored and analyzed to facilitate enhanced planning for future missions. Specifically, minimizing the lifetime of the spent stage without jeopardizing the operational requirements. The orbital evolution along a GTO is sensitive to the initial conditions. An accurate simulation of the process requires a good estimate of the initial state. The semimajor axis of the initial orbit is often better known than the eccentricity, because the semimajor axis is directly related to the orbital period, which is an easily measurable parameter. Therefore, the eccentricity is treated as an uncertain parameter in the present study. The ballistic coefficient B m=CDA kg=m 2 is also treated as an uncertain parameter [8]. This coefficient depends on the mass of the objectm, the drag coefficient CD, and the effective area A. The response surface methodology (RSM), or response surface approximation, is a collection of mathematical and statistical techniques developed by Box and Wilson [9]. It is useful for modeling and analysis of problems where the response is influenced by several variables [10] and the objective is to optimize the response. The methodology is commonly employed in various fields, including agriculture and manufacturing [11]. Earlier investigations [12] predict the lifetime of a sample rocket body by combining the RSM with a genetic algorithm (GA) to estimate the eccentricity and ballistic drag coefficient. The estimation process employs the twoline elements (TLEs) for up to 250 days. The TLE consists of two lines of formatted text. It provides the orbital elements that describe the orbit of Earth satellites at a specified epoch. It also provides the first and second time derivatives of the mean motion and the ballistic drag terms of the satellite. The osculating Keplerian elements from the TLEs are produced by removing the longand short-periodic variations using simplified general perturbations (SGP4/SDP4) orbit theories [13]. SGP4 theory includes secular effects of J2, J4, and J 2 ; long-periodic effects of J2 and J3; and short-periodic effects of J2 truncated to zeroth order in eccentricity. For an orbit with a time period less than 225min, SGP4 theory is employed to obtain the state vector at a specified epoch. Otherwise, SDP4 theory is used to obtain the state vector. SDP4 theory includes all the terms of SGP4 plus the Earth’s tesseral terms (2, 2), (3, 2), (5, 2), (4, 4), (5, 4), and the firstorder lunisolar point-mass gravity effects. The state vectors consisting of position and velocity, are obtained from the TLEs using the SDP4 theory. These state vectors are used in the Numerical Prediction of Orbital Events (NPOE) software to obtain the osculating and mean orbital elements at the initial state for the purpose of orbit propagation. The reentry bounds are computed by considering variations of up to 10% in the ballistic coefficient, the solar fluxF10:7, and geomagnetic indexAp. The actual reentry time is found to be well within the bounds. In this Note, the reentry time of the cryogenic stage of the Indian Geostationary Satellite LaunchVehicle GSLV-F01/CS§ is carried out as an optimal estimation problem. The RSM with GA are applied to determine the optimal estimates of B and e. The investigation employs TLEs, determined 165 days before the reentry epoch, 24 November 2007, 19:30 [Greenwich Mean Time (GMT)]. The decay location is characterized by longitude of 5 N, a latitude of 2 E, and an orbital inclination of 19.3 . An accurate reentry time prediction of the cryogenic stage is made seven days before its reentry. Themethodology selected offers an improvement over leastsquares method. The study also shows that the object must have tumbled during the last day in the orbit.