Evolution of the rotational motion of space debris acted upon by eddy current torque

Abstract

Finding analytical solutions to the eddy current torque produced by a conducting body rotating within a magnetic field is arduous. In this paper, the finite difference method is adopted to solve numerically the boundary problem regarding the distributions of eddy currents in determining eddy current torque. Through analysis of the solutions, this paper presents the expression of eddy current torque that applies to a model of arbitrary shape rotating around an arbitrary axis. The parameters of the physical properties of the rigid body are integrated into the eddy current torque tensor, the features of which are analogous to the inertia tensor. The elements in the tensor are constants for a specific rigid body; thus, in the expression, the torque is associated only with the relative angular velocity and magnetic field. The expression is used to investigate the evolution of the rotation of space debris subjected to eddy current torque, through numerical integration with the angular velocity of the variation of the geomagnetic field, which is assumed a dipole. The results explain the observed phenomenon of change in the spin decay rate. Moreover, the effects of gravity-gradient torque and orbit precession cause the self-spin of the space debris to resonate with the orbital motion and ultimately, to reach a steady state.

Tiangong-1’s accelerated self-spin before reentry

“…Within the known major external torques in space, the gravity gradient torque generally acts as a conservative torque and does not have a secular effect on the angular momentum (Lin et al 2016;Crenshaw and Fitzpatrick 1968;Holland and Sperling 1969;Hitzl and Breakwell 1971). The eddy current torque and magnetic torque are dissipative torques that decrease the rotational speed (Smith 1962;Williams and Meadows 1978;Praly et al 2012;Lin and Zhao 2015). Light pressure torque may theoretically have a slight acceleration effect under specific orbit and attitude configurations (Kucharski et al 2016), but no secular effect was found in our numerical test on Tiangong-1.…”

“…When the fourth artificial satellite in human history, Vanguard I, was launched in 1958, its rotational speed was found to decay exponentially with time, which was suggested to be the effect of eddy current torque (Wilson 1959;LaPaz and Wilson 1960). After decades of research, eddy current torque has been verified as the main dissipation factor that causes a decrease in spacecraft rotational speed (Smith 1962;Williams and Meadows 1978;Praly et al 2012;Lin and Zhao 2015). However, some observations of individual rockets show that they have increasing rotational speed (Meeus 1971), and some satellites begin to spin after losing attitude control.…”

Long-term attitude dynamics of space debris in Sun-synchronous orbits: Cassini cycles and chaotic stabilization

“…This assumption holds for rocket bodies, which are the primary target of this study. As in prior research (Gomez and Walker (2015); Lin and Zhao (2015); Praly et al (2012)) when modeling the rotational dynamics with respect to object's center of mass, we shall take into account gravity gradient torque M G and torque due to eddy currents M EC .…”

“…The model we use in our study comprises the same key factors as in (Praly et al (2012); Gomez and Walker (2015)) -gravity gradient torque and the torque due to eddy currents. As did Lin and Zhao (2015) we also take into account the orbit precession, which is responsible for remarkable dynamical effects unexamined in previous studies. Besides that, when calculating the torque due to eddy currents we employ a more accurate formula for eddy currents torque proposed in Golubkov (1972) and Martynenko (1985), which includes terms describing the influence of orbital motion that are considered small for fast rotations and are often neglected.…”

“…Most ADR techniques depend substantially on the character of the debris object's rotational dynamics, hence much effort has been spent lately to determine the rotation parameters through groundbased observations (Koshkin et al (2016); Kucharski et al (2014); Lemmens et al (2013); Šilha et al (2017); Santoni et al (2013); Yanagisawa and Kurosaki (2012)). At the same time, much attention has been paid to studying space debris rotational dynamics theoretically (Gomez and Walker (2015); Lin and Zhao (2015); Ojakangas et al (2012); Praly et al (2012); Albuja et al (2015); Sagnieres and Sharf (2017)).…”