A study of the errors of the averaged models in the restricted three-body problem in a short time scale

Abstract

The objective of the present research is to study the accuracy of the double-and single-averaged models that are usually considered to predict the motion of spacecrafts or celestial bodies that have their motion perturbed by a third-body. Those two models are compared with each other and then validated against the complete elliptic restricted threebody problem. Those models are developed to give a faster but general behaviour of the motion of the perturbed body in a medium or longer time scale. The researches performed here verify the accuracy of those methods for shorter time scales by showing the differences in terms of the values of the inclination and eccentricity of the perturbed body predicted by those models. Those differences are calculated both at every instant of time and as an integral over the time. The use of the integral along the time for the errors is a new form to study those differences and show a more complete comparison of the accuracy of those approximations, completing the instantaneous picture given by the usual approach of looking at the instantaneous measurement. If the value of the integral is divided by the time integration, the mean error is obtained. The results show that the single-averaged model is better in the short time scale and the difference among those models is smaller when predicting the eccentricity than the inclination. The effects of the time scale is verified by varying the study to values from 2 days up to 50 revolutions of the Moon (1,366 days). Another important point found in the present paper is the range of the eccentricities of the perturbing body that accelerates the dynamics.

Studying the lifetime of orbits around Moons in elliptic motion

“…The equations of motion used in the present research were studied in details in Domingos et al (2014). The present paper summarizes the final results, in the form used here to perform the numerical integrations that evaluate the lifetime of the spacecraft.…”

“…Different initial eccentricities (0.0, 0.05, 0.10, 0.15 and 0.20) are considered for the orbit of the moon, which means that, in a system fixed in the moon, those are the eccentricities of the disturbing body, the planet. Those choices are made based on Domingos et al (2013Domingos et al ( , 2014, which showed that averaged models have good accuracy up to those values. The system used as an example for the present studies has the same physical data (masses and distances) of the Earth-Moon system, but considers possible eccentricities for the orbit of the secondary body around the primary.…”

“…Domingos et al (2008) made a detailed development of the disturbing potential and the average process. The equations of motion used here are the same ones used in Domingos et al (2008Domingos et al ( , 2014, which are the papers that developed the force models, but they did not consider the study of lifetimes of spacecrafts. The model is based in an expansion in Legendre's polynomials, considering that the semi-major axis of the orbit of the disturbing body is much larger than the one of the spacecraft.…”

“…This model is shown in detail next, but the main goal is to eliminate the dependence of the perturbations with the true anomaly of the spacecraft, so generating results that are independent of the initial conditions of the orbit. Domingos et al (2014) showed that this type of approximation is very good for eccentricities in the range of 0.0-0.2, so this limit is used. The end of the lifetime is assumed to occur when there is a collision of the spacecraft with the surface of the moon.…”

Mapping Natural Orbits around Io

“…After that, ref. [18] used the solution obtained by [19] for the three-body elliptical problem and obtained the lifetime for the Earth-Moon system as a function of the initial eccentricities and inclinations of the probe. In [20], lifetime maps were obtained for the Callisto-Jupiter system assuming different values for the eccentricity of the perturbing body, also taking into account the J 2 and the order 2 of the sectoral flattening coefficient, C 22 , of Callisto.…”