Single Average Model Research Articles

Analysis of the orbital evolution of space debris using a solar sail and natural forces

Since the launch of the first artificial satellite around the Earth up to the present date, various non-functional objects are orbiting the Earth, such as deactivated satellites, satellite components, rocket bodies, among others. These objects can collide with operational satellites or other debris, generating a cloud of smaller particles. Space agencies are concerned about the current scenario of the space environment. If we continue as we are the number of objects in orbit will make it difficult to operate safely in the space environment. So space exploration may become unsustainable (ESA (2020)). In this work, the orbital evolution of these objects that are located in the geostationary orbit (GEO) is analyzed. In the mathematical model we consider the main disturbing forces, such as the direct solar radiation pressure (SRP), the perturbation of the third body (Sun and Moon) and the Earth oblateness. The SRP is the most relevant perturbation for objects with larger area-mass ratio. Knowing this, the possibility of using a solar sail is considered to help to clean the space environment. The main natural environmental perturbations that act in the orbit of the debris are considered in the dynamics. Such forces acting in the solar sail can force the growth of the eccentricity of these objects in the GEO orbit. Several authors have presented models of the solar radiation pressure considering the single-averaged model. But, doing a literature research, we found that the authors consider the Earth around the Sun in a circular and inclined orbit. Our contribution to the SRP model is in developing a different approach from other authors, where we consider the Sun in an elliptical and inclined orbit, which is valid for other bodies in the solar system when the eccentricity cannot be neglected. The expression of the SRP is developed up to the second order. We found that the first-order term is much superior to the second-order term, so the quadrupole term can be neglected. Another contribution is the approach to identify the initial conditions of the perigee argument (g) and the longitude of the ascending node (h), where some values of the (g,h) plane contribute to amplify the eccentricity growth. In the numerical simulations we consider real data from space debris removed from the site Stuff in Space. The solar sail helps to clean up the space environment using a propulsion system that uses the Sun itself, a clean and abundant energy source, unlike chemical propellants, to contribute to the sustainability of space exploration.

A semi-analytical approach using the single and double averaged methods and the Lidov–Kozai mechanism

An analysis of the orbital motion of artificial satellites around Mercury is presented taking into account its non-sphericity (J 2 , J 3 , C 22 ) and the perturbation of the third body. The disturbing potential due to the third body is developed in circular and inclined orbit. The double-averaged method should be used with caution in some situations where the averaging is applied at different timescales. In this work, a study is presented considering this observation for orbits around Mercury. When the mean anomaly of the Sun is eliminated, the idea is that all effects whose periods below 88 days are neglected. As the rotation of Mercury is about 58.6 days, this means that the perturbation due to the C 22 term must also be neglected. However, since the C 22 term is important and should be taken into account, then terms longer than 58.6 days should also be preserved. In other words, keeping the C 22 term with a period of 58.6 days means that the solar terms with the longest period (88 days) will be maintained here. The single-averaged method is applied to eliminate only the mean anomaly of the spacecraft. A comparison between the single and double averaged models is presented. We found that for the case of Mercury the two models are in agreement, but the single-averaged model is more realistic because it keeps more terms in the disturbing potential. Several types of resonances can be analyzed starting of the single-averaged potential. Considering our single-averaged disturbing potential, the terms due to Lidov–Kozai resonance were isolated to make a qualitative analysis considering the libration and circulation regions in the diagram, eccentricity versus argument of the pericenter.

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

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 three-body 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.

A Comparison of Averaged and Full Models to Study the Third-Body Perturbation

The effects of a third-body travelling in a circular orbit around a main body on a massless satellite that is orbiting the same main body are studied under two averaged models, single and double, where expansions of the disturbing function are made, and the full restricted circular three-body problem. The goal is to compare the behavior of these two averaged models against the full problem for long-term effects, in order to have some knowledge of their differences. The single averaged model eliminates the terms due to the short period of the spacecraft. The double average is taken over the mean motion of the satellite and the mean motion of the disturbing body, so removing both short period terms. As an example of the methods, an artificial satellite around the Earth perturbed by the Moon is used. A detailed study of the effects of different initial conditions in the orbit of the spacecraft is made.

Sequencing procedure for balancing the workloads variations in case of mixed model assembly system with multiple secondary feeder lines

In order to increase flexibility and reduce costs, assembly systems are changing from simple mixed-model assembly lines, to multi-lines systems, where some components are assembled in secondary lines, called feeder lines, connected to the main line by a ‘pull philosophy’. The production of different models in such a complex multi-lines assembly system, where the tasks to perform could be very different from model to model, impacts the production with very high workload time variations, with the consequence of lack of productivity. If operators can perform tasks on different stations on both the feeder and the main lines these time variations can generally be absorbed. But if working across the stations is not possible (closed stations), the balancing of these workload time variations becomes critical. In such contexts the consolidated approach, that allows to configure mixed-model-multi lines assembly system with a single average model representative of the whole mix, can be very limited and is unable to bring substantial results. This paper aims to introduce an innovative sequencing model for mixed-model-multi-lines system, in the case of closed stations, in which it is possible to obtain, after a first long term configuration, a short term balancing, using an appropriate sequencing, for a given production mix and characteristics of the assembly system. The proposed procedure is applied to un-paced assembly lines and validated through simulation and industrial applications.

Low-altitude, near-polar and near-circular orbits around Europa

The dynamics of orbits around planetary satellites, taking into account the gravitational attraction of a third-body and the non-uniform distribution of mass of the planetary satellite, is studied. The Hamiltonian considered is explicitly time-dependent. Conditions for frozen orbits are presented. Low-altitude, near-polar orbits, very desirable for scientific missions to study planetary satellites such as the Jupiter’s moon Europa, are analyzed. Lifetimes for these orbits are computed through the single and double averaged method. Comparison between the results obtained by the single and double averaged method is presented. The single-averaged model is more realistic, since it does not eliminate the term due to the equatorial ellipticity of the planetary satellite as done by the double-averaged problem. Considering the single-averaged method, we found unstable frozen orbits where the satellite does not impact with the surface of Europa for at least 200days. We present an approach using the unaveraged disturbing potential to analyze the effects of these terms in the amplitude of the eccentricity.

Third-Body Perturbation Using a Single Averaged Model: Application in Nonsingular Variables

The Lagrange's planetary equations written in terms of the classical orbital elements have the disadvantage of singularities in eccentricity and inclination. These singularities are due to the mathematical model used and do not have physical reasons. In this paper, we studied the third-body perturbation using a single averaged model in nonsingular variables. The goal is to develop a semianalytical study of the perturbation caused in a spacecraft by a third body using a single averaged model to eliminate short-period terms caused by the motion of the spacecraft. This is valid if no resonance occurs with the moon or the sun. Several plots show the time histories of the Keplerian elements of equatorial and circular orbits, which are the situations with singularities. In this paper, the expansions are limited only to second order in eccentricity and for the ratio of the semimajor axis of the perturbing and perturbed bodies and to the fourth order for the inclination.