In the 1600s, Johannes Kepler proposed an elegant model for planetary motions—all planets move in elliptical orbits. Each orbit is the result of the gravitational attraction exerted by another body on an object in motion.

“If you drop an object from a certain height, it will fall towards the center of the earth starting from zero velocity, and it will acquire speed as a result of the gravitational acceleration,” explained Dr. Elena Fantino, Assistant Professor of Aerospace Engineering at Khalifa University. “If you give that same object enough velocity in a ‘forwards direction’ instead, it will fly in a circular orbit around the center of the earth. The speed that the object has, determines the type of orbit that it will follow.”

However, over time, orbits will degrade. Dr. Fantino has published an article in the journal Astronomy and Astrophysics investigating the formulae and equations used to determine changes in orbits. Predicting the orbit of various astronomical objects seen from Earth is crucial to various modern day applications, especially traditional celestial navigation, which is still used as a backup to the Global Positioning System, or GPS.

Orbital elements change over time due to perturbations, where an orbit is subject to forces other than the gravitational attraction of a single other massive body. The Moon’s orbit around the Earth, for example, is affected by the gravitational force of the Sun. Other forces can include a third body or resistance from an atmosphere.

In celestial mechanics, a Keplerian orbit is the motion of one body relative to another; it only considers the gravitational attraction of two bodies and in most applications, there is a large central body and a smaller body in orbit around it. The Sun and its orbiting planets of our Solar System form Keplerian orbits.

For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy, but the two objects in a Keplerian orbit are not alone. Gravitational forces from other objects can cause perturbations to an orbit.

In astronomy and celestial navigation, the trajectory of naturally occurring astronomical objects in the sky is given as an ‘ephemeris’. An ephemeris is a piece of information that allows users to calculate the positions of planets and their satellites, asteroids, or comets at virtually any time. They cover past positions and future, generated through several accurate observations and theories arising from celestial mechanics. But uncertainty remains, the greatest of which are caused by unmodelled perturbations.

“Perturbed Keplerian motion is a multi-scale problem, where orbital elements evolve slowly when compared to the change of time of ephemeris, whose fast evolution is determined by the rate of variation of the mean anomaly,” explained Dr. Fantino. “Keplerian orbits are commonly represented by sets of so-called orbital elements, a collection of six geometrical parameters which fully and unambiguously define an orbit. These six parameters are constants of motion; in other words, they do not change with time. When a perturbation is acting, the orbital elements change and these changes express the evolution of the orbit. The way in which the orbital elements change depends on the characteristics of the perturbation. These rates or modes of variation can differ considerably and, in particular, astrodynamicists classify the variations as slow or fast. Many effects can be understood and analyzed by isolating the slowly-varying terms and neglecting the rapidly varying components (which typically are the angles, like the mean anomaly), in this way saving computing time. This separation can be accomplished by analytical theories and not by numerical techniques, such as the Cowell method, which is just numerical integration of the equations of motion of an object, perturbations included. Numerical techniques simply approximate numerically the solution of the whole set of differential equations (acceleration equals the sum of all terms that generate it) and cannot separate, discard or isolate terms.”

Dr. Fantino proposed an analytical theory to isolate or consider the slowly-varying part of the third-body perturbation and remove the fast component.

“Usual integration schemes that look for the separation of fast and slow frequencies of motion may be superior to the simpler integration done with the Cowell method,” said Dr. Fantino. “The description of the long-term dynamics of highly elliptic orbits under third-body perturbations may require an expansion of the disturbing function in series of the semi-major axes ratio up to higher orders.”

Vectorial formulation has proved useful in the case of third-body perturbations and may be an efficient alternative to classical formulations when orbits are highly elliptic, such as in extrasolar planetary systems, artificial satellite theory, and in hierarchical n-body systems in general.

“When approximating the long-term dynamics of a system under third-body perturbations, the vectorial approach is much more simple compared with classical expansions based in trigonometric terms,” explained Dr. Fantino.

Source: Khalifa University