1.0  History/Background

The ancients.  The history of the study of the motion of bodies in the heavens is very closely aligned with the development of physical laws. “In a sense the story of physics properly begins with Greek astronomy, because the Greeks were the first to try to understand and explain the movements of the stars and planets…”¹ Astronomy dominated scientific study, and the early Greeks accumulated a vast amount of observational data. Aristarchus of Samos (310 – 230 B.C.) was believed to be the first to offer a sun-centered view of the universe, but the view of an earth-centered universe prevailed generally. Aristarchus also made a serious attempt to determine the relative distances between the earth and moon and between the earth and sun. 

Hipparchus was perhaps the greatest ancient Greek astronomical observer, but we may not have known of much of his work if it hadn’t been for the writings of Ptolomy who lived in Egypt. (100-170 A.D.). In his book Almagest, Ptolomy developed a theory of epicycles in an earth-centered solar system to explain the apparent motions of the Sun, Moon, and planets. Epicycles used a combination of circles, a smaller moving on a larger, to explain the motion that we now know as elliptic. It wasn’t until the turn of the seventeenth century that planetary motion could be explained with mathematical accuracy.¹

There was little progress in astronomy/physics as far as orbital mechanics is concerned between the ancients and the fifteenth century. Copernicus (1473–1543) took up the task, and the heliocentric (sun-centered) universe became much more widely accepted.  He focused on circular orbits for the planets and advocated his own version of epicycles to explain their motion. At the end of that century our understanding of orbital motion took a big step forward. Modern orbital mechanics began when orbital motion was quantified by Johannes Kepler with accurate equations for time as a function of elliptic orbital position.

The history of mathematics parallels in many ways the history of astronomy. Euclid was one very influential Greek mathematician.  He is famous for his Elements (about 300 B.C.), a thirteen volume work that summarized all the mathematical knowledge of ancient Greece. It had tremendous influence, and Euclidean three-dimensional geometry was accepted as the geometrical framework on which the laws of nature were formulated. Newtonian mechanics incorporated Euclidean geometry into its theoretical structure. Euclid’s geometric structure remained unchallenged until the nineteenth century when geometry transitioned into non-Euclidean. 

Apollonius of Perge (262-200 B.C.) contributed significantly to mathematics related to planetary motion. He also developed a theory of conic sections that was remarkable for the time. There is evidence that Euclid studied the conic sections, but it is the writings of Apollonius that survived as the first evidence of knowledge of these curves. It wasn’t until Kepler’s work in the seventeenth century A.D. that elliptic paths were associated with orbital motion. Each of the conic sections is a potential path for a two-body orbit. 

Modern orbital mechanics.  At the end of the sixteenth century Kepler met Tycho Brahe, and their combined dedication and efforts led to the establishment of laws and equations that describe planetary motion. Brahe had spent twenty years meticulously collecting and organizing data from astronomical observations and documenting the positions of planets at the time observed. At the time of Brahe's death in 1601, Kepler acquired this data and applied his mathematical skill to the task of explaining the planetary orbital paths and time-position relationships. The ultimate product of this work provided the foundation for mathematical equations of orbital motion that are still applied today. The fundamental relationships were expressed in Kepler's laws, the third of which was published in 1618:

1. Each planet moves in an ellipse around the sun, which is at one of the foci of the ellipse.

2. The line from the sun to a planet sweeps out equal areas in equal times.

3. The square of a planet's period is equal to the cube of its mean distance from the sun.

Kepler then proceeded to derive the equation that holds for any body in an elliptic orbit giving the time-position relationship. The derivation applies his second law and expresses mathematically the sweeping out of "equal areas in equal times". The method uses a geometric construction and area relationships. Sir Isaac Newton (1642-1727) provided a refined understanding of gravitational force, the forces involved in changing the momentum of a body, and other fundamental principles of classical mechanics. He discovered that two bodies have a gravitational force of attracton proportional to the product of their masses and inversely proportional to the square of the distance between them. This law and principles governing the motion of bodies were published in 1687. These principles are stated in three laws that are frequently quoted in introductory discussions on the subject of orbital mechanics:

1. Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.

2. The rate of change of momentum is proportional to the force impressed and is in the same direction as that force.

3. To every action there is always opposed an equal reaction.

Newton made profound contributions to mathematics and science. He published a paper on the discovery of the calculus in 1704.  Though Leibnitz made an independent discovery of calculus at about the same time, Newton often is credited with being the first. We will see that this branch of mathematics is ideal for the derivation of orbital motion equations. Kepler did not have the benefit of calculus when he derived his equation.

Other contributors to classical mechanics, and more specifically orbital mechanics, added the principles of conservation of energy and conservation of angular momentum. These and many methods for solving Kepler's equations have contributed to the efficiency and speed of solving orbital motion problems. However, Kepler's approach remains the fundamental source for methods by which all orbital time-displacement problems are solved. Many problems require that a satellite's position be predicted at some future time, and problems of this nature must be solved by a numerical method. These methods apply a repetitious sequence of iterative calculations (successive approximations) until a solution of sufficient accuracy is found. Computers and clever methods of applying their rapid calculation capabilities have made this situation manageable, but Kepler's transcendental equation limits our capability to solve some problems. 

In other pages on this site I emphasize the analysis of linear trajectories, and offer that they are neglected in textbooks on the subject of orbital mechanics. I can point specifically to the point of departure between the inclusion and exclusion of linear trajectories in the general equation development. It seems to me that an almost universal assumption is made that all orbits have a rotational component. This is evident in a cross multiplication step in deriving the equation for specific angular momentum and then the continuance of the use of that angular momentum in the derivation of subsequent equations. A very important development is often called the trajectory equation applicable to each type of orbit. It is identically equal to the general equation for all conic sections. These derivation steps are correct for general orbital motion, but they preclude development of the linear trajectory equations which are special cases of the general orbits. We can develop the linear equations when we recognize that two-body motion with no rotational component is real.

Reference

1. Motz, M. and Weaver, J. H., The Story of Physics, Avon Books, New York, 1989.