Appendix A
Basic Principles

 

Kepler developed his well know equation for time as a function of elliptic orbital position with a system that we know as the two-body system, or we describe the fundamental problem as the two-body problem.  Similarly, we develop time-displacement equations for parabolic and hyperbolic orbits with the two-body system.  It is important, therefore, that we list and understand the assumptions that are made for the two-body problem before the equations of motion are developed.  We assume: A-1


1.  The mass of the satellite is negligible compared to that of the attracting body.
2.  The coordinate system chosen for a particular problem is inertial
     (the geocentric equatorial system for Earth orbits, the heliocentric system for interplanetary satellites, etc.).
3.  The satellite and attracting body are spherically symmetric with uniform density.
4.  No other [independent] forces act on the system except for the gravitational force that
     acts along a line joining the centers of the two bodies.

     [Two dependent forces will be analyzed, centrifugal and inertial.]


Expanding on the notion of inertial coordinates systems, according to Einstein, "The laws of the mechanics of Galilei-Newton can be regarded as valid only for a Galileian system of coordiantes."A-2   All mechanics in this treatise is that of Galilei-Newton described in Galileian systems of coordinates.  We will not explore very high velocity problems that would take us out of these systems.  Any discussion of relativity is that which was understood long before Einstein; motion that is measured in one inertial reference frame is unchanged when measured in another inertial reference frame.  We will manifest this principle by fixing a reference frame to an orbiting body rather than always using the center of the primary body as the origin of our reference frame.

Having described the two-body problem, let's consider it in a slightly different way, as a satellite in a gravitational central force field (CFF).  The attracting body no longer has a physical presence, but just serves to establish the CFF.  We then have no inhibitions about considering satellite motion down to the center of the field where r = 0.  This point is a singularity, but our interest is mathematical, not physical.  We will then develop equations that are typically not seen in the literature.


Let's explain two conventions that are used commonly in the study of orbital mechanics: 1) the mass of the satellite is one (actually in most equations it is multiplied times all terms and therefore will cancel), 2) derivatives with respect to time are denoted by dots above a variable.  For example, two dots above r denote the second derivative of r with respect to time, or acceleration in the r direction.  These conventions allow us to expedite equation development and representation.


A circular orbit, mathematically the simplest of all possible orbital trajectories, is characterized by constant radius, constant force, and constant angular velocity.  Calculations of position versus time in a circular orbital are therefore trivial if we have a point that can be used for an initial angular displacement angle.

We will consider simultaneous radial and rotational motion in a satellite's trajectory while the gravitational force varies with the change in radial displacement.  The first step in mathematical complexity in orbital motion is that in which the satellite motion is radial only.  Sir Issac Newton gave us the law of gravitation, and with it we can quantify the force relationship between a satellite and its attracting body.  He also stated the important relationship that the rate of change of momentum is proportional to the force impressed and is in the same direction as that force.  (1) Sum forces.  If our trajectory has no rotational component,  we have a linear relationship with the sum of the gravitational force and the acceleration that is also the inertial force.  The value of mu for the parent body is applied, the change of momentum is the acceleration of the satellite in the r direction, and we write

Eq. A-1.

These terms are vectors, but the next step will lead us to scalar terms.  This system is similar to any linear F = ma problem with m being unity.  (2) Multiply by velocity.  If each term is dot multiplied by the velocity in the radial direction, we have a power equation.  (3) Integrate with respect to time.  The result is the conservation of energy equation


Eq. A-2.


Using knowledge of the general orbits, we assign values to the constant for elliptic, parabolic and hyperbolic trajectories of


Eq. A-3

respectively.


(4) Time-displacement integration.  With one more step we can  write an equation for time as a function of position.  Rewrite the conservation of energy equation and take the square root.  Then separate variables and integrate with respect to time.  The result is an equation for time as a function of position.  As an example the linear parabolic integral is


Eq. A-4.


The sequence of steps (1) through (4) is the same with the inclusion of the rotational force term in the first step.  We now have an approach that follows the same steps for the entire family of orbital trajectories to produce time-displacement equations.  For the linear parabolic trajectory we integrate and apply the initial condition that r = r0 when t = 0 to write the equation

Eq. A-5.

This method produces algebraic equations for the parabolic orbits, a transcendental equation for elliptic orbits that is identical to Keper's equation, and a transcendental equation for hyperbolic orbits.  The hyperbolic equation has trigonometric terms instead of hyperbolic terms, but it can be shown to be mathematically identical to the classical equation.


CFF