Appendix D
It’s Mechanics

Principles or methods related to the classical approach to orbital mechanics on this site are generally taken from Fundamentals of Astrodynamics by Bate, Mueller and White. I will refer to this source as BMW. It has been used as a text since the early 1970’s, and it has a broad readership. The presentation of the derivation of time-displacement equations in this book are unlikely to be challenged by experts in the field.

I. Rectilinear Motion. Let’s begin with Newton’s laws since these are fundamental to the mechanics that will apply.

Newton’s First Law states:

        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.

Compare this to conservation of linear momentum from The Concise Oxford Dictionary of Mathematics by Christopher Clapham (1996). The definition reads:

        When the total force acting on a system is zero, the linear momentum of the system remains constant for all time.

and

        The linear momentum of a particle is the product of its mass and its velocity. It is a vector quantity.

Conserving linear momentum and continuing uniform motion in a straight line are stating the same thing. We are not changing the mass of any body that we are concerned with, so the velocity vector defines the linear momentum vector.

What do we have so far? It appears that we have “one-body” motion. It is simple, yet it is difficult to describe in our universe. There is no natural frame of reference except for a single straight line. To what do we relate position or velocity? My topic is two-body motion, so let’s move on.

Newton’s Second Law states:

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

The rate of change of momentum is involved in a number of ways in orbital mechanics. A spacecraft changes its momentum with the thrust of propulsion. As it does that some of its mass is consumed to produce the thrust. We will let that go for now and just deal with bodies with constant mass and no propulsive activity. I'll refer to this as an inertial force equal to the mass times the derivative of the velocity with respect to time. We can analyze the component of the motion of a body in the direction of the applied force if we can quantify that force.

Newton’s Third Law states:

        To every action there is always opposed an equal reaction.

In two body orbits this means that each body exerts the same force on the other, the force of gravity. So we need one more law to begin the analysis.

Newton’s law of Universal Gravitation states:

        Any two bodies attract one another with a force proportional to the product of their masses
        and inversely proportional to the square of the distance between them.

                                                                            (D-1)

In this equation G is the universal gravitational constant, m₁ is the mass of the parent body, m₂ is the mass of the satellite, and r is the distance between their centers. By convention the mass of the satellite is set equal to one for simplification, and this is consistent in subsequent derived equations.  Each gravitational force field is defined by the mass of the parent body.  The equation above is typically written by substituting the Greek letter mu for the product of G and the mass of the parent body. 

Now we have enough to derive a set of equations for linear orbital motion, and we will do that. Each of the three equations depends on the total energy of the two-body system, which can have a positive, negative or zero value. However, we would like to include in our equations for general motion those orbits in which the satellite actually goes around the parent body. In other words, the orbits so well developed in the literature. We must develop another principle, and it is simple in concept. What can we say if the center of the parent body and the direction of the force of gravity do not lie along the line of the velocity vector described above?

II. Rotational Motion. Now we have the situation of rotational motion in a central force field, and we extend the concept of linear angular momentum. The Concise Oxford Dictionary of Mathematics defines angular momentum as:

        The moment of the linear momentum about the [axis or point of rotation]

The concept of a moment is now introduced. The moment of a force acting on a lever arm is the component of that force perpendicular to the lever arm times r, the distance from the center of rotation to the point of the applied force. Similarly, angular momentum is the component of linear momentum (in our case, velocity) perpendicular to the radius times r. If you are familiar with vector operations this is r x v, or cross-multiplying the velocity vector with the radial vector. The magnitude of this cross product is the radius times the velocity component perpendicular to r, or the radius squared times the angular velocity. According to the principle of conservation of angular momentum, this quantity remains constant if no force acts to change it. In other words, it is conserved if no torque is applied. No torque is applied in a two-body problem because the only forces acting, gravity and inertial forces, are aligned with the radius at all times. Another consequence of this principle is that the motion stays in a plane. Angular momentum is a vector directed normal to the plane of motion, and it does not change in magnitude or direction. Its symbol is h, and we can evaluate it at one point in the orbit and will know it for the entire orbit.  The magnitude of h is expressed as (see BMW, p 26)

                                                                         (D-2)

  When we have rotational motion there is acceleration due to the rotation of the velocity vector normal to r. This is the centrifugal force or centripetal acceleration. It always opposes the gravitational force. Its magnitude is the normal velocity squared divided by r, or r times the angular velocity squared. Caution is appropriate with this force. The Microsoft Encarta College Dictionary (MECD) defines it as:

        An apparent force that seems to pull a rotating or spinning object away from a center [italics added].

This is a real force.  Referring to it as an apparent force seems to refer to its accelerative quality.  It is the vector cross product −w x (w x r), where w is the rotation vector, according to Classical Mechanics (2^nd Ed.) by Corben and Stehle. It is always aligned with the radius vector in the plus r direction. Einstein told us in the equivalence principle that we cannot sense the difference between a force, like gravity, and acceleration. If one is in an enclosed elevator feeling one g, he or she cannot tell if the elevator is at rest in one g or in space somewhere accelerating at that rate. We can apply this acceleration as a force and include it in a summation of forces acting on a satellite.  Its magnitude is

                                                                (D-3)

We have one more force (acceleration) that applies. It is an inertial force and is the rate of change of momentum of the satellite in response to the sum of the other forces (accelerations) acting on it. Its magnitude is the second derivative of r with respect to time. It is always aligned with the radius vector and its direction can be positive or negative with respect to r.

                                                                                (D-4)

These three forces can be added algebraically because they are always aligned with the radius vector. 

Another principle that is applicable to any system that does not have work crossing its boundaries is the conservation of energy. For its definition I’ll choose the MECD.

Conservation of energy is defined as:

        The principle that the amount of energy in an isolated system remains the same,
        even though the form of the energy may change [within the system].

The change in potential energy is defined as the work done moving a body against a force through a given distance. In this case it is the integral of the gravitational force times dr, and we integrate from a given radius to infinity, the zero level of potential energy. The potential energy at r is a negative gravitational constant divided by r. According to the principle the sum of kinetic and potential energy is a constant. Kinetic energy is half the radial velocity squared plus half the velocity normal to the radius squared.