Appendix C
Do the Math

An Introduction to the Mathematics and methods of Astrodynamics by Richard H. Battin covers the subject very thoroughly. However, I think a good grasp of math through algebra, trigonometry and calculus is sufficient for an understanding of motion equations. The derivations will take you into some interesting areas of the integration tables, and some problems will involve spherical trigonometry. The need to find the roots to a cubic polynomial also arises in some problems. So you may want to dust off some of those math skills you learned in high school and the first couple of years of college.

Certain conventions are adopted that serve as shortcuts in writing and developing equations.  One is that the mass of the satellite in a two-body problem is equated to one.  This gives accurate results unless the mass is changing, as it does when a spacecraft propulsion system is operating.  If the total mass is important in a problem, multiply the actual satellite mass times the mass of unity.  A calculus related shotcut is applied extensively in orbital mechinics as well as many other fields of study.  It is the use of dots above a variable to denote differentiation with respect to time.

Motion studies involve vectors.  The radius from the force field center to a satellite, the satellite velocity and momentum, the forces acting on the satellite and the associated accelerations, are all vectors.  The literature gives thorough analyses of satellite motion with vector notation and operations.  However, this web site takes some shortcuts in these operations.  Explanations are given at appropriate places, but justification for this apparent lack of mathematical rigor is offered here.  The fundamental basis for the derivation of all equations of motion is the summation of the forces acting on the satellite: gravitational, inertial, and centrifugal.  These are always aligned with the radius vector, and their algebraic sum and vector sum are equal.  Operating on these forces to write a power equation gives scalar quantities, and subsequent operations to write energy equations also gives scalar quantities.  Models developed from this sequence of operations display velocities with appropriate vector relationships, confirming and justifying the approach.