4.0 Simple Orbits
Circular
Orbits. We can arrange the trajectories in order of
complexity from the simplest mathematical time-displacement equations
to the most difficult to solve. This does not mean the lowest
order of terms, but rather the calculations required to find a
solution. We will see that elliptic and hyperbolic trajectories
have transcendental equations. Finding time as a function of
position is straightforward, but the reverse can only be solved with a
numerical technique that will converge on an acceptable solution after
several iterations. The simple orbits then are those that have
algebraic time-displacement equations with closed form solutions.
The simplest trajectory is a circle. A circular orbit is a
special case of the elliptic orbits in which the eccentricity is
zero. The major axis of the ellipse disappears in this special
case, but we use the same symbol as the semi-major axis, a,
for the radius. Since we have no major axis there is no unique
point that is a useful datum for the initial angle. We must
choose some point unique to the orbit, such as crossing the equator or
the prime meridian. With this selection we can write the equation
for the constant angular displacement as
A satellite in a circular orbit has a
constant gravitational force, constant kinetic energy, and constant
potential energy.
The term 
will be seen in the
time-displacement equations for all orbital trajectories.
Therefore, we can relate the displacement of all orbits to a circular
orbit with a radius of a
applying the value of a for
the orbit of interest. This term has been given the name mean
anomaly.
Parabolic Orbits. Since
parabolic orbits have a total energy value of zero, they have a special
quality that allows us to separate variables in the angular momentum
equation and integrate to give an algebraic equation. The general
parabolic equation is third order, so we must solve for position as a
function of time with the cubic solution. This does give an exact
solution without the necessity to use a numerical method though.
Before we consider general parabolic motion, let's consider an even
simpler special case. The linear parabolic trajectory is the next
in order of mathematical complexity. Since it has no rotational
component term, the conservation of energy equation is simply
We can take the square root, separate
variables, and integrate to give the time-displacement equation
if we let r = r0 when t = 0.