The purpose of this page is to develop the formulas relating conic
section parameters that we will need for orbit determination and description.
Equations will be developed from basic principles of geometry and dynamics.
First we will develop the equations common to all conic sections from
geometric principles.
The starting point will be the polar equation of a conic
This equation can be used to describe all possible conic geometries
with proper choice of the parameters p and
e. The semiparameter, p ,
determines the size, and the eccentricity, e ,
determines the shape.
Superimposed on a Cartesian coordinate system, the graph of this equation will have one focus at the origin. The closest point to the origin, periapsis, lies on the positive x - axis and the graph opens from that point to the left in the direction of the negative x - axis.
The graph lies in a plane (the perifocal plane) and every point on the
graph can be defined by an ordered pair of polar coordinates
( r , 
).
is an angle
measured from the positive axis and r is the distance
from the origin. We can see from the conic equation and the diagram that
the distance from the origin to the periapsis distance,
r , is
When a and c exist (ellipse
and hyperbola), a is the distance between the two
vertices and c is the distance between the two foci.
then
With a little algebra we can show that
A useful expression for the eccentricity as a function of the dynamic
quatities energy and angular momentum may be derived from the previously
obtained relations
Then
Some expressions specific to the type of conic are now developed.
ELLIPSE
From the expression relating the semi major axis, a ,
to the semi minor axis, b ,
we can derive
The ellipse is the only conic section with a periodic orbit. An expression for the period is derived in Kepler
Since a > 0 the energy,
, of an elliptical orbit is defined
to be negative.
CIRCLE
A circular orbit is a special case of an elliptical
orbit where r = a .
As a result the circular velocity,
,
of the orbiting body remains constant.
PARABOLA
A parabola is the transition conic between the ellipse
and hyperbola and has e = 1 .
As such
A parabolic orbit defines the trajectory of a body that is traveling
at escape velocity,
. Escape velocity
is the velocity required by an orbiting body at distance,
r , for it to descelerate to a velocity of
exactly zero at
r = 
HYPERBOLA
The energy of a hyperbolic orbit is defined to be positive
by our convention. The energy above zero will manifest as
velocity above what is required for escape velocity.
The hyperbolic excess speed, , is the speed of an object
as it approaches an infinite range from the central mass.
If velocity, v , is given to a body at distance,
r , then at
r =
I have put together a package of programs that will analyze any conic section.
Directions are included in the .zip file.