We studied how our energetic predictions for the orbits due to a
gravitational central force manifest themselves in the geometry of our
solutions 
. For 
( 
), we verified
that our solution has both minima and maxima for r, so we get a bound
orbit 
. This orbit will turn out to be an ellipse with
the origin at one focus. For E=0 and 
(e=1 and 
respectively),
we verified that the maximum 
goes to infinity, so we get a
scattering orbit where r can escape to infinity. These orbits will
turn out to be left-opening parabolas and hyperbolas respectively,
each with the origin at one focus.
Then looked at the orbits in more detail. For , we calculated
(from our solution) 
, 
, and
. We found the orbit had both its closest and furthest points
of approach on the x-axis, consistent with an elliptical
shape. Assuming that the shape is elliptical, our values for , 
and determine all the
parameters of the ellipse, as we calculated. It is then a matter of
algebra to show that our solution obeys the defining equation of an
ellipse with those parameters (that is, that each point on the
ellipse has the same total distance 2a from the two foci).
We then discussed the scattering solutions. We found that they have a
closest approach at 
, where 
is positive. r becomes
infinite at 
, 
. Thus (for 
increasing in time), our interaction proceeds
as follows: the planet comes in from infinite r at angle 
(in
the third, or lower left, quadrant), comes in to a closest approach at
with positive x, and departs out to infinite r at angle
(in the second, or upper left, quadrant). For e=0, 
, so the orbit is asymptotically horizontal; otherwise
it opens up at fixed angle.
We calculated 
at . From this it is again only
algebraic to show that, for e=1, our solution obeys a parabolic
equation, with a focus at the origin and directrix given by the line
x= 1/C. For , more algebra shows that our orbit is the left branch
of a hyperbola, whose parameters we displayed. We discussed the
asymptotes to the hyperbola, which approximate our orbit as r starts out
from or returns to infinity.
--
KB