Orbit Equation - Categorization of Orbits

Categorization of Orbits

Consider orbits which are at one point horizontal, near the surface of the Earth. For increasing speeds at this point the orbits are subsequently:

• part of an ellipse with vertical major axis, with the center of the Earth as the far focus (throwing a stone, sub-orbital spaceflight, ballistic missile)
• a circle just above the surface of the Earth (Low Earth orbit)
• an ellipse with vertical major axis, with the center of the Earth as the near focus
• a parabola
• a hyperbola

Note that in the sequence above, and increase monotonically, but first decreases from 1 to 0, then increases from 0 to infinity. The reversal is when the center of the Earth changes from apoapsis to periapsis (the other focus starts near the surface and passes the center of the Earth). We have

Extending this to orbits which are horizontal at another height, and orbits of which the extrapolation is horizontal below the surface of the Earth, we get a categorization of all orbits, except the radial trajectories, for which, by the way, the orbit equation can not be used. In this categorization ellipses are considered twice, so for ellipses with both sides above the surface one can restrict oneself to taking the side which is lower as the reference side, while for ellipses of which only one side is above the surface, taking that side.