Two Body Conic Orbits

The orbit of any body (be it a comet, planet, asteroid, or a space craft) about the Sun will have the same shape that one gets by the intersection of a plane with a cone. This is true in general, not just for a body orbiting the Sun.

• If the plane intersects a vertical cone at an angle that is between 0^o and 90^o then the resulting line of the intersection is that of an ellipse - a bound elliptical orbit.
• If the plane intersects the cone horizontally then the resulting orbits is that of a circle - a special case of an ellipse whose eccentricity is zero.
• If the plane intersects the cone vertically (at 90^o) then the resulting orbit is that of a parabola - an escape velocity orbit.
• If the plane intersects the cone at an angle greater than angle greater than 90^o then the resulting line of the intersection is that of a hyperbola - the orbit of a body that has a velocity greater that the escape velocity.
• The shape of the orbit is determined by the Gravitational Mechanical Energy of the orbiting body. If ME < 0 then the orbit will be elliptical or circular. If ME = 0 then the body will will follow a parabolic escape orbit. If ME > 0 then the body will follow a hyperbolic path.

Elliptical Orbits Table

0 < e < 1 a > 0 c > 0

Circular Orbits Table

e = 0 a = r c = 0

r = constant r = rp = a

---

Parabolic Orbits Table

e = 1 a = c =

---

Hyperbolic Orbits Table

e > 1 a < c

---