Conic Sections - General Equations
These general formulae govern all types of orbital motion in the gravitational two body problem,
including both bound and unbound conics. More specialized formulae, valid only for certain
types of orbits, can be derived from these.
Specific Specific Distance Speed Pericenter
Energy Angular Momentum Distance
C=−GM
2ah=pGMa(1 −e2)r=a(1−e2)
1+ecos fv=sGMµ2
r−1
a¶q=a(1 −e)
Conic Sections - Equations for Specific Orbits
Energy determines whether an orbit is bound or not. Circles and ellipses are the only bound
orbits; parabolas and hyperbolas are the only unbound ones. Note that e= 1 orbits (rectilinear
or straight-line orbits) may be elliptical, parabolic, or hyperbolic.
Bound Orbits (C < 0) Unbound Orbits (C≥0)
Circle Ellipse Parabola Hyperbola
Semimajor Axis: a=r a > 0a→ ±∞ a < 0
Eccentricity: e= 0 0 < e ≤1e= 1 e≥1
Distance r=a r ≥a(1 −e)r≥a(1 −e)r≥a(1 −e)
r≤a(1 + e)r→ ∞ r→ ∞
Speed: v=qGM
av≤rGM(1+e)
a(1−e)v=q2GM
rv≤rGM(1+e)
a(1−e)
v≥rGM(1−e)
a(1+e)v(r→ ∞) = 0 v(r→ ∞)→q−GM
a
