linguistic:translation, quality:correction
Elliptical orbits

If 0 < e < 1 {\displaystyle 0<e<1}, then the denominator of the equation of free orbits varies with the true anomaly θ {\displaystyle \theta }, but remains positive, never becoming zero. Therefore, the relative position vector remains bounded, having its smallest magnitude at periapsis r p {\displaystyle r_{p}}, which is given by:

The maximum value r {\displaystyle r} is reached when θ = 180 ∘ {\displaystyle \theta =180^{\circ }}. This point is called the apoapsis, and its radial coordinate, denoted r a {\displaystyle r_{a}}, is

Let 2 a {\displaystyle 2a} be the distance measured along the apse line from periapsis P {\displaystyle P} to apoapsis A {\displaystyle A}, as illustrated in the equation below:

Substituting the equations above, we get:

a is the semimajor axis of the ellipse. Solving for p {\displaystyle p}, and substituting the result in the conic section curve formula above, we get: