In orbital mechanics, the universal variable formulation is a method used to solve the two-body Kepler problem. It is a generalized form of Kepler's Equation, extending it to apply not only to elliptic orbits, but also parabolic and hyperbolic orbits common for spacecraft departing from a planetary orbit. It is also applicable to ejection of small bodies in Solar System from the vicinity of massive planets, during which processes the approximating two-body orbits can have widely varying eccentricities, almost always e ≥ 1 .
Introduction[edit]
A common problem in orbital mechanics is the following: Given a body in an orbit and a fixed original time find the position of the body at some later time For elliptical orbits with a reasonably small eccentricity, solving Kepler's Equation by methods like Newton's method gives excellent results. However, as the orbit approaches an escape trajectory, it becomes more and more eccentric, convergence of numerical iteration may become unusably sluggish, or fail to converge at all for e ≥ 1 .[1][2] Furthermore, Kepler's equation cannot be directly applied to parabolic and hyperbolic orbits, since it specifically is tailored to elliptic orbits.
Derivation[edit]
Although equations similar to Kepler's equation can be derived for parabolic and hyperbolic orbits, it is more convenient to introduce a new independent variable to take the place of the eccentric anomaly and having a single equation that can be solved regardless of the eccentricity of the orbit. The new variable is defined by the following differential equation:
- where is the system gravitational scaling constant,
is regularized by applying this change of variables that yields:[2]
where is some t.b.d. constant vector and is the orbital energy, defined by
where and are the position and velocity respectively at time and and are the position and velocity, respectively, at arbitrary initial time