Universal variable formulation explained

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

Introduction

A common problem in orbital mechanics is the following: Given a body in an orbit and a fixed original time

t _{o ,} find the position of the body at some later time t ~. 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 ^{[1] [2]} Furthermore, Kepler's equation cannot be directly applied to parabolic and hyperbolic orbits, since it specifically is tailored to elliptic orbits.

Derivation

E , and having a single equation that can be solved regardless of the eccentricity of the orbit. The new variable s is defined by the following differential equation:$$\backslash frac\; =\; \backslash frac$$where r \equiv r(t) is the time-dependent scalar distance to the center of attraction. (In all of the following formulas, carefully note the distinction between scalars r , in italics, and vectors r , in upright bold.)The fundamental equation(\operatorname d ^{2} r)/(\operatorname d t ^{2}) + \mu (r)/(~ r ^{3}) = 0 , where \mu \equiv G \left( m _{1} + m _{2} \right) is the system gravitational scaling constant,is regularized by applying this change of variables that yields:^[2]

$$\backslash frac\; +\; \backslash alpha\backslash \; \backslash mathbf\; =\; -\; \backslash mathbf\backslash$$where

P is some t.b.d. constant vector and \alpha is the orbital energy, defined by$$\backslash alpha\; \backslash equiv\; \backslash frac\; ~.$$The equation is the same as the equation for the harmonic oscillator, a well-known equation in both physics and mathematics. Taking the derivative again, we eliminate the constant vector P , at the price of getting a third-degree differential equation:$$\backslash \; \backslash frac\; +\; \backslash alpha\backslash frac\; =\; \backslash mathbf\backslash$$The family of solutions to this differential equation^[2] are for convenience written symbolically in terms of the three functions s c _{1\left( \alpha} s ^{2 \right) ,} s ^{2} c _{2\left( \alpha} s ^{2 \right) ,} and s ^{3} c _{3\left( \alpha} s ^{2 \right) ;} where the functions c _{k(x) ,} called Stumpff functions, which are truncated generalizations of sine and cosine series. Applying this results in:^[2] $$\backslash \; t\; -\; t\_\backslash mathsf\; =\; r\_\backslash mathsf\backslash \; s\backslash \; c\_1\backslash !\backslash !\backslash left(\backslash \; \backslash alpha\; s^2\backslash \; \backslash right)\; +\; r\_\backslash mathsf\; \backslash frac\backslash \; s^2\; c\_2\backslash !\backslash !\backslash left(\backslash \; \backslash alpha\; s^2\backslash \; \backslash right)\; +\; \backslash mu\; \backslash \; s^3\; c\_3\backslash !\backslash !\backslash left(\backslash \; \backslash alpha\; s^2\backslash \; \backslash right)\backslash$$which is the universal variable formulation of Kepler's equation. This equation can now be solved numerically using a root-finding algorithm such as Newton's method or Laguerre's method for a given time t to yield s , which in turn is used to compute the f and g functions:The values of the f and g functions determine the position of the body at the time t :$$\backslash \; \backslash mathbf\; =\; \backslash mathbf\_\backslash mathsf\backslash \; f(s)\; +\; \backslash mathbf\_\backslash mathsf\backslash \; g(s)\backslash$$In addition the velocity of the body at time t can be found using • f (s) and • g (s) as follows:$$\backslash \; \backslash mathbf\; =\; \backslash mathbf\_\backslash mathsf\backslash \; \backslash dot(s)\; +\; \backslash mathbf\_\backslash mathsf\backslash \; \backslash dot(s)\backslash$$

where

r and v are the position and velocity respectively at time t , and r _{o} and v _{o} are the position and velocity, respectively, at arbitrary initial time t _{o} ~.

Notes and References

1. Book: Eduard L. . Stiefel . Gerhard . Scheifele . 1971 . Linear and Regular Celestial Mechanics: Perturbed two-body motion numerical methods canonical theory . Springer-Verlag .
2. Book: Danby, J.M.A. . 1988 . Fundamentals of Celestial Mechanics . 2nd . Willmann-Bell . 0943396204 .