Initial value problem of space dynamics in universal Stumpff anomaly

In this paper, the initial value problem of space dynamics in universal Stumpff anomaly \(\psi\) is set up and developed in analytical and computational approach.For the analytical expansions, the linear independence of the functions \(\mathrm{U}_{{ j}} { (\psi;\varsigma)}\); \({j=0,1,2,3}\) are proved. The differential and recurrence equations satisfied by them and their relations with the elementary functions are given. The universal Kepler equation and its validations for different conic orbits are established together with the Lagrangian coefficients. Efficient representations of these functions are developed in terms of the continued fractions.

For the computational developments we consider the following items:

1. 1.
   Top-down algorithm for continued fraction evaluation.
    
2. 2.
   One-point iteration formulae.
    
3. 3.
   Determination of the coefficients of Kepler’s equation.
    
4. 4.
   Derivatives of Kepler’s equation of any integer order.
    
5. 5.
   Determination of the initial guess for the solution of the universal Kepler equation.
    

Finally we give summary on the computational design for the initial value problem of space dynamics in universal Stumpff anomaly. This design based on the solution of the universal Kepler’s equation by an iterative schemes of quadratic up to any desired order \(\ell\).