Universal Variables

One formulation of Kepler’s equation that works for every conic: universal anomaly and Stumpff functions (Curtis section 3.7).

Inverse Semimajor Axis

• From the vis-viva relation , use its reciprocal instead:
• ellipse, parabola, hyperbola: no division by zero for the parabola

Universal Anomaly

• The universal anomaly reduces to a scaled version of each conic’s own anomaly:
  • : eccentric anomaly, : parabolic anomaly , : hyperbolic anomaly (see Kepler’s Equation)
• Define

Stumpff Functions

• Series definitions:
• and are related to the circular and hyperbolic trig functions as follows ():
• : ellipse (circular functions), : hyperbola (hyperbolic functions), : parabola

Universal Kepler’s Equation

Universal Kepler's Equation

• Curtis (3.49)
• , : radius and radial velocity at the initial time; : time since then
• Solve for given (no special cases for the conic type)

• Used again in the universal form of the Lagrange coefficients and in Battin’s state transition matrix