Partial Derivatives of Numerical Ephemerides with Respect to Parameters of the Dynamical System

For a numerical solution obtained by numerically integrating the differential equations of motion of a celestial body, its partial derivatives with respect to parameters such as initial position and velocity, gravitational constant and the mass of the body can not be acquired as an analytical solution pattern. A set of differential equations for this kind of partial derivatives is deduced in this paper. The initial conditions for the set of differential equations are given together. The use of these partial derivatives to fit lunar laser ranging data is also discussed.