- The paper derives high order analytic state transition tensors of the equations governing the motion of a particle in the presence of a central attracting body with an inverse square law of attraction. The perturbation tensors are the high order sensitivity generalizations of the more familiar fundamental perturbation (state transition) matrix governing the two body problem. The formal state transition tensors are presented up to third order. Preliminary derivation steps are shown as to enable the calculation of the state transition tensors using the formal expressions when F and G functions (Lagrange Coefficients) are expressed as functions of trigonometric and universal functions. Applications of the analytic tensors derived in the paper to trajectory optimization and estimation are outlined. A second order Halley like correction to the Newton's method is presented (for general multivariable vector function solutions) and the state transition tensors derived are used in the method to improve the convergence properties of an iteration scheme to solve the classical Lambert two point boundary value problem.