Regularization in orbital mechanics. Theory and practice (Q2408683)

The regularization as a procedure to improve accuracy of analytic and mostly numerical calculations in celestial mechanics were known long time ago. Singularities of differential equations of planetary motions due to commensurabilities of planetary mean motions (known as great inequalities and small devisors problem) were particularly solved by proper chose of variables (anomaly Delaunay, Delaunay, (1860) Hansen (1853)). Singularities due to small eccentricities were eliminated or at least suppressed by use known Lagrange variables, etc. The terminology `regularization' was appeard later after papers Levi-Chivita (1903, 1904) and the regularization itself was born from necessity of avoiding the singularities of a close encounter in the three body problem. Later during design and of operation of a space mission, it was recognized that the proper dynamical regime was far from the singularity and regularization might not seem useful. But it was soon realized that transformations that worked so well in the three body problem exhibited important advantages in the two-body problem, too. The main application of regularization for the last 60 years has been the development of improved schemes for numerical integration. The book consists of an introduction. (Current challenges in space exploration) and three parts. Part. I. (Regularization). Ch. 2. Theoretical aspecs of regularization; Ch. 3. The Kustaanheimo-Stiefel space and Hopf fibration; Ch. 4. Dromo formulation; Ch. 5. Dedicated formulation: Propagating hyperbolic orbits; Ch. 6. Evaluating the numerical performance. Part. II. ( Applications.) Ch. 7. The theory of asynchronous relative motion; Ch. 8. Universal and regular solutions for relative motion; Ch. 9. Generalized logarithmic spirals: A new analytic solution with continuous trust; Ch. 10. Lambert's problem with generalized logarithmic Spirals; Ch. 11. Low-trust trajectory design with controlled generalized logarithmic spirals; Ch. 12. Nonconservative extension of Keplerian integrals and new families of orbits. Part III. (Appendices.) A. Hypercomplex numbers; B. Formulations in PERFORM; C. Stumpff functions; D. Inverse transformations; E. Elliptic integrals and elliptic functions; F. Controlled generalized logarithmic spirals; G. Dynamics in Seiffert's spherical spirals. References 320 items. Doubtlessly the monograph will be useful for graduate and postgraduate students and researches working in the area of celestial mechanics and astrodynamics.