Joseph O’Leary - General Relativistic and Post-Newtonian Dynamics for Near-Earth Objects and Solar System Bodies

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The next generation of near-Earth satellite systems will require a detailed mapping of the Earths gravitational field combined with the development of accurate predictive mathematical modelling. Accordingly, future space scientists will be required to have an understanding and knowledge of practical general relativity, as suggested by the International Astronomical Union, that all astronomical problems should be formulated within the framework of general relativity. Hence, all future mission planning tools and orbit determination software suites must address this issue. Astronomical problems posed within the framework of general relativity present a formidable task both in terms of shear complexity and depth of the gravitational field equations and do not submit to a comparable analytical treatment as their classical Newtonian counterparts. This text is designed for those future space scientists requiring a working knowledge of the practical aspects of general relativity.

Einsteins general theory of relativity was published over a century ago and provides an accurate description of gravitation giving rise to a myriad of interesting and peculiar phenomena in our universe, such as the bending of light through gravitational lensing, the slowing of clocks in gravitational fields and the recently detected ripples in spacetime due to cataclysmic astrophysical events such as the coalescence of dense stellar objects.

The practical importance of general relativity is evident for the description of exotic compact astronomical objects such as neutron stars and black holes, for which classical Newtonian gravity does not capture the qualitative behaviour of such strong-field phenomena. While fundamental in fields such as cosmology and gravitational wave astronomy; general relativity might be perceived by some as somewhat esoteric and academic when describing the interaction of bodies in the weak-field arena of the solar system. However, this could not be further from the truth which is the central theme of this book.

Given the increased accuracy requirements in fields such as astrometry and geodesy, and owing to the rapid improvements in time and frequency technology, Einsteins general theory of relativity can take such matters into account for those space missions requiring highly accurate orbit information and practically all observation techniques. The general relativistic problems in this book have both theoretical and practical applications and demonstrate the confluence of applied mathematics, general relativity and space engineering through examination of problems of space science.

The first part of the book begins with the exact vacuum solution to the Einstein field equations of gravitation by Karl Schwarzschild which was discovered shortly after the publication of Einsteins general relativity. Birkhoffs theorem states that all spherically symmetric solutions are static and asymptotically flat, which in consequence, means that any vacuum solution exhibiting spherical symmetry is equivalent to that of Schwarzschild. While this is known to practitioners of general relativity, the Schwarzschild solution contains coordinate singularities and requires additional transformations to achieve maximal analytic coverage of the underlying spacetime. Chapter explores Birkhoffs theorem and derives an alternative, singularity-free solution to the Einstein field equation exhibiting an additional degree of freedom, by directly solving the resulting system of non-linear differential equations.

The second part of this book is dedicated to the gravitational interaction between objects in the solar system using the so-called post-Newtonian approximation. The post-Newtonian regime has been coined unreasonably effective in describing gravitational physics and is an invaluable tool in modern astrodynamics and geodesy. Following an introduction to the post-Newtonian framework in Chap. , the author explores three post-Newtonian dynamical problems, namely: energy integrals for spacecraft in the local geocentric frame, the post-Newtonian Kepler and two-body problems and the relativistic Euler or two-centre problem.