This volume is designed as an introductory text and reference book for graduate students, researchers and practitioners in the fields of astronomy, astrodynamics, satellite systems, space sciences and astrophysics. The purpose of the book is to emphasize the similarities between celestial mechanics and astrodynamics, and to present recent advances in these two fields so that the reader can understand the inter-relations and mutual influences. The juxtaposition of celestial mechanics and astrodynamics is a unique approach that is expected to be a refreshing attempt to discuss both the mechanics of space flight and the dynamics of celestial objects. "Celestial Mechanics and Astrodynamics: Theory and Practice" also presents the main challenges and future prospects for the two fields in an elaborate, comprehensive and rigorous manner. The book presents homogenous and fluent discussions of the key problems, rendering a portrayal of recent advances in the field together with some basic concepts and essential infrastructure in orbital mechanics. The text contains introductory material followed by a gradual development of ideas interweaved to yield a coherent presentation of advanced topics.

This edited volume charts the history of celestial navigation over the course of five centuries. Written by a group of historians and scientists, it analyzes how competing navigation systems, technologies, and institutions emerged and developed, with a focus on the major players in the US and the UK. The history covers the founding of the Royal Observatory; the first printing of a Nautical Almanac; the founding of the US and UK Nautical Almanac Offices; the creation of international standards for reference systems and astronomical constants; and the impact of 20th century technology on the field, among other topics. Additionally, the volume analyzes the present role and status of celestial navigation, particularly with respect to modern radio and satellite navigation systems. With its diverse authorship and nontechnical language, this book will appeal to any reader interested in the history of science, technology, astronomy, and navigation over the ages.

"Fundamental Astronomy and Solar System Dynamics," a program of invited papers honoring Professor Walter Fricke, who for thirty years has been Director of the Astronomisches Rechen lnstitut in Heidelberg, was held at the Thompson Conference Center of the University of Texas at Austin on Wednesday 27 March 1985 on the occasion of his seventieth birthday and retirement as Director of ARl. Professor Fricke's contributions to astronomy encompass the areas of galactic dynamics, radial velocities, stellar statistics. the fundamental reference system and the constant of precession. Participants were welcomed to the Uni versi ty of Texas by Professor J. Parker Lamb, Chairman of the Department of Aerospace Engineering and Engineering Mechanics. The presentations ranged from discussions of astrometric problems concerned with the reference system, the constant of precession, major and minor planet observations, planetary ephemerides and lunar and satellite laser ranging, to a study of disc galaxies in massive halos. The program concluded with a review of Professor Fricke's career. The three sessions were chaired by Victor G. Szebehely, Carol A. Williams and Jay H. Lieske. The participants in this meeting, and in the Division on Dynamical Astronomy meeting that followed, were happy that Professor Fricke was able to attend. His presence at these meetings, as well as his thoughtful comments, were greatly appreciated. We are pleased to acknowledge the support of the Center for Space Research, the Department of Astronomy and the Department of Aerospace Engineering and Engineering Mechanics of the University of Texas at Austin.

Astrometry is on the threshold of great changes due to the fact that this decade, alone, is witnessing an improvement of stellar positions equivalent to the total improvement of the previous two centuries. The Hipparcos Satellite has concluded its observations, and the catalog is in preparation. Preliminary results assure that the Hipparcos catalog will provide positions, parallaxes and annual proper motions for over 100,000 stars with accuracies of 1.5 milliarcseconds. In addition, the Tycho catalog will provide positions of about 30 milliarcseconds accuracy for over 1 million stars, and annual proper motions with 3 milliarcsecond accuracy will subsequently be ob tained by means of first epoch positions from the Astrographic Catalog. Optical interferometers on the ground are beginning operation, and these instruments can provide observational accuracies of approximately one milliarcsecond. Also, the traditional reference frame based on the Fun damental Catalog of bright stars is being replaced by the extragalactic ref erence frame, based on radio sources with accuracies of one milliarcsecond. Thus, astrometry will change from a fundamental reference frame defined in terms of the dynamical reference frame of the solar system with accuracies of 100 milliarcseconds to a space-fixed, extragalactic reference frame with accuracies of one milliarcsecond. Future astrometric observations should be in the 1 -100 milliarcsecond accuracy range. There are a number of concepts for future astrometric instruments in space. Most of these can provide sub-milliarcsecond astrometric accuracies."

The uses of time in astronomy - from pointing telescopes, coordinating and processing observations, predicting ephemerides, cultures, religious practices, history, businesses, determining Earth orientation, analyzing time-series data and in many other ways - represent a broad sample of how time is used throughout human society and in space. Time and its reciprocal, frequency, is the most accurately measurable quantity and often an important path to the frontiers of science. But the future of timekeeping is changing with the development of optical frequency standards and the resulting challenges of distributing time at ever higher precision, with the possibility of timescales based on pulsars, and with the inclusion of higher-order relativistic effects. The definition of the second will likely be changed before the end of this decade, and its realization will increase in accuracy; the definition of the day is no longer obvious. The variability of the Earth's rotation presents challenges of understanding and prediction. In this symposium speakers took a closer look at time in astronomy, other sciences, cultures, and business as a defining element of modern civilization. The symposium aimed to set the stage for future timekeeping standards, infrastructure, and engineering best practices for astronomers and the broader society. At the same time the program was cognizant of the rich history from Harrison's chronometer to today's atomic clocks and pulsar observations. The theoreticians and engineers of time were brought together with the educators and historians of science, enriching the understanding of time among both experts and the public.

This volume contains papers presented at the US/European Celestial Mecha nics Workshop organized by the Astronomical Observatory of Adam Mickiewicz University in Poznan, Poland and held in Poznan, from 3 to 7 July 2000. The purpose of the workshop was to identify future research in celestial mech anics and encourage collaboration among scientists from eastem and westem coun tries. There was a full program of invited and contributed presentations on selected subjects and each day ended with a discussion period on a general subject in celestial mechanics. The discussion topics and the leaders were: Resonances and Chaos-A. Morbidelli; Artificial Satellite Orbits-K. T. Alfriend; Near Earth Ob jects - K. Muinonen; Small Solar System Bodies - I. Williams; and Summary - P. K. Seidelmann. The goal of the discussions was to identify what we did not know and how we might further our knowledge. The size of the meeting and the language differences somewhat limited the real discussion, but, due to the excellence of the different discussion leaders, each of these sessions was very interesting and productive. Celestial Mechanics and Astrometry are both small fields within the general subject of Astronomy. There is also an overlap and relationship between these fields and Astrodynamics. The amount of interaction depends on the interest and efforts of individual scientists."

Astrometry encompasses all that is necessary to measure the positions and motions of celestial bodies: observational techniques, instrumentation, processing and analysis of observational data, and reference systems and frames, as well as the resulting astronomical phenomena. It is fundamental to all other fields of astronomy, from the focusing of telescopes to navigation and guidance systems and distance and motion determinations for astrophysics. Starting from basic principles, this work provides the principles of astrometry at milli- and micro-arcsecond accuracies and will be an invaluable reference for graduate students and research astronomers.

This edited volume charts the history of celestial navigation over the course of five centuries. Written by a group of historians and scientists, it analyzes how competing navigation systems, technologies, and institutions emerged and developed, with a focus on the major players in the US and the UK. The history covers the founding of the Royal Observatory; the first printing of a Nautical Almanac; the founding of the US and UK Nautical Almanac Offices; the creation of international standards for reference systems and astronomical constants; and the impact of 20th century technology on the field, among other topics. Additionally, the volume analyzes the present role and status of celestial navigation, particularly with respect to modern radio and satellite navigation systems. With its diverse authorship and nontechnical language, this book will appeal to any reader interested in the history of science, technology, astronomy, and navigation over the ages.

This volume is designed as an introductory text and reference book for graduate students, researchers and practitioners in the fields of astronomy, astrodynamics, satellite systems, space sciences and astrophysics. The purpose of the book is to emphasize the similarities between celestial mechanics and astrodynamics, and to present recent advances in these two fields so that the reader can understand the inter-relations and mutual influences. The juxtaposition of celestial mechanics and astrodynamics is a unique approach that is expected to be a refreshing attempt to discuss both the mechanics of space flight and the dynamics of celestial objects. "Celestial Mechanics and Astrodynamics: Theory and Practice" also presents the main challenges and future prospects for the two fields in an elaborate, comprehensive and rigorous manner. The book presents homogenous and fluent discussions of the key problems, rendering a portrayal of recent advances in the field together with some basic concepts and essential infrastructure in orbital mechanics. The text contains introductory material followed by a gradual development of ideas interweaved to yield a coherent presentation of advanced topics.

"Fundamental Astronomy and Solar System Dynamics", a program of invited papers honoring Professor Walter Fricke, who for thirty years has been Director of the Astronomisches Rechen lnstitut in Heidelberg, was held at the Thompson Conference Center of the University of Texas at Austin on Wednesday 27 March 1985 on the occasion of his seventieth birthday and retirement as Director of ARl. Professor Fricke's contributions to astronomy encompass the areas of galactic dynamics, radial velocities, stellar statistics. the fundamental reference system and the constant of precession. Participants were welcomed to the Uni versi ty of Texas by Professor J. Parker Lamb, Chairman of the Department of Aerospace Engineering and Engineering Mechanics. The presentations ranged from discussions of astrometric problems concerned with the reference system, the constant of precession, major and minor planet observations, planetary ephemerides and lunar and satellite laser ranging, to a study of disc galaxies in massive halos. The program concluded with a review of Professor Fricke's career. The three sessions were chaired by Victor G. Szebehely, Carol A. Williams and Jay H. Lieske. The participants in this meeting, and in the Division on Dynamical Astronomy meeting that followed, were happy that Professor Fricke was able to attend. His presence at these meetings, as well as his thoughtful comments, were greatly appreciated. We are pleased to acknowledge the support of the Center for Space Research, the Department of Astronomy and the Department of Aerospace Engineering and Engineering Mechanics of the University of Texas at Austin.

decision and investigative process of the scientist, they can also wri te the paragra}ils of scientific resul ts. Once a canputer research activity progresses from intuitive guesses to a systematic study, the canputer program can free the scientist to pursue new original ideas. '!his concept is illustrated in the ptper by Hut. Sussnan says "Knowledge" is knowing what not to canpute. His main message is that we must change from trying to maximize the efficiency of the canputer to maximizing the effectiveness of the scientist. The canputer is relatively cheap, it is the trained scientist who is most valuable. In the second keynote address, R. L. Duncombe (University of Texas) described the "Early Applications of Canputer Technology to Dynamical Astronany. " His talk traced the worldwide introduction of punched card equipnent and their applications through the use of the first electronic canputers. Sussman and Duncombe, therefore, provided a forecast and review of the use of computers in dynamical astronany. This proceedings is intended to provide a consolidated general view of the ptst, present, and future of the Applications of Canputer Technology to Dynamical Astronany. Of necessity this requires that the ptpers be rela ti vely short and provide references to the more canplete discussions. Thanks to the representa tion from so many different countries the proceedings give a picture of the activities worldwide. The Scientific Organizing Comnittee included K. B. IIlatnagar, V. A. Brtmberg, G. Contopoulos, A. Depri t , J. Henrard, J. Kovalevsky, Y.

I. In this second volume, we continue at first the study of non homogeneous boundary value problems for particular classes of evolu tion equations. 1 In Chapter 4 , we study parabolic operators by the method of Agranovitch-Vishik [lJ; this is step (i) (Introduction to Volume I, Section 4), i.e. the study of regularity. The next steps: (ii) transposition, (iii) interpolation, are similar in principle to those of Chapter 2, but involve rather considerable additional technical difficulties. In Chapter 5, we study hyperbolic operators or operators well defined in thesense of Petrowski or Schroedinger. Our regularity results (step (i)) seem to be new. Steps (ii) and (iii) are all3.logous to those of the parabolic case, except for certain technical differences. In Chapter 6, the results of Chapter'> 4 and 5 are applied to the study of optimal control problems for systems governed by evolution equations, when the control appears in the boundary conditions (so that non-homogeneous boundary value problems are the basic tool of this theory). Another type of application, to the characterization of "all" well-posed problems for the operators in question, is given in the Ap pendix. Still other applications, for example to numerical analysis, will be given in Volume 3.

1. We describe, at first in a very formaI manner, our essential aim. n Let m be an op en subset of R , with boundary am. In m and on am we introduce, respectively, linear differential operators P and Qj' 0 ~ i ~ 'V. By "non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be given in function space s F and G , F being a space" on m" and the G/ s spaces" on am" ; j we seek u in a function space u/t "on m" satisfying (1) Pu = f in m, (2) Qju = gj on am, 0 ~ i ~ 'v"])). Qj may be identically zero on part of am, so that the number of boundary conditions may depend on the part of am considered 2. We take as "working hypothesis" that, for fEF and gjEG , j the problem (1), (2) admits a unique solution u E U/t, which depends 3 continuously on the data . But for alllinear probIems, there is a large number of choiees for the space s u/t and {F; G} (naturally linke d together). j Generally speaking, our aim is to determine families of spaces 'ft and {F; G}, associated in a "natural" way with problem (1), (2) and con j venient for applications, and also all possible choiees for u/t and {F; G} j in these families.

1. Our essential objective is the study of the linear, non-homogeneous problems: (1) Pu = I in CD, an open set in RN, (2) fQjtl = gj on am (boundary of m), lor on a subset of the boundm"J am 1 v, where Pis a linear differential operator in m and where the Q/s are linear differential operators on am. In Volumes 1 and 2, we studied, for particular c1asses of systems {P, Qj}, problem (1), (2) in c1asses of Sobolev spaces (in general constructed starting from P) of positive integer or (by interpolation) non-integer order; then, by transposition, in c1asses of Sobolev spaces of negative order, until, by passage to the limit on the order, we reached the spaces of distributions of finite order. In this volume, we study the analogous problems in spaces of inlinitely dilferentiable or analytic Itlnctions or of Gevrey-type I~mctions and by duality, in spaces 01 distribtltions, of analytic Itlnctionals or of Gevrey- type ultra-distributions. In this manner, we obtain a c1ear vision (at least we hope so) of the various possible formulations of the boundary value problems (1), (2) for the systems {P, Qj} considered here.

In the twenty-first century, we take the means to measure time for granted, without contemplating the sophisticated concepts on which our time scales are based. This volume presents the evolution of concepts of time and methods of time keeping up to the present day. It outlines the progression of time based on sundials, water clocks, and the Earth's rotation, to time measurement using pendulum clocks, quartz crystal clocks, and atomic frequency standards. Time scales created as a result of these improvements in technology and the development of general and special relativity are explained. This second edition has been updated throughout to describe twentieth- and twenty-first-century advances and discusses the redefinition of SI units and the future of UTC. A new chapter on time and cosmology has been added. This broad-ranging reference benefits a diverse readership, including historians, scientists, engineers, educators, and it is accessible to general readers.

Astrometry encompasses all that is necessary to provide the positions and motions of celestial bodies. This includes observational techniques, instrumentation, processing and analysis of observational data, reference systems and frames, and the resulting astronomical phenomena. Astrometry is fundamental to all other fields of astronomy, from the pointing of telescopes, to navigation and guidance systems, to distance and motion determinations for astrophysics. In the last few decades, new observational techniques have enabled improvements in accuracy by orders of magnitude. Starting from basic principles, this book provides the fundamentals for this new astrometry at milli- and micro-arcsecond accuracies. Topics include: basics of general relativity; co-ordinate systems; vectors, tensors, quaternions, and observational uncertainties; determination and use of the celestial and terrestrial reference systems and frames; applications of new observational techniques; present and future star catalogues and double star astrometry. This comprehensive reference will be invaluable for graduate students and research astronomers.

This volume contains papers presented at the US/European Celestial Mecha nics Workshop organized by the Astronomical Observatory of Adam Mickiewicz University in Poznan, Poland and held in Poznan, from 3 to 7 July 2000. The purpose of the workshop was to identify future research in celestial mech anics and encourage collaboration among scientists from eastem and westem coun tries. There was a full program of invited and contributed presentations on selected subjects and each day ended with a discussion period on a general subject in celestial mechanics. The discussion topics and the leaders were: Resonances and Chaos-A. Morbidelli; Artificial Satellite Orbits-K. T. Alfriend; Near Earth Ob jects - K. Muinonen; Small Solar System Bodies - I. Williams; and Summary - P. K. Seidelmann. The goal of the discussions was to identify what we did not know and how we might further our knowledge. The size of the meeting and the language differences somewhat limited the real discussion, but, due to the excellence of the different discussion leaders, each of these sessions was very interesting and productive. Celestial Mechanics and Astrometry are both small fields within the general subject of Astronomy. There is also an overlap and relationship between these fields and Astrodynamics. The amount of interaction depends on the interest and efforts of individual scientists."

The Explanatory Supplement to the Astronomical Almanac offers explanatory material, supplemental information, and detailed descriptions of the computational models and algorithms used to produce The Astronomical Almanac, which is an annual publication prepared jointly by the US Naval Observatory and Her Majesty's Nautical Almanac Office in the UK. Like The Astronomical Almanac, The Explanatory Supplement provides detailed coverage of modern positional astronomy. Chapters are devoted to the celestial and terrestrial reference frames, orbital ephemerides, precession, nutation, Earth rotation, and coordinate transformations. These topics have undergone substantial revisions since the last edition was published in 1992. Astronomical positions are intertwined with timescales and relativity in The Astronomical Almanac, so related chapters are provided in The Explanatory Supplement. The Astronomical Almanac also includes information on lunar and solar eclipses, physical ephemerides of solar system bodies, and calendars, so The Explanatory Supplement expounds upon each of these topics as well. The book is written at a technical, but non-expert level. As such, it provides an important reference for a full range of users including astronomers, engineers, navigators, surveyors, space scientists, and educators.