11R6. Essays on the Motion of Celestial Bodies. - VV Beletsky (Keldysh Inst of Appl Math, Russian Acad of Sci, Miusskaja Sq 4, Moscow, 125047, Russia). Birkhauser Verlag AG, Basel, Switzerland. 2001. 372 pp. ISBN 3-7643-5866-1. $169.00.

Reviewed by FH Lutze (Dept of Aerospace and Ocean Eng, VPI, Blacksburg VA 24061-0203).

These “Essays” are the second edition of a publication that first appeared in Russia in 1972.

The first edition was reprinted in four different languages, the last one appearing in French in 1986. Since that time, a considerable amount of material has been added to the work, leading to a second edition that appeared in 1999. The English version was published in 2001 and is the subject of this review. Readers who have worked in the area of celestial mechanics and artificial satellite motion will be quite familiar with the name VV Beletsky and the quality of the work that he has produced. His classic work on the “Motion of an Artificial Satellite About its Center of Mass” (NASA-TT-F-429, TT-67-51366, 1967) and with EM Levine the work “Dynamics of Space Tether Systems” (Advances in the Astronautical Sciences, Vol 83, 1993) are indicative of the thoroughness and delightful writing style that is characteristic of his work. The essay in this volume are written in a more informal manner and convey the joy and passion that the author has for his work.

As stated by the author in the preface, the one major feature in common for the problems discussed is that they are all interesting. The problems discussed range from fun problems to important ones, and from old well-known problems to new problems of current interest. In each case, care and preciseness associated with this author prevails. However, do not be misled into thinking that these essays are just derivations of old and new results. On the contrary, these essays present results with little derivation, relying on references to provide the details. It allows the reader, under the tutelage of a seasoned dynamicist, to step back and look at the problem from various viewpoints, without getting lost in the details. One of the objectives of these essays as stated in the preface is to “help its readers become aware, even to a small extent, of how astonishing and rich in events and phenomena the mechanics of space flight is.” It is this reviewer’s opinion that this goal is met.

It will be impossible to give each essay its justice. The titles of each essay sound quite routine and do not reflect the complete story of what is presented. Here we will look at a few of the essays to try and give the flavor of this book. Each essay starts with a quote. The quote for the first essay is “Dear Fagot, show us something simple for the start.” M Bulgakov, The Master and Margarita. The title of the first essay is “On the Unperturbed and Perturbed Motion of a Satellite, with a Digression on Asymptotic Methods of Nonlinear Mechanics.” It essentially covers the material in a first-level graduate astrodynamics course. One of the first points that is made is that we are “lucky” that we only have a single star system rather than a multiple star system where integrating the equations of motion might not be so easy! The second essay, “On the Rebirth of an Old Problem, or what Happens if two Masses are Placed at a Purely Imaginary Distance from one Another,” starts with the quote “… and the more he looked at the bell-rope, the more he felt that he had seen something like it, somewhere else, sometime before.” AA Milne, The World of Pooh. The essay deals with the problem of the motion of a particle about two fixed Newtonian centers. He writes the equations of motion, expands them, and compares the resulting terms with those of a nonspherical earth that were presented in essay one. He then proceeds to introduce several ways to attack the problem and adds much insight from experience.

The third essay is titled “Yet Another Reincarnation of an Old Problem” and discusses the solution to the problem of motion in an inverse square gravitational field with an additional constant force in a constant direction. Some of the results presented are very unexpected. The fourth essay, “The Motion of the Worlds,” discusses resonances in the solar system and its stability. In this essay a conjecture by AM Molchanov is presented that says, “Oscillating systems that have reached evolutionary maturity are unavoidably resonant, and their structure is given by a set of integers.” There is also an interesting section that indicates that if the moon had a high inclination, say 90°, it would have fallen into the Earth. The fifth essay deals with the three-body problem, patched conics and galaxy models. The sixth essay is the longest and is titled “They are Waltzing in Orbits.” This essay is the longest and reflects a lot of the work done by Beletsky on the motion satellites about their mass centers. Included are Moon-Earth, Mercury-Sun motions in 1:1 and 3:2 rotational resonances, respectively. In addition there is further discussion on stability and magnetic torques acting on satellites. The seventh essay, “A Spiral to Space,” deals with low thrust problems and spiral escape from the earth. The eighth essay is “The Full Force of the Sun Blows in the Sails,” and looks at the solar sailing problem, comparing some analytical results with AC Clark’s science fiction story, “The Wind From the Sun.” The ninth essay, “The Gravity Flyer,” is one of the more “off the wall” essays. Because a finite-size spaceship’s center of gravity is not located in the same orbit as it would be if all the mass were concentrated at one point, energy can be pumped into the orbit by pulsing the spaceship from a point to its full deployment and back again with the appropriate period. By so doing, you can eventually escape the attracting body. Of interest here is a full reporting of comments on this idea by various scientists, some of which indicated Beletsky’s ideas were “… based simply on ignorance… .” The tenth essay, “Interplanetary Flights: Low Thrusts for High Goals,” considered low-thrust interplanetary orbits for which optimization of the thrust direction takes an important role. The eleventh essay, “Relative Motion of Orbiting Bodies,” deals with a problem that is of current interest with regard to formation flying. In addition he looks at a version of the problem of two satellites attached with a tether. Also included is an analysis of a particle cloud. The final essay is “Cosmic Pinwheel.” It discusses and explains the motion of the Proton satellites that had solar panels fixed like a pinwheel about the axis of the satellite. This caused the satellite to spin up or spin down, causing unusual attitude behavior that was observed first, and then explained by applying basic physics.

Essay on the Motion of Celestial Bodies is extremely well written, although there are a few problems in the translation, they do not interfere with the spirit of the author. The technical figures are excellent and in addition there are several cartoon like figures in each essay reflecting the basic idea of the essay. It would help a lot if the reader has a solid background in space mechanics, although the author claims it is not necessary. The beauty of the book and the treatment of the subject would be missed by a general reader. This is not a textbook; nor is it a reference book, it is just a fun book to read that will give the reader new ways to think about old problems and some new ideas for attacking more recent problems of interest. Each essay has its own set of references and the book has a complete list of referenced authors with the page(s) on which they are referenced. Many of the references are from the Russian literature, so it provides a good review of work done in that country in the 1960s and 1970s. I would strongly recommend this book, as an addition to one’s library, although I think the price will discourage most people ($170) from purchasing it.