The Dynamics of Natural Satellites of the Planets

By Nikolay Emelyanov

The Dynamics of Natural Satellites of the Planets is an accessible reference for understanding the celestial mechanics of planetary moons through the lens of both theory and observation. Based on decades of research by the author, the book utilizes state-of-the-art observations of the natural satellites in the solar system to establish models, measurements and calculations to better understand the theory of the satellite movement and dynamics. It presents an extensive set of study methods and results on the motion of natural satellites of the planets and includes reviews and references to related publication for further explanation.

By relating observations to numerical theory, the book serves as a quick and comprehensive reference for applying the theory of orbital dynamics to observational data on orbits and physical properties of the natural satellites in order to formulate state-of-the-art explanations and models, particularly for determining the parameters of satellite motion.

• Combines astronomy and celestial mechanics, providing astrometric data from observations to inform methods and models for predicting natural satellite dynamics
• Includes both theory and observation in one place and presents new models based on observations
• Organized into small sections, each providing specific measurements, calculations or models, making it a quick and comprehensive reference

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 16, 2020
• ISBN: 9780128227121

Nikolay Emelyanov

Nikolay V. Emelyanov is Professor of Physical and Mathematical Sciences for the Sternberg State Astronomical Institute at Lomonosov Moscow State University and Chief of the Celestial Mechanics department, a position he has held since 1992. Author of over 100 scientific works, including two monographs in Russian. His research interests focus on satellite motion and natural satellites of the planets. His doctoral dissertation in 1986 was on “The theory of motion of distant satellites: Building an analytical theory of motion and differential refinement of the orbits of artificial Earth satellites using computers.” The theory of satellite motion he created was the most advanced in the world. It has found applications in refining satellite orbits based on laser observations. Since 1996, he has been participating with the Paris Institute of Celestial Mechanics to create a database of natural planetary satellites in the framework of international programs.

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1: Objectives, current problems and general approach to the study of the dynamics of satellites

Abstract

In this chapter, the reader first meets with the author. This is a very important point. Therefore, here is a general view of the author on the whole topic of the book. Here, the specific features of the science of the dynamics of the natural satellites (sometimes referred to as moons in the literature) of the planets will be explained, and the objectives of this entire scientific field will be indicated, as well as what it is useful for.

For a good understanding, we always need a common language, in particular a common terminology. Therefore, in this first chapter, the basic concepts that the author uses are given. Individual scientific communities develop their own definitions, sometimes even their own slang. The author uses here the definitions and concepts that his teachers used, and which are usually used in lectures at the astronomical department of the Faculty of Physics of Moscow State University.

The chapter then presents the main methodological approach. The whole subsequent content of the book is based on this approach. The book attempts to provide a comprehensive description of the dynamics of the satellites of the planets. However, the main objective is to build motion models based on observations. In many other tasks, researchers become users of these models.

Keywords

Celestial mechanics; basic concepts; general approach; model of motion; observations

1.1 Introduction

The understanding that the vast Universe extends in all directions has always worried mankind. This causes a double desire. Firstly, it would be nice to understand our place in the boundless space and the infinite diversity of the world. People often experience a slight discomfort from the lack of an answer to such a question. At the same time, a desire arises to extract benefits from the Cosmos to satisfy ever-increasing needs. People are even more worried when they discover a threat to their lives by the forces of nature. Nothing scares us so much as an incomprehensible phenomenon. It is surprisingly easy to reassure people by explaining terrible phenomena, even with not quite familiar words. The information that at least someone understands the processes of nature returns us to the usual comfort of everyday life. That is why we should be grateful to the few people who work to save us from painful questions about space and fate.

Since ancient times, people have thought about the influence of celestial bodies on terrestrial life. Attempts to compare celestial phenomena with the fate of man were made by both scientists and investigative individuals not being scientists. However, at all times, a very unreliable result was obtained time and again. As for the fate of the celestial bodies themselves, astronomers and mathematicians have long calculated the surprisingly stable nature of their movement. The sizes and shapes of the orbits of the planets, or the slopes of the axes of their rotation, have not changed much even at cosmogonic time intervals.

Natural scientists and philosophers have come to the conclusion that the main reason for the existence of a Cosmic Mind in the Universe is the function of cognition. Led by reason life is characterized by a desire to understand and explain what is happening as regards the phenomenon.

At any stage of cognition of the Universe, we already have a more or less adequate model for it. New, more accurate observations may lead to a model mismatch with reality. At most, the required adjustment of the model is restored by clarifying the known parameters of the motion or the state of celestial bodies. Sometimes it is necessary to significantly improve theories, the model-constructing technique, or the calculation methods. This process is unconsciously aimed at discovering new, unexplained phenomena. At some stage, it is possible to get this much-needed food for the Mind, but this is always preceded by the colossal work of scientists—observers, theorists, and calculators. The motion models of celestial bodies are also valuable in that they allow us to predict their location at any time in the past or future.

A theory only makes sense when it is not only by abstract constructions in the imagination of a fascinated theoretician, but also one needs well-established procedures that regularly serve the purposes of practical knowledge of nature. One of the main tools in this respect is practical celestial mechanics. It is practical celestial mechanics that gives us the most complete and accurate knowledge of the dynamics of planetary satellites.

1.2 Celestial mechanics—the basis for studying the dynamics of planetary satellites

Celestial mechanics is the branch of science that studies the movements of celestial bodies under the action of natural forces.

The subject of celestial mechanics is the mechanical form of the motion of matter.

The objects of research are all kinds of material formations, from the smallest particles of cosmic dust to colossal systems such as star clusters, galaxies and clusters of galaxies.

The purpose of celestial mechanics is to study the laws of nature that govern the mechanical movements of celestial bodies.

For all natural sciences, celestial mechanics plays the role of a foundation, without which the study of the Universe and the exploration of the Cosmos are unthinkable. The significance of celestial mechanics for life on Earth is to gain knowledge about the motion of celestial bodies and the near Cosmos to better meet the needs of mankind and to result in protection from the forces of nature. The theory of motion of artificial satellites of Earth allows for the use of spacecraft for communication and research of terrestrial resources. The theory of motion of asteroids, comets and meteors gives an assessment of the danger of these bodies entering the atmosphere and falling to the Earth's surface. Studies of the motions of the bodies of the solar system made it possible to create a fundamental reference frame—a model of the inertial system implemented by celestial mechanics and astrometry in the form of astronomical yearbooks and the fundamental star catalogues.

In the development of celestial mechanics many of the most effective methods of mathematical physics and computational mathematics arose, took shape and were furthered.

As an example (and by no means the only one!), we can indicate methods for the numerical integration of differential equations describing various natural phenomena and man-made processes. Having arisen in celestial mechanics, these and other numerical methods are widely used in science and technology. In the 17–18th centuries, with the solution of astronomical problems by the methods of celestial mechanics, essentially all theoretical physics began.

Not only the theory of systems of ordinary differential equations, as it occurred in the last century, is predominant, but, in fact, the entire set of modern tools of applied mathematics is used by modern celestial mechanics to model the movements of space objects.

1.3 Objectives of studying the dynamics of planetary satellites

The primary objective of research into the dynamics of Solar System bodies is the determination of parameters of motion of planets and their satellites. This objective is relevant to the perennial challenge of mankind: expanding and exploring our habitat. Satellites of major planets are the most suitable targets for unmanned and manned landing missions. Research of the structure and dynamics of Solar System bodies is an integral part of dynamical astronomy. The methods of celestial mechanics and astrometric observations are used in this research. Interplanetary navigation, which attracted the interest of scientists in the second half of the 20th century, is a new problem of the dynamics of Solar System bodies.

The general approach to studying the dynamics of celestial bodies consists in developing models of motion and ephemerides of planets, asteroids, and planetary satellites. Such models are built based on the general laws of nature, the physical parameters of celestial bodies, and, most importantly, observations. Advanced mathematical and computational techniques are used in the process. Ephemerides are the end result of this research and incorporate the entire body of knowledge on the dynamics of Solar System bodies.

Ephemerides are used to determine the physical properties of celestial bodies and to study the origins and evolution of the Solar System. They are also needed to prepare and launch space missions to other planets and help discover new celestial bodies. In the middle of the 19th century, Urbain Le Verrier had used ephemerides to predict the existence of the then unknown planet Neptune, and new planets and satellites are still being discovered this way. Therefore, one may conclude that ephemerides also serve as a research tool, since they incorporate all the available data on the motion of planets and satellites.

The results and conclusions of celestial mechanics are visibly and invisibly present in many other areas of science and human practice.

1.4 Basic concepts of celestial mechanics and astrometry

We establish some basic concepts of practical celestial mechanics and astrometry, with which we will operate in the following presentation.

The objects of our research are the planets and satellites of the Solar System. Thus, we operate with models of celestial bodies, which in nature do not exist, but which to a certain extent differ little from the behaviors of real celestial bodies. Examples of such objects are a material point and an absolutely solid homogeneous body bounded by the surface of a triaxial ellipsoid.

Laws of motion. The real manifestation of the motion of celestial bodies is a change in their relative position, which is determined by the mutual distances. To set the motion of a system of celestial bodies, one should set the law of change in their mutual distances in time. The mathematical description of the laws of motion are these or other functions of time.

For a convenient representation of the motion of celestial bodies, we use the concepts of a reference frame, coordinate system and time scale. The abstract concept of a coordinate system is somehow connected with real celestial bodies. Examples include the Greenwich meridian on Earth or extragalactic radio sources. The abstract concept of a time scale is associated with real physical processes. Examples include Earth's rotation or electromagnetic radiation from an atom.

Laws of interaction. The basis for studying the motion of celestial bodies is the laws of physics that are strictly established from observations, which describe the interactions of bodies or the effects on them of the environment in which they move. The mathematical form of the laws of interaction of celestial bodies are ordinary differential equations, while the mutual distances between celestial bodies or their coordinates satisfy these equations.

Mechanical model. In celestial mechanics, the concept of a mechanical model is used. The model is described by the composition of moving objects and their properties, by specifying the forces acting on the individual components of the model. Mechanical models are used either for an approximate description of the motions of celestial bodies or as a basis for the development of more accurate methods for describing their motions.

The task of practical celestial mechanics is the creation and study of various mechanical models, as well as the study and description of the motion of real celestial bodies.

A mechanical model, being, as a rule, an approximate description of the motions of a system of real celestial bodies, can fundamentally differ from them. In particular, the properties of bodies in the model may not correspond to reality, and the laws of the acting forces can be specified in a special way. Examples include the motion of a system of material points in which celestial bodies are dimensionless, or a restricted three-body problem that does not satisfy Newton's third law.

Observations. Measured values. The source of our knowledge of celestial bodies is observation. In observing, we cannot be content with stating the fact of the presence of a celestial body in the sky. During astronomical observations, measurements of various quantities are carried out using a variety of instruments. Unlike abstract coordinates, the measured value is always the real one. It is formed in the measuring device. Astronomers deal with a wide variety of instruments and measured values. Examples are the angles of rotation of the telescope axis relative to the vertical line and the meridian plane, the distance between images of celestial bodies on photographic plates, the time interval between the flash of the laser rangefinder and the fixation of the light pulse reflected from the celestial body, the background intensity from a single pixel of a semiconductor light detector, and the difference in recordings of the signal from a space radio source at two radio telescopes.

Accuracy of observations. Instruments usually have measurement errors. Note that the mysteries of the processes occurring in measuring instruments leave us only with the opportunity to build hypotheses regarding measurement errors. The magnitude of the error of an individual measurement is never known. Often we assume that the errors are purely random, and we consider various statistical characteristics of the errors. Mostly, we use the concept of the most probable root-mean-square error. The structural properties of measuring instruments sometimes make it possible to approximately establish the accuracy of measurements. In the general case, we are talking about the accuracy of observations.

Time. Variation of the measured value in time is due to the motion of celestial bodies. Measurement is performed at some point in time. This time point is counted by the clock of the observatory. In practical celestial mechanics, a specific time of measurement is always ascribed to a measurable quantity.

Time is an abstract concept and some instruments are needed to measure it. However, any device has its own measurement error. First, time was measured by the angle of rotation of the Earth. Such a time was called universal was and designated as UT (Universal Time). When discrepancies between the theory of the motion of the Moon and observations were discovered, it became clear that the Earth rotates unevenly, and time has become the standard, as an independent variable in the theory of motion of the Moon. Time, measured by observations of the Moon, was called ephemeris time and was denoted ET. However, the accuracy of the observations of the Moon is still limited. The search for a more accurate time meter led to an atomic clock. This time sensor is now the most accurate. Time, averaged over several of the most accurate atomic clocks in the world, is called international atomic time and is designated as IAT (International Atomic Time).

In the future, we will talk about observations of celestial bodies, always assuming that one or another measured value is received at a certain point in time: measurement time.

The accuracy of astronomical measurements has already reached such a level that the inadequacy of classical Newtonian mechanics for describing the observed motion of celestial bodies has become noticeable. In a more accurate theory of general relativity, time passes differently at any two points in space. To connect different time scales, it is necessary to take into account the motion of bodies and their masses.

Motion parameters. When we study planets and satellites, stars and galaxies, we boldly assume that some parameters inherent in celestial bodies and their motion remain constant all the time. These include the mass, size and shape of bodies, orbit parameters and many other quantities. These parameters cannot be directly measured using existing instruments. However, their meanings really manifest themselves in the observed motion of celestial bodies. In the future we will call such quantities motion parameters of celestial bodies.

Coordinate systems. Measured quantities do not give visual representations of the configuration of the system of celestial bodies and are even less suitable for expressing general laws of motion. A convenient means of describing the spatial arrangement of bodies and directions of celestial bodies is the use of coordinate systems. When we talk about the position of the star or about the orientation of the body in a certain coordinate system, we mean the abstract coordinate axes in space and imaginary lines in the sky. Coordinate systems are chosen so as to give a clear idea of the laws and properties of the motion of celestial bodies.

The choice of a coordinate system is due to the convenience of describing and studying the motion of a particular celestial body. The origin and coordinate axes are associated either with the details of the object, for example, the Earth's Greenwich meridian, or with its dynamic properties, for example, with the principal axes of inertia of the body, or with the properties of motion, for example, with the rotation axis of the body, or with the position of the body at some time point, or we may choose a coordinate system in another particular way.

Mostly, a system of rectangular or Cartesian coordinates is used, its origin is denoted by the letter O, and the axes by the letters x, y, and z. The system of spherical coordinates is often used with the designation of the central distance by the letter r, the latitude by the letter φ and the longitude by the letter λ.

We refer to any coordinate systems with an origin located at the observation point as topocentric coordinate systems. In addition, we associate the axes of the topocentric system with the vertical line and the local meridian. When the origin of the coordinate system is placed at the mass center of the Earth, we are talking about geocentric coordinate systems.

The laws of motion of celestial bodies are the dependences of the coordinates of bodies on time and motion parameters. Dependencies can take many forms. At most, analytical functions are used that describe the explicit dependence of the coordinates on time. In some cases, the dependence is given in implicit form, then the coordinates are obtained by calculations with formulas by way of successive approximations. The law of motion can take the form of numerical tables in which the coordinates of celestial bodies are given for a number of fixed points in time, usually defined with some constant step. With such a numerical specification of the law of motion, the dependence of the coordinates on the motion parameters of the celestial body is lost. In this case, it is difficult to analyze the properties of motion, and we are limited to the time interval for which the coordinates were calculated.

The coordinates of celestial bodies are abstract concepts. They cannot be measured by any instruments. Coordinate systems are modeled using formulas and algorithms and form a constituent part of the motion model of celestial bodies.

A model of motion of a celestial body. We do not know exactly how the celestial bodies are arranged and by what exact laws they move. Therefore, we have to be content with the study of motion models, putting forward the bold hypothesis that our models differ little from reality.

In the general case, by a model of motion of a celestial body we will mean a certain construction that allows us to determine the values of the measured quantity at any given time instants for known values of the parameters of motion.

Implementations of the model of motion of a celestial body can have very different forms. These can be mathematical formulas, written manually on paper or published as printed material. These can be printed numeric tables of coordinate values. Currently, both formulas and tables are displayed in computer memory units. In this case, the formulas are converted into calculation algorithms, and the tables are available to computational programs that solve certain problems. Even in the era of powerful computing technology, the coordinates of the principal celestial bodies calculated for several years in advance are created and printed in the form of astronomical yearbooks in several world research centers.

Where do our ideas of the laws of motion of celestial bodies come from? In ancient times, they were established almost empirically from simple observations. Now, of course, the laws of motion are found in the process of solving differential equations of motion relative to the coordinates of celestial bodies. These equations are compiled on the basis of strictly established laws of physics, which describe the interactions of bodies or the effects on them of the environment in which they move. This is done as part of a mechanical model. All factors affecting the movement of each body of the system and included in the model under consideration are clearly fixed. The set of constructs of the laws of motion of celestial bodies, as well as its result, the laws of motion themselves, are called the theory of motion. This is what celestial mechanics addresses.

In the vast majority of problems of celestial mechanics, it is impossible to obtain an exact solution of the equations of motion. One has to be content with either an approximate solution of the exact equations, or an exact solution of the approximate equations. Both analytical and numerical methods for solving differential equations are used. In both cases, the solution has an error. This error can be more or less reliably estimated using the theory itself.

The accuracy of the motion model of a celestial body. The initial data for the model of motion of a celestial body are motion parameters, which, in turn, are known with some error. This error will also affect the accuracy of the pre-calculation of the coordinates of the celestial body and the accuracy of the pre-calculation of the measured value. Furthermore, we will talk about the model accuracy, implying an error in the calculation of the measured value. In this case, we separate two sources of this error: the proximity of the obtained solution of the motion equations and the inaccuracy of the motion parameters. The error of the solution of the motion equations will also be called the error of the calculations or the error of the method. When we talk about the accuracy of the theory of motion of a celestial body, it is always necessary to clarify whether the inaccuracy of the motion parameters is included in the error of the theory or is in the accuracy of the theory under the assumption of absolutely accurate parameters.

Research methods. From other astronomical disciplines celestial mechanics differs only in research methods, among which are analytical, numerical and qualitative approaches.

Analytical methods make it possible to obtain a set of analytical relationships that allow us to calculate the approximate positions and velocities of celestial bodies at given time points, omitting its values at any intermediate time points. A feature of analytical methods is the great complexity and growing bulkiness of the calculations. In addition, analytical methods make it impossible to assess the properties of the studied motions at very large time intervals. Another drawback is that analytical methods are not applicable to all objects.

The limitations inherent in analytical methods do not apply to numerical methods, which are suitable for calculating the motions of any celestial bodies and their systems with a predetermined accuracy. With the use of powerful computers in scientific research, the previously considered excessive laboriousness of numerical methods has ceased to be an obstacle to their application. But they have their own Achilles' heel—this is the steady accumulation of error with increase in the integration interval, while rigorous estimates of the growth of this error are impossible. Another drawback of these methods is the numerical form of presenting the results and the inevitability of calculating the intermediate stages, although often the goal of the study is the final configuration after integration.

Qualitative methods of celestial mechanics make it possible to judge the properties of the movements of celestial bodies without full integration of (analytical or numerical) differential equations.

Analytical, numerical and qualitative methods continue to be applied in modern practical celestial mechanics, and the beauty and high efficiency of analytical methods are successfully combined with the simplicity and universality of numerical methods, and all this is complemented by the cosmogonic importance of the conclusions obtained by qualitative research methods.

1.5 General approach to studying the dynamics of planets and satellites based on observations

A general approach to studying the dynamics of planets and satellites is the construction of a model of motion based on observations. It is the model of motion that is needed for the practical knowledge of nature.

Fig. 1.1 shows a scheme for studying the dynamics of Solar System bodies based on observations. At any stage of research, we fix the composition of the studied system of celestial bodies. The laws of the interaction of bodies (gravitational attraction, resistance of the medium), currently established, allow for writing down the differential equations of motion. Using analytical methods, one can find a general solution of the equations of motion. After substituting the values of arbitrary constants (motion parameters) into this general solution, we obtain the required model of motion of the system of celestial bodies. When we solve equations of motion by methods of numerical integration under known initial conditions (motion parameters), we also obtain a model of motion of a system of celestial bodies. Some preliminary values of motion parameters are usually known from previous studies. To construct a model of motion, the values of the physical parameters entering the equations of motion through the laws of interaction (for example, the mass of bodies) will also be required.

Figure 1.1 Scheme of studying the dynamics of celestial bodies.

The main procedure for studying the dynamics of celestial bodies is to refine the model based on observations. Observations give us the values of the measured quantities. Call them measured values. On the other hand, we have a motion model that serves to pre-calculate measured values. We can calculate the measured values precisely at the times of observation. Results are called calculated measured values. Values different in origin of the same entity will differ from each other. We denote this difference of values in Fig. 1.1 symbolically by O-C (O for observatum, C for calculatum). The difference is a natural result, since it contains an error of observation and an error in the model of motion of a celestial body. However, in some cases, the differences O-C will exceed the model error and the observation error. New, more accurate observations reveal a model mismatch with reality. In these cases, the mismatch is attributed to the simplest and most probable cause—the inaccuracy of the accepted values of the motion parameters of the celestial body. A process called refinement of motion parameters from observations is included in this case (see Parameter refinement methods in Fig. 1.1). Mostly, the required agreement between the theory and observations is achieved by refining the parameters, and the differences O-C again fall within the errors of the model and observations.

In some rare cases, the theory cannot be reconciled with observations—the differences O-C remain significant. Then we have to improve methods for solving the equations of motion and calculation methods. This is the most laborious part of celestial mechanics. The factors affecting the motion of each celestial body are being reconsidered. New, more accurate formulas of the theory are derived. As a result, the formulas become longer and more complex. In addition, more accurate calculation methods are being developed and applied. As a result, the required computing time is significantly increased.

In even rarer cases, the mismatch of the theory with observations remains significant, no matter how hard the researchers try to refine the motion parameters and improve the motion model. As a result of the generalization of facts, testing of new hypotheses and higher tension of intelligence, a discovery is made. Previously unknown celestial bodies or new laws of the interaction of the known bodies may be discovered. In such a situation, our general ideas about the world around us expand significantly. A generalization of the basic laws of nature is made.

The scheme presented here, like any scheme, is meagre and limited, it only in general terms reflects the mixture of scientific research and the accumulation of facts, fantasies and errors.

Note that the described process also has a purely practical focus. The model of motion of celestial bodies is the basis for tracking possible dangers threatening from outer cosmic space. The model of the motion of celestial bodies is also directly used for the design and support of flights of automatic and manned near-Earth and interplanetary vehicles—artificial celestial bodies.

1.6 Special properties of necessary observations

The motion of most real and imaginary celestial bodies is of the form of the circulations of some bodies around others. The proper rotations of celestial bodies are also being studied. The revolution or rotation of the body is described by an angle whose magnitude monotonically increases in time. Let us consider in more detail how these processes are determined from observations.

The orbital revolution angle or the rotation angle of a celestial body is conventionally called the longitude and denoted here by λis approximately constant.

Advancement can be achieved by increasing the accuracy of the observations. In this way, the discovery either of a new property of a known celestial body, or of a new planet or satellites, can occur. Let us illustrate this with an example.

Let us assume that we have built a good model of motion and with its help calculated the so-called O-C, the differences between the observed and the theoretical orbital longitude values. If measurements are inaccurate, i.e. in the presence of observation errors, a plot of these differences may look like the one shown in Fig. 1.2a, where noise is the only apparent component.

Figure 1.2 Examples of O-C residuals of the orbital longitude of a celestial body at different levels of accuracy of observations.

Let us assume that progress in observational techniques provided an opportunity to improve the accuracy of observations and suppress noise. A certain pattern then emerges (Fig. 1.2b), and sinusoidal variation of the O-C differences becomes clearly visible when observations get even more accurate (Fig. 1.2c). This signal helps one to determine the factors that were left unaccounted for by the theory.

The orbital motion of celestial bodies is distinctive in that the orbital longitude increases monotonically with time. If one removes the function of the theoretical variation of orbital longitude from its observed values, a plot similar to the one in Fig. 1.3a may be obtained. Again, there is nothing interesting in it. If past and new observations of the celestial body under study are added to the data presented in Fig. 1.3a, the plot in Fig. 1.3b is obtained. It can be seen that the longitude varies almost quadratically in time. This effect may be induced by the unaccounted dissipation of the mechanical energy of a celestial body, which, in its turn, may be attributed, for example, to tidal forces.

Figure 1.3 Examples of O-C residuals of the orbital longitude of a celestial body at different time intervals.

It is now clear that the observation time interval should also be expanded in order to make progress in this field. At some stage, this may lead to the discovery of new