Mechanics is a branch of science and technology where we deal with various kinds of motions of bodies. Based on the observations for the motions of bodies, Newton put forth three laws, often called the Newton's laws of motion. In the present chapter, we shall discuss about them. While discussing the Newton's laws of motion, we shall assume that the atmosphere around us is not imposing any force on the moving bodies. That is, we shall assume that the atmosphere is not present. For expressing the motion, we require some reference frame and a coordinate system.

Inertial reference frame

For describing motion of a body, we require a reference frame. Newton considered an absolute reference frame and a universal time. For an absolute reference frame, its origin as well as its axes are absolutely fixed. Distances (displacements) of bodies are measured with reference to this reference frame.

The reference frame in which Newton's laws of motion are valid is known as an inertial reference frame. In fact such a reference frame does not exist, as no material body is at absolute rest and therefore we cannot have an absolute reference frame. Even a reference frame attached to the earth is not an absolute one, as the earth revolves around the sun, as well as it spins about its own axis. Further, sun also moves in our galaxy, the Milky Way. In fact, no body in the universe is at the absolute rest.

Fundamentals of Mechanics is the culmination of our years of experience in research and teaching the subject. It takes into account our close observations of student responses. Furthermore, it is a sincere attempt to make the study of mechanics enjoyable and arouse interest in the subject. Mechanics, as a subject of physics, is taught at BSc and BTech level. Professor Suresh Chandra conceptualized the book to fulfill requirements of science and engineering students of this level. The other two authors put in their extensive effort to give it a shape of a comprehensive textbook. There are chapters on vectors, laws of motion, conservation laws, inverse-square-law forces, harmonic oscillator, theory of relativity and non-inertial reference frames to match the title with the syllabuses of mechanics at different universities. The chapters are supplemented by ample number of exercises and diagrams which would help in better understanding of the subject. Multiple-choice questions and problems are provided at the end of each chapter to test progress after a lesson is taught.

It was only with the help, support and encouragement of a number of people that this project could come to fruition. First of all, we would like to express sincere thanks to our colleagues for their constructive criticism. Suresh Chandra is thankful to the authorities of the Lovely Professional University for their constant support. We also thank our family members for their kind cooperation and express our gratitude to Cambridge University Press India Pvt. Ltd. for publishing this book. Any suggestion for improvement of the book will be greatly appreciated and may be addressed through the publishers.

We have seen in preceding chapters that molecular lines are excellent tracers of interstellar gas, of star-forming regions, and of the interactions of stars on their environments in the Milky Way Galaxy and in external galaxies. Observations of molecular emissions, supported by detailed modelling, allow a rather complete physical description to be made of the regions where these molecules are located, even when the galaxies are not spatially resolved. But what about pregalactic situations in the Universe? These include some of the most active areas of research in modern astronomy. Did molecules have a role to play in pregalactic astronomy, and if so could molecular emissions help to trace processes occurring very early in the evolution of the Universe? When did molecular processes begin to play an important role? What are the best tracers of the first galaxies in the Universe?

In this chapter we show that molecules were likely to be present from the era of recombination after the Big Bang and certainly played an important role in the formation of protogalaxies and of the first stars. Whether molecules generated detectable signatures of those very early events is problematic, at least with our present range of astronomical instrumentation. However, it seems likely that we shall soon find molecular signatures of the post-recombination era. Once the first stars appeared and seeded their environments with metallicity, the formation of the first galaxies was modulated by molecules and it should certainly be possible to trace their formation using molecular emissions.

The preceding chapters in this book have demonstrated that to trace particular astronomical features in the Milky Way or in external galaxies by using molecular line emissions, the astronomer needs to choose lines corresponding to appropriate transitions. The transitions to use will, obviously, be those whose upper levels are readily populated in the gas that is to be observed. In many situations, the most important excitation mechanism to the upper level is collisional, and H2 is often the main collisional partner.

For example, we have seen that the CO(1–0) transition is appropriate for searching for and detecting cold neutral gas with a kinetic temperature of ∼10 K, where the number density of hydrogen molecules is ∼103 cm−3. However, observations of radiation emitted in this transition cannot reveal, say, the presence of either cold or warm gas at a density of, say, ∼105 cm−3, because collisional de-excitation of the upper level occurs before radiation in the (1–0) line can occur. Therefore, to observe gas at higher densities, observers must use more highly excited CO lines that have larger spontaneous radiation probabilities (assuming that these highly excited levels are sufficiently populated at the prevailing temperature). Alternatively, observers may use a line from some other molecular species that has more appropriate fundamental properties for the physical conditions in the gas to be observed. Of course, as we have seen in Chapters 8 and 9, complications introduced by high optical depths in the lines observed may also make it difficult to infer physical properties in the observed regions. The simple physics in the above arguments is encapsulated in the concept of critical density (see Section 2.3).

Studying the interstellar medium of the Milky Way Galaxy gives us the opportunity of identifying in detail the various components of the medium. The equivalent components in distant galaxies may be unresolved, but contribute to the overall emission. We show in Chapter 6 how to deal with emission from unresolved regions. In this chapter we consider the various distinct types of region in the Milky Way that can be explored through molecular line absorptions and emissions. We show that the chemistry in each of these molecular regions is dominated by one or more of the chemical drivers discussed in Chapter 3. The sensitivity of the chemistry to particular physical parameters, discussed in Chapter 4, may be an important concern in some cases. For most molecular regions, we identify a well-known example of each type, which is not necessarily typical but is one in which the consequences of the chemical driver are prominently displayed. We also list some molecular tracers useful in describing the physical conditions in these different situations. We emphasise in particular the tracers of density and temperature for Milky Way conditions. The aim of this chapter is to show how tracer molecules can reveal the nature, origin, and evolution of many types of region in the Milky Way. Tracers of conditions in galaxies external to the Milky Way are considered in Chapter 6.

In Chapter 8 we covered the basic formulae and recipes that astronomers use to derive physical quantities from molecular observations. These simple LTE analyses provide observers with rough estimates of the density and temperature of the gas at equilibrium. However, molecular observations can also provide much further insight into the physical conditions and the history and dynamics of the gas if interpreted with the right tools. In this chapter we describe the chemical and radiative transfer models that have been developed over many years and we show how a careful use of such tools makes molecules into powerful diagnostics of the evolution and distribution of molecular gas in the interstellar medium. It is now possible for the observer to use well-established modelling codes to exploit the information contained in the observational data and to determine a rather complete description of the observed interstellar material. This chapter discusses the inputs required and the outputs expected from such models.

Chemical Modelling

Owing to the large range of densities and temperatures present in the interstellar medium, significant changes in the energetics and dynamics of the gas can occur, leading to large variations in the chemical abundances. For decades now, chemical simulations (based on the processes described in Chapter 3) have provided astrochemists with predictions of molecular abundances as a function of the physical conditions. However, the interpretation of chemical models is not a trivial task and demands a detailed knowledge of the way the chemical model is developed.