The mathematician John von Neumann once urged the information theorist Claude Shannon to assign the name entropy to the measure of uncertainty Shannon had been investigating. After all, a structurally identical measure with the name entropy had long been an element of statistical mechanics. Furthermore, “No one really knows what entropy really is, so in a debate you will always have the advantage.” Most of us love clever one-liners and allow each other to bend the truth in making them. But von Neumann was wrong about entropy. Many people have understood the concept of entropy since it was first discovered 150 years ago.

Actually, scientists have no choice but to understand entropy because the concept describes an important aspect of reality. We know how to calculate and how to measure the entropy of a physical system. We know how to use entropy to solve problems and to place limits on processes. We understand the role of entropy in thermodynamics and in statistical mechanics. We also understand the parallelism between the entropy of physics and chemistry and the entropy of information theory.

But von Neumann’s witticism contains a kernel of truth: entropy is difficult, if not impossible, to visualize. Consider that we are able to invest the concept of the energy of a rod of iron with meaning by imagining the rod broken into its smallest parts, atoms of iron, and comparing the energy of an iron atom to that of a macroscopic, massive object attached to a network of springs that model the interactions of the atom with its nearest neighbors. The object’s energy is then the sum of its kinetic and potential energies – types of energy that can be studied in elementary physics laboratories. Finally, the energy of the entire system is the sum of the energy of its parts.

Messages and message sources

Information technologies are as old as the first recorded messages, but not until the twentieth century did engineers and scientists begin to quantify something they called information. Yet the word information poorly describes the concept the first information theorists quantified. Of course specialists have every right to select words in common use and give them new meaning. Isaac Newton, for instance, defined force and work in ways useful in his theory of dynamics. But a well-chosen name is one whose special, technical meaning does not clash with its range of common meanings. Curiously, the information of information theory violates this commonsense rule.

Compare the opening phrase of Dickens’s A Tale of Two Cities: It was the best of times, it was the worst of times … with the sequence of 50 letters, spaces, and a comma: eon jhktsiwnsho d ri nwfnn ti losabt,tob euffr te … taken from the tenth position on the first 50 pages of the same book. To me the first is richly associative; the second means nothing. The first has meaning and form; the second does not. Yet these two phrases could be said to carry the same information content because they have the same source. Each is a sequence of 50 characters taken from English text.

Even before taking an astronomy class, most people have a sense of how gravity works. No mathematics is needed to understand the idea that every mass attracts every other mass and that gravity is the force that causes apples to fall from trees. But what if you want to know how much you'd weigh on Saturn's moon Titan, or why the Moon doesn't come crashing down onto the Earth, or how it can possibly be true that you're tugging on the Earth exactly as hard as the Earth is tugging on you? The best way to answer questions like that is to gain a practical understanding of Newton's Law of Gravity and related principles.

This chapter is designed to help you achieve that understanding. It begins with an overview of Newton's Law of Gravity, in which you'll find a detailed explanation of the meaning of each term. You'll also find plenty of examples showing how to use this law – with or without a calculator. Later sections of this chapter deal with Newton's Laws of Motion as well as Kepler's Laws. And like every chapter in this book, this one is modular. So, if you're solid on gravity but would like a review of Newton's Third Law, you can skip to that section and dive right in.

This chapter reviews four important mathematical concepts and techniques that will be helpful in many quantitative problems you're likely to encounter in a college-level introductory astronomy course or textbook. As with all the chapters in the book, you can read the sections within this chapter in any order, or you can skip them entirely if you're already comfortable with this material. But if you're working through one of the later chapters and you find that you're uncertain about some aspect of unit conversion, the ratio method, rate problems, or scientific notation, you can turn back to the relevant section of this chapter.

Units and unit conversions

One of the most powerful tools you can use in solving problems and in checking your solutions is to consistently include units in your calculations. As you may have noticed, among the first things that physics and astronomy professors look for when checking students' work is whether the units of the answer make sense. Students who become adept at problem-solving develop the habit of checking this for themselves.

Understanding units is important not just in science, but in everyday life as well. That's because units are all around you, giving meaning to the numbers that precede them. Telling someone “I have a dozen” is meaningless. A dozen what? Bagels? Minutes to live? Spouses? If you hope to communicate information about quantities to others, numbers alone are insufficient. Nearly every number must have units to define its meaning. So a very good habit to start building mastery is to always include the units of any number you write down.

In astronomy, virtually all of the information that we can learn about the Universe comes from various forms of light. Since planets, stars, and other objects in space are so far away and our ability to travel in space is rudimentary, we must glean as much information as possible from their light. Therefore, it behooves you to understand how light works and what kind of information it carries.

A great deal of the information in light from astronomical objects can be derived from the spectrum of that light. You can read about astronomical spectra in the first section of this chapter, and the later sections discuss some of the techniques astronomers use to interpret spectra.

Light and spectrum fundamentals

The most fundamental property that distinguishes one type of light from another is its color. This section introduces the concept of a spectrum as a graphical presentation of the brightness of different colors in light, and you'll learn how to translate between various quantitative properties associated with the color of light. Light behaves both as waves and particles, and you'll see how the properties related to color can be used to describe both the electromagnetic-wave and the photon-particle aspects of light. If you'd like to understand why light is called an electromagnetic wave and exactly what's doing the waving in light, you can find additional resources about the nature of light on the book's website.

Two of the most popular topics in astronomy classes are black holes and cosmology. Both of these subjects can be somewhat abstract, hard to visualize, and quite mathematical, giving them a mystique which likely contributes to their popularity. Precisely because some of the objects and processes are hard to visualize, the mathematical foundations of these topics are a valuable source of insight into their nature. So for these topics, even more than for other topics for which you have physical intuition on your side, it behooves you to understand the mathematics.

This chapter deals with “limiting cases” by investigating the mathematical ramifications of taking one variable to an extreme, such as allowing the radius of an object to shrink to zero or permitting time to run to infinity. The physical manifestations of these mathematical limiting cases lead to the most exotic concepts in astronomy: black holes, which are singularities of mass; and cosmology, which deals with the history and fate of the Universe. The chapter draws upon many of the tools discussed previously in this book, including units, solving equations using ratios and the absolute method, gravity, light, and graph interpretation. Black holes and cosmology bookend the entire range of possible sizes, from the infinitesimally small to the unimaginably immense, and are well worth the investment of time it takes to understand their mathematical foundations.

Before diving into black holes, you should make sure you have a solid understanding of the concepts and equations related to density and escape speed. Those are the subjects of the first two sections of this chapter, so if you're already comfortable with those topics, you can jump ahead to Section 6.3.

This book has one purpose: to help you understand and apply the mathematics used in college-level astronomy. The authors have instructed several thousand students in introductory astronomy courses at large and small universities, and in our experience a common response to the question “How's the course going for you?” is “I'm doing fine with the concepts, but I'm struggling with the math.” If you're a student in that situation, or if you're a life-long learner who'd like to be able to delve more deeply into the many wonderful astronomy books and articles in bookstores and on-line, this book is here to help.

We want to be clear that this book is not intended to be your first exposure to astronomy, and it is not a comprehensive treatment of the many topics you can find in traditional astronomy textbooks. Instead, it provides a detailed treatment of selected topics that our students have found to be mathematically challenging. We have endeavored to provide just enough context for those topics to help foster deeper understanding, to explain the meaning of important mathematical relationships, and most of all to provide lots of illustrative examples.

We've also tried to design this book in a way that supports its use as a supplemental text. You'll notice that the format is modular, so you can go right to the topic of interest. If you're solid on gravity but uncertain of how to use the radiation laws, you can skip Chapter 2 and dive right into Section 3.2 of Chapter 3.

While it remains true that “entropy is not a localized, microscale phenomenon at which we can point, even in our imaginations, and say, ‘Look! There is entropy’” and that, “if we insist on trying to understand a subject in ways inconsistent with its nature, we will be disappointed,” the eight chapters of this guide have prepared us to give a constructive answer to the question “What is entropy?”

Any short description of entropy will necessarily be figurative. After all, one task of a figure of speech is to transfer a complex meaning from an extended description to a word or short phrase. In fact, we have already considered several figurative descriptions of entropy that are appropriate in special contexts: transformation content, disorder, uncertainty, spread in phase space, and missing information. Transformation content was Clausius’s way of referring to how the entropy function indicates the direction in which an isolated system may evolve. Spread in phase space, while appropriate for statistical systems, depends upon familiarity with the technical concept of phase space.

Disorder has long been a popular synonym for entropy. But recently order and disorder as describing low and high entropy systems have fallen into disfavor. This is because scientists have become fascinated with isolated systems that generate apparent order from apparent disorder. For instance, consider a thoroughly shaken bottle of water and olive oil. When left undisturbed, the water and olive oil begin to separate into distinct layers with the less dense olive oil on top. Yet even in this process the entropy of the oil–water system increases. Thus, while order and disorder are suggestive, they can mislead.