1.3 Astronomers Use Mathematics to Find Patterns

Scientific thinking allows scientists to make predictions. Once a pattern has been observed, such as the daily rising and setting of the Sun, scientists can predict what will happen next. Making an accurate prediction typically means making a numerical prediction. To do that, scientists need math.

The rhythms of nature produce patterns in our lives, and those patterns give us clues about the nature of the physical world. The Sun rises, sets, and then rises again at predictable times and in predictable locations. Spring turns into summer, summer turns into autumn, autumn turns into winter, and winter turns into spring again. Astronomers identify and characterize those patterns and use them to understand the world around us. As shown in Figure 1.6, the visible star patterns in the sky change predictably with the seasons. You will learn many other examples of patterns in the sky in Chapter 2.

Figure 1.6 Since ancient times, people recognized that patterns in the sky change with the seasons. Those and other patterns shape our lives. These star maps show the sky in the Northern Hemisphere during each season. Find the constellation Ursa Major in each star map to see a pattern over the course of the year.

Astronomers use mathematics to analyze patterns of all kinds and to communicate complex material compactly and accurately. Many people find mathematics to be a major obstacle that prevents them from appreciating the beauty and elegance of the world as seen through the eyes of a scientist. In this book, we strive to explain any necessary math in everyday language. We describe what equations mean and help you use them in a way that allows you to connect scientific concepts to the world. You will learn to appreciate the beauty of mathematics if you accept the challenge and make an honest effort to understand the mathematical material. Working It Out 1.1 introduces units and scientific notation. Units are not only necessary to understand relationships between numbers (how large is an object a meter in diameter compared to one that is a kilometer in diameter?), but they also provide a convenient way to check your own work! Numbers in astronomy are typically so large that we must express them in scientific notation. Working It Out 1.2 reviews some basics of mathematical tools and graphs. In modern society, the ability to read and interpret graphs is indispensable, whether they are graphs of natural systems or graphs of systems like the stock market. You will need the tools in these two boxes as you learn astronomy, whether or not your course is quantitative. But more importantly, you will need these tools in your later life, whether or not you ever pick up another astronomy book. In addition to these tools, Appendix 1 at the back of the book gathers together some essential mathematics, and Appendix 2 collects the physical constants of nature. Other appendixes contain data tables with key information about planets, moons, and stars.

working it out 1.2

Reading a Graph

Scientists often convey complex information and mathematical patterns in graphical form. Graphs typically have two axes: a horizontal axis (the x-axis) and a vertical axis (the y-axis). The x-axis usually shows an independent variable, the one a researcher might have control over in an experiment, whereas the y-axis shows the dependent variable, which is typically the variable a researcher is studying.

Graphs can take different shapes. Suppose we plot the distance a car travels over time, as shown in Figure 1.7a. In a linear graph, each interval on an axis represents the same-sized step. Each step on the horizontal axis of the graph in Figure 1.7a represents 5 minutes. Each step on the vertical axis represents a distance of 5 km traveled by the car. Data are plotted on the graph, with one dot for each observation; for example, after 20 minutes, the car has traveled 20 km.

Drawing a line through those data indicates the trend (or relationship) of the data. To understand what the trend means, scientists often find the slope of the line, which is the relationship of the line’s rise along the y-axis to its movement along the x-axis. To find the slope, we look at the change between two points on the vertical axis divided by the change between the same two points on the horizontal axis. For example, finding the slope of the line gives

There, the trend tells us that the car is traveling at 1 kilometer per minute (km/min), or 60 kilometers per hour (km/h).

Many observations of natural processes do not yield a straight line on a graph. When you catch a cold, for example, you feel fine when you get up in the morning at 7 a.m. At 9 a.m. you feel a little tired. By 11 a.m. you have a bit of a sore throat or a sniffle and think, “I wonder whether I’m getting sick,” and by 1 p.m., you have a runny nose, congestion, fever, and chills. The illness progresses faster with each hour as time goes by. That process is exponential because the virus that has infected you reproduces exponentially.

For the sake of this discussion, suppose that the virus produces one copy of itself each time it invades a cell. (Viruses actually produce 1000–10,000 copies each time they invade a cell, so the exponential curve is much steeper.) One virus infects a cell and multiplies, so now two viruses exist—the original and a copy. Those viruses invade two new cells, and each one produces a copy. Now four viruses are there. After the next cell invasion, we have eight. Then 16, 32, 64, 128, 256, 512, 1024, 2048, and so on. That behavior is plotted in Figure 1.7b.

Seeing what’s happening in the early stages of an exponential curve can be difficult because the later numbers are so much larger than the earlier ones. That’s why we sometimes plot that type of data logarithmically, by putting the logarithm (roughly the exponent of the 10 in scientific notation) of the data on the vertical axis, as shown in Figure 1.7c. Now each step on the axis represents 10 times as many viruses as the previous step. Even though we draw all the steps the same size on the page, they represent different-sized steps in the data (for example, the number of viruses). We often use that technique in astronomy because it has a second, related advantage: very large variations in the data can easily fit on the same graph.

a. The relationship between time and distance traveled.

b. The relationship between time and number of viruses.

c. The data in part b. plotted logarithmically.

Figure 1.7 Graphs such as these show relationships between quantities.

Each time you see a graph, you should first understand the axes—what data are plotted on the graph? Then you should check whether the axes are linear or logarithmic. Finally, you can look at the actual data or lines in the graph to understand how the system behaves.

CHECK YOUR UNDERSTANDING 1.3

When you see a pattern in nature, it is usually evidence of: (a) a theory’s being displayed; (b) a breakdown of random clustering; (c) an underlying physical law; (d) a coincidence.

AnswerAnswer

c