# 14.1 The Distribution of Stars in Space

The Milky Way. Click here for original source URL

The night sky blazes with starlight. Go to a site far from any cities, and you can see over 6000 stars scattered across the sky. Long before the invention of the telescope, people could plainly see a band of light arching across midnight skies during certain seasons. Myths and legends have been molded around this prominent feature of the night sky. Nearly 2500 years ago, the Greek philosopher Democritus correctly attributed this glow to a mass of unresolved stars, which came to be called the Via Lactea, or Milky Way. The Milky Way was familiar to all prehistoric humans. Sadly, the spread of urban light has made it an unfamiliar sight to most people.

The Infrared Milky Way: GLIMPSE. Click here for original source URL.

In 1610 Galileo turned his telescope on the Milky Way and confirmed the ancient idea of Democritus that the filmy glow was due to a vast number of unresolved stars. He recognized that if the stars were like the Sun, they must be at vast distances, and the range of star brightness within the Milky Way implied a large range of distances. The telescope opened up a sense of the third dimension: depth. We call an entire set of stars that is held together by gravity a galaxy. The Milky Way galaxy contains the Sun and all the other stars in the night sky, and has its greatest concentration of stars in one strip of the sky.

How far away are all these stars? Does the distribution of stars have an end? What is the role of the Sun in this great assemblage? These questions get to the heart of an important issue — our place in the larger universe. Astronomers have built up a picture of the Milky Way Galaxy, working up in scale. It starts with a description of the immediate environment of stars, showing that most stars have companions. There is also a thin medium between stars. Astronomers can also identify groups and clusters of stars. All this information is combined to define the architecture of the Milky Way galaxy.

The stars seem to be arrayed above our heads on a two-dimensional backdrop. Measuring the third dimension of distance is a great challenge. Stars differ in absolute brightness by factors of a million or more, so a dwarf star might be 1000 times nearer than a supergiant of the same apparent brightness. Absolute brightness or luminosity is given by L = d^{2}F, where d is distance and F is flux or apparent brightness. Remember that L is the true brightness of an object, or the number of photons it emits each second, while F is the brightness we measure on Earth, or the number of photons per second we collect with our telescope.

Apparent brightness is a good measure of distance if we can identify stars of the same luminosity. If luminosity is a constant, then F ∝ d^{-2}. This is a statement of the familiar square law inverse. The excellent correlation between F and d means that the flux can be used to calculate distance. But if stars have a larger range in luminosity, apparent brightness is a much poorer indicator of distance. Finally, think of the more realistic situation that an astronomer might face when stars of every type are measured. The range of luminosity on the H-R diagram is so large that any correlation between apparent brightness and distance has been washed out. The appearance of a star gives no useful measure of its distance.

Astronomers can measure distances to some stars directly, using the method of trigonometric parallax. Unfortunately, most of the stars that are visible through a small telescope are too remote for a parallax to be measured. However, it turns out that we can learn something about distance simply by observing the way that stars are distributed on the plane of the sky. Go out on a dark night and you will see that some stars appear very close to each other. Are these pairs just caused by the chance alignment of two stars at different distances, or are they connected in some way? As early as 1767, John Michell, who was the father of the idea of black holes, decided that there were too many alignments to be caused by chance. He believed that the stars in each pair were close enough in space to orbit each other by gravity. Michelle was an extraordinary thinker, far ahead of his time in many ways. As well as coming up with the concept of black holes, he was first to understand that earthquakes travel as waves in rock and first to develop artificial magnets.

Uniform, Random, and Clustered Distributions of Stars. Click here for original source URL.

To understand how John Michelle deduced that some stars are double, start by understanding two types of distribution: uniform and random. Imagine that stars are distributed in only two dimensions, on a plane. A uniform distribution refers to objects that are separated by equal distances. A uniform distribution of stars would be spread out in a regular grid with equal distances between each one. The distance from any star to its nearest neighbor is always the same. A uniform distribution may be realistic to describe the way atoms are laid out in a crystal, but it is not realistic to describe the way stars are laid out in space.

Distribution of 14,000 nearby Sun-like stars in space. Click here for original source URL.

For a more realistic situation, consider a random distribution. A random distribution refers to objects that are separated by random distances. In this case, the distance between a star and its nearest neighbor star can vary quite widely. However, the average distance between stars is the same as for a uniform distribution. To see that this must be true, remember that the number of stars and the total area are unchanged, so the average spacing is unchanged too. For any particular star, we can ask what the probability is that it will have a neighbor within a certain distance. The average spacing is the distance where the probability is 0.5, so half of the stars will have a nearest neighbor closer than the average spacing and half will have a nearest neighbor farther than the average spacing. The probability of a star having a neighbor within any distance is proportional to the area considered, or the distance squared. In a random distribution, it is certainly possible for stars to be very close to each other, but it does not happen very often.

Stars in the real universe show clustering. A clustered distribution refers to objects that are separated by distances that tend to be smaller than for a random distribution. When stars are clustered on the plane of the sky, the nearest neighbors tend to be separated by small angles. Clustering is revealed by the higher probability of any star having a neighbor at a small separation, compared to a random distribution. Our example uses distributions in two dimensions but the same argument works for a three-dimensional distribution as well. We have described a statistical measure of clustering based on a distribution of nearest neighbor distances. In other words, we cannot claim that a particular pair of stars with small separation is clustered, because small separations will occur by chance in a random distribution as well. However, the statistical approach is very powerful because it allows astronomers to detect departures from a random distribution that are quite subtle. We don’t need a statistical test to tell us that a large clump of stars is clustered; it is obvious to the eye.

Here is John Michell’s reasoning on double stars. He measured the angles that separated each star in the sky from its nearest neighbor. He then calculated what angles he would measure if the same number of stars were distributed randomly on the sky. He compared the two distributions. Mitchell detected a clear excess of stars with close companions compared with what he would have expected from a random distribution. He concluded that some stars were physically associated — held together in one region of space by gravity. In particular, he identified many pairs of stars, which often differed greatly in apparent brightness, which must be at the same distance. Through his detective work, Michell provided the first clues to how stars are distributed in space.