# Chapter 17: The Stars: A Celestial Census

Stars are not forever, but have a finite lifespan -- typically billions of years long, but still finite. If the lifespan is billions of years, how can we expect to learn about the different stages a star goes through? The answer is to study a large sample of stars at various points in their lives and piece together the life cycle of a single star from this. The authors of the text compare this to learning about the human lifecycle by studying a large sample of humans at various stages of their lives, and piecing together the life cycle of a single human.

## 17.1 A Stellar Census

We will begin by taking a census of stars. Taking a census of stars is alot like taking a census of people, if you want to be sure that the census is unbiased, you must work hard to count everyone! For stars, this means trying not to bias the census by just counting the most easily visible stars and missing the fainter, harder to find stars.

### Small Is Beautiful -- Or at Least More Common

When our bias towards seeing very luminous (bright) stars is accounted for, we find that most stars are rather dim, 100 to 10,000 times less luminous than our Sun. There are actually relatively few stars that are much brighter than our Sun. This is strange because most of the stars visible to the naked eye are much, much brighter than the Sun! This is just the result of a sampling bias, in this case a bias towards the most visible stars. To do an unbiased survey, we must be very careful to also look for the dim stars, those that may be invisible to the naked eye.

### The Density of Stars in Space

So, what is the density of stars, at least the density in the neighborhood of the Sun? We can determine this from the count of stars within a given distance of Earth. There are 59 stars that lie within 16LY of Earth. Doing the math, we find that there is one star in a volume of 300LY3. This also means that the average distance between stars in our neighborhood is 7LY.

## Other Census Questions

Now that we have a simple count of stars we want to measure additional quantities. When the U.S. takes a census, they don't simply count people, they collect additional information as well, information such as income, health, type of residence, marital and family status, location, ... We should do the same for stars. So, what types of things can we measure about stars? Mass, size, temperature, composition, age, motion, and rotation is at least a partial list of measurable quantities.

## 17.2 Stellar Masses

How can we measure the mass of a star?

The answer involves using a trick. To understand the trick, let me point out that the mass of the Sun is determined by careful measurements of the orbits of the planets. This is not as easy as it sounds because the mass of the Sun is so much larger than the mass of the planets. It is much easier to determine masses if two bodies in orbit are comparable in mass.

It turns out that there are many pairs of stars that orbit each other. The pairs of stars are called binary stars. If two stars happen to be close together from our line of sight, but widely separated in space, they are called double stars. Binary stars refers to a pair that are physically close, so close that they orbit each other by their gravitational attraction.

Not all stars are visible, as we will discuss in a later chapter. For now though, we will limit ourselves to binary pairs where both stars shine. A visual binary is one where both stars can be seen. In a spectroscopic binary, the stars are too close to separate by eye, but by looking at the spectra, we can see that the light is from two objects orbiting each other.

I've been saying that the two stars orbit each other. Now what does that mean?

Because the two stars may have about the same mass, we can't claim that one is fixed and the other orbits it. Imagine that both have exactly the same mass, then which one is fixed and which is moving? (Proof by showing that the opposite leads to an absurdity.)

Demonstrate the idea of center of mass.

In a binary star system, both stars orbit about their common center of mass.

### Masses from the Orbits of Binary Stars

The masses of binary stars are determined using the modified version of Kepler's law:

D3 = (M1 + M2)P2
By careful measurement, astronomers can determine both the period of the orbit (the period is the same for both stars), and their average separation (the harder quantity to determine). This determines the sum of the masses. The ratio of the masses is known from the characteristics of the orbit, and this is enough to get each of the masses separately.

### The Range of Stellar Masses

How large and small can star's be?

At the large end, stars with masses 100 times that of the Sun are known, and there may be stars with up to 200 times the mass of the Sun (200 Msun). Based on calculations, we beleive that a true star must have a mass greater than 1/12 the mass of the Sun, 1/12 Msun.

### Failed Stars

Objects with a mass between 1/100 Msun and 1/12 Msun may have briefly produced energy by nuclear reactions. They are not viable as stars for a substantial period of time, but since they may have produced energy by nuclear reaction, they're not quite planets. These objects are classified as brown dwarfs.

Objects with a mass less than 1/100 Msun are planets. Jupiter has a mass of about 1/1000 Msun and is definitely a planet. Other planets have been found around stars, and we will discuss them in a later chapter.

### The Mass-Luminosity Relation

It is interesting to look at the relationship between mass and luminosity (Fig. 17.10 in the second edition and 17.8 in the third edition). We find that, in general, more luminous stars are also more massive, and that there is a definite relationship between luminosity and mass. That is, if we know the mass of a star, the relationship will tell us the luminosity, or vice versa.

There are a few stars for which this relationship doesn't work. These exceptions are the white dwarf and giant stars. They are very interesting in their own right, and will be discussed in chapter 21.

## 17.3 Diameters of Stars

Stars are extremely distant objects, just points of light even in our best telescopes. (Don't be fooled by some pictures where some stars seem bigger than others, that's just an effect arising from their relative brightness.) How then can we measure the diameters of stars?

### Stars Whose Light Is Blocked by the Moon

The answer is to again use a little trick. Instead of trying to directly measure the diameter, we try to measure the time it takes for an intervening object (the Moon for instance) to block all the light from the star. Since we know how fast the Moon moves, we can then determine how wide the object appears in the sky, and the precision achievable is much better than a direct telescope measurement. A drawback is that only a few stars can be measured this way.

### Eclipsing Binary Stars

Some binary systems are oriented so that, as seen from Earth, one star blocks all or part of the other during their orbit. Such an arrangement can easily occur. When one star eclipses the other, the brightness of the pair changes. By measuring the time it takes for the brightness to change, and determining the size and period of the orbit, we can calculate the diameter of one (or possibly both) stars (Fig. 17.11 in the second edition and Figs. 17.9 and 17.10 in the third edition).

The star Algol in the constellation Perseus is a famous example of an eclipsing binary. The regular changes in brightness were known for centuries. It wasn't until the 18th century that it was suggested that Algol was really an eclipsing binary. About 100 years later measurements could be made to verify that hypothesis.

### Stellar Diameters

Most of the measurements of stellar diameters are as we would expect from the ratio of the star's mass to our Sun's. There are a few outstanding exceptions, known as red giants. Red giant stars are some of the most luminous stars in the sky, but have very low surface temperatures, making them red in color. They also have incredibly large diameters. The giant star Betelgeuse has a diameter about the size of the orbit of Jupiter (not the size of the planet, but the size of its orbit around the Sun!).

## 17.4 The H-R Diagram

One of the most important relationships for stars is discovered by plotting the luminosity of a star versus its surface temperature (as determined by its spectral class). Such plots were first made by Hertzsprung and Russell in the early 20th century and are known as H-R diagrams. They are important because they lead to an understanding of the lifecycle of stars. We will use H-R diagrams in a number of the upcoming chapters.

### Features of the H-R Diagram

Figure 17.13 (Fig. 17.15 in the second edition) shows an H-R diagram for nearby stars. The vertical axis is a logarithmic luminosity scale and the horizontal axis is the spectral class (with the corresponding temperature shown below). This is the traditional way to make an H-R diagram.

First, we see that stars are not distributed randomly over the diagram. Most of the stars lie in a band running from the upper left to the lower right, known as the main sequence. Our Sun lies on the main sequence, as well as Alpha Centauri A, Sirius, Vega, Barnard's Star, and Proxima Centauri. This band represents a relationship between temperature and luminosity. Tell me the temperature of a main sequence star and I can tell you it's luminosity.

Not all the stars lie on the main sequence. A number of stars lie above the main sequence, generally having a high luminosity, yet much cooler than the main sequence stars. How can a star be both highly luminous and relatively cool? The answer is that they are extremely large in size, the giants or supergiants. Since the luminosity depends on the total energy output of the star, a cool but very large star can have a high luminosity.

Other stars lie below the main sequence. These stars are very hot but have a low luminosity. Again, how can a very hot star have a low luminosity? The answer is that it has a small surface area, meaning that it is a small compact object. These stars are known as white dwarfs -- white because they are hot and dwarfs because they are small.

After a brief hiatus to discuss how to measure the distances to stars, and what lies in the space between stars, we will come back to the H-R diagram in our discussion of the life cycle of stars. We will learn that stars begin their lives on the main sequence. Probably 90% of a star's life is spent "on the main sequence". Towards the end of their life, stars move off the main sequence. The large hot stars can become giants or supergiants, eventually ending their lives in massive explosions called super novae. Somewhat smaller stars become white dwarfs. Much smaller stars simply fade away.