# Chapter 18: Celestial Distances

I've been telling you that stars are very distant objects. The closest are several light years away, and the most distant are more than 8 billion light years away. How do we measure these distances?

The answer is complex; there is not one way to measure distances, but there are many that work in different cases. We will discuss what some of these are and how they are used.

## 18.1 Fundamental Units of Distance

The standard for measuring distance is the meter, defined as the distance light travels in 1/299,792,458.2 seconds, exactly. While this is very accurate, it is not a very practical unit for astronomy. Therefore, we use different units of distance. Two of these you've already seen, the astronomical unit (AU) defined as the average distance between the Earth and the Sun, and the light year (LY) defined as the distance light travels in one year.

### Distances Within the Solar System

The measurement of the distances to stars begins by determining distances within our own solar system. These distances are now known very accurately thanks to radar measurements of distances between the Earth and various planets, namely Venus, Mercury, Mars, the satellites of Jupiter, and the rings of Saturn. Thanks to all these measurements, the length of an AU is known to within one part in a billion.

## 18.2 Surveying the Stars

Now how do we turn this accurate knowledge of distances within our solar system into distances to stars?

### Triangulation

The distance to the nearest stars is measured by triangulation. Triangulation is a common technique for measuring distances on Earth; it is used by surveyors and GPS hardware. To see how triangulation allows us to measure the distance to stars, look at Figure 18.3.

Triangulation is based on the properties of triangles. In Figure 18.3, we know the distance from the Sun to point B (1 AU), and the angle at the corner occupied by the Sun (90°). A measurement of the angle P allows us to determine the length of the dotted line from the Sun to the star. The angle P is called the parallax. The reason to use the angle P and not twice P is because the distance from the Sun to B is 1 AU whereas the distance from A to B is 2 AU.

### Units of Stellar Distance

The distance from the Sun to B is called the baseline of the measurement. With a baseline of 1 AU, a parallax of 1 arcsec (1/3600 of a degree) corresponds to a distance called a parsec (pc) short for parallax arcsec. A distance of 1 parsec equals 3.26 LY, and 1LY = 0.31pc. The parsec is not as common as it once was. The light-year is preferred as it stresses the relationship between distance and time.

### The Nearest Stars

The nearest stars were some of the first to have their distances measured by their parallax. The nearest star to Earth is Proxima Centauri, a distance of 4.3 LY. Close by is a binary pair known as Alpha Centauri. All three stars (Proxima plus the two stars in Alpha) are in the Southern hemisphere. The closest star in the northern hemisphere is Sirius, just over 8 LY from Earth.

### The Extent of Parallax Measurements

Parallax measurements are difficult, and only possible for the nearest stars. The Hipparcos satellite was instrumental in extending parallax measurements from stars within about 60 LY to those out to about 300 LY from Earth.

## 18.3 Variable Stars: One Key to Cosmic Distances

### Standard Bulbs

If we know that some stars have the same luminosity, then we can calculate their distance based on their apparent brightness. More generally, if the luminosity of a star can be determined independently of its distance, then we can calculate its distance based on its apparent brightness. After much searching, some types of stars were found with characteristics that allow astronomers to determine their luminosity independently of their distance.

### Variable Stars

While most stars, like our Sun, have nearly constant luminosities, the luminosity of some stars varies substantially. These stars are known as variable stars. (Note that it takes work to separate binary stars from variable stars, but there are clues in the spectra to help.) We use a light curve to display how the brightness varies over time. If the pattern repeats at regular intervals, then the star is periodic, and the period is given by the time it takes for the pattern to repeat.

### Pulsating Variables

There are two special types of variable stars for which we can obtain distances. They are called cepheids and RR Lyrae variables. The two types are grouped under the single heading of pulsating variables.

We say these are pulsating because not only do we observe a change in brightness, but along with it is a doppler shift of the lines in the spectrum consistent with an expansion and contraction of the star, and a change in color consistent with a change in the temperature of its surface. That is, these stars are, for unknown reasons, undergoing regular expansions and contractions, and the overall luminosity changes in a corresponding fashion.

### Cepheid Variables

The first type of pulsating variables are the cepheid variables. The name comes from the first cepheid discovered, Delta Cephei.

Cepheids have periods of from 3 to 50 days, and the variation of their luminosity is from a few percent to a factor of 10. The cepheids have luminosities that are 1000 to 10,000 times that of the Sun, i.e. they are large stars. Polaris, the north star, is a cepheid variable, though the amount by which its luminosity varies is decreasing with time. Eventually it will no longer be a cepheid.

### The Period-Luminosity Relationship

As I stated earlier, the point is to find stars whose luminosity can be determined by some means independent of the brightness-distance-luminosity relationship. In 1908, Henrietta Leavitt discovered a relationship between the period of cepheid variables and their luminosity. The argument goes like this:

• There are a few hundred cepheids in the Large and Small Magellanic Clouds (LMC and SMC).
• The stars in the LMC and SMC are all about the same distance from Earth (within a few percent).
• Therefore, the relative luminosity of stars in the LMC and SMC is the same as their relative brightness (if one star is twice as bright as another, then it is also twice as luminous).
• Henrietta Leavitt observed that the brighter cepheids in the LMC/SMC have longer periods.
• Therefore, the luminosity of a cepheid is related to its period.
• The relationship can be calibrated by measuring a few nearby cepheids in our galaxy whose distance can be determined from parallax.
Cepheids can be observed in distant galaxies with the Hubble telescope and other modern instruments, so now we have a way to measure distances not only within the Milky Way, but to other galaxies as well.

### RR Lyrae Stars

The RR Lyrae stars are named for the star RR Lyrae, the best known of this class. RR Lyrae are less luminous than cepheids (so not as useful for measuring distant galaxies) but more numerous. They are useful for measuring distances within our galaxy.

RR Lyrae all have periods of less than a day, and the change in their brightness is less than a factor of two. Astronomers discovered that they all have about the same luminosity, about 50 Lsun. In this sense, they are like true standard bulbs. Figure 18.10 shows the relationship between period and luminosity for RR Lyrae and cepheids.

## 18.4 The H-R Diagram and Cosmic Distances

### Distances from Spectral Types

It is also possible to use the H-R diagram to determine the luminosity of a star based on its temperature and other characteristics. From Figure 18.12 (18.11 in the second edition) we see that if we have a G type star, then it could be a main sequence star with a luminosity of about 1Lsun or it could be a giant with a luminosity of 100Lsun. The difference can be discerned by details of the star's spectra. Recall that the pressure at the surface of a star can be learned from the absorption lines in the spectrum, and that giants have a lower pressure than smaller main sequence stars. Based on such measurements, a star is assigned to one of 6 luminosity classes.

Therefore, from detailed measurements of a star's spectrum astronomers can determine its spectral class (temperature) and luminosity class (size or type of star). This information uniquely identifies a star's location on the H-R diagram, from which we can directly read off the luminosity.

Unlike the technique for finding distances for variable stars, this technique can be used with any star. Just note that when astronomers talk about measuring the distances of stars, they don't normally mention the accuracy of the measurement. All the techniques mentioned here are accurate to 10% at best. Often the accuracy is much worse, maybe only 50%.