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# 14.17 Distances to Groups of Stars

A parsec is the distance from theÂ Sunto anÂ astronomical objectÂ which has aparallaxÂ angle of oneÂ arcsecond. (1 AU and 1 pc are not to scale (1 pc = 206265 AU)). Click here for original source URL

In the late 18th century, William Herschel’s ambition was to map out the size and shape of the Milky Way. He was hampered because he had no way of measuring stellar distance. Apparent brightness alone is a very poor indicator of distance because stars have such a wide range of intrinsic properties. The most direct measure of stellar distance is parallax. In a parallax measurement, the diameter of Earth’s orbit of the Sun is used as the base of a triangle to make a geometric measurement of distance. By using the data gathered by the European Space Agency’s Hipparcos satellite launched in 1989, astronomers can reliably calculate distances to just over one hundred parsecs. The Hyades cluster, at about 47 pc, is pivotal in the galactic distance scale. The cluster can be measured not only by the parallax technique, but also by a variety of methods, which allows a more reliable determination of distance.

Pleiades, also known as the Seven Sisters, in infrared light. Click here for original source URL.

Astronomers have devised varied and sometimes ingenious methods to determine the distances to individual stars and groups. Here are details of some of the most important methods. First, we can use the collective properties of stars to determine the distances to open clusters. For example, we might use the angular size of a cluster to infer its distance. If all clusters have the same physical size, then simple geometry tells us that D ∝ 1/ Θ , where D is the distance to the cluster and Θ is the angle subtended by the cluster. Astronomers test the method with the Hyades, where parallax provides a direct measure of distance. If the Hyades subtends an angle of 7.5° at a distance of 42 pc, then a cluster that subtends only 1/2° must be (7.5/0.5) × 42 = 630 pc away. The problem with this method is that open clusters have a wide range in physical size, ranging from 4 pc for the Pleiades to 16 pc for h Persei. The same is true of globular clusters. So the error in using angular size to predict distance can be as large as a factor of four. This would be like saying that all humans, from newborns to adults, are about 6 feet (or 72 inches) tall.  For any given human you would be making a similar maximum error — newborns are on average 20 inches tall which is about 4 times smaller than 72 inches.

Relative distances can be measured if we have spectra for the stars and can place them on an H-R diagram. We can then compare this H-R diagram of a more distant cluster to the H-R diagram for the Hyades. This technique is called main sequence fitting. Each cluster has a concentration of stars that defines its main sequence on an H-R diagram. The main sequence of each cluster has the same shape. The amount the two main sequences have to be shifted vertically to "fit" on top of each other gives the apparent brightness difference between the two clusters. The apparent brightness difference is related to the relative distance by the inverse square law. Using the inverse square law and the known distance to the Hyades, the distance of the fainter cluster can be calculated easily. For example, if main sequence stars in M 67 are 2000 times fainter than main sequence stars of the same spectral type in the Hyades, then M 67 is √ 2000 x 42 = 1900 pc away. Main sequence fitting gives relative distances with an accuracy of about 20 to 30%. This would be like saying that all humans between the age of 20 and 30 are the same height. Since the intrinsic range in heights you were sampling was smaller (between about 5 feet and 7 feet tall), you would have similar accuracy.

One very important technique uses the properties of variable stars to measure their distance. Variable stars vary in brightness over periods ranging from less than an hour to several years. One type of variable called a Cepheid has certain properties that allow distance to be easily calculated. Cepheid variables have periods of 1 to 50 days. There is also a tight relationship between the Cepheid’s period and its luminosity, called the period-luminosity relation. Once you measure the period of one of them, say 5 days, you know the luminosity of the star. Astronomers use images taken nightly over a period of several months to measure the periods of Cepheids.

Variable stars, such as Cepheid variables and RR Lyrae variables, are very luminous, so they can be used to measure the distances to very remote regions. RR Lyraes are 100 times as luminous as the Sun, so they can be seen out to 10 times the distance we could see a Sun-like star. Cepheids range up to 10,000 times the Sun’s luminosity, so they are visible out to 100 times the distance we could detect a star like the Sun. Cepheids are also rare, so a large collection of stars will contain very few. Luckily, they reveal themselves easily by their variations since the vast majority of stars do not vary in brightness. The use of variable stars to measure distance assumes that stars in one part of the Milky Way function the same as stars in another part of the Milky Way. In other words, we assume that the physics that leads to stellar pulsation is universal. There is no way to directly test this assertion.

Color-magnitude diagram of the globular cluster M55. Click here for original source URL.

Hertzsprung-Russell diagram showing color and size of stars. Click here for original source URL.

Astronomers can use Cepheid variables to measure the distance of a cluster in a process requiring several steps. First, a Cepheid in the cluster must be located and its period and type measured. Then its luminosity must be read from the appropriate period-luminosity diagram. Finally, the Cepheid’s luminosity is combined with the apparent brightness to calculate the distance. Few astronomers are equipped to carry out all the necessary observations, such as measuring light variations and periods, determining spectral properties, and measuring interstellar reddening and extinction. Thus astronomers have specialized. Some study periods of variable stars; some make measures of absolute brightness; some study interstellar reddening. The simple statement that a cluster is "this many" parsecs from the Earth may represent years of work by many researchers.

Before 1930, astronomers calculated distances under the assumption that all intervening interstellar space is transparent. However, early estimates of the distance, brightness, and size of clusters gave inconsistent results. In 1930 Robert Trumpler demonstrated that it was wrong to assume that space is clear. He showed that diffuse interstellar dust dims stars and clusters that are more than a few dozen parsecs away. The intensity of starlight falls off more rapidly than we would predict by the inverse square law. Since we see a star as dimmer than it truly is, we will overestimate its distance. This very important systematic error must be understood if we are to correctly map out the Milky Way. The obscuration is not the same in every direction because interstellar material is inhomogeneous. Fortunately, the total amount of dimming can be estimated by measuring the amount of interstellar reddening, or color change caused by the dust. The more reddening is observed, the greater the degree of dimming. Once the extinction is measured and taken into account, distances can be accurately measured if the luminosity of any star or class of stars in the cluster is known.