The distances to the nearest stars can be measured by trigonometric parallax. A star with a parallax of one arc second (1) is 1 parsec—about 3.3 light years—away.
Stars have real motion through space as well as apparent motion as Earth orbits the Sun. A star's proper motion., its true motion across the sky, is a measure of the star's velocity perpendicular to our line of sight. The star's radial velocity—along the line of sight—is measured by the Doppler shift of its spectral lines.
Only a few stars are large enough and close enough that their radii can be measured directly. The sizes of most stars are estimated indirectly through the radius—luminosity—temperature relationship. Stars are categorized as dwarfs comparable in size to or smaller than the Sun, giants up to 100 times larger than the Sun, and supergiants more than 100 times larger than the Sun. In addition to "normal" stars such as the Sun, two other important classes of star are red giants, which are large, cool, and luminous, and white dwarfs, which are small, hot, and faint.
The absolute brightness of a star is equivalent to its luminosity. The apparent brightness of a star is the rate at which energy from the star reaches unit area of a detector. Apparent brightness falls off as the inverse square of the distance. Optical astronomers use the magnitude scale to express and compare stellar brightnesses. The greater the magnitude, the fainter the star; a difference of 5 magnitudes corresponds to a factor of 100 in brightness. Apparent magnitude is a measure of apparent brightness. The absolute magnitude of a star is the apparent magnitude it would have if placed at a standard distance of 10 pc from the viewer. It is a measure of the star's luminosity.
Astronomers often measure the temperatures of stars by measuring their brightnesses through two or more optical filters, then fitting a blackbody curve to the results. The color index of a star is the ratio of its brightnesses measured through two standard filters. The measurement of the amount of starlight received through each of a set of filters is called photometry.
Astronomers classify stars according to the absorption lines in their spectra. The lines seen in the spectrum of a given star depend mainly on its temperature, and spectroscopic observations of stars provide an accurate means of determining both stellar temperatures and stellar composition. The standard stellar spectral classes, in order of decreasing temperature, are O B A F G K M.
A plot of stellar luminosities versus stellar spectral classes (or temperatures) is called an H—R diagram, or a color—magnitude diagram. About 90 percent of all stars plotted on an H—R diagram lie on the main sequence, which stretches from hot, bright blue supergiants and blue giants, through intermediate stars such as the Sun, to cool, faint red dwarfs. Most main-sequence stars are red dwarfs; blue giants are quite rare. About 9 percent of stars are in the white dwarf region, and the remaining 1 percent are in the red giant region.
By careful spectroscopic observations, astronomers can determine a star's luminosity class, allowing them to distinguish main-sequence stars from red giants or white dwarfs of the same spectral type (or color). Once a star is known to be on the main sequence, measurement of its spectral type allows its luminosity to be estimated and its distance to be measured. This method of distance determination, which is valid for stars up to several thousand parsecs from Earth, is called spectroscopic parallax.
Most stars are not isolated in space but instead orbit other stars in binary-star systems. In a visual binary, both stars can be seen and their orbit charted. In a spectroscopic binary, the stars cannot be resolved, but their orbital motion can be detected spectroscopically. In an eclipsing binary, the orbit is oriented in such a way that one star periodically passes in front of the other as seen from Earth and dims the light we receive. The binary's light curve is a plot of its apparent brightness as a function of time.
Studies of binary stars often allow stellar masses to be measured. The mass of a star determines its size, temperature, and brightness. Fairly well defined mass—radius and mass—luminosity relations exist for main-sequence stars. Hot blue giants are much more massive than the Sun; cool red dwarfs are much less massive. The lifetime of a star can be estimated by dividing its mass by its luminosity. High-mass stars burn their fuel rapidly and have much shorter lifetimes than the Sun. Low-mass stars consume their fuel slowly and may remain on the main sequence for trillions of years.
Many stars are found in compact groups known as star clusters. Open clusters, with a few hundred to a few thousand stars, are found mostly in the plane of the Milky Way. They typically contain many bright blue stars, indicating that they formed relatively recently. Globular clusters are found mainly away from the Milky Way plane and may contain millions of stars. They include no main-sequence stars much more massive than the Sun, indicating that they formed long ago. Globular clusters are believed to date from the formation of our Galaxy.
1. One parsec is a little over 200,000 A.U. (Hint)
2. There are no stars within 1 pc of the Sun. (Hint)
3. Parallax can be used to measure stellar distances out to about 1000 pc. (Hint)
4. Most stars have radii between 0.1 and 10 times the radius of the Sun. (Hint)
5. Star A appears brighter than star B, as seen from Earth. Therefore, star A must be closer to Earth than star B. (Hint)
6. Star A and star B have the same absolute brightness (luminosity), but star B is twice as distant as star A. Therefore, star A appears four times brighter than star B. (Hint)
7. A magnitude 5 star looks brighter than a magnitude 2 star. (Hint)
8. Differences among stellar spectra are mainly due to differences in composition. (Hint)
9. Cool stars have very strong lines of hydrogen in their spectra. (Hint)
10. A G9 star is cooler than a G5 star. (Hint)
11. Red dwarfs lie in the lower left part of the H-R diagram. (Hint)
12. The brightest stars visible in the night sky are all found in the upper part of the H-R diagram. (Hint)
13. In a spectroscopic binary, the orbital motion of the component stars appears as variations in their radial velocities. (Hint)
14. Once a binary is recognized, it is always possible to determine the masses of both components. (Hint)
15. It is impossible to have a 1-billion-year-old O- or B-type main-sequence star. (Hint)
1. Parallax measurements of the distances to the nearest stars use a baseline of _____. (Hint)
2. The radial velocity of a star is determined by observing its _____ and using the _____ effect. (Hint)
3. To determine the true space velocity of a star, its _____ , radial velocity, and _____ must all be measured. (Hint)
4. The radius of a star can be indirectly determined if the _____ and _____ of the star are known. (Hint)
5. The smallest stars normally plotted on the H—R diagram are _____. (Hint)
6. The largest stars normally plotted on the H—R diagram are _____. (Hint)
7. Observations of stars through B and V filters are used to determine stellar _____. (Hint)
8. The hottest stars show little evidence of hydrogen in their spectra because hydrogen is mostly _____ at these temperatures. (Hint)
9. The coolest stars show little evidence of hydrogen in their spectra because hydrogen is mostly _____ at these temperatures. (Hint)
10. The Sun has a spectral type of _____. (Hint)
11. The H—R diagram is a plot of _____ on the horizontal scale versus _____ on the vertical scale. (Hint)
12. The band of stars extending from the top left of the H—R diagram to its bottom right is known as the _____. (Hint)
13. _____ star systems are important for providing measurements of stellar masses. (Hint)
14. Going from spectral type O to M along the main sequence, stellar masses _____. (Hint)
15. The main-sequence lifetimes of high-mass stars are much _____ than the main-sequence lifetimes of low-mass stars. (Hint)
1. How is parallax used to measure the distances to stars? (Hint)
2. What is a parsec? Compare it with the astronomical unit. (Hint)
3. Explain two ways in which a star's real space motion translates into motion observable from Earth. (Hint)
4. Describe some characteristics of red giant and white dwarf stars. (Hint)
5. What is the difference between absolute and apparent brightness? (Hint)
6. How do astronomers measure the temperatures of stars? (Hint)
7. Briefly describe how stars are classified according to their spectral characteristics. (Hint)
8. What information is needed to plot a star on the Hertzsprung—Russell diagram? (Hint)
9. What is the main sequence? What basic property of a star determines where it lies on the main sequence? (Hint)
10. How are distances determined using spectroscopic parallax? (Hint)
11. Why does the H—R diagram constructed using the brightest stars differ so much from the diagram constructed using the nearest stars? (Hint)
12. Which stars are most common in the Galaxy? Why don't we see many of them in H—R diagrams? (Hint)
13. How can stellar masses be determined by observing binary star systems? (Hint)
14. High-mass stars start off with much more fuel than low-mass stars. Why don't high-mass stars live longer? (Hint)
15. Compare and contrast the properties of open star clusters and globular star clusters. (Hint)
1. How far away is the star Spica, whose parallax is 0.013"?(Hint) What would Spica's parallax be if it were measured from an observatory on Neptune's moon Triton as Neptune orbited the Sun? (Hint)
2. A star lying 20 pc from the Sun has proper motion of 0.5"/yr. (Hint) What is its transverse velocity? If the star's spectral lines are observed to be redshifted by 0.01 percent, calculate the magnitude of its three-dimensional velocity relative to the Sun. (Hint)
3. (a) What is the luminosity of a star having three times the radius of the Sun and a surface temperature of 10,000 K? (b) A certain star has a temperature twice that of the Sun and a luminosity 64 times greater than the solar value. What is its radius, in solar units? (Hint)
4. Two stars—A and B, of luminosities 0.5 and 4.5 times the luminosity of the Sun, respectively—are observed to have the same apparent brightness. Which one is more distant, and how much farther away is it than the other? (Hint)
5. Calculate the solar energy flux (energy received per unit area per unit time) at a distance of 10 pc from the Sun. (Hint)
6. Astronomical objects visible to the naked eye range in apparent brightness from faint sixth-magnitude stars to the Sun, with magnitude —27. What is the range in energy flux corresponding to this magnitude range? (Hint)
7. A star has apparent magnitude 7.5 and absolute magnitude 2.5. How far away is it? (Hint)
8. Two stars in an eclipsing spectroscopic binary are observed to have an orbital period of 10 days. Further observations reveal that the orbit is circular, with a separation of 0.5 A.U., and that one star is 1.5 times the mass of the other. What are the masses of the stars? (Hint)
9. Given that the Sun's lifetime is about 10 billion years, estimate the life expectancy of (a) a 0.2—solar mass, 0.01—solar luminosity red dwarf (b) A 3—solar mass, 30—solar luminosity star, (c) A 10—solar mass, 1000—solar luminosity blue giant. (Hint)
10. Assuming the mass—luminosity relation shown in Figure 17.22, estimate the mass of the faintest main-sequence star in Omega Centauri that could be observed by (a) a typical 1-m telescope and (b) the Hubble Space Telescope. (See also Figure 17.9.) (Hint)
1. Every winter, you can find an astronomy lesson in the evening sky. The Winter Circle is an asterism—or pattern of stars—made up six bright stars in five different constellations: Sirius, Rigel, Betelgeuse, Aldebaran, Capella, and Procyon. These stars span nearly the entire range of colors (and therefore temperatures) possible for normal stars. Rigel is a B star. Sirius is an A. Procyon is an F star. Capella is a G star. Aldebaran a K star. Betelgeuse is an M star. The color differences of these stars are easy to see. Why do you suppose there is no O star in the Winter Circle?
2. Summer is a good time to search with binoculars for open star clusters. Open clusters are generally found in the plane of the Galaxy. If you can see the hazy band of the Milky Way arcing across your night sky—in other words, if you are far from city lights and looking at an appropriate time of night and year—you can simply sweep with your binoculars along the Milky Way. Numerous "clumps" of stars will pop into view. Many will turn out to be open star clusters.
3. Globular star clusters are harder to find. They are intrinsically larger, but they are also much farther away and therefore appear smaller in the sky. The most famous globular cluster visible from the Northern Hemisphere is M13 in the constellation Hercules, visible on spring and summer evenings. This cluster contains half a million or so of the Galaxy's most ancient stars. It may be glimpsed in binoculars as a little ball of light, located about one-third of the way from the star Eta to the star Zeta in the Keystone asterism of the constellation Hercules. Telescopes reveal this cluster as a magnificent, symmetrical grouping of stars.
