1. Chapter 17 Measuring the Stars
Chapter 17 opener. The total number of stars, even in our local neighborhood, is virtually beyond our ability to count. Relatively few of them have been studied in detail. Yet astronomers have learned a great deal about stars in general, and their ranges of properties—their masses, their luminosities, even their ages and (as we shall see in later chapters) their destinies. Here, the Very Large Telescope in Chile has imaged the magnificent star cluster NGC 3603, a rich group of more than a thousand stars roughly 20,000 light-years away and probably less than a million years old. (EUROPEAN SOUTHERN OBSERVATORY)

2. Units of Chapter 17 17.1 The Solar Neighborhood
Naming the Stars
17.2 Luminosity and Apparent Brightness
17.3 Stellar Temperatures
More on the Magnitude Scale
17.4 Stellar Sizes
Estimating Stellar Radii
17.5 The Hertzsprung-Russell Diagram

3. Units of Chapter 17 (cont.)
17.6 Extending the Cosmic Distance Scale
17.7 Stellar Masses
Measuring Stellar Masses in Binary Stars
17.8 Mass and Other Stellar Properties

4. 17.1 The Solar Neighborhood
Remember that stellar distances can be measured using parallax:
Figure Stellar Parallax (a) For observations of stars made 6 months apart, the baseline is twice the Earth–Sun distance, or 2 AU. Compare with Figure 1.31, which shows the same geometry, but on a much smaller scale. (b) The parallactic shift (greatly exaggerated by the white arrow in this drawing) is usually measured photographically, as illustrated by the red star seen here. Images of the same region of the sky made at different times of the year are used to determine a star’s apparent movement relative to background stars.

5. 17.1 The Solar Neighborhood
Nearest star to the Sun: Proxima Centauri, which is a member of the three-star system Alpha Centauri complex
Model of distances:
Sun is a marble, Earth is a grain of sand orbiting 1 m away
Nearest star is another marble 270 km away
Solar system extends about 50 m from Sun; rest of distance to nearest star is basically empty

6. 17.1 The Solar Neighborhood
The 30 closest stars to the Sun:
Figure The Solar Neighborhood A plot of the 30 closest stars to the Sun, projected so as to reveal their three-dimensional relationships. The circular gridlines represent distances from the Sun in the Galactic plane; the vertical lines denote distances perpendicular to that plane. Notice that many stars are members of multiple-star systems. All lie within 4 pc (about 13 light-years) of Earth.

7. 17.1 The Solar Neighborhood
Next nearest neighbor: Barnard’s Star
Barnard’s Star has the largest proper motion of any star—proper motion is the actual shift of the star in the sky, after correcting for parallax
These pictures were taken 22 years apart:
Figure Proper Motion A comparison of two photographic plates taken 22 years apart shows evidence of real spatial motion for Barnard’s Star (denoted by an arrow). (Harvard College Observatory)

8. 17.1 The Solar Neighborhood
Actual motion of the Alpha Centauri complex:
Figure Real Spatial Motion The motion of the Alpha Centauri star system drawn relative to our solar system. The transverse component of the velocity has been determined by observing the system’s proper motion. The radial component is measured by using the Doppler shift of lines in Alpha Centauri’s spectrum. The true spatial velocity, indicated by the red arrow, results from the combination of the two.

9. Discovery 17-1: Naming the Stars
Naming stars:
Brightest stars were known to, and named by, the ancients (Procyon)
In 1604, stars within a constellation were ranked in order of brightness and labeled with Greek letters (Alpha Centauri)
In the early 18th century, stars were numbered from west to east in a constellation (61 Cygni)

10. Discovery 17-1: Naming the Stars
As more and more stars were discovered, different naming schemes were developed (G51-15, Lacaille 8760, S 2398)
Now, new stars are simply labeled by their celestial coordinates

11. 17.2 Luminosity and Apparent Brightness
Luminosity, or absolute brightness, is a measure of the total power radiated by a star.
Apparent brightness is how bright a star appears when viewed from Earth; it depends on the absolute brightness but also on the distance of the star:

12. 17.2 Luminosity and Apparent Brightness
Therefore, two stars that appear equally bright might be a closer, dimmer star and a farther, brighter one:
Figure Luminosity Two stars A and B of different luminosities can appear equally bright to an observer on Earth if the brighter star B is more distant than the fainter star A.

13. 17.2 Luminosity and Apparent Brightness
Apparent luminosity is measured using a magnitude scale, which is related to our perception.
It is a logarithmic scale; a change of 5 in magnitude corresponds to a change of a factor of 100 in apparent brightness.
It is also inverted—larger magnitudes are dimmer.
Figure Apparent Magnitude This graph illustrates the apparent magnitudes of some astronomical objects and the limiting magnitudes (that is, the faintest magnitudes attainable) of some telescopes used to observe them. The original magnitude scale was defined so that the brightest stars in the night sky have magnitude 1, whereas the faintest stars visible to the naked eye have magnitude 6. The scale has since been extended to cover much brighter and much fainter objects. An increase of 1 in apparent magnitude corresponds to a decrease in apparent brightness by a factor of approximately 2.5.

14. 17.2 Luminosity and Apparent Brightness
If we know a star’s apparent magnitude and its distance from us, we can calculate its absolute luminosity.
Figure Inverse-Square Law As light moves away from a source such as a star, it steadily dilutes while spreading over progressively larger surface areas (depicted here as sections of spherical shells). Thus, the amount of radiation received by a detector (the source’s apparent brightness) varies inversely as the square of its distance from the source.

15. 17.3 Stellar Temperatures
The color of a star is indicative of its temperature. Red stars are relatively cool, while blue ones are hotter.
Figure Star Colors (a) The different colors of the stars comprising the constellation Orion are easily distinguished in this photograph taken by a wide-field camera attached to a small telescope. The bright red star at the upper left is Betelgeuse (a); the bright blue-white star at the lower right is Rigel (b). (Compare with Figure 1.8 and the large opening photo for Chapter 1.) The scale of the photograph is about 20ϒacross. (b) An incredibly rich field of colorful stars, this time in the direction of the center of the Milky Way. Here, the field of view is just 2 arc minutes across—much smaller than in (a). The image is heavily populated with stars of many different temperatures. (J Sanford/Astrostock-Sanford; NASA)

16. 17.3 Stellar Temperatures
The radiation from stars is blackbody radiation; as the blackbody curve is not symmetric, observations at two wavelengths are enough to define the temperature.
Figure Blackbody Curves The locations of the B (blue) and V (visual) filters, with blackbody curves for three different temperatures. Star (a) is very hot— 10,000 K—so its B intensity is greater than its V intensity (as is actually the case for Rigel in Figure 17.8a). Star (b) has roughly equal B and V readings and so appears white. Its temperature is about 10,000 K. Star (c) is red; its V intensity greatly exceeds the B value, and its temperature is 3000 K (much as for Betelgeuse in Figure 17.8a).

17. 17.3 Stellar Temperatures
Stellar spectra are much more informative than the blackbody curves.
There are seven general categories of stellar spectra, corresponding to different temperatures.
From highest to lowest, those categories are:
O B A F G K M

18. 17.3 Stellar Temperatures Here are their spectra:
Figure Stellar Spectra Comparison of spectra observed for seven different stars having a range of surface temperatures. These are not actual spectra, which are messy and complex, but simplified artists’ renderings illustrating a few spectral features. The spectra of the hottest stars, at the top, show lines of helium and multiply ionized heavy elements. In the coolest stars, at the bottom, helium lines are absent, but lines of neutral atoms and molecules are plentiful. At intermediate temperatures, hydrogen lines are strongest. All seven stars have about the same chemical composition.

19. 17.3 Stellar Temperatures
Characteristics of the spectral classifications:

20. More Precisely 17-1: More on the Magnitude Scale
Converting from magnitude to luminosity in solar units: This graph allows us to perform this conversion simply by reading horizontally. A reduction of 5 in magnitude corresponds to an increase in a factor of 100 in luminosity, as it should.

21. 17.4 Stellar Sizes
A few very large, very close stars can be imaged directly using speckle interferometry. This is Betelgeuse.
Figure Betelgeuse (a) The swollen star Betelgeuse (shown here in false color) is close enough for us to resolve its size directly, along with some surface features thought to be storms similar to those that occur on the Sun. Betelgeuse is such a huge star (about 600 times the size of the Sun) that its photosphere exceeds the size of Mars’s orbit. Most of the surface features discernible here are larger than the entire Sun. (The dark lines are added contours of constant brightness, not part of the star itself.) (b) An ultraviolet view of Betelgeuse, in false color, as seen by a European camera onboard the Hubble Space Telescope, shows more clearly just how large this huge star is. (NOAO; NASA/ESA)

22. 17.4 Stellar Sizes
For the vast majority of stars that cannot be imaged directly, size must be calculated knowing the luminosity and temperature:
Giant stars have radii between 10 and 100 times the Sun’s
Dwarf stars have radii equal to, or less than, the Sun’s
Supergiant stars have radii more than 100 times the Sun’s

23. 17.4 Stellar Sizes Stellar radii vary widely:
Figure Stellar Sizes Star sizes vary greatly. Shown here are the estimated sizes of several well-known stars, including a few of those discussed in this chapter. Only part of the red-giant star Antares can be shown on this scale, and the supergiant Betelgeuse would fill the entire page.

24. More Precisely 17-2: Estimating Stellar Radii
Combining the Stefan-Boltzmann law for the power per unit area emitted by a blackbody as a function of temperature with the formula for the area of a sphere gives the total luminosity:
If we measure luminosity, radius, and temperature in solar units, we can write
L = R2T4

25. 17.5 The Hertzsprung-Russell Diagram
The H-R diagram plots stellar luminosity against surface temperature.
This is an H-R diagram of a few prominent stars:
Figure H–R Diagram of Well-Known Stars A plot of luminosity against surface temperature (or spectral class), known as an H–R diagram, is a useful way to compare stars. Plotted here are the data for some stars mentioned earlier in the text. The Sun, of course, has a luminosity of 1 solar unit. Its temperature, read off the bottom scale, is 5800 K—that of a G-type star. Similarly, the B-type star Rigel, at top left, has a temperature of about 11,000 K and a luminosity more than 10,000 times that of the Sun. The M-type star Proxima Centauri, at bottom right, has a temperature of about 3000 K and a luminosity less than 1/10,000 that of the Sun. (See also Overlay 1 of the acetate insert.)

26. 17.5 The Hertzsprung-Russell Diagram
Once many stars are plotted on an H-R diagram, a pattern begins to form:
These are the 80 closest stars to us; note the dashed lines of constant radius.
The darkened curve is called the main sequence, as this is where most stars are.
Also indicated is the white dwarf region; these stars are hot but not very luminous, as they are quite small.
Figure H–R Diagram of Nearby Stars Most stars have properties within the long, thin, shaded region of the H–R diagram known as the main sequence. The points plotted here are for stars lying within about 5 pc of the Sun. Each dashed diagonal line corresponds to a constant stellar radius, so that stellar size can be indicated on the same diagram as stellar luminosity and temperature.

27. 17.5 The Hertzsprung-Russell Diagram
An H-R diagram of the 100 brightest stars looks quite different:
These stars are all more luminous than the Sun. Two new categories appear here—the red giants and the blue giants.
Clearly, the brightest stars in the sky appear bright because of their enormous luminosities, not their proximity.
Figure H–R Diagram of Brightest Stars An H–R diagram for the 100 brightest stars in the sky is biased in favor of the most luminous stars—which appear toward the upper left—because we can see them more easily than we can the faintest stars. (Compare with Figure 17.14, which shows only the closest stars.)

28. 17.5 The Hertzsprung-Russell Diagram
This is an H-R plot of about 20,000 stars. The main sequence is clear, as is the red giant region.
About 90% of stars lie on the main sequence; 9% are red giants and 1% are white dwarfs.
Figure Hipparcos H–R Diagram This simplified version of the most complete H–R diagram ever compiled represents more than 20,000 data points, as measured by the European Hipparcos spacecraft for stars within a few hundred parsecs of the Sun.

29. 17.6 Extending the Cosmic Distance Scale
Spectroscopic parallax: Has nothing to do with parallax, but does use spectroscopy in finding the distance to a star.
Measure the star’s apparent magnitude and spectral class
Use spectral class to estimate luminosity
Apply inverse-square law to find distance

30. 17.6 Extending the Cosmic Distance Scale
Spectroscopic parallax can extend the cosmic distance scale to several thousand parsecs:
Figure Stellar Distances Knowledge of a star’s luminosity and apparent brightness can yield an estimate of its distance. Astronomers use this third rung on the distance ladder, called spectroscopic parallax, to measure distances as far out as individual stars can be clearly discerned—several thousand parsecs.

31. 17.6 Extending the Cosmic Distance Scale
The spectroscopic parallax calculation can be misleading if the star is not on the main sequence. The width of spectral lines can be used to define luminosity classes:
Figure Luminosity Classes (a) Approximate locations of the standard stellar luminosity classes in the H–R diagram. The widths of absorption lines also provide information on the density of a star’s atmosphere. The denser atmosphere of a main-sequence K-type star has broader lines (c) than a giant star of the same spectral class (b).

32. 17.6 Extending the Cosmic Distance Scale
In this way, giants and supergiants can be distinguished from main-sequence stars

33. 17.7 Stellar Masses Determination of stellar masses:
Many stars are in binary pairs; measurement of their orbital motion allows determination of the masses of the stars.
Visual binaries can be measured directly. This is Kruger 60:
Figure Visual Binary The period and separation of a binary-star system can be observed directly if each star is clearly seen. At the left is an orbital diagram for the double star Kruger 60; at the right are actual photographs taken in some of the years indicated. (Harvard College Observatory)

34. 17.7 Stellar Masses
Spectroscopic binaries can be measured using their Doppler shifts:
Figure Spectroscopic Binary Properties of binary stars can be determined indirectly by measuring the periodic Doppler shift of one star relative to the other as they move in their orbits. The spectra diagrammed here show a so-called single-line system, in which only one spectrum (from the brighter component) is visible. Reference lab spectra are also shown at the top and bottom at left. An observer of the binary stars at right would see the spectrum redshifted for the visible component moving away (as at top right). Likewise, the spectrum is blueshifted for motion toward the observer (as at bottom right).

35. 17.7 Stellar Masses
Finally, eclipsing binaries can be measured using the changes in luminosity.
Figure Eclipsing Binary If the two stars in a binary-star system happen to eclipse one another, additional information on their radii and masses can be obtained by observing the periodic decrease in starlight as one star passes in front of the other.

36. 17.7 Stellar Masses
Mass is the main determinant of where a star will be on the Main Sequence.
Figure Stellar Masses More than any other stellar property, mass determines a star’s position on the main sequence. Low-mass stars are cool and faint; they lie at the bottom of the main sequence. Very massive stars are hot and bright; they lie at the top of the main sequence.

37. More Precisely 17-3: Measuring Stellar Masses in Binary Stars
In order to measure stellar masses in a binary star, the period and semimajor axis of the orbit must be measured. Once this is done, Kepler’s third law gives the sum of the masses of the two stars. Then the relative speeds of the two stars can be measured using the Doppler effect; the speed will be inversely proportional to the mass. This allows us to calculate the mass of each star.

38. 17.8 Mass and Other Stellar Properties
This pie chart shows the distribution of stellar masses. The more massive stars are much rarer than the least massive.
Figure Stellar Mass Distribution The range of masses of main-sequence stars, as determined from careful measurement of stars in the solar neighborhood.

39. 17.8 Mass and Other Stellar Properties
Mass is correlated with radius and is very strongly correlated with luminosity:
Figure Stellar Radii and Luminosities (a) Dependence of stellar radius on mass for main-sequence stars; actual measurements are plotted here. The radius increases nearly in proportion to the mass over much of the range (as indicated by the straight line drawn through the data points). (b) Dependence of main-sequence luminosity on mass. The luminosity increases roughly as the fourth power of the mass (indicated again by the straight line).

40. 17.8 Mass and Other Stellar Properties
Mass is also related to stellar lifetime:
Using the mass–luminosity relationship:

41. 17.8 Mass and Other Stellar Properties
So the most massive stars have the shortest lifetimes—they have a lot of fuel but burn it at a very rapid pace.
On the other hand, small red dwarfs burn their fuel extremely slowly and can have lifetimes of a trillion years or more.

42. Summary of Chapter 17
Can measure distances to nearby stars using parallax
Apparent magnitude is related to apparent brightness
Absolute magnitude is a measure of the power output of the star
Spectral analysis has led to the defining of seven spectral classes of stars
Stellar radii can be calculated if distance and luminosity are known

43. Summary of Chapter 17 (cont.)
In addition to “normal” stars, there are also red giants, red supergiants, blue giants, blue supergiants, red dwarfs, and white dwarfs
Luminosity class can distinguish giant star from main-sequence one in the same spectral class
If spectrum is measured, can find luminosity; combining this with apparent brightness allows distance to be calculated

44. Summary of Chapter 17 (cont.)
Measurements of binary-star systems allow stellar masses to be measured directly
Mass is well correlated with radius and luminosity
Stellar lifetimes depend on mass; the more the mass, the shorter the lifetime