Chapter 16--Properties of Stars

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luminosity if we can first measure its distance or to calculate a star’s distance if we somehow know its luminosity. (The luminosity–distance formula is strictly correct only if interstellar dust does not absorb or scatter the starlight along its path to Earth.) Although watts are the standard units for luminosity, it’s often more meaningful to describe stellar luminosities in comparison to the Sun by using units of solar luminosity: L Sun 3.8 10 26 watts. For example, Proxima Centauri, the nearest of the three stars in the Alpha Centauri system and hence the nearest star besides our Sun, is only about 0.0006 times as luminous as the Sun, or 0.0006L Sun . Betelgeuse, the bright left-shoulder star of Orion, has a luminosity of 38,000L Sun , meaning that it is 38,000 times more luminous than the Sun. Measuring Apparent Brightness We can measure a star’s apparent brightness by using a detector, such as a CCD, that records how much energy strikes its light-sensitive surface each second. For example, such a detector would record an apparent brightness of 2.7 10 8 watt per square meter from Alpha Centauri A. The only difficulties involved in measuring apparent brightness are making sure the detector is properly calibrated and, for ground-based telescopes, taking into account the absorption of light by Earth’s atmosphere. No detector can record light of all wavelengths, so we necessarily measure apparent brightness in only some small range of the complete spectrum. For example, the human eye is sensitive to visible light but does not respond to ultraviolet or infrared photons. Thus, when we perceive a star’s brightness, our eyes are measuring the apparent brightness only in the visible region of the spectrum. We can derive the luminosity–distance formula by extending the idea illustrated in Figure 16.2. Suppose we are located a distance d from a star with luminosity L. The apparent brightness of the star is the power per unit area that we receive at our distance d. We can find this apparent brightness by imagining that we are part of a giant sphere with radius d, similar to any one of the three spheres in Figure 16.2. The surface area of this giant sphere is 4p d 2 , and the star’s entire luminosity L must pass through this surface area. (The surface area of any sphere is 4p radius 2 .) Thus, the apparent brightness at distance d is the power per unit area passing through the sphere: apparent brightness L 4p d 2 This is our luminosity–distance formula. 524 part V • Stellar Alchemy When we measure the apparent brightness in visible light, we can calculate only the star’s visible-light luminosity. Similarly, when we observe a star with a spaceborne X-ray telescope, we measure only the apparent brightness in X rays and can calculate only the star’s X-ray luminosity. We will use the terms total luminosity and total apparent brightness to describe the luminosity and apparent brightness we would measure if we could detect photons across the entire electromagnetic spectrum. (Astronomers refer to the total luminosity as the bolometric luminosity.) Measuring Distance Through Stellar Parallax Once we have measured a star’s apparent brightness, the next step in determining its luminosity is to measure its distance. The most direct way to measure the distances to stars is with stellar parallax, the small annual shifts in a star’s apparent position caused by Earth’s motion around the Sun [Section 2.6]. Recall that you can observe parallax of your finger by holding it at arm’s length and looking at it alternately with first one eye closed and then the other. Astronomers measure stellar parallax by comparing observations of a nearby star made 6 months apart (Figure 16.3). The nearby star appears to shift against the background of more distant stars because we are observing it from two opposite points of Earth’s orbit. The star’s parallax angle is defined as half the star’s annual back-and-forth shift. Measuring stellar parallax is difficult because stars are so far away, making their parallax angles very small. Even the nearest star, Proxima Centauri, has a parallax angle of only 0.77 arcsecond. For increasingly distant stars, the parallax angles quickly become too small to measure even with our highest-resolution telescopes. Current technology Mathematical Insight 16.1 The Luminosity–Distance Formula star’s luminosity surface area of imaginary sphere Example: What is the Sun’s apparent brightness as seen from Earth? Solution: The Sun’s luminosity is LSun 3.8 1026 watts, and Earth’s distance from the Sun is d 1.5 1011 meters. Thus, the Sun’s apparent brightness is: L 4p d 2 1.3 103 watts/m2 3.8 1026 watts 4p (1.5 1011 m) 2 The Sun’s apparent brightness is about 1,300 watts per square meter at Earth’s distance. It is the maximum power per unit area that could be collected by a detector on Earth that directly faces the Sun, such as a solar power (or photovoltaic) cell. In reality, solar collectors usually collect less power because Earth’s atmosphere absorbs some sunlight, particularly when it is cloudy.
Every January, we see this: July d 1 AU nearby star distant stars Figure 16.3 Parallax makes the apparent position of a nearby star shift back and forth with respect to distant stars over the course of each year. If we measure the parallax angle p in arcseconds, the distance d to the star in parsecs is 1 p . The angle in this figure is greatly exaggerated: All stars have parallax angles of less than 1 arcsecond. p Not to scale Every July, we see this: January allows us to measure parallax only for stars within a few hundred light-years—not much farther than what we call our local solar neighborhood in the vast, 100,000-light-yeardiameter Milky Way Galaxy. Here is one of several ways to derive the formula relating a star’s distance and parallax angle. Figure 16.3 shows that the parallax angle p is part of a right triangle, the side opposite p is the Earth– Sun distance of 1 AU, and the hypotenuse is the distance d to the object. You may recall that the sine of an angle in a right triangle is the length of its opposite side divided by the length of the hypotenuse. In this case, we find: sin p 1AU d If we solve for d, the formula becomes: d 1 AU sin p By definition, 1 parsec is the distance to an object with a parallax angle of 1 arcsecond (1), or 1/3,600 degree (because that 1° 60 and 160). Substituting these numbers into the parallax formula and using a calculator to find that sin 14.84814 106 ,we get: 1 AU 1 AU 1 pc sin 1 4.84814 106 206,265 AU That is, 1 parsec 206,265 AU, which is equivalent to 3.09 1013 length of opposite side length of hypotenuse km or 3.26 light-years. (Recall that 1 AU 149.6 million km.) By definition, the distance to an object with a parallax angle of 1 arcsecond is 1 parsec,abbreviated pc.(The word parsec comes from the words parallax and arcsecond.) With a little geometry and Figure 16.3 (see Mathematical Insight 16.2), it is possible to show that: 1 pc 3.26 light-years 3.09 10 13 km If we use units of arcseconds for the parallax angle, a simple formula allows us to calculate distances in parsecs: 1 d (in parsecs) p (in arcseconds) For example, the distance to a star with a parallax angle of 1 2 arcsecond is 2 parsecs, the distance to a star with 1 a parallax angle of 10 arcsecond is 10 parsecs, and the dis- 1 tance to a star with a parallax angle of 100 arcsecond is 100 parsecs. Astronomers often express distances in parsecs or light-years interchangeably. You can convert quickly between them by remembering that 1 pc 3.26 light-years. Thus, 10 parsecs is about 32.6 light-years; 1,000 parsecs, or 1 kiloparsec (1 kpc), is about 3,260 light-years; and 1 million parsecs, or 1 megaparsec (1 Mpc), is about 3.26 million light-years. Enough stars have measurable parallax to give us a fairly good sample of the many different types of stars. For example, we know of more than 300 stars within about 33 light-years (10 parsecs) of the Sun. About half are binary star systems consisting of two orbiting stars or Mathematical Insight 16.2 The Parallax Formula We need one more fact from geometry to derive the parallax formula given in the text. As long as the parallax angle, p, is small, sin p is proportional to p. For example, sin 2 is twice as large as sin 1, and sin 1 2 is half as large as sin 1.(You can verify these examples with your calculator.) Thus, if we use 1 2 instead of 1 for the parallax angle in the formula above, we get a distance of 2 pc 1 instead of 1 pc. Similarly, if we use a parallax angle of ,we 10 get a distance of 10 pc. Generalizing, we get the simple parallax formula given in the text: 1 d (in parsecs) p (in arcseconds) Example: Sirius, the brightest star in our night sky, has a measured parallax angle of 0.379.How far away is Sirius in parsecs? In light-years? Solution: From the formula, the distance to Sirius in parsecs is: 1 d (in pc) 2.64 pc 0.379 Because 1 pc 3.26 light-years, this distance is equivalent to: 2.64 pc 3.26 light-years 8.60 light-years pc chapter 16 • Properties of Stars 525
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Chapter 16--Properties of Stars

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