Greek astronomers, from Hipparchus in the second century b.c.e to Ptolemy in the second century c.e., undertook the classification of stars strictly by evaluating how bright they appear to be relative to each other. This comparison made sense because back then astronomers assumed the stars were all at the same distance from us, and, therefore, differences in brightness due to different stellar distances from Earth were not expected. Classification regardless of distance is still useful to help us navigate around the night sky, and so we will study it now. In Section 10-3 we will add the effects of different distances on how bright stars appear from Earth.

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The brightnesses of stars measured without regard to their distances from Earth are called apparent magnitudes, denoted by lowercase m. The brightest stars were originally said to be of first magnitude, and their apparent magnitudes were designated m = +1. Those stars that appeared to be about half as bright as first-magnitude stars were said to be second-magnitude stars (designated m = +2), and so forth, down to sixth-magnitude stars, the dimmest ones visible to the unaided eye. (Greek astronomers did not try to classify the Sun’s dazzling brightness in this scheme.) Because stars do not appear with discrete levels of brightness, this system has noninteger magnitudes as well, such as +3.5 or +4.8.

More quantitative methods of classifying stars were developed in the mid-nineteenth century. While maintaining the basic idea that the apparent magnitudes of brighter objects are smaller numbers than the apparent magnitudes of dimmer ones, the scale used today begins by giving the star Vega an apparent magnitude of 0.0. Because some bodies are brighter than Vega, astronomers assign negative numbers to the apparent magnitudes of the very brightest objects. Sirius, the brightest star in the night sky, has an apparent magnitude of m = −1.44. Figure 10-2a shows Sirius along with the apparent magnitudes of some of the stars in Orion. With this convention we can describe other bright objects in the sky, such as the Sun, the Moon, comets, and planets. At its brightest, Venus shines with an apparent magnitude of m = −4.4, the full Moon has an apparent magnitude of m = −12.6, and the Sun has an apparent magnitude of m = −26.7. Remember: Objects with negative apparent magnitudes appear brighter than those with positive apparent magnitudes—the more negative, the brighter.

Astronomers also extended the magnitude scale so that dimmer stars, visible only through telescopes, could be included in the magnitude system. For example, the dimmest stars visible through a pair of powerful binoculars have an apparent magnitude of about m = +10. Time-exposure photographs through telescopes reveal even dimmer stars. The Keck telescopes and the Hubble Space Telescope, among others, image stars as dim as magnitude m = +30. Figure 10-2b illustrates the modern apparent magnitude scale. Similarly, apparent magnitudes from entire groups of stars, such as distant galaxies, can be measured.

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Knowing, as we now do, that stars are at different distances from us, the apparent magnitudes do not directly reveal fundamental stellar properties. All other things being equal, the closer of two identical stars appears brighter to us (has a smaller apparent magnitude) than the farther star. We take the different distances into account with either of two measures: absolute magnitude and luminosity.

To determine the total energy emitted by each star in space, astronomers need to take into account that different stars are at different distances from Earth. They do this by calculating the brightness each star would have if they were all at the same distance from here. Knowing how far away a star really is and how bright it appears (its apparent magnitude), we can calculate how bright it would be at any distance. Absolute magnitude, M, is the brightness each star would have at a distance of 10 pc (32.6 ly). Unfortunately, absolute magnitudes have the same counterintuitive numeric scale as apparent magnitudes.

To understand the relationship between the apparent magnitude and the absolute magnitude, we need to know how the brightness of an object changes with distance. Suppose we observe two identical stars, one twice as far away as the other. How much dimmer will the farther one appear to be, as seen from Earth? Light moving outward from a source spreads over increasingly larger areas of space and, therefore, its brightness decreases with increasing distance. The inverse-square law provides the rule for just how quickly the brightness of an object changes with distance.

Imagine light moving out from a star (Figure 10-3a). Start with a small square area of light that has moved out a distance d = 1. The light in that square has a certain brightness. When the same square of light has gone twice as far (d = 2), you can see on the figure that it has become 4 times larger. The light in each of the same-sized squares at d = 2 contains one-quarter of the photons that were in the single square at d = 1. Therefore, each small square at d = 2 is one-quarter as bright as the same-sized square at d = 1. Similarly, when the square of light has moved to d = 3, there are now nine squares the same size as the original, each one-ninth as bright as the original square at d = 1. For example, the Sun emits 3.83 × 10²⁶ W (watts) of power (compared to 100 W for a bright incandescent home lightbulb). The Sun’s power is spread over a wider area as it travels through space. When it passes Mercury, sunlight provides 9140 watts per square meter (W/m²) of space. That same energy has spread out so much that by the time it passes us at 1 AU from the Sun, it provides only 1370 W/m². You see the same effect every day when you look at a car’s headlights at different distances (Figure 10-3b).

This inverse-square law can be summarized mathematically as follows:

Returning to our two identical stars, one twice as far away as the other, the farther one’s brightness decreases to , or , of the brightness of its closer twin by the time the starlight gets to us.

Absolute magnitudes are usually determined by calculations based on apparent magnitudes. Of course, we cannot just move a star to 10 pc distance and then remeasure its apparent magnitude. However, we can calculate the absolute magnitude of a nearby star: We first measure its apparent magnitude and then we find its distance by measuring its parallax angle. Combining these numbers, as shown in Discovery 10-2: The Distance–Magnitude Relationship, gives the star’s absolute magnitude.

Knowing the Sun’s true distance and its apparent magnitude, we can use the inverse-square law to determine how bright it would appear at 10 pc. At that distance, it would have an apparent magnitude of m = +4.8. Therefore, the absolute magnitude of the Sun is M = +4.8.

Because absolute magnitudes tell astronomers how bright stars are compared with one another, this information can be used to evaluate models of stellar evolution, which we discuss in the next few chapters. Absolute magnitudes range from roughly M = −10 for the brightest stars to M = +17 for the dimmest. Although absolute magnitudes give us comparisons between the energy outputs of stars, we also need to know the total energy they release. The total amount of electromagnetic power (energy emitted each second) is called a star’s luminosity. The greater the luminosity, the intrinsically brighter the star is compared to stars with lower luminosities. Therefore, the smaller or more negative a star’s absolute magnitude, the greater its luminosity. For convenience, stellar luminosities are expressed in multiples of the Sun’s luminosity, denoted L_⊙. As we saw earlier, this value is about 3.83 × 10²⁶ W. The intrinsically brightest stars (M = −10) have luminosities of 10⁶ L_⊙. In other words, each of these stars has the energy output of a million Suns. The dimmest stars (M = +17) have luminosities of 10⁻⁵ L_⊙. We will provide both luminosity and absolute magnitude data about stars in the chapters that follow.

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