Luminosity is an intrinsic property of a starit does not depend in any way on the location or motion of the observer. It is sometimes referred to as the stars absolute brightness. However, when we look at a star, we see not its luminosity, but rather its apparent brightnessthe amount of energy striking unit area of some light-sensitive surface or device (such as a human eye or a CCD chip) per unit time. In this section, we discuss how these important quantities are related to one another.
Another Inverse-Square Law
Figure 10.4 shows light leaving a star and traveling through space. Moving outward, the radiation passes through imaginary spheres of increasing radius surrounding the source. The amount of radiation leaving the star per unit timethe stars luminosityis constant, so the farther the light travels from the source, the less energy passes through each unit of area. Think of the energy as being spread out over an ever-larger area, and therefore spread more thinly, or "diluted," as it expands into space. Because the area of a sphere grows as the square of the radius, the energy per unit areathe stars apparent brightnessis inversely proportional to the square of the distance from the star. Doubling the distance from a star makes it appear 22, or four, times dimmer. Tripling the distance reduces the apparent brightness by a factor of 32, or nine, and so on.
Of course, the stars luminosity also affects its apparent brightness. Doubling the luminosity doubles the energy crossing any spherical shell surrounding the star and hence doubles the apparent brightness. The apparent brightness of a star is therefore directly proportional to the stars luminosity and inversely proportional to the square of its distance:
Determining a stars luminosity is a twofold task. First, the astronomer must determine the stars apparent brightness by measuring the amount of energy detected through a telescope in a given amount of time. Second, the stars distance must be measuredby parallax for nearby stars and by other means (to be discussed later) for more distant stars. The luminosity can then be found using the inverse-square law. Note that this is basically the same reasoning we used earlier in our discussion of how astronomers measure the solar luminosity (in our new terminology, the solar constant is just the apparent brightness of the Sun). (Sec. 9.1)
Instead of measuring apparent brightness in SI units (for example, watts per square meter W/m2, the unit in which we expressed the solar constant in Section 9.1), optical astronomers find it more convenient to work in terms of a construct called the magnitude scale. This scale dates back to the second century B.C., when the Greek astronomer Hipparchus ranked the naked-eye stars into six groups. The brightest stars were categorized as first magnitude. The next brightest stars were labeled second magnitude, and so on, down to the faintest stars visible to the naked eye, which were classified as sixth magnitude. The range one (brightest) through six (faintest) spanned all the stars known to the ancients. Notice that a large magnitude means a faint star.
When astronomers began using telescopes with sophisticated detectors to measure the light received from stars, they quickly discovered two important facts about the magnitude scale. First, the one through six magnitude range defined by Hipparchus spans about a factor of 100 in apparent brightnessa first-magnitude star is approximately 100 times brighter than a sixth-magnitude star. Second, the characteristics of the human eye are such that a change of one magnitude corresponds to a factor of about 2.5 in apparent brightness. In other words, to the human eye a first-magnitude star is roughly 2.5 times brighter than a second-magnitude star, which is roughly 2.5 times brighter than a third-magnitude star, and so on. (By combining factors of 2.5, we confirm that a first-magnitude star is indeed (2.5)5 100 times brighter than a sixth-magnitude star.)
Apparent magnitude measures a stars apparent brightness when seen at the stars actual distance from the Sun. To compare intrinsic, or absolute, properties of stars, however, astronomers imagine looking at all stars from a standard distance of 10 pc. (There is no particular reason to use 10 pcit is simply convenient.) A stars absolute magnitude is its apparent magnitude when viewed from a distance of 10 pc. Because distance is fixed in this definition, absolute magnitude is a measure of a stars absolute brightness, or luminosity. The Suns absolute magnitude is 4.8. In other words, if the Sun were moved to a distance of 10 pc from Earth, it would be only a little brighter than the faintest stars visible in the night sky. As discussed further in More Precisely 10-1, the numerical difference between a stars absolute and apparent magnitudes is a measure of the distance to the star.
Two stars are observed to have the same apparent magnitude. Based on this information, what, if anything, can be said about their luminosities?