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3. STELLAR BRIGHTNESS

After position, the second obvious property of stars is their brightness. In about 150 BC, Hipparcos made the first known catalog of stars. He listed about 5000 stars and ranked them according to brightness. He gave the brightest stars (about 25 of them) the rank of 1st magnitude. Then he classified the rest 2nd through 5th magnitude, with 5th magnitude stars designating the faintest stars that he could see on a moonless night. Today, astronomers still use the magnitude system to define the brightness of stars, but they have refined the system to about 3 significant figures accuracy. Thus, the stars that Hipparcos classified as first magnitude now range in magnitude from about m = -1.47 for the brightest Star, Sirius, to a star such as Deneb (a-Cygni) with m = 1.26. The symbol m is called the apparent magnitude.

The magnitude scale is described below. You don't need to memorize it. Just remember this: if the magnitude increases by adding 5, the actual brightness decreases by a factor of 100. That means, for example, that a star with m = 6.5 is 100 times fainter than a star with magnitude 1.5, and a star with magnitude 11.5 is 100 times fainter than a star with magnitude 6.5. So, a star with magnitude 11.5 is (100 x 100) = 10,000 times fainter than a star with magnitude 1.5.

The brightness (B), or magnitude, of a star is an extrinsic property, because it depends not only on the star's actual light output, or luminosity (L), but also on the distance of the star. Imagine two stars, Alpha and Beta, having equal luminosity. If Beta is the same distance from the Sun as Alpha, it will appear equally bright as Alpha. But if Beta is twice as far, it will be 1/4 as bright as Alpha. Three times as far, 1/9 as bright, and so on. This decrease in brightness with increasing distance (D) is called the inverse square law, which is expressed by the equation B = L/(4p D²). See if you can answer this: suppose that you have two identical stars, Alpha and Beta, with equal luminosity. Alpha has apparent magnitude m = 0. Beta is ten times more distant from the Sun than Alpha. What is the apparent magnitude of Beta?

If we know the distance of a star (say, as a result of measuring its parallax), we can infer its luminosity from its observed brightness and the inverse square law: L = 4p D²B. Thus, we can infer an intrinsic quantity, L, which describes a property of the star itself, from two extrinsic quantities, D, and B, which we can measure.

Besides for distance, another factor can cause the brightness of a star to decrease, more than the inverse square law would predict. That is absorption by interstellar dust. Suppose again two identical stars, Alpha and Beta, having the same luminosity and both at the same distance. But now suppose that Beta lies behind a dust cloud and Alpha doesn't. Beta will be fainter because the dust absorbs part some of its light. Thus a star's brightness (an extrinsic property) depends not only on its luminosity (an intrinsic property), but also on two other factors, its distance and whether or not a dust cloud happens to lie between the star and the Sun.

We can tell whether a star is dimmed as a result of obscuration by interstellar dust, because the dust makes the star appear redder than it actually is. That happens because the dust absorbs more blue light than red light. We see this effect on Earth, especially during a desert sunset. The Sun is very red when it sets because the dust in the Earth's atmosphere absorbs the blue sunlight and lets the red sunlight through.

To understand how stars work, astronomers want to measure intrinsic properties, such as luminosity (L). If we can measure a star's parallax (p), we can calculate its distance, D = 1/p. Then, if we measure its brightness (B), and if we know that the absorption due to dust is negligible, we can calculate its luminosity (L) from the inverse square law above. (In fact, there is very little interstellar dust within about 200 pc of the Sun, so for nearby stars we don't have to worry about this absorption.) So, we can infer the luminosity of thousands of nearby stars from this procedure.

There is one more term you need to know: Absolute Magnitude. Absolute magnitude (denoted capital M) is a measure of luminosity (L), while apparent magnitude (denoted m) is a measure of brightness (B). The important point to remember about the definition of absolute magnitude is this: the absolute magnitude is the apparent magnitude that a star would have if it were at a distance of 10 pc (with no absorption by interstellar dust). If the star is at a distance greater than 10 pc, its absolute magnitude is less than its apparent magnitude (see table of the brightest stars in the sky). For example, the star Arcturus is at a distance D = 11 pc, and has apparent magnitude m = -0.06. The absolute magnitude of Arcturus is M = -0.3, only a small amount (-0.24) less than m. But now take a -Cen, with m = -0.01, almost equally bright as Arcturus. But since a-Cen (D = 1.3 pc) is much closer than Arcturus, its absolute magnitude (M = 4.4) is quite a bit greater than that of Arcturus (M = -0.3). That means that a-Cen is much (about 75 times) less luminous than Arcturus. In fact, a-Cen is just 1.6 times more luminous than the Sun.

Here are some simple mathematical formulas that define the magnitude scale:

Definitions	Conversions	Formula
L: Luminosity in solar units	Given M, find L	L = 10^0.4(5-M)
M: Absolute Magnitude	Given L, find M	M = 5 - 2.5 log L
m: apparent magnitude	Given M and D, find m	m = M + 5 log (D/10)
D: distance in parsecs	Given M and m, find D	D = 10^1+(m-M)/5

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Last modified February 3, 2001
Copyright by Richard McCray