The Devil's In the Magnitudes: Part 1

With roots in antiquity, the magnitude system can cause fits and gnashing of teeth. But with a little introduction, it becomes a wonderful and essential tool.

by James C. White II

Reprinted from Mercury, May.-Jun. 1998.

With roots in antiquity, the magnitude system can cause fits and gnashing of teeth. But with a little introduction, it becomes a wonderful and essential tool.

All one needs is an evening sky, and then the accounting begins. "Oh, but there's a bright one, darling." Stars appear as luminous dots, but those dots have different brightnesses. Is this a real difference or merely a perception problem?

A little over two millennia ago, a Greek named Hipparchus stood on his balcony and compared stars' brightnesses. From his position on the island of Rhodes, the heavens must have seemed gaspingly deep and full of stars. Hipparchus was an excellent observer and pushed his vision to the very limit by ranking stars according to how bright they appeared to him. The brightest stars he placed in the category first magnitude, the next brightest in the second magnitude bin, and so on. The very faintest stars visible to Hipparchus were categorized as sixth magnitude. This was the birth of our modern magnitude system, and note from the beginning that the fainter the star, the larger its magnitude number.

Figure 1 Brighter stars in the constellation Leo. Shown by nine of the brighter stars are the stars' magnitudes. Illustration courtesy of author.

In the 18th and l9th centuries, after telescopes were fully installed in astronomical observations, a neat fact was discovered. It turns out that Hipparchus's 5-magnitude scale spans a factor of 100 in the amount of fight we receive from a star–specifically, a factor of 100 in the energy falling on a photographic plate, a CCD detector, or your eye per unit area per unit time.

Norman Pogson said, in 1856, that we should define a difference of 5 magnitudes to be exacta difference of 100 in energy flux. It might seem that leaving the old, backwardseeming system alone is a good idea, but there's a problem. Hipparchus's magnitude bins are too broad: Both Sirius and Vega are classed as first magnitude objects, but we know that Sirius is, in fact, quite a bit brighter than Vega. The old system was adequate, but tweaking like this, and allowing other than whole-number magnitudes, was required.

A little ciphering shows that on Pogson's scale a difference of one magnitude corresponds to a brightness ratio of the fifth root of 100, about 2.512. And you know what? This is just about what your eye tells you–one magnitude does, indeed, correspond to an energy flux difference of about 2.5! I know all this is not neat looking, but it is neat in principle. Consider two stars, one of 3rd magnitude and the other a dimmer 6th magnitude. But how much dimmer? Well, there is a difference of three magnitudes between them, and that amounts to an energy-flux ratio of (2.512)3 = 15.85. Hence, even though there's a difference of three magnitudes between the stars, we can now say directly that the fainter one is about 16 times fainter. All this leads to a good relation between magnitude and energy-flux: (2.512)^m is the ratio of energy flux for two objects that differ in magnitudes by m.

To use two bright examples, consider again Sirius at -1.46 mag and Vega at 0.03 mag. m for the two is 0.03 - (-1.46) = 1.49. And with this we see that (2.512)^1.49 = 3.94. This is great! 5 We can say that Sirius is about four times brighter than Vega. They both appear bright, but this enables us to quantify that difference. With our current magnitude system, we can quite easily put numbers to those apparent differences between objects' brightnesses. But we are only beginning. Just think of how bright a 60-watt light bulb appears from 2 meters away. Now, will that bulb appear to be equally bright from a distance of one kilometer? This is what we must address next time, because distance, you see, matters.

Project Guidelines
It is impossible for us to ignore differences in brightness. We are comparators, measurers of everything. And this project is an introduction to the world of magnitudes and stellar brightness. The items you will need are the SkyCharts from this Mercury or those from Sky & Telescope or Astronomy magazines, paper, pencil, binoculars, and a (gulp!) calculator. All the skycharts available from these sources have legends that include magnitude scales for the charts. My mission for you is to get outside and pay attention to stars'brightnesses. I'm giving you a start by including here a "magnitude chart" for the constellation Leo. Compare the stars on the chart with what you see in the sky. As my grand mother used to say, "This'll learn ya," to recognize subtle brightness differences between stars. And finally, let's exercise the computer in all of us. Use the magnitudes provided on the Leo chart, and inferred from the scales on the formal sky charts, to calculate real differences in energy flux. As an example, consider the difference between the Leo star with magnitude 1.35 (named Regulus) and the one with magnitude 2.14 (Denebola). That's a m of 0.79. Thus, Regulus is (2.512)^0.79, or about two times brighter than Denebola.

JAMES C. WHITE II is editor of Mercury and a professor in the Physics and Astronomy Department at Middle Tennessee State University He admits that his old brightness scale of "dim," "bright," and "it hurts," is too coarse for good astronomical research. His email address is jwhite@physics.mtsu.edu.

Copyright 1998, 2001 by Astronomical Society of the Pacific, all rights reserved.