Magnitudes
Published on Jun 11, 2007 at 4:22 pm.
4 Comments.
Filed under astronomy.
In my last post, I mentioned the term “magnitude” several times. From the context, even if you don’t know anything about astronomy, you can tell that magnitude is a measure of how bright an astronomical object appears. But, what sort of measure? How is it quantified?
The concept of magnitudes goes all the way back to the 2nd Century B.C., to a Greek named Hipparchus. Born in Nicaea, Hipparchus lived most of his life in Rhodes. Hipparchus came up with a system of ranking stars by brightness. Now, we don’t really know exactly how Hipparchus’ system worked, because his writings have not survived. Recently, a carving of one of his star charts was found, but still we have nothing about his magnitude system. What we do have, though, is Ptolemy’s explanation of magnitudes written a few hundred years later, based on Hipparchus’ work. A few other secondary sources exist, but very little of Hipparchus’ original writings. There have been a number of attempts to reconstruct Hipparchus’ actual system, but to my knowledge that has never really been done. Still, in my astronomy classes, I attribute the system that we use today to Hipparchus, though it has been through several small changes since then.
Basically, Hipparchus grouped stars by order of brightness. The first group were the brightest stars. Second group consisted of the second brightest stars. The third group was the third brightest, and so forth. I have read different sources that have said that Hipparchus had five or six groups. It is believed by many that Ptolemy extended Hipparchus’ work somewhat in this area. These groups became magnitudes. So, first magnitude stars are brighter than second magnitude, and second magnitude stars are brighter than third magnitude stars. This has the effect of making smaller magnitudes brighter than larger magnitudes. This confuses my students no end! Ptolemy set the dimmest stars at sixth magnitude. Later astronomers, though, noted that not all first magnitude stars are equally bright. Some of them, like Vega, were as much brighter than other first magnitude stars as first magnitude stars are brighter than second magnitude stars. So, a new magnitude was eventually added: zeroth magnitude. But, the stars Canopus and Sirius are even brighter yet. So, they were given a magnitude of -1. Jupiter gets even brighter that that, and Venus even brighter than Jupiter. So, the magnitude system can be extended negative. And, of course, with binoculars or telescopes, you can see dimmer objects, so the magnitude scale can be extended past magnitude six for dimmer objects.
However, as astronomy became more quantitative, astronomers wanted to measure magnitudes rather than just guessing. And, it quickly became apparent that not all second magnitude stars are equally bright. Some are really magnitude 1.8, and others are magnitude 2.2, and so forth. So, a new scale had to be determined.
Under the old system, the range of stellar brightnesses from magnitude 1 to magnitude 6 was close to a factor of 100. So, it was decided to define a difference of 5 magnitudes as a difference of 100 in brightness. Now, comes an interesting thing. That means that a magnitude 1 star is 100 times brighter than a magnitude 6 star. A magnitude 6 star is 100 times brighter than a magnitude 11 star, and a magnitude 11 star is 100 times brighter than a magnitude 16 star. So, how much brighter is a magnitude 1 star from a magnitude 11 star, or perhaps a magnitude 16 star? My students’ first guess is normally that from magnitude 1 to magnitude 11 is ten magnitudes, and since 5 magnitudes is 100 times difference, then 10 magnitudes must be 200 times difference. Wrong! Remember, the magnitude 6 star is 100 times brighter than magnitude 11. Magnitude 1 is 100 times brighter than magnitude 6. So, that means it is 100 times brighter than something 100 times brighter. In other words, the magnitude 1 star is
100×100=10000
times brighter than the magnitude 11 star! Similarly, the magnitude 1 star is one million times brighter than a magnitude 16 star! So, what is going on here? It works this way because your eye picks up light roughly logarithmically.
But, how much brighter is a magnitude 3 star from a magnitude 6 star? That is a difference of 3 magnitudes. So, would that mean that it is 3/5 of 100, or 60 times difference? Uh, no. Logarithms don’t work that way. Instead, we need to think back to the definition of what a difference of 5 magnitudes is: a difference of 100 times brightness. So that means whatever a difference of 1 magnitude is, then the difference in brightness of five of them, 100, is what you get when you multiply five of them together. Hmm. That means that the difference of 1 magnitude must be the fifth root of one hundred. That turns out to be about 2.512. So, a difference of 3 magnitudes is going to be (2.512)3 = 15.9 times difference in brightness.
You don’t really need a whole number exponent. As long as you have the magnitudes, you can raise 2.512 to whatever power the difference in magnitudes comes out to be using a modern calculator. The ratio of brightnesses (symbol L) between two celestial bodies can be given by the equation:
So, in my last post, I said that Venus is currently magnitude -4.2 and Jupiter is -2.6. Now, I want to know how much brighter that makes Venus than Jupiter. The difference in magnitude is 1.6. Plugging that in for (m2-m1) in the equation above, you get a difference in brightness of just under 4.4. So, Venus appears nearly 4.4 times brighter than Jupiter.
But, how do we even know the fractional magnitudes to begin with? A couple hundred years ago, astronomers decided to make the star Vega the reference star. Vega was defined to be magnitude zero. So, once you have a reference, then you can figure out any other magnitude after measuring the difference in brightnesses using the formula:
Eventually, the Vega standard was dropped once magnitudes had become set. Now, modern measurements of Vega indicate that it really has magnitude +0.03, but that is still pretty close to zero!
So, that is what astronomers mean when they talk about magnitudes. I should point out that this really refers to a very special case, what we call apparent visual magnitude. We now have the ability to measure how bright an object is in non-visual wavelengths, such as infrared and ultraviolet. And, some objects emit more light in those wavelengths than they do in visual light. And, of course, the instrument that you are using to measure the light may be more sensitive to some wavelengths than to others, or may have a different range of wavelengths than the human eye. So, there are corrections often made to the magnitudes measured. There is even a theoretical value called the bolometric magnitude that would be the magnitude of an object if all of its radiation were measured somehow. But, for most of you, simply the most basic case of apparent visual magnitude is all that you care about.
-Astroprof
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