I originally published this post to my now-inactive blog, so if you know me from back when the Old Content Archive was ftlofacts, you’ve read this already.
If you know me from Science at Your Doorstep, then read on! It’s all new material!
I have to warn you, this is an astronomy post but it’s very mathy. If you love math, stick around—I’m glad to have you! If math makes you want to puke, then please, just click elsewhere; I’ve got plenty other posts for you. I’d hate to put you off astronomy entirely.
Ok, we’re talking about astronomy here. We all knew math would come along eventually, right?
Last time around, we talked about the way stars are named and classified based on what constellations they’re in and how bright they are compared to the other stars in that constellation.
For example, here’s Orion. I’ve shown you guys this image before. As I’ve explained before, this is a depiction of Orion as a way to map out the sky, not as a picture in the heavens. As you can see, many of the stars within Orion are accompanied by little squiggles.
Those little squiggles—as I have also explained before—are Greek letters, used to rank the stars according to brightness within any one constellation. Star charts have the added benefit of being visual diagrams, and can rank brightness according to the size in which the stars are drawn.
But I’ve also mentioned before that stars are all at varying distances from us. Some are quite close—Proxima Centauri is 4.2 light-years away—and some are insanely far. Yet some of the farthest stars, such as Sirius, are the brightest in the sky. And some of the closest ones—like Proxima Centauri—are some of the dimmest.
Astronomers have a way to calculate most things. How do they calculate the apparent intensities of these stars?
Note my words. We’re talking about apparent brightness and intensity right now. Later on, I’ll get into stars’ actual brightness, but for now we’re just talking about how bright they appear to be in the sky.
Naturally, actual brightness will have a much greater range. If a star as far away as Sirius can appear so much brighter than a star as close as Proxima Centauri, then imagine how much more energy it must be putting out from that far away.
We’re going to talk about stars’ intensities in terms of flux, rather than brightness. Flux is, according to my textbook, a measure of the light energy from a star that hits one square meter in one second.
So, we’re not talking about how much energy is actually being emitted. We’re talking about how much of it we see at any one time.
Naturally, there’s an equation for it…
…because when isn’t there an equation for a phenomenon concerning physics?
So let’s break this equation down.
M1 and M2 refer to the magnitudes of the two stars, according to the magnitude scale. If you missed my last post, I’ll replicate the magnitude scale here.
The measurement 2.5 results from a calculation. Astronomers wanted to make this equation agree with the magnitude scale above, even though it’s been around since pretty much the dawn of astronomy.
It’s simply easier to stick with the familiar, I suppose. Many of the stars were already classified according to this scale, so why change anything and have to reclassify everything?
In order to make the scale and the equation agree, astronomers agreed that if two stars differ by five magnitudes—say, a star of 10 mv and a star of 15 mv—then the ratio of their intensities is exactly 100.
What exactly does that mean?
Okay, so let’s reflect back on primary school. I know it was a while ago for most people 😉 What exactly is a ratio?
It’s a fraction. For example, if you have the ratio 1:2, it can also be written as “1 to 2” or, case in point, 1/2.
The fraction in this case is I2/I1, one star’s intensity—that is, how bright it appears to be—divided by the other. So, if you divide one star’s intensity by the other and those two stars are 5 magnitudes apart, you get 100.
To turn this into an equation, we need to know what the ratio of intensities for stars that differ by one magnitude would be. Otherwise, the equation would only work for stars whose intensity ratio is multiples of five. And we want to be more precise than that.
Easy. Take the fifth root of 100. The answer is 2.512, roughly.
What does that mean, anyway? Basically, whichever star appears brighter will appear 2.512 times brighter than the other. And remember, we’re dealing with stars that are exactly one magnitude apart.
Now to break down the rest of the equation…
I’m not going to explain logarithms, except to say that they are the inverse operations of exponential functions. If that makes any sense to you, that’s cool. If not, trust the equation.
I’ve already explained I2/I1—it’s the intensity ratio between the two stars we’re talking about.
Let’s assume, if you’re still here reading this post by now, that you’re actually somewhat interested in using this equation. Most calculators these days allow you to input logarithms (logs for short).
This calculator is pretty advanced, but it makes the job easy. The “log” button is right next to the 7.
If you do end up using this equation, simply isolate the variable you want to solve for and use that log key.
Here’s another less advanced Texas Instruments calculator. (Sorry, can you tell I love the brand? I honestly don’t have another example to show you.)
Here, you can find the “log” button near the top of the keypad, right under the “2nd” button.
And that’s it for this post. Sorry it’s a little shorter than usual—but I figured I didn’t really need to do a full astronomy post just on math and calculations. Honestly, this isn’t the fun part. Though it is pretty fun. I love how astronomy is a measurable science, rather than the philosophy it was when it started out centuries ago.
But that is a discussion for another time. For now, keep an eye out for my post on the celestial sphere! I figured at least one non-mathy post was the least I could do for you guys 😉