3e - 1. Photometry Basics

In one of my earlier posts I discussed the information that can be gleaned from the HR diagram, but what I didn't really discuss is how such diagrams are really constructed. I mentioned that it's necessary to either pick stars that you know the distance to (so you can correct for dimming due to distance) or stars in a cluster that are all at the same distance (so there is no variation).

But what I didn't mention is precisely how astronomers go about getting the information for the brightness and the temperature. If you've read my post on the HR diagram, I've mentioned one way: You find the peak emission and use Wien's Law to get the temperature.

But doing that requires getting the spectra of the star. And therein lies the problem. To get the spectra of a star, you have to pass it through a prism (or a diffraction grating as is more typically the case). But this means that instead of having a single dot on your image plane, you're going to have a band. And if you have lots of stars, you'll have lots of bands. And if you have lots of bands, they'll overlap and make a mess. So spectroscopy is slow because you have to do one star at a time. There's been some ways to get around this by using fiber optic cables at the image plane to intercept the light and send it to lots of different spectrographs, but such things have to be individually set up for every field you want to look at. What a pain!

And that's not the only thing that makes spectroscopy slow. Since you're spreading out the light that you're getting, this also means that the amount of photons hitting at any one point will be less. Your image gets fainter the more you spread it out into the rainbow!

So there's two major things that make doing spectroscopy slow work. It's good and necessary for getting things like the chemical composition, but if we really just want to make an HR diagram, isn't there a quicker way?

YES!

And that method is known as photometry. The trick of this is that instead of looking at all the wavelengths, it picks out just a few important ones and does the work that way.

The reason this works is that stars tend to behave pretty close to what's known as a "blackbody". What this means is that it gives off radiation in a certain way with a peak wavelength dependent on it's temperature. They're described by Planck's Law (setting the first derivative equal to zero and doing a few substitutions gives Wein's Law). But what's really important is that the wavelength or the color of the peak is determined by the temperature. It's easiest to explain with a diagram:

As you can see, the hotter stars peak off in the blue region, and have a greater luminosity. The cooler a star gets, the more red it's peak emission and the less energy it gives off (which is shown by the area under the curve or the first integral of the blackbody equation).

Like I've already said, through spectroscopy, you can get that entire blackbody curve (with the superimposed absorption and emission features). That's great, aside from the slow part. But how does only looking at a few specific points on that curve tell you what you need to know?

This is most easily demonstrated by example, so I'll just jump right in with the most common photometric system, the Johnson/Cousin system. This system has five filters:

U: Ultraviolet
B: Blue
V: Visual (green-yellow)
R: Red
I: Infrared

Each of these filters only allows light from a narrow range of wavelengths (known as a bandpass) to get to the detector. It's essentially looking at a series of five points along those blackbody curves I just showed.

So let's take that first blackbody curve, the one for a hot star, and put the filters on it (note: For some reason I didn't show the R filter. Also, the units are removed since for the purposes of illustration, all we need is a qualitative effect).

In this image, we can see that the filter doesn't intersect at equal luminosities. In the U filter, it's pretty low. It gets higher in B (which it should since we already said, it's a hot star), and then gets lower in the V, and is lower still in the I.

Now consider what happens if you take the difference between the luminosities of two filters. Most commonly done is the B-V so we'll use that for the example. If you do this, the B is greater than the V, so if you subtract it, you'll get a positive number. But remember, this is in luminosity and astronomers work in the magnitude system in which brighter stars are smaller numbers. It's backwards. So when talking in terms of magnitudes, a blue star will have a negative B-V.

So let's now look at a cool star on the same filters:

Here, if we try the same thing, and take the B-V luminosities, the star is brighter in the V than it is in B, so if you're talking about luminosity and take B-V, you'll get a negative number. Flip that around for magnitudes, and you get a positive number.

So already, you should be able to see the trend. In terms of magnitudes, a negative B-V means a hot star. A positive B-V means a cooler star. The more negative you get, the hotter it gets. The more positive you get, the cooler it gets.

This is great! It fixes both problems we had earlier. It doesn't spread the starlight out, so images don't take any longer to expose. Nor does it require you to put each star through a slit. You can do an entire field of stars at once! And this is, for the most part, exactly what I did for my research 2 summers ago. Using this basic principle, I worked out the H-R diagram for NGC 7142 (except that when we use photometry to replace the temperature, we call it a "color-magnitude diagram" or CMD). Of course, there were a few nuances that made it a bit more difficult than what I've just described.

But instead of going into all that now, I'll save that for another post.