Astronomy 103, Fall, 2006

[Return to the Dark Matter Lab]

Dark Matter Lab : Making an Integrated Surface Brightness Profile

As you can tell, figure 2 does not have any data points. It is simply a model of how the brightness of the galaxy falls off with distance. (The reason the curve doesn't fall off is because of the integrated nature of the plot. It is cummulative, so at every radius you are seeing all the light emitted from inside that radius. This was done to make our data consistent with the masses we find using Newton's equations.)

The plot is really a curve. The equation that describes the curve is:

where h=3.87 kpc, and Σ0=6.725×107 solar luminosities/kpc2.

What do h and Σ0 stand for?

h is some "scale height." The galaxy doesn't have a hard edge, so it's difficult to identify a true hard "radius" for the size of the disk. The "scale height" represents the radius over which the light drops by a certain amount. It is analogous to the "half-life" of a radioactive element. It should be comparable (within a factor of a few of) the size of the disk as you see it in an image.

Σ0 is the peak surface brightness at the center of the galaxy.

(Aren't you glad you asked?)

What does that mean?

Let's start from the beginning.

You have your galaxy, NGC 2742:

NGC 2742

Obviously, it gets dimmer towards the edge. We are dealing with surface brightness here, which is how much light comes from some area of the galaxy. It is typically assumed that surface brightness can be modeled with a decaying exponential. That means that going out one particular radius, the brightness in each same sized area patch dims as you move outwards. It starts from some high point, Σ₀. The point at which it's decayed to 1/e of it's orginal value is the radius h (the scale height). Mathematically, we say:

To create our integrated surface brightness profile, we need to integrate that function over radius. Since we're taking account all the brightness inside, we also need to take into account the fact that the galaxy is a disk. That means integrating over the entire circle (thus the 2πr in the below integral).

(Now I bet you're really glad you asked!)

If you know more math than I do, you will see that this can be integrated by parts and the result is:

(Sorry about all the math, that's just the easiest way to explain the curve in this case.) (Which means, of course, that for many readers, it wasn't explained at all! Unless you really know what an integral means, and you think about it.... Someday we'll try to write this out so that somebody who isn't comfortable with integrals can understand it. As it is right now, you're stuck with the lab as we inherited it.) (It may comfort you to know that we spend the better part of a lecture in Astro 253 just on the meaning of the concept "surface brightess," so if this whole thing is not obvious to you at the moment, it's not necessarily a reflection on you....)

In particular, Σ0 comes from the fact that we assume the absolute magnitude of the center of every spiral galaxy is -21.7 mags/arcsec2. This seems odd, but it has been noticed through repetitive observation that the center of spiral galaxies are all quite close to the same surface brightness (see Persic and Salucci, 1988 if you don't believe me) (that is, if you can figure out exactly what reference the original author meant). You need to convert magnitudes to surface brightness units and arc seconds to kpc. Since the value is an absolute magnitude, that means it's the brightness 10 pc away. I'll leave as an exercise for the student using the magnitude equation to convert magnitudes to luminosities (yeah, right), and using the small angle approximation to convert arc seconds to kpcs. I did get tripped up on the units the first time I did it.

(And if you followed that last paragraph – good for you! If you didn't, don't worry about it too much, because it assumes way more than we've covered in this class.)

It sounds like you cheated

Yep, we took the easy way out. Andrew West claims that you could use pictures from the Sloan Digital Sky Survey (SDSS) to create a surface brightness profile. While I admit that it wouldn't be hard to just plot how the brightness falls off with area (thereby getting real values for our two parameters), it would be a trigonometric mess to try to photometrically integrate over the concentric rings to find an integrated surface brightness profile. There might be some simple trick that aleviates the problem I'm talking about. Anyways, it was reasonably easy to look on NED to find the reference mentioned above.

(And if you followed that last paragraph... well, yay.)