Recall Wien's law and the role it played in the plotting of spectral energy distributions. Also, recall what information one can extrapolate from an SED. How might we use this to our advantage when studying circumstellar disks. Let's begin with a simple picture of a circumstellar disk. Let's assume that it is geometrically thin. Let's also assume that the dust emits as a blackbody. The cartoon below should help us visualize this.
Don't worry about the two plots right now at the top of the image, just look at the cartoon itself. What you are looking at is a cartoon of a central star surrounded by a thin disk of dust and gas. The redder rings signify hotter dust while the browner rings signify cooler dust. Notice that the hotter dust is closer to the surface of the star because the star is the primary heat source of the disk. That may be obvious, but because the star is the only object heating the dust and gas in the disk, it is easy to imagine that every radius corresponds to a unique temperature. In other words, for every distance from the surface of the star, the dust will be a different and unique temperature. Now we know from Wien's Law that each temperature emits at a characteristic peak wavelength. So if temperature depends on radius and the wavelength of peak emission depends on temperature, we can simply say that each radius emits at a characteristic peak wavelength.
Let's think more in depth about the fact a particular radius corresponds to a specific temperature. Mathematically we say that the temperature is a function of radius. In other words, there is is a mathematical equation that depends on radius which governs the temperature of the dust and gas in the disk. You can imagine that there are a wide range of factors that would effect this type of function or "temperature distribution" as they are normally called. Everything from the size of the dust particles to the type of material the dust particles are made of affects how the temperature distribution will behave.
In its most basic form, a temperature distribution is a simple power law:
The variable, p, is the value that is of most concern because it changes as the properties of the dust particles do. The value of "p" will always be greater than 0 and less than or equal to 2, but usually less than 1. The largest factor that effects the value of "p" is the size of the grains. Smaller grains characteristically have "flatter" temperature distributions corresponding to a smaller values of "p", while larger grains have "steeper" temperature distributions corresponding to larger values of "p".
Now let's take a look at those two plots at the top of the image. The plot on the left is an example of a temperature distribution. It is a log-log plot so the power law shows up as a straight line. The plot on the right is an example SED brought on by this temperature distribution. The yellow line represents the SED of the star while the red line represents the SED of the disk and the star added together. This SED is a log-log plot also in units of microns so 0 corresponds to 1 micron, 1 corresponds to 10 microns, 2 corresponds to 100 microns, and so on. Notice that the excess emission occurs in the infrared.
Now imagine that the temperature distribution is different. Remember that the Planck function is a function of temperature so the SED on the right would also look different. So if the temperature distributions depends on the properties of the dust, we can use different temperature distributions that make different assumptions about the properties of the dust (mostly based on size) in a circumstellar disk and see what the SEDs look like. This process is called modeling.
Modeling is a very important process that basically allows us to compare observations to what we think might be going on in a given physical system. When we make models of circumstellar disks, we make certain assumptions about the properties of the disk and the grains that compose it. We then compare these models with observations taken from telescopes and satellites. The telescopes measure the amount of energy being emitted from a star (and also a disk that might be surrounding it) at a particular wavelength. This data is then plotted on an SED where we can see how well our models fit the data. Take a look at the plots on the left. Also, click here to view another simpler plot that illustrates how this process works. The data points are observations taken at different telescopes and the solid and dotted lines are models that make different assumptions about the disk size, shape, and the properties of the dust grains. We can assume different things about the disk in order to make the model fit the data. This does not mean that the disk, for sure, has the properties of the model, it is just one possibility. Many of the parameters play off on each other so that many models fit the same data.
Take another look at the SED in the pop-up window. The asterisks and crosses are data points taken of the same circumstellar disk at different wavelengths. The instruments that took the measurements are labeled in the legend. The red, purple, and blue lines are all SED models of circumstellar disks that make certain assumptions. The black line is the SED of the star. All of the models assume that the grains are very large. The only things that change are the positions of the inner and outer radii (Ro and Gi respectively). Notice that small but significant differences in the position and quantity of dust in different disk models do not make a lot of difference in the appearance of the SED. So now we begin to see some of the limitations of this type of modeling. First, there is some ambiguity in that two models making two totally different sets of assumptions might both fit the data perfectly well. And second, small changes in some of the disk properties (more specifically the positions of the inner and outer radii) do not necessarily change the appearance of the SED significantly.