Details on the dwarf planet size calculations

One of the stumbling blocks to understanding the true numbers of dwarf planets has been that measurement of the true sizes of these distant bodies has been difficult. The only thing that we can measure well for most of these bodies is simply the brightness. And while bigger objects tend to be brighter, it is also true that objects which simply have more reflective surfaces will be brighter too. If we knew how reflective each of the surfaces was, however, we could use the brightness to infer the actual size.

After nearly 20 years of studying the Kuiper belt, we actually know enough to come up with a good estimate of an objects reflectivity (its "albedo") knowing nothing more than where it is, how bright it is, and what color it is. While these estimates will not be perfect, there are adequate for an initial exploration of sizes and albedos of bodies in the outer solar system.

The largest objects

A handful of objects in the Kuiper belt have had their albedos measured. We will use those to help our estimates. The most complete database comes from a chapter by John Stansberry et al. in The Solar System Beyond Neptune , which is a compliation of results from the Spitzer Space Telescope. Stansberry find that objects that are intrinsically brighter also have higher albedo. Intrinsic brightness is measured by "absolute magnitude" which is defined as how bright the object would look to you if it were in the position of the earth and you were sitting on the sun. While it sounds odd (and perhaps a bit uncomfortable), it is a nice way to compare the intrinsic brightness of objects without regard for how far they are away from you or from the sun. Everything is the same in this system. And, since it is an astronomical magnitude system, small numbers mean brighter, and every 5 magnitudes is a factor of 100 in brightness (blame the Greeks).

If you look at albedo vs. absolute magnitude, you see a very clear trend that brighter objects (smaller absolute magnitudes) have higher albedos. We can use this trend to predict albedos of bright objects whose albedo we have measured. We make the simple assumption that for objects larger than 400km, albedo is a power law which connects albedos of 5% at 400 km and 20% at 900 km. Mathematically, this is written by setting two constants, m=log10(.05/.20)/log10(400./900.), c=.05/400., and then calculating albedo as, albedo=[c^(1/m)*1369*10^(-h/5.)]^[m*2/(m+2)] (deriving the formula is an exercise for the student....). The result of this formula can be seen by the red line on the figure, which does a pretty good job fitting the data for the largest objects.

Smaller objects

For objects fainter than an absolute magnitude of about 5, no systematic trend appears with magnitude. There is, however, a trend with color. Small Centaurs and possibly Kuiper belt objects appear to divide into a class of very similarly neutraly colored objects (often called "blue" but they are not really blue, just less red than the others) and a more dispersed class of red objects (often called "red" because they are). If you look at the albedos of the smaller objects as a function of color gradient (where values between 0 and 20 generally are classified as blue and values above 20 are classified as red) you see that the blue objects generally have lower albedos than the red objects. We will use this to estimate albedos for objects smaller than 400 km. If they are blue, we will estimate their albedos to be 4.4% (the median of the measured blue values), while if they are red we will estimate them to be 8% (the median of the measured red values). If they have no color estimate, we will assume them to have the 8% albedo of red objects simply because that makes for a conservative estimate of the size.

Special objects

Two classes of objects are known to be special and have unique albedos. Members of Haumea's collisional family have nearly pure water ice surfaces and extremely high albedos of ~70%. And members of the cold classical Kuiper belt appear to have unique high albedos of ~20% (this number is very poorly constrained, however).

Summary

The table of dwarf planet candidates uses measured values when available and otherwise uses the above estimates. The comments column indicates the source of the albedo. Measured refers to a direct measurement through a stellar occulatation or through the technique discussed in the Stansberry paper above. Large means that the albedo was calculated from the formula for large KBOs. Blue or Red means that the albedo was assumed from color as above. Cold classical or Haumea family means that the object is one of these special classes with special albedos. Finally, a blank entry in the column means that object is small with an unknown color and is not a Haumea family member or a cold classical object. The 8% assumed albedo is a conservative estimate.