Additional information

We are impressed by the direct Hubble Space Telescope size measurement of Xena by Mike Brown and his team (Brown et al. in press). Measuring the size of such a tiny and moving object is a very difficult task. We were pleased to learn that our claim that 2003UB313 is larger than Pluto, was confirmed. However, we were surprised by the smaller diameter implied by the new measurements and the implied high albedo (reflectivity). Although an albedo of 90% has never been observed for Kuiper Belt Objects (which typically show 10-20%), it is not totally implausible, as Brown et al. suggest in their article.

The question naturally arises why our size measurement based on the detection of thermal emission (2600 - 3400 km diameter, a 68% confidence level range typically used, corresponding to one standard deviation in statistical terms) is in apparent conflict with the HST measurement (2300 - 2500 km, 68% confidence). One should realize though that the measurements are consistent at the 1.3 standard deviation (80%) confidence limit of both measurements, i.e. in each measurement there is a 10% chance that the diameter is around 2530 km.

One way to even better reconcile both measurements is to lower the assumed ratio between the bolometric (also called "Bond") albedo, A, and the optical "geometric" albedo, p, from the value we had adopted, q=A/p=0.9, to q=0.7 (q is also called the "phase integral"). In this case, the size derived from the thermal measurement is reduced by 100 km, so that both measurements would agree within the 68% confidence limits at ca. 2500 km.

What does it mean to reduce the value of q? This parameter is not very well constrained, neither theoretically nor empirically from observations of solar system objects. It relates two albedos: the bolometric Bond albedo specifies what fraction of the total incident radiation energy is reflected, whereas the geometric albedo, p (usually measured in the optical red) specifies what fraction of the red sunlight is reflected toward the observer. Both values are usually not equal. For many materials, the reflectivity depends on wavelength, so that the albedo as averaged over all wavelengths, A, is different from that at any particular wavelength. This is a rather complicated issue since the back-reflectivity of a tilted material surface also depends much on the surface roughness. Consider, e.g., a highly reflective surface such as a metal plate. Placing a smooth metal film on a sphere results in a very low average albedo for this sphere, because only few rays reflect toward the observer - despite the reflective nature of the material. A ragged or porous material such as snow on the other hand would reflect back well even if viewed at some angle.

A wide range of values of q has been measured for solar system objects, ranging from 0.2 to over 1.2. We had chosen 0.9 because this is an average value found for Pluto, which we thought to be an appropriate analogon. The "small" diameter implied by the HST measurement may suggest that q is indeed smaller than 0.9, and it thereby tells us something interesting about the optical property of the surface.

Besides this ratio of albedos we should point out at least other possible sources of uncertainty in UB313's size measurement. Two important assumptions were made when deriving a size for UB313 from the HST measurements: a specific radial brightness profile on the optical "disk" (the "center-to-limb function"), and second, the implicit assumption that the star observed for comparison with UB313 is a single star.

For the radial brightness profile Brown et al. assumed a resonable best guess, i.e. that measured for Triton. Using a different profile, e.g. flat instead of a limb-darkened, or, in the opposite extreme, a reflective smooth film as mentioned above, would have a very large effect on the derived size, an effect much bigger than the statistical measurement uncertainty of 100 km. Brown and collaborators did use the best reasonable guess they could make (just as we used q=0.9 assuming UB313 is similar to Pluto) - which may be correct, or may not.

Assuming the reference star to be single is a good guess, but is far from certain. Roughly half of all stars are multiple systems, and a faint field star such as the one chosen for the measurement is likely to be distant enough (more than 1000 light years) that the most likely separation of any companion star is comparable to the apparent size of UB313, of order 10 milli-arcseconds. There is thus a small but not negligible chance that the reference star image was slightly extended, which would result in too small a diameter derived for UB313 when comparing the respective images.

Given the inherent uncertainties in both methods, the actual size of 2003 UB313 is not established beyond reasonable doubt by either the MAMBO or HST measurements. Both may be right, in which case we learn something rather interesting about the surface property of this exotic object. Precise measurements of the diameter (and with this the albedo) of UB313 will eventually be possible with powerful ALMA radio interferometer to be installed in the Chilean desert, or with the James Webb Space Telescope .

The most distant object yet known in the solar system turns out to be a rich source of puzzeling information, and an exciting challenge.