Encyclopedia of the Solar System 2nd ed

Kuiper Belt Objects: Physical Studies 609

between absolute magnitude, Hv, and apparent magnitude,
V, is given by

Hv=V−5 log(r)+ 2 .5 log[(1−G) 1 (α)+G 2 (α)],

where the last term of the equation is an empirical phase
function that describes how Hvof an object varies with
phase angle. G=0.15 and 1 and 2 given by

i(α)=exp

[

−Ai

(

tan

1

2

α

)Bi]

where i=1 and 2, A 1 =3.33, B 1 =0.63, A 2 =1.87, and
B 2 =1.22 seem most appropriate for KBOs. Hvvalues for
discovered KBOs and Centaurs range from about−1to15.
Table 2 lists KBOs with the brightest Hvvalues.

6. Diameter

Size is among the most fundamental physical properties of
an astronomical object, yet we are only beginning to get
accurate diameter measurements for KBOs and Centaurs.
The most direct way to measure the diameter of a KBO, D
(in km), is to measure its angular diameter,θ(in arc sec),
and geocentric distance,(in AU). Geometry gives

D= 727 θ.

Unfortunately, KBOs and Centaurs have sufficiently
small values for D and sufficiently large values forthat
the resulting values forθare too small for measurement
even by the HST. Michael Brown pushed the HST to its
limits and measuredθ= 0. 0343 ± 0 .0014 arc sec for the
KBO Eris, which was at a geocentric distance of 96.4 AU
at the time of their observations. They found a diameter of
2400 ±100 km for Eris, making it slightly larger than Pluto,
D=2302 km.
For KBOs and Centaurs withθtoo small for measure-
ment, it is possible to estimate their diameters from their
brightness,

pD^2 = 9 x 1016 r^2 ^2100 .4(m−V),

where, as before, r is the heliocentric distance in AU and
is the geocentric distance in AU, m is the V-band brightness
of the Sun (− 26 .74), V is the brightness of the KBO, p is
the albedo of the object, and

=[(1−G) 1 (α)+G 2 (α)].

Since Jupiter-family comets come from Centaurs and
the Kuiper Belt, most KBO diameter estimates assume an

albedo similar to albedo measurements for a handful of

Jupiter-family comets, i.e., p=0.04. Diameter estimates

from V magnitudes for about 100 objects range between

D=25 km for 2003 BH 91 to D=2400 km for Eris. KBO

and Centaur object diameters on the scale of Figure 1 range

from the tiniest specks to Pluto.

The assumption of a comet-like albedo, although rea-

sonable, is dangerous because Jupiter-family comets come

much closer to the Sun than KBOs and Centaur objects.

The frequent close proximity of short-period comets to the

Sun results in the sublimation of H 2 O ice and produces sur-

faces largely covered by a dark, refractory-rich, lag deposit.

The surfaces of Jupiter-family comets may have chemical

and physical properties quite different from the surfaces of

Centaurs and KBOs. Charon has a relatively large albedo

of 0.37. If we assume that a KBO has p=0.04, but it is ac-

tually has p=0.4, we will estimate a diameter that is more

than three times too large. Measurements of albedos are

essential for accurate measurements of KBO and Centaur

object diameters.

7. Albedo

By measuring the brightness of sunlightreflectedfrom a

KBO at visible wavelengths and the brightness of heatemit-

tedby the same KBO at thermal infrared wavelengths,

it is possible to disentangle albedo from diameter, and

thereby measure separate values for both quantities. The

Spitzer Space Telescope, an infrared telescope in orbit

about the Sun, is enabling John Stansberry of the Univer-

sity of Arizona, Dale Cruikshank of NASA’s Ames Research

Center, William Grundy of Lowell Observatory, and John

Spencer of Southwest Research Institute to observe much

fainter levels of heat from KBOs and Centaurs than is pos-

sible with telescopes on the Earth. As a result of their work,

we have accurate diameters and albedos for more than a

dozen KBOs (Table 2).

8. Brightness Variation

KBOs and Centaurs may have weak internal constitutions

(i.e., rubble pile type interiors) due to fracturing by past

impacts between objects. In other words, it is possible that

KBOs and Centaurs are nearly strengthless bodies, held

together primarily by their own self-gravity. If so, then some

objects may deform from spheres into triaxial ellipsoids with

axes a>b>c as a result of their rotation.

The rotation of an ellipsoid can result in periodic varia-

tion of its projected area on the sky and hence a periodic

variation of the sunlight it reflects and its brightness (Fig. 5).

Monitoring such a brightness variation can result in a wealth

of physical data about the object (e.g., its period of rotation,

shape, and perhaps even its density and porosity).