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\def\runningtitletext{Kuiper Belt}
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\title{PHYSICAL NATURE OF THE KUIPER BELT}

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\name{DAVID JEWITT}
\affiliation{Institute for Astronomy, University of Hawaii
}
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\leftline{and}
\name{JANE LUU}
\affiliation{Sterrewacht, Leiden University
}
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%\affiliation{From Wherever Else}
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%Version for review: 98-Sep-21
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%Invited review for the Protostars and Planets IV meeting, held 1998 July
%5-12, Santa Barbara, California.
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\abstract{
Recent ground-based observations have unveiled a large number of
bodies in orbit beyond Neptune, in a region now widely known as the
Kuiper (or, less commonly, Edgeworth-Kuiper) Belt.  About $10^5$
Kuiper Belt Objects (KBOs) with diameters larger than 100 km exist in 
the 30 AU to 50 AU trans-Neptunian region. Their combined mass is about 
10\% of that of Earth.  The orbits of KBOs fall into at least three 
distinct dynamical
classes (the "Classical" objects, the Plutinos and the "Scattered"
objects).  Each throws light on physical processes operating in the
solar system prior to and during the formation of the planets.  The
Kuiper Belt is significant both as the likely source of the
short-period comets (and the dynamically intermediate Centaurs), and
as a repository of the solar system's most primitive (least thermally
processed) material.  KBOs show an unexpected and presently
unexplained diversity of surface colors, possibly reflecting intrinsic
compositional variations and transient resurfacing by impacts.  The
present-day Kuiper Belt is probably the surviving remnant of a once
much more massive ($10 M_{\oplus}$?) preplanetary disk.  It is very
likely that collisions and disk-planet interactions played a major 
role in shaping this early precursor.  While the collisional production 
of dust is presently
modest ($ \sim 10^3$ kg s$^{-1}$), and the optical depth small
($ \sim 10^{-7}$), the early Belt was probably very dusty and
may have sustained a disk analogous to those reported around some
nearby main-sequence stars.}

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\mainsection{I.~~History}
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The Kuiper Belt has a long and mostly speculative history, dating back
at least to papers by Edgeworth (1943, 1949) and Kuiper (1951).  These
authors noticed that prevailing theories of planetary accretion
provided no reason to expect that the planetary system should end at
the orbit of Neptune (or of Pluto, then believed to be a massive and
bona fide planet in its own right).  Edgeworth believed that the
comets might be derived from the trans-Plutonian region (MacFarland
1996), an idea that is now widely accepted.  Kuiper argued that
kilometer-sized comets would grow in the trans-Neptunian region on
timescales of $10^9$ yrs, and that Pluto was responsible for
scattering these objects to the Oort Cloud.  Subsequently, these comets
would be sent back to the planetary region by stellar perturbations to
appear as long-period comets (those having periods $>$ 200 yrs), according to
Oort's (1950) prescription.  These ideas were further developed by
other researchers, including Whipple (1964), who used anomalies in
the motion of Neptune to infer the mass of trans-Neptunian objects
(the anomalies are now known to be due to systematic astrometric
errors).  Later workers realised that the observed flux of
short-period comets was too large to be derived from planetary
perturbations of long-period comets falling from the Oort Cloud (Joss
1973), raising the possibility of separate sources for the long-period
and short-period comets.  The Kuiper Belt was explicitly advanced as
the source of short-period comets by Fernandez (1980).  
Numerical work by Duncan et al. (1988) confirmed that the
short-period comets could not be captured from the spherical Oort 
Cloud, providing indirect support for the existence of a belt-like
cometary source.

Observationally, the search for objects in the outer solar system
was impeded by their expected faintness. The flux of sunlight scattered 
from a solid 
body varies approximately as  $p_R r^2 R^{-4}$, where $p_R$ is the geometric
albedo, $r$ is the radius and $R$ is the heliocentric 
distance.  Objects of
a given size and albedo are $10^4$ times fainter when beyond Neptune 
($R$ = 30 AU)
than when in the main asteroid belt ($R$ = 3 AU).  This simple
fact provided a daunting and, for many years, insurmountable 
obstacle to the direct observation of small bodies in the
outer solar system.  Kowal (1989), for example, used the Palomar Schmidt 
to conduct a 6000 square degree photographic survey of the
ecliptic to red magnitude 19.5 but found nothing beyond the planetary
region.  The turning point came with the application of large-format,
high quantum efficiency charge-coupled device (CCD) detectors in
the late 1980's.  Although the first CCD survey was also
negative (Luu and Jewitt 1988) the new technology spurred increasing 
observational effort (Levison and Duncan 1990; Cochran et al. 1991; Tyson et al. 
1992), culminating with the detection of the first Kuiper Belt object (1992 QB1)
on 1992 August 30, using a $2048 \times 2048$ pixel CCD on
the University of Hawaii 2.2-m telescope (Jewitt and Luu 1993).
Intensive surveys since 1992 have increased the
observational sample to about 160 objects at the time of writing (Jewitt and 
Luu 1995, Irwin,
Tremaine and Zytkow 1995, Williams et al. 1995, Jewitt, Luu and Chen
1996, Jewitt, Luu and Trujillo 1998, Luu and Jewitt 1998, Gladman et
al. 1998, Chiang and Brown 1999). Our knowledge of the Kuiper Belt is based primarily on
these 160 objects, most particularly on the 54 whose orbital elements
are known with confidence from astrometry at more than one opposition.
An updated list of these orbits is maintained by the Minor Planet
Center at the web site
http://cfa-www.harvard.edu/cfa/ps/lists/TNOs.html, while updated
information on
Kuiper Belt research is maintained at
http://www.ifa.hawaii.edu/faculty/jewitt/kb.html.

The present chapter aims to present the observational status
of the Kuiper Belt, and is organized as follows.  In Section II, we discuss the
optical surveys, in Section III we consider dust production in the Kuiper
Belt, and in Section IV we remark on the dynamically related Centaur
class.  Physical properties are the subject of Section V. We end with a
list of topical questions in Section VI.  
	
\mainsection{II.~~Results from the  Optical Surveys}

Figure 1 shows a plan view of the Kuiper Belt based on survey observations
up to mid-1999.  Evidence for dynamical sub-structure in the Kuiper Belt is
provided in Figure 2, which shows orbital semi-major axis $a$ vs. 
eccentricity $e$ for
the known KBOs.  The KBOs display a highly non-uniform distribution of
orbital elements in $a-e$ space, leading to the definition of three
distinct dynamical groups within the Kuiper Belt.

%%%%%%%%% FIGURE 1 goes here %%%%%%%%%

\subsection{A.~~Classical Objects}

About 2/3rd of the well-observed KBOs have semi-major axes $a \ge 42$
AU and perihelia $q > 35$ AU and together these characteristics define
the "classical" Kuiper Belt.  Numerical simulations suggest that the
orbits are stable over periods comparable to the age of the planetary
system (Holman and Wisdom 1993; Duncan, Levison and Budd 1995;
Morbidelli, Thomas and Moons 1995), essentially because the perihelia are
well separated from Neptune.  The classical KBOs possess modest
eccentricities but can have inclinations as large as $32^{\circ}$
(Table 1).
Both the inclination and eccentricity distributions are wider than
expected on the basis of long-term simulations that account only for
planetary perturbations (e.g., Figs. 3 and 4 of Holman and Wisdom 
1993 show typical eccentricities and inclinations of only 0.02 and
1 degree, respectively), implying excitation by another mechanism.

%%%%%%%%% Table 1 goes here.%%%%%%%%%

\subsection{B.~~Resonant Objects (Plutinos)}

Approximately 1/3rd of KBOs appear clustered near the 3:2 mean-motion resonance
with Neptune at $a = 39.4$ AU (Fig. 2).  These objects have large
eccentricities ($0.1 \le e \le 0.34$) and inclinations up
to $\sim 20 deg$ (Table 1).  Many of them cross the
orbit of Neptune (the critical eccentricity for crossing is 
$e = 0.24$), and would be
subject to strong Neptune perturbations were they not dynamically
protected by the resonance.  In this regard, the 3:2 objects resemble
Pluto, which is located within the resonance (plotted in Fig. 2 with
an X) and which is also Neptune-crossing. Like Pluto, the "Plutinos" (a
name chosen to symbolise this remarkable dynamical similarity) are
aphelion librators, which reach perihelion when near $\pm 90$ deg. from
Neptune. Pluto itself occupies several other resonances, notably the
"argument of perihelion libration", which maintains its perihelion at
high ecliptic latitude and therefore further minimizes Neptune
perturbations (e.g., Malhotra and Williams 1998).  It is not yet known
what fraction of the Plutinos might occupy these other resonances because the sky-plane coverage of published surveys remains very small.  The
upper limit to the eccentricity of the Plutinos (the most eccentric is
1996 TP$_{66}$, with $e = 0.34$, $q = 26.4$ AU) is presumably set by
destabilizing perturbations from Uranus.  There is a deficiency of Plutinos
with $e \le 0.1$.

The apparent fraction of Plutinos is over-estimated as a result of
observational bias (the Plutinos have smaller orbits than classical
KBOs and so appear brighter, and are easier to detect, at a given size).  
Bias-corrected
statistics suggest that the "true Plutino fraction" is closer to 10\%
to 15\% (JLT98).  Thus, the Plutinos appear to be a significant but
distinct minority component of the Kuiper Belt.  Other mean-motion
resonances are also populated (Fig. 2), notably the 2:1 resonance at 47.6
AU (1996 TR$_{66}$ and 1997 SZ${10}$), the 4:3 at 36.4 AU (1995
DA$_2$), and the second order 5:3 resonance at 42.3 AU (1994 JS).

The mechanism by which the resonances were populated has not yet been
firmly established.  The leading hypothesis invokes radial migration
of the planets due to the exchange of angular momentum with
planetesimals scattered from the protoplanetary disk.  Simulations 
suggest that planetesimals scattered
among the planets lead to an asymmetry that drives Saturn, Uranus and
Neptune away from the sun.  Jupiter provides the ultimate source of
the angular momentum and, accordingly, its orbit contracts (Fernandez and Ip
1984, Malhotra 1995, Hahn and Malhotra 1999).  As Neptune migrated outward, its mean-motion
resonances would have swept through the Kuiper Belt collecting objects.  Malhotra
(1995) finds especially efficient trapping of KBOs in the 3:2 and 2:1
resonances. The trapping efficiency is high (of order unity) for
initial orbits of low-eccentricity and inclination, and for slow and
smooth planetary migration.  The relative populations of the
resonances depend partly, in this resonance sweeping model, on
the rate of migration.  This raises the intriguing possibility that
measurements of the population ratios may allow an estimate of the
rate at which Neptune's orbit expanded.  One problem for the resonance
sweeping hypothesis, at least in its simplest form, is that 
most known KBOs are not trapped in mean-motion resonances. For
example, the classical KBOs lie in a region that should have been 
swept clean by the outwardly migrating 2:1 resonance.  It is likely
that effects such as the stochastic migration of Neptune in response
to the ejection of massive planetesimals would decrease the trapping
efficiency, possibly to levels consistent with the data.  



%%%%%%%%%%%%%%Figure 2 goes here.%%%%%%%%%%%%%%%%


\subsection{C.~~ Scattered Objects}

Four KBOs in the known sample possess large, eccentric, inclined orbits with 
perihelia near 35 AU (Fig 3).  The prototype "Scattered Kuiper Belt Object" 
is 1996 TL$_{66}$ (Luu {\it et al.} 1997), with $a = 85$ AU, $e = 0.59$, 
$i = 24 deg$.  These objects may represent a chaotic swarm of bodies 
scattered outwards by Neptune in the early stages of the solar system 
(Torbett 1989; Ip and Fernandez 1991;
Duncan and Levison 1997).  With perihelia near 35 AU, Neptune is able
to exert weak dynamical control.  The number of Scattered KBOs
is highly uncertain.  A population of order $6 \times
10^3$ (500-km diameter or larger) is suggested by the discovery of
1996 TL$_{66}$ (Luu et al. 1997), but this estimate is good at best to
order of magnitude.  The SKBOs appear rare in flux-limited surveys
because their large, eccentric orbits render them invisible except
when near perihelion.  In absolute numbers the SKBOs may dominate the
trans-Neptunian region.

%%%%%%%%%%%%%%Figure 3 goes here.%%%%%%%%%%%%%%%%


\subsection{D.~~Cumulative Luminosity Function
}

 The cumulative luminosity function (CLF) of the Kuiper Belt is shown
in Figure 4.  In the Figure we have omitted measurements based on
photographic plates (principally the survey by Kowal 1989 and the
photographic portion of Luu and Jewitt 1988).  Photographic surveys
are difficult to calibrate and require long integrations that permit
excessive trailing loss even on slow moving objects.  The Hubble Space
Telescope (HST) measurement (Cochran et al. 1995) at red magnitude
28.1 is controversial and deserves special mention.  The central issue
is that the HST images are under-sampled, so that astronomical objects
and cosmic rays cannot be readily distinguished by their morphology
alone.  Consequently, the KBOs must be sought against a large background
due to cosmic rays.  The measurement has been criticized by Brown et al. 
(1997) on the basis that noise in the 
data would prevent the detection of objects at the claimed limiting 
magnitude (but see Cochran et al. 1998).  Furthermore, the authors 
searched the HST data only for Plutino-type orbits.  Inclusion of 
other dynamical classes
would, by analogy with the ground-based surveys, increase the derived
surface density by a factor of order 3.  For these reasons the
usefulness of the HST measurement is presently not clear.

The CLF is well described by a power law relation of the form 

$$log[\Sigma(m_R)] = \alpha (m_R - m_0)            \eqno(1)$$

\noindent where $\Sigma(m_R)$ is the cumulative surface density measured
to limiting magnitude $m_R$, and $a$ and $m_0$ are constants.  
Fits to the CLF give $\alpha = 0.58 \pm 0.05$, $m_0 = 23.27 \pm
0.11$ ($20 \le m_R \le 25$; JLT98), $\alpha = 0.54 \pm 0.04$, $m_0 =
23.20 \pm 0.10$ ($20 \le m_R \le 26.6$; LJ98),  $\alpha = 
0.76 ^{+0.10}_{-0.11}$, $m_0 = 23.40 ^{+0.20}_{-0.18}$
($20 \le m_R \le 26$; Gladman et al. 1998) and 
$\alpha = 0.52 \pm 0.02$, $m_0 = 23.50 \pm 0.06$ ($20 \le m_R \le 27$;
Chiang and Brown 1999).  
Gladman et al. (1998) observed that least-squares fits are not 
appropriate when the data are cumulative and the uncertainties are Poissonian, 
as opposed to Gaussian. The slope they derived using a maximum
likelihood method differs from those found
by other investigators using least-squares, but Chiang and Brown (1999) 
showed that this difference is due to their selective use of only a 
sub-set of the available data, and not to the fitting method.
In any case, differences among the fitted parameters are at
the $2 \sigma$ level for $\alpha$ and $m_0$, and are therefore 
statistically insignificant.  
Furthermore, the assumption that the uncertainties are Poissonian in distribution 
is probably not correct.  We already know, for example, that the 
azimuthal
distribution of Plutinos is non-uniform (Malhotra 1995, JLT98),
meaning that systematic sky-plane surface density variations are
folded into the survey data, but have not been taken into
account. More survey observations, especially at $m_R \le 20$ and $m_R
\ge 26$, will soon resolve these ambiguities.

%%%%%%%%%  Fig. 4 goes here %%%%%%%%%%%%%%%%%%%%%

\subsection{E.~~Inclination distribution}

The apparent
Full Width at Half Maximum (FWHM) of the belt is $\sim 10^{\circ}$
(JLC96, c.f. Fig. 5), the most extreme inclination being 
$39.1^{\circ}$ for 1999 CZ$_{118}$ (not plotted in Fig. 5 because
it is still a single opposition object).  However, this must be 
considered as a lower limit to the width of the intrinsic 
inclination distribution, because of the effects of observational 
bias.  Objects of high orbital 
inclination spend a larger fraction of each orbit far from the 
ecliptic than do objects of small inclination.  Existing surveys, 
which have mainly been focused on the ecliptic, are therefore biassed 
against finding objects of high inclination.  

Models which attempt to correct for the inclination bias so far
do not yield unique values of the intrinsic inclination distribution
because of the limited sample size.  The most secure result from these
models is that the intrinsic inclination distribution must be very
broad, perhaps 2 to 3 times the apparent FWHM (JLC96), in order to
give rise to the apparent distribution.  An immediate implication of
the broad inclination distribution is that the velocity dispersion
among KBOs is $\Delta V \approx $1 -- 1.5 km s$^{-1}$.  Collisions
occur with specific energies that are
high enough to guarantee erosion of all but the largest objects (Stern
1995). Indeed, it has been suggested that the nuclei of short-period 
comets are collisionally produced fragments of large KBOs (Farinella 
and Davis 1996).

%%%%%%%%%%%%%%%%%%%%% Fig. 5 goes here %%%%%%%%%%%%%%%%%%%%%%%

Numerical simulations show that the inclinations of resonant objects
can be pumped to maximal values near $20^{\circ}$ (Malhotra 1996).
This is in good agreement with the maximum inclination among the known
Plutinos, namely $19.5^{\circ}$ (1995 QZ9).  What is more
surprising is that the classical
KBOs have a mean inclination formally consistent with the Plutinos,
and that the most highly inclined KBOs are found outside of mean motion
resonances (Fig. 5).  The origins of the high inclinations of classical KBOs
are not understood.  
One possibility is that the classical Belt might have been stirred
by a small number of bodies of Earth- and sub-Earth mass, scattered
outwards by Neptune and now lost from the planetary system (Morbidelli
and Valsecchi 1997; Petit et al.  1998).  While interesting in its own
right (and possibly even correct!) it has not been shown how the
massive scatterers could excite the classical Belt without
simultaneously depopulating the mean motion resonances.  Some KBOs could
be scattered into resonances (the fractional volume of phase space occupied by
resonances is $\approx$ 7\%) albeit with low probability of long term capture
(Levison and Stern 1995).


\subsection{F.~~Size distribution and Statistics}
The optical surveys are flux-limited rather than volume-limited,
leading to a bias towards large (bright) objects near the inner edge
of the belt and against small (faint) objects far away. As a result,
the size distribution of KBOs cannot be measured directly, but must 
be inferred from the CLF using a model of the spatial distribution.
We consider the size distribution of KBOs as represented by a
differential power law, $n(r)dr = \Gamma r^{-q} dr$, where $r$ [km] is
radius and $\Gamma$ and $q$ are constants.  Assuming a fixed albedo
0.04 for all objects and an inverse square radial distance
distribution, Monte Carlo simulations give best-fit values $\Gamma =
3.8 \times 10^{10}$ and $q \sim 4$ (JLT98; Luu and Jewitt 1998).  In a
$q = 4$ distribution, the cumulative number of objects with radius $>
r$ is $N(r) = \Gamma/(3 r^3)$.  The objects discovered in ground-based
surveys number $N$(50 km) $\sim 10^5$.  The distribution
allows $N$(1000 km) $\approx$ 10, i.e., ten Pluto-scale objects,
all but one as yet undiscovered.  At the
other end of the size distribution, the effective radius of the
well-studied short-period comets is in the range 1~km to 5~km.  The
fitted distribution allows $N$(5 km) $ = 1 \times 10^8$, while $N$(1
km) $\sim 10^{10}$.  Clearly, the Kuiper Belt is likely to hold
large numbers of undetected small bodies.  Models of the growth of
KBOs (Kenyon and Luu 1999) give $q = 3.5$ ($r < 0.3 - 3$ km) 
and $q = 4$ ($r > 1 - 3$ km), in excellent agreement with inferences
from the gradient of the CLF.

The total mass of a $q = 4$ distribution is

$$ M = {4 \times 10^9 \pi \rho \Gamma \over 3} \ \Bigl({0.04 \over
p_R} \Bigr)^{3/2} \ {\rm ln} \Bigl({r_{max} \over r_{min}}
\Bigr)         \eqno(2)$$

\noindent where $\rho$ [kg m$^{-3}$] is the bulk density and $p_R$ is the
red geometric albedo.  The bulk density is uncertain to within a
factor of a few.  Cometary nuclei may provide useful analogs of small
KBOs.  They have bulk density $\rho \approx 500$ to 1000 kg
m$^{-3}$. Conversely, Pluto and Charon are examples of
large KBOs: they have $\rho \sim 2000$ kg m$^{-3}$ (Foust et al.
1997).  Taking $\rho = 1000$ kg m$^{-3}$ as a plausible average
effective density, and with $p_R = 0.04$, we find that the observable
KBOs, with $r_{min} = 50$ km and $r_{max} = 1150$ km (the radius of
Pluto), have mass $M \sim 5 \times 10^{23}$ kg ($0.1~M_{\oplus}$). A
comparable mass is contained in bodies smaller than 50 km radius in
this distribution; however the size distribution of the smaller bodies
is observationally unconstrained, and their combined mass could
potentially be larger if the distribution is steeper than $q = 4$.  
Higher geometric albedos would decrease the
mass estimate.  With $p_R = 0.4$, for example, the mass given by
Eq. (2) is reduced by a factor 30. The optical survey data thus
suggest a mass within the 30 AU -- 50 AU zone that is a few tenths of
an Earth mass, but which is quite uncertain.  The optical mass is
smaller than the dynamical limit ($\sim 1~M_{\oplus}$) set by Hamid et
al. (1968), and far smaller than the mass expected by Kuiper (1951)
based on an extrapolation of the surface mass density of the solar
system.  Significantly, the modern Kuiper Belt contains too little
mass for KBOs to have grown in the $\sim 10^8$ yrs available prior to
the disruptive emergence of Neptune (Stern and Colwell 1997; Kenyon
and Luu 1998, 1999).  If the observed KBOs formed {\it in situ}, then
the original mass must have been 100 times the present mass.  Scattering by Neptune
(Holman and Wisdom 1993) appears incapable of clearing the Kuiper
Belt, except in the region close to the planet.  Self-destruction
by collisional grinding has been suggested (Stern 1996) and partially
modelled (Kenyon and Luu 1999).  KBOs larger than $\sim 100$ km 
are largely immune to collisional disruption (Farinella and Davis 1996).
The ground-based KBOs are thus likely to be true survivors from the
earliest days of the solar system.
  
\subsection{G.~~ Radial Extent of the Belt}

Depletion of the Kuiper Belt by Neptune perturbations is largely
restricted to the $30 \le R \le 42$ AU range (Holman and Wisdom 1993;
Duncan et al.  1995), suggesting that the more distant parts of the
Belt might be more densely populated than the region so far observed.
Using an $R^{-2}$ extrapolation of the surface mass density of the
solar nebula, Stern (1996) has estimated that the density of the
Kuiper Belt beyond 50 AU might be 100 times higher than in the 30 $\le
R \le 50$ AU annulus.  On the other hand, it is not obvious that
the Kuiper Belt necessarily extends to very large radii.  An $R^{-2}$ 
disk with a finite mass must be truncated at some definite outer 
radius, and
several physical processes are known which could provide an edge.
Empirically, two of six protoplanetary disks seen in silhouette
against the Orion Nebula have diameters $\le 100$ AU (McCaughrean and
O'Dell 1996), and are thus comparable in scale to the known Kuiper
Belt.  These disks are thought to have been tidally truncated during
close stellar encounters in an early, dense stellar environment.

The semimajor axes of the orbits of known KBOs span the range 35 AU to
about 115 AU, while the heliocentric distances at which KBOs
have been discovered are restricted to the more limited range 26 AU to
about 50 AU. Clearly, observational selection discriminates against
the discovery of distant objects because they are faint.  However,
detailed Monte Carlo simulations (Fig. 6) show that, with the
parameters of our own Mauna Kea CCD surveys, we should expect to have
discovered substantial numbers of classical KBOs beyond 50 AU, whereas none has
yet been found (Dones 1997; JLT98).  This result may be explained in
several ways.  First, the maximum object size ($r_{max}$) might be a
decreasing function of semimajor axis.  This would seem physically
plausible as a consequence of lower surface density at larger
distances, resulting in longer timescales for growth in the
protoplanetary disk.  However, the surface density in a $R^{- 2}$ disk at
45 AU (where we see many objects) is only 50\% larger than at 55 AU
(where we see none), and it is hard to imagine that $r_{max}$ could be so
sensitive to the local surface density.  Second, the size distribution 
might be steeper than $q = 4$, so
reducing the number of large objects visible beyond 50 AU (Gladman et
al. 1998).  Third, the Kuiper Belt surface density might decline
faster than $R^{-2}$ beyond the observed region.  In fact, our data
are consistent with a discrete outer edge to the classical belt at $a
\sim 50$ AU, although we remain uncomfortable with this interpretation.
Only the scattered KBOs occupy a volume that is clearly more extended than
the known classical belt.

%%%%%%%%% Figure 6 goes here.%%%%%%%%% 

\mainsection{III.~~Dust and the Relation to Circumstellar Disks}

Collisions among KBOs generate dust.  Unfortunately, in a $q = 4$
distribution the collisional cross-section is dominated by the smallest
(optically invisible) KBOs, and the dust production rate is
therefore difficult to assess with confidence.  A
crude upper limit, $\dot M_+ \sim 10^7$ kg s$^{-1}$, may be
obtained by dividing the current mass of the Belt ($\sim 0.1~
M_{\oplus}$) by the 4.5 Gyr age of the solar system.  A model-dependent
lower limit is set by erosion of KBOs by interstellar dust grains,
with potential production rates estimated at $\dot M_-
\sim 10^2$ to $10^4$ kg s$^{-1}$ (Yamamoto and Mukai 1998).  

Surprisingly, Kuiper Belt dust may already have been detected.
The Voyager 1 and 2 spacecraft recorded significant dust impact 
rates when beyond Neptune's orbit (Gurnett et al. 1997).  The 
Voyager experimenters did not themselves cite
the Kuiper Belt as the source of dust.  However, in the absence of 
other identifiable sources in the trans-Neptunian region, this seems 
to us a plausible hypothesis. 

The reported average number density of $\mu$m-sized grains 
is $N_1 = 2 \times 10^{-8}$ m$^{-3}$ (Gurnett et al. 1997).
The normal optical depth due to these grains is of order
$\tau \sim N_1 Q_s \pi a^2 H$, where $a$ is the grain radius, $Q_s$ is
the dimensionless scattering efficiency, and $H$ is the vertical
thickness of the Kuiper Belt.  We take $Q_s \sim 1, a \sim 1
\mu$m, and $H \sim 10$ AU to find $\tau \sim 10^{-7}$.  This is
much smaller than the optical depths of dust disks around other nearby
main-sequence stars.  For example, the $\beta$ Pic and $\alpha$ Lyr
dust disks have normal optical depths $\tau \sim 10^{-3}$ and $\tau
\sim 10^{-5}$, respectively (Backman and Paresce 1993).

We take the average mass of a grain as $m_d \sim 2 \times
10^{-14}$ kg (c.f. Gurnett et al. 1997), and the volume of the Kuiper
Belt (represented as an annular slab with inner and outer radii 30 AU
and 50 AU, respectively, and a thickness of 10 AU) as $V \sim 2 \times
10^{38}$ m$^3$.  The total mass of dust in micron-sized particles is
then $M_d \sim m_d N_1 V \sim 8 \times 10^{16}$ kg
($10^{-8}~M_{\oplus}$).  With a $\tau_c
\sim 10^6$ yr lifetime against collisional destruction (Jewitt
and Luu 1997), the implied production rate in micron-sized particles
is of order $M_d/\tau_c \sim 3 \times 10^3$ kg s$^{-1}$.  This constitutes  a
realistic estimate provided the dust size distribution extends to
particles no larger than those measured by the plasma wave analyzers,
as would be the case for dust grains produced by the small interstellar 
dust impacts. It is intriguing to observe that the empirical dust 
production rate is within the range of estimates for interstellar 
erosion (Yamamoto and Mukai 1998).

On the other hand, if the dust is produced by collisional grinding
among KBOs, the size distribution must extend to sizes much larger
than those measured by Voyager, and the mass production rate should be
corrected for the presence of larger but less abundant particles.  The
plasma wave analyser is sensitive to only a narrow range of particle
sizes: impacts below the threshold mass $1.2 \times 10^{-14}$ kg
(corresponding to particle radius $1.4~
\mu$m at unit density) do not excite measurable plasma waves (Gurnett
et al. 1997) while substantially larger particles are too rare to be
counted.  We assume that the Voyager spacecraft detected particles in the
radius range $1.4 \le a \le 10~\mu$m, and adopt a differential power
law size distribution with index -3.5 (Dohnanyi 1969) in order to
estimate the mass in larger particles.  The mass and mass production
rates must then be augmented by a factor

$$ f = {{\int^{a_{max}}_{1.4} a^{-1/2} \rm d a} \over
{\int^{10}_{1.4} a^{-1/2} \rm d a}} \approx {{\sqrt a_{max}} \over
{2}}		 \eqno(3)$$

\noindent where $a_{max} >> 1.4~\mu$m is the maximum particle size
expressed in $\mu$m.  From Eq. (3) we find $f \sim 16$ for $a_{max} =
1$ mm, $f \sim 500$ for $a_{max} = 1$ m and $f \sim 1.6 \times 10^4$
for $a_{max} = 1$ km.  The debris production rates therefore approach
$5 \times 10^4$ kg s$^{-1}$ for millimeter-sized particles, $1.5
\times 10^5$ kg s$^{-1}$ for meter-sized particles and $5 \times 10^7$
kg s$^{-1}$ for km-sized particles.  Clearly, these values involve
huge extrapolations from the Voyager dust detections and are highly
uncertain.  Nevertheless, they seem broadly compatible with our crude
upper limit to the dust production rate, and with at least the lower
estimates ($10^6$ kg s$^{-1}$) of Stern (1996). 

Perhaps the principal unexplained feature of the Voyager data is that
no impacts were detected beyond 51 AU for Voyager 1 and 33 AU for
Voyager 2.  If future detailed models of dust dynamics cannot explain
this cut-off, then it is possible that the detected dust grains have
another source.  In this case, the above mass and mass production
rates must be regarded as upper limits to the collisionally produced
Kuiper Belt dust content.

The survival of a 42 cm-diameter fuel tank on the spacecraft Pioneer
10 during its 10-year flight through the Kuiper Belt (at relative
velocity 12 km s$^{-1}$) sets an independent but less tightly constraining
limit to the dust content of the Kuiper Belt (Anderson et al. 1998).


%%%%%%%%%Table 2 goes here.%%%%%%%%%

Three other opportunities exist for the detection of Kuiper Belt
 dust.

First, one might search for diffuse thermal emission from Kuiper Belt
dust.  From an observational point of view, the detection of diffuse
Kuiper Belt dust is complicated by the foreground Zodiacal Cloud, as
well as by the background due to galactic dust.  Backman et al. (1995)
used COBE observations at 140 $\mu$m and 240 $\mu$m wavelength to set
an upper limit to the mass of Kuiper Belt dust $M_d < 10^{-5}
M_{\oplus}$.  This limit, which is itself quite model-dependent, is
compatible with $10^{-8} M_{\oplus}$ estimated above from the Voyager
dust impacts.

Second, recent impacts in the Kuiper Belt should produce observable
optical (Alcock and Hut 1993) and infrared (Stern 1996) dust
signatures.  Consider a body of initial radius $a_0$ that is
pulverized by impact into particles of mean radius $a$.  The resulting
dust cloud will contain $(a_0/a)^3$ particles of combined geometric
cross-section $C \sim a^2 (a_0/a)^3$.  The cloud will expand and
brighten in reflected light until it reaches the critical radius $a_C
= a\ (a_0/a)^{3/2}$, at which the optical depth is unity.  This
expansion takes a time $t = a_C/v$, where $v$ is the mean speed of
ejection of the dust.  For example, with $a_0$ = 1 km, $a = 1~\mu$m
and $v \sim 1$ km s$^{-1}$ we find $C = 10^{15}$ m$^2$, $a_C = 3
\times 10^7$ m (corresponding to angular radius 1" at 40 AU distance)
and $t \sim 1$ day.  Such a debris cloud would reach peak apparent red
magnitude $ \sim 7$ on a timescale of 1 day and thereafter expand into
a diffuse cloud of steadily decreasing surface brightness.  After 100
days, the cloud would have angular diameter $\sim 100$", a mean
surface brightness of order 17 mag. arcsec$^{-2}$, and would still be
an easy observational target with 50 to 100 times the surface
brightness of the
moonless night sky. The timescale for radiation pressure to deflect
particles in the expanding cloud is $t_{rp} \sim v/(\beta g)$, where
$\beta \sim 1$ for micron-sized grains and $g = 4 \times 10^{-6}$ m
s$^{-2}$ is the solar gravity at 40 AU.  We find $t_{rp} \sim 10$ yr,
meaning that no perceptible radiation pressure swept tail would grow.
The number of such spherical, expanding debris clouds
visible at one time is highly uncertain.  Estimates range from a few to many
thousands (Alcock and Hut 1993; Stern 1996).  No convincing examples
have yet been reported.

Third, the Poynting-Robertson and plasma-drag forces may carry Kuiper
Belt dust particles into the inner solar system, where direct samples
might be found in existing interplanetary dust particle collections
(Flynn 1996).  The dust lifetime against collisional destruction by
interstellar impacts ($\sim 10^6$ yr; Jewitt and Luu 1996) is short
compared to the $\sim 10^7$ yr transport time (Liou et al. 1996) for
particle sizes from $0.1 \le a \le 100~\mu$m.  Therefore, most Kuiper
Belt particles will be collisionally destroyed while en-route to the
Earth.  On the other hand, the rate of production of Zodiacal dust is
only $\sim 10^4$ kg s$^{-1}$ (Leinert et al. 1983), so that even small
amounts of surviving Kuiper Belt debris might contribute measurably to
the Zodiacal dust complex (c.f. Table 2).

Some nearby stars possess disks in which the dust lifetimes (to Poynting-
Robertson drag and collisional shattering) are short compared to the
stellar age (Smith and Terrile 1984, Jayawardhana et al. 1998, Koerner et al.
1998, Trilling and Brown 1998, Schneider et al. 1999).  In these disks, 
dust might be generated by collisional grinding of larger, parent bodies, 
just as in the Kuiper Belt.  Images of HR 4796A show a 70 AU radius,
ring-like morphology that is highly suggestive of our own Kuiper Belt.
The famous disk of $\beta$ Pictoris extends to $\sim$ 1000 AU.
As mentioned above, the quantities of dust (estimated from the scattering
optical depths) are 3 to 4 orders of magnitude larger than the dust
content of the Kuiper Belt.  It is tempting but conjectural to associate 
this difference with a collisionally more active phase in the lives of 
circumstellar debris rings.  Future observations with sensitive ground-
and space-based coronagraphs will presumably show that Kuiper Belts are
common if not ubiquitous around other stars.


\mainsection{IV.~~Relation to the Short-Period Comets and
Centaurs}

Objects that are dislodged from the Kuiper Belt as a result of intrinsic
dynamical instabilities (Duncan et al. 1995) or mutual gravitational scattering
(Ip and Fernandez 1997) eventually fall under the gravitational control
of the gas giant planets.  The prime candidates for these ex-KBOs are
the "Centaurs", whose orbits intersect those of the gas giant planets
and which are consequently highly chaotic and short-lived.  Their
eventual fate is (1) to be scattered close to the Sun, where they
sublimate and are recorded as short-period comets, (2) to be scattered
out of the solar system, or (3) to impact a planet or the sun.  A
useful practical definition is that the Centaurs are objects having
both perihelion distance and semi-major axis between the orbits of
Jupiter (5 AU) and Neptune (30 AU).  By this definition, there are 11
known Centaurs (Table 3).  Three (P/Schwassmann-Wachmann 1, P/Oterma and
2060 Chiron) display comae at least part of the time indicating their 
volatile-rich cometary nature (P/Schwassmann-Wachmann 1 is a prodigious
source of carbon monoxide, indicating formation at temperature $< 50 K$, 
Senay and Jewitt 1994).  
The remaining six have been stellar in appearance in all
observations to date, but presumably contain buried ice.  The Centaurs 
are important because they provide
the dynamical link between the KBOs and the short-period comets and,
on the more practical side, they are much closer (brighter) than the
typical KBOs, and therefore make much easier targets for physical
study.

The populations of Kuiper Belt Objects, Centaurs and short-period
comets should be in the approximate ratio $n_{KBO}:n_C:n_{SPC} \sim
t_{KBO}:t_C:t_{SPC}$.  Here, $t_{KBO} \sim 10^{10}$ yr, $t_C \sim
10^6$ to $10^7$ yr (Asher and Steel 1993; Hahn and Bailey 1990; Dones
et al.  1997; Levison and Duncan 1997) and $t_{SPC} \sim 10^5$ yr
(Levison and Duncan 1994) are the respective dynamical lifetimes.
With $10^5$ KBOs larger than 100 km in diameter, we would expect to
find $n_{KBO}:n_C:n_{SPC} \sim 10^5:(10^2$ -- $10^1): 1$. Currently,
only 3 Centaurs with diameters $> 100$ km are known (1997 CU$_{26}$,
Chiron and Pholus; Table 3) compared with 10 to 100 expected. The 
majority of such objects should be located just interior to 
Neptune (Levison and Duncan 1997) and sky
surveys sensitive to these objects are far from complete (the
most telling evidence of incompleteness is that the largest Centaur,
1997 CU$_{26}$, is also one of the most recently discovered).  There is
presently no short-period comet with diameter near 100 km.

%%%%%%%%%Table 3 goes here.%%%%%%%%%


\mainsection{V.~~Physical Properties}

\subsection{A.~~Albedos}

The apparent optical brightness of a body viewed in reflected sunlight
is proportional to the product of the geometric albedo with the
cross-section, and given by

$$ p_R r^2 \phi(\alpha) = 2.25 \times 10^{22} R^2 \Delta ^2
10^{0.4(m_{\odot} - m_R)}		 \eqno(4)$$

Here, $p_R$ is the geometric albedo, $\alpha$ is the phase
(Earth-object-Sun) angle, the heliocentric and geocentric distances,
$R$ and $\Delta$, are expressed in AU, while $m_{\odot}$ and $m_R$ are
the apparent magnitudes of the sun and Kuiper Belt object,
respectively.  In Eq. (4) $\phi(\alpha)$ is a phase function that
describes the angular dependence of the scattered light, reasonably
approximated by $\phi(\alpha) = 10^{-0.04\alpha}$. At the small phase
angles attained by KBOs ($\alpha < 2^{\circ}$), $\phi(\alpha) = 1$
provides a good first approximation.

 The thermal flux density is given by 

$$S_{\nu} = {\epsilon _{\nu} B_{\nu} (T) \pi r^2 \over {\Delta ^2}}	
\eqno(5)$$

\noindent where $\epsilon_{\nu}$ is the emissivity, and $B_{\nu}(T)$ [W
m$^{-2}$ sr$^{-1}$ Hz$^{-1}$] is the Planck function. The effective
surface temperature $T$[K], is a complicated function of the
emissivity, Bond albedo and thermal diffusivity of the body, as well
as of the heliocentric distance and rotation vector (Spencer et
al. 1989).  Eqs. (4) and (5) contain too many unknowns to allow a
unique solution for radius or albedo.  Nevertheless, measurements of
main-belt asteroids have shown that useful estimates for radius and
albedo may be obtained from Eqs. (4) and (5) using judiciously 
assumed values of the unknown parameters.

In the case of the KBOs, thermal emission has yet to be detected, and
the albedos remain unknown.  At $T \sim 50$ K, the Planck
maximum falls near $60 \mu$m, a wavelength which is inaccessible from
the ground.  Marginal ($\le 3 \sigma$) detections of 1993 SC and 1996
TL$_{66}$ obtained using the ISO satellite  imply low (few percent) 
albedos (Nick Thomas, private communication).  Future orbiting 
infrared satellites (e.g., SIRTF) may be able to detect thermal 
radiation from KBOs with greater significance.  Rayleigh-Jeans 
emission ($800~\mu$m wavelength) may also be detected using 
submillimeter detectors on ground based interferometers (such
as the Submillimeter Array nearing completion on Mauna Kea).

In principle, occultations of background stars might be used
to determine the sizes of KBOs (c.f. Bailey 1976).  Optical
observations would then give the albedos directly, through Eq. (4).
Astrometry of KBOs is presently good at the $\pm 0.3$" level, while
the largest angular diameters subtended by KBOs are of order 0.02"
(for 1996 TO$_{66}$).  Consequently, meaningful predictions of
occultations by KBOs cannot yet be made and, to date, no occultation
observation has been attempted.  The Taiwanese-American Occultation
Survey (TAOS) experiment should detect occultations by previously unseen KBOs
within the next 5 years. However, most of these objects will be so
small that optical detections will be impossible, and the albedos will
remain unmeasured.

All KBO diameters reported in the literature have been derived under
the assumption of a red geometric albedo $p_R = 0.04$.  This is not
unreasonable, given that both short-period comets and Centaurs
probably originated in the Kuiper Belt, and these objects have red
geometric albedos mostly in the range $0.02 \le p_R \le 0.05$.  The
existence of a wide diversity of colors (Luu and Jewitt 1996; Jewitt
and Luu 1998; Tegler and Romanishin 1998) suggests that different KBOs
very possibly could have different albedos. On the other hand, the
extreme color differences between Centaurs 2060 Chiron, 5145 Pholus
and 1997 CU$_{26}$ are not matched by a wide range in the measured albedos
(0.14 +0.06/-0.03, $0.044 \pm 0.013$ and $0.045 \pm 0.010$,
respectively; Jewitt and Kalas 1998).

  
\subsection{B.~~Colors}

Most KBOs are too faint for spectroscopy and observers have resorted
to broadband colors as low-resolution (but higher in
signal-to-noise) substitutes.  Optical colors for $\sim 20$ KBOs are
now available (Luu and Jewitt 1996; Green et al. 1997; Tegler and
Romanishin 1998), as well as near-infrared colors for a handful of
KBOs (Jewitt and Luu 1998).  

%%%%%%%%%Figure 7 goes here%%%%%%%%%

Surprisingly, the KBOs (Luu and Jewitt 1996; Green et al.  1997;
Jewitt and Luu 1998) and Centaurs (Romanishin et al. 1997; Davies et
al. 1998) exhibit a wide spread of colors, ranging from nearly neutral
to very red (Fig. 7a).  Tegler and Romanishin (1998) confirm this
diversity of KBO colors but further report that the B-V and V-R 
colors of KBOs and Centaurs are bimodally distributed (Fig. 7b),
suggesting the existence of two distinct surface types.  The 
optical-infrared V-J index varies widely among KBOs, but is
not obviously bimodal (Jewitt and Luu 1998).
A statistically significant (3 $\sigma$) but unexpected and 
unexplained correlation between the absolute red magnitude and 
V-J has also been reported (Jewitt and Luu 1998).  

Fig. 7 shows that the KBOs and Centaurs are indistinguishable in
broadband color-color plots, consistent with a common origin.
However, their color distributions are unmatched by those of comet
nuclei and asteroids (Fig. 8), thanks to some extremely red KBO and
Centaur surface materials (V-R $> 0.6$) that seem to be rare or absent among
comets and asteroids.  The absence of extremely red material on the
cometary nuclei is consistent with progressive destruction, or burial,
of irradiation mantle formed in the Kuiper Belt (c.f. Cruikshank et
al. 1998), but might also be an observational artifact, since few 
nuclei have yet been measured.


%%%%%%%%%Figure 8 goes here%%%%%%%%%


Laboratory experiments suggest that KBOs might be covered by
"irradiation mantles," produced by high energy particle irradiation of
surface ices.  When carbon-containing ices (e.g., methane or CO) are
irradiated by high energy particles (cosmic rays and UV photons), this
irradiation leads to the selective loss of hydrogen while encouraging
the formation of complex carbon compounds (e.g., see Moroz et al. 1998
and references therein).  The result is a solid residue (having a 
column density of $\sim 10^3 kg m^{-2}$, corresponding to a
thickness $\sim 1 m$ in solid ice) that is dark due to its complex 
carbon compounds. Exactly
how the color and albedo change with time depends on the initial
composition and the irradiation fluence (Moroz et al.  1998).  The
composition of the mantle is quite different from the interior, which
retains pristine ice. The mantle thickness and strength are uncertain,
but may be sufficient to survive upon the object's first entry into
the inner solar system.

\subsection{C.~~Spectra}

 Wilson et al. (1994) were able to fit the optical spectrum of 
Centaur 5145 Pholus
with "Titan tholins," chemically complex hydrocarbon mixtures produced
by UV irradiation of simple ices.  Cruikshank et al. (1998) obtained a
fit for both the optical and near-infrared spectra of Pholus with
a model consisting of carbon black (61.5\%) and an intimate mixture
(38.5\%) of olivine, Titan tholin, H$_2$O ice, and CH$_3$OH ice
(Fig. 9).  Water ice has been reported on Centaur 1997 CU$_{26}$ 
(Brown et al. 1998; Brown and Koresco 1998) although its surface is very dark
($p_R = 0.045 \pm 0.010$; Jewitt and Kalas 1998).  On the other hand,
2060 Chiron is spectrally bland (Luu et al. 1994).  Spectral
fitting models are non-unique, but there is consensus that organics of
some type are present on the surface of Pholus, while the
identification of water on CU$_{26}$ also appears secure.  Color 
diversity among the KBOs is thus matched by spectral 
and compositional diversity among the Centaurs.

Spectra of only three KBOs have been reported.  The smoothed reflectance
spectrum of 1993 SC (Brown et al. 1997) shows prominent absorptions
that have been qualitatively interpreted as hydrocarbon features
(Fig. 10).  In contrast, the spectrum of 1996 TL$_{66}$ is neutral, with no
apparent absorption features (Luu and Jewitt 1998).  The spectrum of
1996 TO66 shows absorptions at 1.5 and 2.0 $\mu$m indicative of water
ice.  The differences among the KBOs are intriguing, but it would be premature
to attempt an interpretation.  Spectra of higher quality are
urgently needed in order to make progress in this field.

%%%%%%%%%Figures 9 and 10 go here%%%%%%%%%

\subsection{D.~~Origin of Color Diversity}

 The diversity of the colors is a surprise in the sense that
all KBOs should be exposed to cosmic rays and thus should
have irradiation mantles with similar colors.  Three simple explanations
have been suggested for the origin of the color diversity (Luu and
Jewitt 1996).  Firstly, it is possible that the KBOs possess
intrinsically different compositions, and that the different colors
are tracers of compositional variations. For example, asteroids in the
Mars-Jupiter belt show different compositions that seem to be related
to their sites and temperatures of formation.  However, the KBOs
probably formed in-situ, in the presence of a very slight radial
temperature gradient ($T \propto R^{-1/2}$).  Temperature differences
across the 35 to 50 AU zone likely amounted to only $\sim 10$ K.  It
is hard to see how strong compositional differences might arise from
such small differences in the formation temperature.  A second
explanation is that occasional collisions in the Kuiper Belt puncture
the irradiation mantles on some objects, excavating craters, and
showering nearby regions with impact debris.  The freshly excavated
material would be unirradiated, and thus should be of a different
composition (more ice-rich, less red?)  than the irradiated mantle.  
This mechanism requires that the timescales for 
resurfacing and for radiation damage of the surface layers are 
comparable, to avoid saturation of the color at one or the other extreme.  
One testable prediction of the resurfacing hypothesis is that KBOs 
should show rotational modulation of albedo and color, corresponding 
to local blanketing by excavated debris. Another is that albedo and 
color should be closely related.  The bimodal color distribution of 
KBOs (Tegler and Romanishin 1998), if independently confirmed, would 
seem to rule out impact resurfacing as an explanation for the color
diversity.  Thirdly, Moroz et al. (1998) have observed that the
color of light reflected from laboratory bitumen samples is a 
function of the effective particle size in the scattering surface.

\subsection{E.~~Structure}

The observed few-hundred km diameter KBOs have thermal diffusion 
times ($\tau \sim r^2 /\kappa$,
where $r [m]$ is radius and $\kappa$ $[m^2 s^{-1}]$ is thermal diffusivity)
comparable to or longer than the age of the solar system.  Energy liberated in these
bodies by the decay of radioactive elements is trapped and must contribute
to an increase in the central temperature of magnitude


$$\Delta T \sim H r^2 / (\kappa c_p)
\eqno(5)$$

\noindent where $H [W kg^{-1}]$ is the power production per unit
mass due to radioactivity and $c_p$ $[J kg^{-1} K^{-1}]$ is the specific
heat capacity of the bulk material.  With $H \sim 10^{-12} W kg^{-1}$ 
(the heating rate measured for carbonaceous chondrite material, diluted
by a factor of 4 to account for the presence of substantial ice),
$\kappa \sim $(1 to 10)$ \times 10^{-7} m^2 s^{-1}$ and $c_p \sim 10^3 J kg^{-1} K^{-1}$,
we estimate $\Delta T [K] \sim $(10 to 100)$ (r/100 [km])^2$. Objects of a few 
100 km radius may
experience internal heating sufficient to mobilise interior volatiles, 
giving rise to a compositionally layered structure in the KBOs.  This,
in turn, would complicate our understanding of the compositional
significance of short-period comets, if these are indeed collisional
fragments from the upper layers of 100 km scale KBO parents.  One
might even expect that the larger KBOs should be predisposed to
out-gas their super-volatiles ($CO, N_2$), giving rise to surface
frosts and systematic variations in albedo and surface color with
radius.  The limiting case, Pluto, has long been known to sport a
high albedo, patchily frosted surface that is probably not typical
of the smaller KBOs.  Outgassing due to mobilization of internal
volatiles might be responsible for some of the color and spectral
variation observed amongst the KBOs.


\mainsection{VI.~~Major Questions}

 We end with ten leading questions about the Kuiper Belt.

\bigskip
\noindent 1) What is the size distribution of KBOs at the smallest ($<
50$ km) and largest ($> 500$ km) scales?  Are the size distributions
of the classical, resonant and scattered object sub-classes equal?  Is
there observational evidence for a cut-off in the size distribution at
large radii?  Is Pluto unique?

\smallskip
\noindent 2) What are the albedos of KBOs?  Is there an albedo-color
relation as expected from collisional resurfacing?  

\smallskip
\noindent 3) What is the origin of the spectral diversity exhibited by
the KBOs?  Are the optical colors bimodal?  Are the colors
compositionally diagnostic or indicative of other effects
(e.g. particle size variations)?

\smallskip
\noindent 4) Can compelling evidence be found for compositional
differences between the Kuiper Belt (short-period) and Oort
Cloud (long-period) comets?  Different temperatures in the 
source zones of the long-period comets (5 - 20 AU) and
the short-period comets (30 - 40 AU) should, for example,
lead to major variations in the abundances of super-volatiles
(e.g. $N_2, CO$) between the two groups, yet no such differences
have been reported.

\smallskip
\noindent 5) Are small (1 to 10 km) KBOs present in numbers sufficient
to supply short-period comets to the inner solar system for 4.5 Gyr?
If the comets originate in the Kuiper Belt, precisely where (in 
chaotic zones associated with resonances, among the scattered
objects, or elsewhere)?

\smallskip
\noindent 6) Was the ancient Kuiper Belt much more massive than the
present-day Kuiper Belt and, if so, how was the original mass
depleted?  What, if anything, can the cratering rates on the outer
planet satellites tell us about the time dependence of the Kuiper Belt
mass? Was the early Kuiper Belt massive enough to collisionally
produce dust in quantities sufficient to rival the dust rings circling
some nearby, main-sequence stars (e.g. $\beta$ Pic)?  

\smallskip
\noindent 7) How common is binarity in the Kuiper Belt?  Presently, only Pluto
 is known to hold a satellite. If Charon's presence indicates a once
 more active collisional regime, then it is reasonable to suppose
 that other KBO binaries await discovery.

\smallskip
\noindent 8) What are the relative populations in the mean-motion
resonances and what do these tell us about the population mechanism
(c.f. Malhotra 1995, 1996)?

\smallskip
\noindent 9) What processes were responsible for the excitation of the
velocity dispersion in the Classical Kuiper Belt?

\smallskip
\noindent 10) What role has been played by collisions in shaping the
present-day Kuiper Belt? Are the nuclei of short-period comets the collision
fragments of larger bodies (c.f. Farinella and Davis 1996)?


\mainsection{Acknowledgements}
This work was supported by grants from NASA's Origins Program.


\vfill\eject
\null

\vskip .5in
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\vfill\eject
\null

\vskip .5in
\centerline{\bf FIGURE CAPTIONS}
\vskip .25in

\caption{Figure 1.\capskip Plan view of the Kuiper Belt showing the 
orbits of the Kuiper Belt objects known as of June 1999.  Orbits of Jupiter
(J) and Neptune (N) are marked, as are the first four scattered KBOs.  
The majority of the KBOs follow orbits in the 30 to 50 AU radius band.
The region shown is 260 AU x 340 AU. Figure prepared by Chad Trujillo,
University of Hawaii.
}

\caption{Figure 2.\capskip Semimajor axis versus eccentricity
for Kuiper Belt Objects.  Filled circles denote multi-opposition orbits. 
Empty circles denote orbits computed from astrometry taken within a single
opposition.  Pluto is marked by X.  Neptune is at ($a$,$e$) = (30,0). 
The dashed lines mark $q = 30$ AU and $q = 35$ AU.  Objects above
the upper line are
Neptune-crossers.  Scattered Kuiper Belt Object 1996 TL$_{66}$ ($a$,$e =
85$ AU, 0.59) is off scale and not plotted.  Vertical lines mark the
locations of mean-motion resonances.}

\caption{Figure 3.\capskip Same as Figure 2 but on a larger scale
to show the scattered Kuiper Belt Objects.  All four scattered KBOs
fall along the perihelion distance q = 30 AU line.
}

\caption{Figure 4.\capskip Luminosity function of the Kuiper
Belt excluding photographic surveys.  The lines show fits to the
data from Luu and Jewitt (1998) and Gladman et al. (1998),
discussed in the text. Chiang and Brown (1999) report a CLF close
to that of Luu and Jewitt.}
 
\caption{Figure 5.\capskip Histogram of inclinations of Kuiper
Belt Objects, plotted with a 5 degree bin size.  The Plutinos and Classical 
KBOs are shown separately.  Only multi-opposition orbits have been 
used in the histogram. The two populations have similar mean 
inclinations. However, the Plutinos are deficient in both high ($i > 20 deg$) and
low ($i < 5 deg$) inclination orbits relative to the Classical objects.}

\caption{Figure 6.\capskip  Heliocentric distance at discovery
vs. apparent red magnitude.  Solid circles mark objects found in
the 8k survey (JLT98, from which this figure is taken).  Crosses
denote simulated objects that passed the survey detection
criteria.  The classical Kuiper Belt is taken to extend to
maximum semi-major axis $a_{max} = 200$ AU.  The KBOs obey a
power law size distribution with index $q = 4.0$, and all have
albedo $p_R = 0.04$.  Plutinos have been included in proportions
needed to yield an apparent Plutino fraction $P_a \approx 0.35$.  The
radial density index in the classical Kuiper Belt is taken to be
$p = 2$ in both models and, in the particular simulation shown
here we have chosen $r_{max} = 500$ km.  Diagonal lines show the
apparent magnitude as a function of heliocentric distance for
KBO radii 50, 100, 200 and 400 km.}  

\caption{Figure 7. \capskip (a).  B-V color plotted against V-R
color for KBOs (filled circles) and Centaurs (empty circles). 
The color of the Sun is marked. Error bars are 1 $\sigma$. Data 
from Luu and Jewitt 1996  
(b). Same as (a) but showing data from Tegler and Romanishin 1998.
Note that many objects are common to both panels.  
}  

\caption {Figure 8.\capskip  Histograms of the V-R color index
computed for five different dynamical classes of object.  KBOs
and Centaurs share a common, large color range.  The nuclei of
comets and the Jovian Trojans display a smaller color dispersion
and seem to lack the extremely red material present on some KBOs
and Centaurs.  The Near Earth Objects, included here only for
reference, are comparatively blue, and quite distinct from the
other classes.  Figure modified from Luu and Jewitt 1996.}

\caption{Figure 9.\capskip Composite optical and near-infrared
spectrum of Centaur 5145 Pholus. The model is an intimate
mixture of olivine (to provide absorption near $1.5 \mu$m), Titan
tholin (the red optical slope), water ice (the $2 \mu$m absorption),
methanol ice (the $2.27 \mu$m absorption), and carbon black
(Cruikshank et al. 1998).}

\caption{Figure 10.\capskip Optical - infrared reflectance
spectra of 1993 SC and 1996 TL$_{66}$.  The 1993 SC spectrum has
been smoothed to a resolution $0.08 \mu$m (by Brown et al.
1997).  The 1996 TL$_{66}$ spectrum is shown unsmoothed (Luu and
Jewitt 1998) and is featureless, ruling out a surface of pure
water ice.  The spectra have been vertically displaced for
clarity.}

 

\bye

