The Shapes and Spins of Kuiper Belt Objects

The Shapes and Spins of Kuiper Belt Objects
The Shapes and Spins
of Kuiper Belt Objects
The Shapes and Spins
of Kuiper Belt Objects
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van de Rector Magnificus Dr. D.D. Breimer,
hoogleraar in de faculteit der Wiskunde en
Natuurwetenschappen en die der Geneeskunde,
volgens besluit van het College voor Promoties
te verdedigen op donderdag 17 februari 2005
klokke 15.15 uur
door
Pedro Bernardino Lacerda Cruz
geboren te Lisboa, Portugal
in 1975
Promotiecommissie
Promotor:
Prof. dr. H.J. Habing
Referent:
Dr. C. Dominik (Universiteit Amsterdam)
Overige leden:
Dr. M. Hogerheijde
Dr. F.P. Israel
Dr. S. Kenyon (Harvard University, USA)
Dr. J. Luu (MIT Lincoln Laboratory, USA)
Prof. dr. G.K. Miley
Prof. dr. A. Quirrenbach
Prof. dr. S. Schlemmer (Universität zu Köln, Germany)
Prof. dr. P.T. de Zeeuw
There are infinite worlds both like and unlike this world of ours.
Epicurus (341–270 B.C.)
cover design: Maria Reis and Pedro Lacerda
vii
Table of contents
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Page
1
.
1
.
7
. 11
. 14
. 16
lightcurves
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
21
22
24
26
28
32
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
35
36
36
38
39
42
45
Chapter 4. Analysis of the rotational properties
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 Observations and Photometry . . . . . . . . . . . .
4.3 Lightcurve Analysis . . . . . . . . . . . . . . . . . .
4.3.1 Can we detect the KBO brightness variation?
4.3.2 Period determination . . . . . . . . . . . . .
4.3.3 Amplitude determination . . . . . . . . . . .
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 1998 SN165 . . . . . . . . . . . . . . . . . . .
4.4.2 1999 DF9 . . . . . . . . . . . . . . . . . . . .
4.4.3 2001 CZ31 . . . . . . . . . . . . . . . . . . . .
4.4.4 Flat Lightcurves . . . . . . . . . . . . . . . .
4.4.5 Other lightcurve measurements . . . . . . . .
4.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Spin period statistics . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
47
48
49
52
53
56
57
58
58
58
60
63
63
63
66
Chapter 1. Introduction
1.1 The origin of comets
1.2 The Kuiper belt . . .
1.3 Kuiper belt objects .
1.4 Thesis summary . . .
1.5 Future prospects . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter 2. Detectability of KBO
2.1 Introduction . . . . . . . . . .
2.2 Definitions and Assumptions .
2.3 “Flat” Lightcurves . . . . . . .
2.4 Detectability of Lightcurves .
2.5 Conclusions . . . . . . . . . .
.
.
.
.
.
Chapter 3. The shape distribution
3.1 Introduction . . . . . . . . . . .
3.2 Observations . . . . . . . . . . .
3.3 Discussion . . . . . . . . . . . .
3.3.1 Gaussian distribution . .
3.3.2 Power law distribution . .
3.4 Summary . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
viii
4.5.2 Lightcurve amplitudes and the shapes of KBOs . . . . . . .
4.5.3 The inner structure of KBOs . . . . . . . . . . . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 5. Origin and evolution of KBO spins
5.1 Motivation . . . . . . . . . . . . . . . . . . . .
5.2 Spin evolution model . . . . . . . . . . . . . .
5.2.1 General description . . . . . . . . . . .
5.2.2 Model Details . . . . . . . . . . . . . . .
5.3 Results and Discussion . . . . . . . . . . . . .
5.3.1 Effect of disruption energy scaling laws
5.3.2 Effect of density . . . . . . . . . . . . .
5.3.3 High angular momentum collisions . . .
5.4 The origin of KBO spin rates . . . . . . . . . .
5.4.1 Anisotropic accretion . . . . . . . . . .
5.5 Limitations and future improvements . . . . .
5.6 Summary and Conclusions . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
67
71
74
79
. 80
. 81
. 81
. 82
. 88
. 89
. 89
. 92
. 94
. 96
. 101
. 101
Nederlandse samenvatting (Dutch summary)
105
Resumo em Português (Portuguese summary)
113
Curriculum Vitae
119
Nawoord / Acknowledgments
120
CHAPTER
1
Introduction
1.1
The origin of comets
B
efore the XVIIth century comets were seen as portents of divine will, sent
by the gods to punish mankind. Newton (1686) showed that the paths of
these wandering celestial objects were actually very well defined, and obeyed the
universal law of gravitation. Newton’s theory, undoubtedly one of the greatest
achievements of human intellect, successfully describes the motions of the moon
around the Earth, of the planets around the Sun, of the Sun around the center
of our galaxy, and so on and so forth.
Making use of Newton’s laws, Halley (1705) proposed that three comet apparitions in 1456, 1531, and 1607 were actually three returns of the same comet.
He predicted that the heavenly body should revisit the inner solar system in
1758. The comet returned around Christmas of 1758, twelve years after Halley’s
death, and has been called Halley’s Comet ever since.
But understanding how comets move was only the first step. Humans are
curious beings, and thereby began questioning. What are comets? Where
do they come from? Aristotle (384–322 BC) was the first man that tried to
explain comets as something physical. He speculated they were atmospheric
phenomena—luminous clouds of gas—that because of their transient character could not be part of the perfect realm of the heavens. Aristotle called
them kometes, which means “wearing long hair” in ancient Greek. As with
most scientific disciplines, the Aristotelian way of thinking survived until the
Renaissance. In the late XVth century natural philosophers recognized that observation is an essential tool in understanding natural phenomena. The works
of Leonardo da Vinci (1452–1519)—who was probably the ultimate observer
of nature—exemplify how much can be learnt from detailed observation of the
surrounding world. The pioneer in accurate astronomical observations was the
Danish astronomer Tycho Brahe. In 1577 he used observations of a comet made
2
Introduction
Figure 1.1 – Histogram of the reciprocal semi-major axes of comet orbits, in astronomical
units. Reproduced from Oort & Schmidt (1951).
from different locations in Europe, to test the hypothesis of Aristotle. The small
observed parallax1 indicated that the comet had to be much further out than
the Earth’s atmosphere—even further than the Moon.
As the number of observed comets increased, statistical analysis of their
orbits became possible. Astronomers have divided the comets into two classes,
according to their orbital period: the short-period comets, with periods shorter
than 200 years, and the long-period comets, with periods longer than 200 years.
The orbits of comets in each class are quite different. Short-period comets have
prograde2 orbits which lie close to the plane where planets move. This plane
is called the ecliptic, and is defined as the plane of the orbit of the Earth. By
contrast, the long-period comets come into the inner regions of the solar system
from all directions—there is no preferred orbital plane. Furthermore, their long
orbital periods indicate that they come from large distances, as a consequence
of Kepler’s 3rd law of orbital motion.
The director of the Sterrewacht Leiden from 1945 to 1970 was Jan Hendrik
Oort. By the time of his appointment, Oort had already made key contributions
to astronomy. He had observationally confirmed, and analytically described the
rotation of the Milky Way3 (Oort 1927), following a hypothesis by Lindblad
(1925), and had made important contributions to the theory of dark matter
(Oort 1940). In the fall of 1948, a PhD student of Oort, van Woerkom, obtained
his doctor degree with a dissertation titled “On the origin of comets”. The work
of his student got Oort pondering on the subject. A little over a year later
he published his conclusions (Oort 1950). The high frequency of comet orbits
with very small reciprocal semi-major axes (see Fig. 1.1) led Oort to propose
1 The apparent difference in position of a body on the sky (relative to the background stars)
as seen from different points of observation.
2 All planets orbit the Sun in the same direction, usually called direct or prograde.
3 The Milky Way is the home galaxy to our solar system; it is a spiral galaxy, containing
some 100 billion (1011 ) stars.
The Shapes and Spins of KBOs
3
the existence of a vast spherical swarm of comets extending to a radius of about
150 000 astronomical units4 . Oort figured that this spherical cloud is occasionally
be perturbed by stars passing close to the Sun. As a result, some comets are
ejected to interstellar space, and some fall into the inner solar system along nearly
parabolic orbits. The latter become the visible comets. The idea prevailed
and the spherical reservoir became known as the “Oort cloud”. Although its
existence cannot be observationally confirmed, the Oort cloud provides the best
explanation of the observed distribution of the orbits of long-period comets.
Until late-1970s it was believed that short-period comets also originate in
the Oort cloud. The evolution of cometary orbits, from randomly-oriented longperiod to prograde low-inclination short-period trajectories, was attributed to
perturbations by the giant planets, particularly Jupiter. It was necessary, however, to demonstrate that such evolution is possible, and that it correctly predicts
the observed number of short-period comets. The work of van Woerkom (1948)
was partly an attempt to show that long-period comets could be brought into
short-period orbits due to perturbations by Jupiter. His theoretical calculations
predicted that this process was a factor ∼20 less efficient than needed to explain
the observed frequency of short-period comets. With the advent of computers
the complex analytical calculations of orbital evolution became complemented by
numerical simulations. Everhart (1972, 1973, 1977), who favoured the idea that
all comets originated from nearly parabolic orbits, used Monte Carlo simulations
to show that a fraction of long-period comets with perihelia close to the orbit
of Jupiter (∼5 AU) could evolve into short-period orbits. As in van Woerkom’s
work, the efficiency of the process was too low. Besides, neither Everhart nor
van Woerkom could convincingly explain the preponderance of prograde orbits
among short-period comets.
Alongside the question of the origin of the short-period comets, there remained the issue of the origin of comets altogether: of where they formed. Theories of a possible interstellar origin had been dismissed by van Woerkom (1948)
on the basis that no comet had ever been found to have a hyperbolic orbit.
Comets must have formed in the Solar System. Important clues to the origin
of comets came from the work of Fred Whipple. He presented a model of the
chemical composition of comets (Whipple 1950) consisting mainly of ices of H 2 O,
NH3 , CH4 , CO2 , CO, and other volatiles, “polluted” by smaller amounts of refractory5 material in the form of dust. This model, popularized by Whipple as
“dirty snowball”, explained the tails and comæ6 of comets upon approaching the
Sun. Due to the temperature increase, the icy material sublimates and becomes
partly ionized—forming the coma and ion tail—and forces solid particles off the
surfaces of comets, which form the dust tail.
4 An
astronomical unit (AU) is the mean distance between the Sun and the Earth.
with a higher (melting) sublimation temperature, here meant to signify rocky
(silicate based) material.
6 Comets usually show an ion (gas) tail pointing away from the Sun, a dust tail slightly
trailing the ion tail, and a luminous halo (coma) surrounding the nucleus (central solid part).
5 Material
4
Introduction
Partly motivated by Whipple’s model, Kuiper (1951) proposed that comets
could have formed in the outer solar system, between 35 and 50 AU. Gerard
Kuiper was born in the Netherlands in 1905. He studied astronomy in Leiden,
where he got his PhD degree in 1933, with a dissertation on binary stars. He
immediately moved to the USA to pursue his studies of multiple star systems.
Later he switched to solar system science, which became his main field of research. Kuiper contributed significantly to the development of planetary science,
both theoretically and observationally. He died in 1973.
Kuiper realized that the volatile-rich composition of comets (as opposed to
the more “rocky” asteroids) was inconsistent with their forming in the inner
region of the solar system. Therefore, Kuiper believed that comets must have
formed far from the Sun. The “nebular model”7 for the formation of the solar
system does not invalidate the formation of “condensations” beyond the orbits
of the known planets. As Kuiper argued, by forming far from the Sun, such
condensations would be smaller and more numerous, due to the lower density of
material, and made up mostly of ices (as Whipple proposed) because of the very
low temperatures. Nevertheless, Kuiper intended to explain the formation of
Oort cloud comets, not short-period comets. Since it is unlikely that there was
enough material at distances ∼100 000 AU from the Sun to support in situ formation of Oort cloud comets, Kuiper speculated that comets must have formed
much closer to the Sun—at about 40 AU. A substantial fraction was scattered
outwards and populated the Oort cloud. Kuiper did not question the idea that
short-period comets were dynamical descendants of long-period comets.
Shortly before Kuiper’s 1951 paper, Kenneth Edgeworth (1880–1972) speculated on the possibility that the outer solar system was occupied by a ring of
small bodies, in his own words, “a vast reservoir of potential comets” (Edgeworth
1949). During his professional career, Edgeworth was an army officer, electrical
engineer, and economist, and only in his retirement years, at the age of 59, began
actively working as an independent theoretical astronomer (McFarland 1996).
Already in 1943, in a paper communicated to the British Astronomical Association, Edgeworth (1943) mentioned that it would be unthinkable that the
cloud from which the solar system formed would be bounded by the orbit of
Pluto. Instead he proposed, and later supported with theoretical calculations
(Edgeworth 1949), a “gradual thinning” of the cloud at greater and greater distances from the Sun; this thinner (less dense) cloud would support the formation
of small bodies. Therefore, outside the orbits of Neptune and Pluto there should
exist a swarm of small bodies. Edgeworth thought that bodies in this swarm
that got displaced into the inner solar system (no mechanism is suggested for
this displacement) would become the visible comets. Edgeworth (1943) further
7 Originally
proposed by Kant (1755) and Laplace (1796), this model supports that the Sun,
the planets, and all the other solar system bodies formed from the gravitational collapse of a
single cloud (nebula) of gas and dust. It has obtained strong observational support in recent
years, and is widely accepted as the most plausible model of formation of planetary systems.
The Shapes and Spins of KBOs
5
speculated that comets, unlike asteroids, are probably “astronomical heaps of
grains with low cohesion”, due to the low formation temperatures; this is very
close to what is known today about the structure of comets. Edgeworth’s work
in astronomy was rarely cited by his contemporaries (including Kuiper). This
has been attributed to his “brusque style of presentation” (McFarland 1996).
On the other hand, at a time when information flowed slower than today, being
an outsider to the astronomical community (Edgeworth was not affiliated to any
research institution) may have contributed to his work remaining unnoticed.
The lack of observational evidence kept the idea of a “comet belt” at the edge
of the solar system in the realm of speculation. Indeed, further (indirect) evidence for the need of such a belt came from theoretical calculations in the 1980s.
Earlier work by Joss (1973) had reinforced the idea that the capture mechanism
of long-period comets into short-period comets by the giant planets was inefficient. In 1980, Julio Fernández presented results of a Monte Carlo simulation
confirming that comets coming from a hypothetical belt beyond Neptune could
produce the observed distribution of short-period comets (Fernandez 1980). Decisive results came nearly a decade later from extensive numerical simulations
by Martin Duncan, Thomas Quinn, and Scott Tremaine. Their calculations
convincingly ruled out that short-period comets could originate in a spherically
symmetric population such as the Oort cloud. The prograde low-inclination
orbits of short-period comets could only be explained if the parent population
had a similar orbital distribution, most likely located in the outer solar system
(Duncan et al. 1988). The authors referred to this parent population as “Kuiper
belt”, acknowledging the hunch of Gerard Kuiper.
Around the same time, sensitive charge-coupled devices (CCDs, electronic
detectors) began replacing photographic plates, as means of registering the light
collected by telescopes. Among many other advantages, such as linearity and
reusability, CCDs are more sensitive than photographic plates, and allow ready
analysis of the collected data using computers. Making use of this new technology, installed at the 2.2 m UH8 telescope atop Mauna Kea (Hawaii), David
Jewitt and Jane Luu began a survey9 of the outer solar system looking for “slow
moving objects”. Expected to lie beyond Neptune, the hypothetical Kuiper belt
objects would take about 250 years to complete a full orbit around the Sun. This
means they must move very slowly against the background stars. Actually, their
apparent movement with respect to the stars is primarily due to the movement
of the Earth around the Sun10 . Therefore, if Jewitt and Luu could identify faint
slow moving objects, they would likely be located deep in the outer solar system.
8 University
of Hawaii, USA.
and Luu actually began the survey in 1987 using photographic plates, but soon
switched to the more sensitive CCDs.
10 Much like the movement of a cow against a background windmill, as seen from a moving
car on a Dutch highway, is due mostly to the movement of the car and not to the movement
of the cow.
9 Jewitt
6
Introduction
In the summer of 1992, after 5 years of persevering, Jewitt and Luu detected a
faint object that seemed to move at the expected pace in four consecutive images.
Accurate measurements of the object’s positions were used to determine it’s orbit: a nearly circular path, at a distance of 40 AU from the Sun. Following the
naming convention of the Minor Planet Center11 (MPC), this object was designated 1992 QB1 . It was the first detection (Jewitt & Luu 1993) of an object with
an orbit entirely outside that of Neptune: a “Kuiper belt object”. 1992 QB 1 was
estimated to have a diameter of about 200 km. About six months later—when
the Earth is on the “other side” of the Sun—Jewitt and Luu found another object, equally beyond Neptune. Since then, nearly 1000 Kuiper belt objects have
been discovered, confirming the predictions of Kuiper, Edgeworth, and others.
The discovery of the Kuiper belt raised doubts about Pluto’s classification as
planet. When Clyde Tombaugh discovered Pluto, in early 1930, he was on a mission to find the planet which was causing perturbations measured in Neptune’s
orbit12 . Tombaugh found an object, and the object was classified as planet. But
Pluto was odd in the context of the outer solar system: it is icy and small, unlike
the outer large gaseous planets, and it has a very elliptical and inclined orbit. In
an inspired—almost prophetic—leaflet of the Astronomical Society of the Pacific
published a few months after Pluto’s discovery, Leonard (1930) wrote:
“... We know that the Sun’s gravitational sphere of control extends
far beyond the orbit of Pluto. Now that a body of the evident dimensions and mass of Pluto has been revealed, is there any reason
to suppose that there are not other, probably similarly constituted,
members revolving around the Sun outside of the orbit of Neptune?
Indeed, it may ultimately be found that the solar system consists of
a number of zones, or families, of planets, one with the other. As a
matter of fact, astronomers have recognized for more than a century
that this system is composed successively of the families of the terrestrial planets, the minor planets, and the giant planets. Is it not
likely that in Pluto there has come to light the first of a series of
ultra-Neptunian bodies, the remaining members of which still await
discovery but which are destined eventually to be detected? ...”
Leonard guessed right. Pluto is the largest known member of the recently
discovered family of “ultra-Neptunian” bodies. Besides being the likely precursors of comets, Kuiper belt objects are believed to be remnants of outer solar
system planetesimals13 . Frozen at the distant edge of the planetary system, they
preserve information about the environment in which the planets formed. The
discovery of the Kuiper belt has helped understand the origins of Pluto and the
short-period comets. But, as is usually the case in science, it has raised a multi11 http://cfa-www.harvard.edu/iau/mpc.html
12 It was later discovered that the observed perturbations in the orbit of Neptune were
actually measurement errors.
13 First aggregates of solid material that merged to form the planets.
The Shapes and Spins of KBOs
7
tude of new questions, thereby opening an entirely new field of research. Observationally speaking, Kuiper belt science is extremely challenging. The brightness of KBOs, due to reflected sunlight, is inversely proportional to roughly the
fourth power of their distance to the Earth. At 40 AU most KBOs are too faint
(hmR i ∼ 23, Trujillo et al. 2001b), and only the largest are accessible to detailed
analysis. Although it is expected that Kuiper belt studies will provide invaluable
information about the history of our planetary system, and even of planetary
systems around other stars, it might take a while before that information can be
gathered and decoded. The process will eventually require spacecraft to be sent
to individual Kuiper belt objects. And that, to the joy of the “curious beings”,
will certainly reveal new surprises begging for an explanation.
The belt and its population of objects still have no unanimously accepted
name. In the first key publications (Duncan et al. 1988; Jewitt & Luu 1993)
it was referred to as “Kuiper belt”, and most astronomers use this name. The
most frequently used alternative is the self-explanatory “Trans-Neptunian belt”.
Some attempts have been made to acknowledge Edgeworth’s contribution, and
use “Edgeworth-Kuiper belt”, but the name is not used very often. Therefore,
in the literature all of the following acronyms are found: KB and KBOs, TNB
and TNOs, and EKB and EKOs. In this thesis the names Kuiper belt (KB) and
Kuiper belt objects (KBOs) will be used.
1.2
The Kuiper belt
The bulk of the Kuiper belt is located beyond the orbit of Neptune, between 30
and 50 astronomical units from the Sun (see Fig. 1.2). The orbit of Neptune
is, by definition, the lower limit to the semi-major axis of the orbits of Kuiper
belt objects; there is no defined upper limit. The belt extends roughly 25 AU
above and below the ecliptic (see Fig. 1.3). All known KBOs orbit the Sun in
the prograde sense. Although most objects follow this regular merry-go-round
pattern, some KBOs have very eccentric and inclined orbits, and only rarely
visit the central region of the belt. These “scattered” objects reach heliocentric
distances of several hundreds of AU.
According to the MPC, almost 1000 KBOs have been detected14 . However,
reliable orbits have only been determined for about half of them. Figure 1.4
shows that the distribution of KBO orbits is not random. The apparent structure has led to a classification of KBOs into 3 dynamically distinct groups: the
Classical KBOs, the Resonant KBOs, and the Scattered KBOs. Table 1.1 lists
the number and mean orbital properties of KBOs belonging to each of these dynamical groups. Only those objects that have been observed for more than one
opposition have been considered. Table 1.1 also shows the number of observed
Centaurs. The main characteristics of these different groups are given below.
14 http://cfa-www.harvard.edu/iau/lists/TNOs.html
and Centaurs.html
8
Introduction
Figure 1.2 – Plan view of the orbits of Kuiper belt objects (grey ellipses). Different dynamical
groups are shown in separate panels. The orbits of Jupiter, Saturn, Uranus, Neptune, and
Pluto are also shown (black ellipses). Black dots (KBOs) and crosses mark the perihelia of all
orbits. The axes are in AU.
Classical KBOs
Making more than half of the known population, the Classical KBOs are the
prototypical group. The first KBO to be discovered, 1992 QB1 , is a Classical
object. CKBOs are selected to have perihelia q > 35 AU and orbital semi-major
axis 42 AU < a < 48 AU (see Figs. 1.2, 1.3, and 1.4). Most have nearly circular
(e < 0.2) and moderately inclined orbits (i < 10◦ ). There is, however, a small
fraction of CKBOs reaching orbital inclinations i ∼ 30◦ . The intrinsic fraction
of CKBOs with inclined orbits may be larger than what is measured, because
of the observational bias towards detecting low-i objects. Numerical simulations
have shown that orbits with q > 42 and e < 0.1 are stable for the age of the solar
The Shapes and Spins of KBOs
9
Figure 1.3 – Same as
Fig. 1.2 but seen from the
side. Axes are in AU.
Table 1.1 – Number and orbital parameters of KBOs and Centaurs.
Dynamical group
N
fobs ?
fint ??
a
hei
[AU]
Classical KBOs
Resonant KBOs
Plutinos (3:2)
Twotinos (2:1)
4:3 resonance
Scattered KBOs
Centaurs
Total (KBOs)
hii
[deg]
267
0.57
0.44
42 · · · 48
0.08
5.3
79
18
4
0.17
0.04
0.01
0.18
0.03
–
∼39.4
∼47.8
∼36.4
0.21
0.23
0.15
9.2
10.0
10.1
101
0.22
0.35
>
30
1−e
30
1−e
0.33
14.6
0.35
12.7
0.17
8.3
47
469
<
1.00
1.00
The data was obtained from the MPC website. Only objects observed
for more than one opposition have been considered.
? Fraction of observed KBOs;
?? Bias-corrected estimate of the intrinsic fraction (Trujillo et al. 2001a).
system (Duncan et al. 1995). Because of their stable dynamical configuration,
the Classical objects are thought to best represent the primordial population.
Resonant KBOs
These objects lie close to mean motion resonances with Neptune. This means
that the quotient of the orbital period of a Resonant KBO and that of Neptune is a ratio of integers. The 3:2 resonance, harbouring ∼80% of the observed
Resonants, is the most populated. Bodies lying in the 3:2 resonance are called
Plutinos, because Pluto itself lies in this resonance. The second most populated
resonance is the 2:1, containing ∼20% of all Resonant KBOs. The 2:1 Resonants
have lately been called Twotinos (Chiang & Jordan 2002). Four objects have
been observed close to the 4:3 resonance. Some Resonants, including Pluto, have
perihelia15 inside the orbit of Neptune (see Figs. 1.2 & 1.4). However, the resonant character of their orbits prevents close encounters. Resonant orbits are also
dynamically stable on Gyr timescales (Duncan et al. 1995). The overabundance
15 Point
of closest approach to the Sun, usually represented by q.
10
Introduction
Figure 1.4 – Orbital eccentricity and inclination
versus semi-major axis of
KBOs.
Symbols are:
(+) Classical KBOs, (¤)
Plutinos, (3) Twotinos,
(4) 4:3 Resonant KBOs,
(·) Scattered KBOs. Gray
vertical lines indicate the
mean motion resonances
with Neptune. Constant
perihelion q = 30 AU (orbit of Neptune) is shown
as a dotted curve.
of Plutinos is understood as evidence of planetary migration (Malhotra 1993,
1995). It is possible that Neptune formed closer to the Sun and migrated outwards to its current location, due to angular momentum exchange with surrounding planetesimals (Fernandez & Ip 1984). As the planet migrated, its mean motion resonances swept through the KB region. Because resonances are more stable they “captured” KBOs, as they swept by. Simulations show that an outward
migration of ∼8 AU on a timescale τ ∼ 107 yr produces the observed distribution
of eccentricities and inclinations of Plutinos (Malhotra 1998; Gomes 2000).
Scattered KBOs
Sometimes referred to as Scattered Disk objects (SDOs), these KBOs have more
eccentric and inclined orbits than the previous two groups. Due to the large
ellipticity of their orbits, some SKBOs spend many Earth centuries outside the
KB region, at large distances from the Sun (see Fig. 1.2). Figure 1.5 shows the
orbital distribution of SKBOs. By definition, SKBOs have perihelia q > 30 AU
(below the dotted line in Fig. 1.5). The reason why all the observed objects
actually have perihelia close to q = 30 AU is that objects with larger q are
very hard to detect. Given the nature of their orbits, SKBOs may represent an
intermediate stage between KBOs and Oort cloud objects.
Centaurs
The Centaurs have perihelia between the orbits of Jupiter and Neptune—they
are sometimes referred to as giant planet crossers. Centaurs are not KBOs: they
are believed to be dynamical transients between KBOs and short-period comets.
The short dynamical lifetimes (1–100 Myr) of Centaur orbits, when compared
to the age of the solar system, implies that the population must be resupplied
on comparable timescales; the most likely source is the KB. Recent numerical
simulations by Tiscareno & Malhotra (2003) show that, due to dynamical inter-
The Shapes and Spins of KBOs
11
Figure 1.5 – Same as Fig. 1.4 for
a broader range of semi-major axes,
to show the distribution of Scattered
KBOs (filled circles) and Centaurs
(open squares). Curves represent constant perihelion q = 30 AU (dotted) and
constant aphelion Q = 30 AU (dashed).
actions with the giant planets, 2/3 of the population is ejected from the solar
system (or enters the Oort cloud), 1/3 become Jupiter Family16 comets, and
a negligible fraction collides with a giant planet. As our understanding of the
structure of KB orbits improves, better simulations of the transition of KBOs
into short-period comets are needed to clarify the role of Centaurs.
1.3
Kuiper belt objects
One thing people always ask, when told about KBOs for the first time is: how
big are they? The answer is: we don’t know. KBOs are not observationally
resolved17 , so their sizes cannot be measured directly. Basic information about
KBOs, such as their sizes and masses, relies on two quantities that are not
known: albedo18 and density. Since KBOs are believed to be progenitors of
short-period comets, these properties are taken to be similar in both families.
As short-period comets, KBOs are expected to have low albedos (A ∼ 0.04)
and densities close to water ice (ρ ∼ 1000 kg m−3 ). With these assumptions, the
observational data can be used to infer, for example, the total number of KBOs,
their size distribution, and the total mass present in the Kuiper belt.
The observed cumulative surface density of KBOs (number of objects per
square degree brighter than a given magnitude) is well fit by an exponential
power law of the form Σ(mR ) = exp [α (mR − m0 )]. The best fit parameters,
α ≈ 0.6 and m0 ≈ 23 mag (Trujillo et al. 2001a; Bernstein et al. 2004), indicate
that 1 KBOs of magnitude 23 can be found per square degree; the number is 4
times higher at each fainter magnitude. The observed cumulative surface density
can be used to infer the size distribution. Assuming the latter can be represented
by a power law, n(r) dr ∝ r −q , the best estimate index is q ≈ 4 (Trujillo et al.
2001a). Note that a different q was used before, to represent perihelion distance.
16 Short-period
comets that have periods P < 20 yr and orbits dominated by Jupiter’s gravity.
the 2 largest known, Pluto and (50000) Quaoar, which have been resolved by HST.
18 The albedo (A) of an object is the fraction (between 0 and 1) of sunlight it reflects.
17 Except
12
Introduction
Recent, deeper observations show that the size distribution may be shallower
(q = 3.0–3.5) at radii below r = 10–100 km (Bernstein et al. 2004). This size
distribution implies that there are roughly 10 000 KBOs with radii larger than
r = 100 km, and about 10 Pluto-size objects (r ∼ 1000 km). Assuming individual body densities ρ ∼ 1000 kg m−3 the total mass of KBOs between 30 AU and
50 AU is MKB = 0.01–0.1 M⊕ , where M⊕ = 6 × 1024 kg is the mass of the Earth.
These measurements agree well with what is predicted by current KBO formation and evolution scenarios; a few examples are cited below. The “minimum
mass solar nebula” (Hayashi 1981; Weidenschilling 1977) estimate of the mass
initially19 present in the KB region is 10 M⊕ , 100 to 1000 times higher than what
is observed. Numerical simulations of KBO accretion show that if this was indeed
the initial mass then several “Plutos” can form in less than 100 Myr (Kenyon &
Luu 1998, 1999). The same simulations produce a power law KBO size distribution of index q ∼ 3.5, at the end of the 100 Myr accretional phase. Subsequently,
erosive collisions between KBOs convert bodies with r . 100 km into smaller
and smaller fragments, some of which may plunge into the inner solar system as
short-period comets (Davis & Farinella 1997). This collisional cascade produces
the observed break in the size distribution at r ∼ 10–100 km (Kenyon & Bromley
2004). A substantial amount of mass is converted to 1–100 µm-size dust grains,
which are blown away from the solar system by solar radiation in a few tens
of million years (Burns et al. 1979; Barge & Pellat 1990). These processes are
believed to have caused the mass depletion (∼ 99%) in the Kuiper belt. The
data also indicate that the KB population is large enough to serve as source of
short-period comets. The observed size distribution predicts that there are 10 10
KBOs larger than 1 km in radius—enough to account for the observed shortperiod comet population (Holman & Wisdom 1993; Levison & Duncan 1997).
KBO albedos can be determined using combined observations in visible and
thermal (infrared) wavelengths. The amount of sunlight reflected by a KBO is
roughly proportional to the product of albedo and cross-section, A × S. Conversely, the fraction of sunlight absorbed by the KBO, which maintains its temperature and is re-emitted at longer (infrared) wavelengths, is proportional to
(1 − A) × S. Measurements in both wavelength ranges permit the determination
of both A and S. At about 40 AU from the Sun, KBOs have surface temperatures of about 50 K. This means their thermal emission peaks at infrared
wavelengths, around 50 µm. Unfortunately the atmosphere is not transparent at
these wavelengths. As a result only the brightest (largest) KBOs can have their
thermal radiation measured from Earth. Observations from space do not suffer
from atmospheric extinction. Table 1.2 lists the KBOs with known albedos. The
measurements by Thomas et al. (2000) have been done from space, using ISO 20 ,
and the size of (50000) Quaoar was measured directly using the High Resolution
Camera of the Hubble Space Telescope. Indeed, KBO albedos seem to be very
19 About
4.5 Gyr ago, when the solar system formed.
Space Observatory
20 Infrared
The Shapes and Spins of KBOs
13
Table 1.2 – KBOs with measured albedo.
Object
Pluto
(50000) Quaoar
(20000) Varuna
1999 TL66
1993 SC
Albedo
r [km]
0.44–0.61
0.09
0.07
0.03
0.02
1150
630
450
320
160
Reference
Stern & Yelle (1999)
Brown & Trujillo (2004)
Jewitt et al. (2001)
Thomas et al. (2000)
Thomas et al. (2000)
low, close to that of short-period comets. Pluto’s extremely high reflectivity is
an exception. The accepted explanation is that Pluto is massive enough to hold
a very thin atmosphere, which can condense on the surface creating a reflective
frost layer. Recently reported observations made with the Spitzer Space Telescope appear to indicate that KBOs may have higher albedos than expected
(Emery et al. 2004). These results are still unpublished.
The chemical compositions of KBOs is poorly known. Even for the largest
KBOs, spectroscopic studies are extremely difficult, due to low signal-to-noise.
For this reason, broadband colours are generally used as a low resolution alternative. KBOs, as a population, have very diverse colours, from blue to very red
(Luu & Jewitt 1996; Jewitt & Luu 1998; Tegler & Romanishin 2000). Statistical
studies show that KBO colours may correlate with orbital inclinations and perihelion distances (e.g., Jewitt & Luu 2001; Doressoundiram et al. 2002; Trujillo &
Brown 2002). Different dynamical groups appear to have distinct colour distributions (Peixinho et al. 2004). The underlying reasons for these trends are not
understood. Comparison between the colours of KBOs and short-period comets
show that the former are redder on average than the latter (Jewitt 2002). This
suggests that comet surfaces have been modified somewhere along the transition
from the KB to their current orbits.
The few existing spectra (optical and near-IR) of KBOs are mostly featureless, although some show a weak 2 µm water ice absorption line (Brown et al.
1999; Jewitt & Luu 2001). Very recent near-infrared spectroscopic observations
of the largest (besides Pluto) known KBO, (50000) Quaoar, have revealed the
presence of water ice with crystalline structure (Jewitt & Luu 2004). This means
the ice (usually at ∼50 K) must have been heated to 110 K. What makes this
finding a puzzle is that, since cosmic radiation destroys the ice molecular bonds
and turns it into amorphous ice in about 107 yr (Strazzulla et al. 1991), the
surface of Quaoar must have somehow been heated in the last 10 million years.
Direct observations of binary KBOs indicate that they may represent about
4% of the known population (Veillet et al. 2002; Noll et al. 2002). The PlutoCharon system is an example of a long known binary KBO. The observed binaries
have separations δ > 0.00 15, and primary-to-secondary mass ratios close to unity.
Existing models of binary KBO formation require a ∼100 times denser environ-
14
Introduction
ment than the present, implying that the binaries must have formed early in the
evolution of the KB (Weidenschilling 2002; Goldreich et al. 2002). Recently, the
discovery of a contact (or very close) binary KBO has been reported (Sheppard
& Jewitt 2004). The authors estimate that the fraction of similar objects in the
Belt may be ∼15%.
The spin states of KBOs can be determined from their “lightcurves”. The
lightcurves are periodical brightness oscillations produced by the varying aspect
of non-spherical KBOs, as they spin. Spherical KBOs, or those whose spin
axis coincides with the line-of-sight, show nearly constant brightness (or“flat”
lightcurves), because their sunlight reflecting area is constant in time. The
period of a KBO lightcurve is a direct measure of the KBO’s spin period, and the
lightcurve peak-to-peak amplitude has information about the shape of the object.
The spin rates of KBOs also place constraints on their bulk densities. In bodies
with no internal strength, the centrifugal acceleration due to rotation must be
balanced by self-gravity, or the body would “fly apart”. Thus by measuring the
spin rate of a KBO we find a lower limit to its bulk density. The shapes and
spins of KBOs potentially carry information about their formation environment,
and their evolution. For example, the larger objects should retain the angular
momentum acquired at formation, while smaller bodies have probably had their
spins and shapes modified by mutual collisions in the last ∼4.5 Gyr.
Rotational data have been used in the past to investigate the evolution and
the physical properties of other minor planets (e.g., asteroids). In the case of
KBOs, such data have only recently become available, although still in meager
amounts. The small brightness variations (typically a few tenths of magnitude)
are difficult to measure for KBOs, and require large collecting areas (large telescopes). Besides, the long timebases needed to accurately determine the periodicity of the variations, are not readily available in the competitive world of time
allocation for usage of large telescopes. Approximately 4.5 years ago, when the
project that led to this thesis started, rotational data had been reported for 10
KBOs. We set out to increase that number, and to use the rotational properties
of KBOs to learn more about their nature. Now, as a combined result of different observational campaigns (Sheppard & Jewitt 2002; Sheppard & Jewitt 2003;
this work), the number of KBO lightcurves is 4 times larger. In this thesis, the
existing rotational data of KBOs are used to investigate some of their physical
properties.
1.4
Thesis summary
My thesis uses rotational data of KBOs to investigate their shapes, spins, and
inner structure. The rotational properties were obtained from analysis of KBO
“lightcurves”. The lightcurves are periodic brightness variations mainly due to
the aspherical shapes of KBOs—as aspherical KBOs spin in space their skyprojected cross-sections vary periodically, and so does the amount of reflected
The Shapes and Spins of KBOs
15
sunlight. The period of a lightcurve is related to the spin period of the KBO
that produces it, and the amplitude of the light variation has information on the
KBO’s shape.
Most KBOs (about 70%) actually show no brightness variations. This can be
because they are not spinning (or spin very slowly), because they are spherical,
or because the spin axis points directly at the observer. In Chapter 2, the
likelyhood of these and other possibilities is discussed, and presented in the form
of a statistical study of the detectability of lightcurves of KBOs. As a result, an
expression is derived that gives the probability of detecting light variations from
a KBO, assuming an a priori shape distribution for the whole population. This
expression can test candidate shape distributions by checking if they reproduce
the observed detection probability, i.e., the fraction of observed KBOs that show
detectable variations.
In Chapter 3, the method developed in Chapter 2 is used to test two possible functional forms for the KBO shape distribution: gaussian and power-law.
The (then) existing database of KBO lightcurves is used to determine the fraction that have detectable light variations. The results show that a power-law
shape distribution gives a better fit to the data. However, a single power-law
distribution does not explain the shapes of the whole population.
Chapter 4 presents our observations of KBO lightcurves. We collected data
for 10 KBOs, 5 of which had not been studied before. Significant light variations
were detected for 3 out of the 10 objects. The periods and amplitudes of the
lightcurves were determined, and combined with the existing database of rotational properties. After adding our observations, lightcurve data exist for a total
of 41 KBOs. Using the conclusions of Chapter 3, we used this data to better
constrain the KBO shape distribution. KBOs were split in two groups, with
diameter smaller and larger than D = 400 km, which were analysed separately.
We find that the shapes of bodies are different in the two groups. The larger
group has a higher fraction of round bodies. A comparison between KBOs and
asteroids shows that KBOs are rounder than asteroids. In the case of asteroids it
has been shown that the ones with diameter D < 100 km have shapes consistent
with collisional fragments. The smallest KBO with a measured lightcurve has a
diameter D = 100 km, so we are still far from being able to compare KBOs and
asteroids in small size ranges. The rotational data were also used to investigate
the inner structure of KBOs. Large KBOs have in principle survived the collisional evolution, but may have been internally fractured by multiple impacts,
and turned into agglomerates of smaller pieces held together mainly by gravity.
If this is true then their shapes are mostly controlled by a balance between gravitational and centrifugal acceleration. In the case of uniform fluids, the shape of
a rotating body depends on the spin rate and on the body’s density. We compare
a diagram KBO spin rates vs. lightcurve amplitudes with what is expected in
the case of perfect fluid behaviour, and with results of N-body simulations of
collisions between ideal “rubble piles”. The results are not conclusive but most
16
Introduction
KBOs are at least consistent with having a rubble pile structure and densities
ρ > 1000 kg m−3 .
In Chapter 5 we investigate how collisions between KBOs have affected
their spins. A model is constructed to simulate the collisional evolution of KBO
spins, in the last 4 Gyr (Note: the best models of KBO formation indicate the
bulk of the population was formed in about 100 Myr). Each simulation follows
a single KBO and calculates the spin rate change and mass change due to each
individual collision. These changes depend on several parameters, which are
tested. Among other things, we find that the spin rates of KBOs with radii
larger than about r = 200 km, have not been changed by collisions—their spin
must be “primordial”. The lightcurve data presented in Chapter 4 show that 4
out of 7 KBOs with r ∼500 km spin with periods of about 15 hours. This raises
the question: what is the origin of the spins of these large KBOs? If these objects
grew by isotropically accreting material, angular momentum conservation should
significantly slow down their spin rates. We estimate how anisotropic does the
accretion need to be to explain the rotation of the large KBOs. It turns out
that an asymmetry of about 10% in the angular momentum contributed by the
accreted particles is enough. But we also found that this is only necessary if
the accreted particles are very small. If they are at least 1/5 of the size of the
growing body, then isotropic accretion can also reproduce the observed spins.
The distribution of spins predicted by these two possibilities is very different. If
KBOs grew from isotropic accretion of large particles, the dispersion of spin rates
should be large, and the spin axes should be randomly oriented. On the other
hand, if KBOs grew by anisotropically accreting small particles, the dispersion in
spin rates should be small, and if this asymmetry exists primarily in the ecliptic
plane then the spin axis of large KBOs would tend to be aligned perpendicularly
to the plane, like most planetary axes. Measurements of the distribution of spin
axis orientations of large KBOs can in principle rule out one of the possibilities.
1.5
Future prospects
Chapter 2 was written before good constraints existed on the fraction of binary KBOs. As more binaries are discovered and the statistics of, for example,
primary-to-secondary size ratios and distance between the two components improves, it becomes possible to account for the probability that a lightcurve is due
to an eclipsing binary. This should be incorporated in the method developed in
the chapter.
It would be extremely useful to construct an “atlas” of lightcurves. The
morphology of a lightcurve reflects the cause of the brightness variation. The
recovery of the cause from the analysis of the morphology is known as lightcurve
inversion. Although the problem of lightcurve inversion is known to be degenerate (Russell 1906), the importance of understanding the morphology of
lightcurves caused by different physical situations cannot be denied. Using ray-
The Shapes and Spins of KBOs
17
tracing software, a host of different physical situations (KBOs of various shapes,
close tidally deformed binaries, objects with surface features, etc.) can be created, “observed”, and used to systematically generate a database of lightcurve
properties. The lightcurves can be analysed in terms of Fourier expansions, to
look for structure in the distribution of the coefficients.
As described in Chapter 4 (§4.5.2), KBOs are more spherical on average than
asteroids. Yanagisawa (2002) has investigated the transfer of angular momentum
by collisions, for spherical as well as ellipsoidal targets. The author calculated
the ratio between the spin-up rate (angular momentum transferred divided by
the moment of inertia of the target) of ellipsoidal targets and the spin-up rate
of spherical targets of the same mass, and concluded that ellipsoidal bodies can
spin up more rapidly than spherical bodies. If this is true, one would expect
a population of rounder objects to have lower spin rates than a population of
more elongated objects, if both have similar collisional evolution histories. The
different shape distributions of KBOs and asteroids could partially justify their
different mean spin rates. A natural extension to the collisional evolution model
presented in Chapter 5 is to consider targets and projectiles with ellipsoidal
shapes, and see how this affects the results.
It is shown in Chapter 5 that the distribution of spin periods of the largest
KBOs is likely to be primordial. If the spins have been caused by accretion
of large planetesimals (comparable, in size, with the growing body) then the
observed distribution of spin periods can constrain the size distribution of the
accreting planetesimals. In Chapter 5 only the simple case where all accreted
planetesimals have a fixed size—function of the size of the growing object—was
considered. More realistic scenarios allowing for a range of accreted planetesimal
sizes should be investigated. The results may serve as an independent check on
the planetesimal size distribution obtained from models of accretion in the KB
(Kenyon & Luu 1998, 1999).
In Chapter 5 the possibility is considered that the rotations of the largest
KBOs were caused by a torque due to the accreted material. It would be interesting to investigate if the dynamics of the particles being accreted into a
large KBO, as it grows in a swarm of smaller planetesimals, can produce such a
torque. This can be done by means of an N-body simulation.
It is essential to increase the database of KBO rotational properties, for
various reasons. With better statistics it will be possible to compare the shapes
and spin states of KBOs in different dynamical groups. The comparison may
serve as a check for different evolution scenarios.
18
Introduction
References
Barge, P. & Pellat, R. 1990, Icarus, 85, 481
Bernstein, G. M., Trilling, D. E., Allen, R. L., Brown, M. E., Holman, M., & Malhotra,
R. 2004, AJ, 128, 1364
Brown, R. H., Cruikshank, D. P., & Pendleton, Y. 1999, ApJ, 519, L101
Brown, M. E. & Trujillo, C. A. 2004, AJ, 127, 2413
Burns, J. A., Lamy, P. L., & Soter, S. 1979, Icarus, 40, 1
Chiang, E. I. & Jordan, A. B. 2002, AJ, 124, 3430
Davis, D. R. & Farinella, P. 1997, Icarus, 125, 50
Doressoundiram, A., Peixinho, N., de Bergh, C., Fornasier, S., Thébault, P., Barucci,
M. A., & Veillet, C. 2002, AJ, 124, 2279
Duncan, M., Quinn, T., & Tremaine, S. 1988, ApJ, 328, L69
Duncan, M. J., Levison, H. F., & Budd, S. M. 1995, AJ, 110, 3073
Edgeworth, K. E. 1943, Journal of the British Astronomical Association, 53, 181
Edgeworth, K. E. 1949, MNRAS, 109, 600
Emery, J. P., Cruikshank, D. P., Van Cleve, J., & Stansberry, J. A. 2004, AAS/Division
for Planetary Sciences Meeting Abstracts, 36,
Everhart, E. 1972, Astrophys. Lett., 10, 131
Everhart, E. 1973, AJ, 78, 329
Everhart, E. 1977, IAU Colloq. 39: Comets, Asteroids, Meteorites: Interrelations,
Evolution and Origins, 99
Fernandez, J. A. 1980, MNRAS, 192, 481
Fernandez, J. A. & Ip, W.-H. 1984, Icarus, 58, 109
Gomes, R. S. 2000, AJ, 120, 2695
Goldreich, P., Lithwick, Y., & Sari, R. 2002, Nature, 420, 643
Halley, E. 1705, A Synopsis of the Astronomy of Comets, Johnson Reprint Corp. New
York, 1972 ,
Hayashi, C. 1981, Progress of Theoretical Physics Supplement, 70, 35
Holman, M. J. & Wisdom, J. 1993, AJ, 105, 1987
Jewitt, D. C. 2002, AJ, 123, 1039
Jewitt, D., Aussel, H., & Evans, A. 2001, Nature, 411, 446
Jewitt, D. & Luu, J. 1993, Nature, 362, 730
Jewitt, D. & Luu, J. 1998, AJ, 115, 1667
—. 2001, AJ, 122, 2099
Jewitt, D. & Luu, J. 2004, Nature, 432, 731
Joss, P. C. 1973, A&A, 25, 271
Kepler, J., Ptolemaeus, C., & Fludd, R. 1619, Lincii Austriae, sumptibus G. Tampachii,
excudebat I. Plancvs, 1619.,
Kant, I. 1755, Allgemeine Naturgeschichte und Theorie des Himmels (Leipzig: Petersen)
Kenyon, S. J. & Bromley, B. C. 2004, AJ, 128, 1916
Kenyon, S. J. & Luu, J. X. 1998, AJ, 115, 2136
Kenyon, S. J. & Luu, J. X. 1999, AJ, 118, 1101
Kuiper, G. P. 1951, Proceedings of a topical symposium, commemorating the 50th
anniversary of the Yerkes Observatory and half a century of progress in astrophysics, New York: McGraw-Hill, 1951, edited by Hynek, J.A., p.357, 357
Laplace, P.-S. 1796, in Exposition du systeme du monde (Paris: de l’Impr. Cercle-
The Shapes and Spins of KBOs
19
Social)
Leonard, F. C. 1930, Leaflet of the Astronomical Society of the Pacific, 1, 121
Levison, H. F. & Duncan, M. J. 1997, Icarus, 127, 13
Lindblad, B. 1925, ApJ, 62, 191
Luu, J. & Jewitt, D. 1996, AJ, 112, 2310
Malhotra, R. 1993, Nature, 365, 819
Malhotra, R. 1995, AJ, 110, 420
Malhotra, R. 1998, Lunar and Planetary Institute Conference Abstracts, 29, 1476
McFarland, J. 1996, Vistas in Astronomy, 40, 343
Newton, I. 1686, Philosophiae naturalis principia mathematica, Joseph Streater. London, 1686
Noll, K. S., Stephens, D. C., Grundy, W. M., Millis, R. L., Spencer, J., Buie, M. W.,
Tegler, S. C., Romanishin, W., & Cruikshank, D. P. 2002, AJ, 124, 3424
Oort, J. H. 1927, Bull. Astron. Inst. Netherlands, 3, 275
Oort, J. H. 1940, ApJ, 91, 273
Oort, J. H. 1950, Bull. Astron. Inst. Netherlands, 11, 91
Oort, J. H. & Schmidt, M. 1951, Bull. Astron. Inst. Netherlands, 11, 259
Peixinho, N., Boehnhardt, H., Belskaya, I., Doressoundiram, A., Barucci, M. A., &
Delsanti, A. 2004, Icarus, 170, 153
Russell, H. N. 1906, ApJ, 24, 1
Sheppard, S. S. & Jewitt, D. C. 2002, AJ, 124, 1757
—. 2003, Earth Moon and Planets, 92, 207
Sheppard, S. S. & Jewitt, D. 2004, AJ, 127, 3023
Stern, S. A. &Yelle, R. V. 1999, Pluto and Charon, Academic Press
Strazzulla, G., Leto, G., Baratta, G. A., & Spinella, F. 1991, J. Geophys. Res., 96,
17547
Tegler, S. C. & Romanishin, W. 2000, Nature, 407, 979
Thomas, N., et al. 2000, ApJ, 534, 446
Tiscareno, M. S. & Malhotra, R. 2003, AJ, 126, 3122
Trujillo, C. A. & Brown, M. E. 2002, ApJ, 566, L125
Trujillo, C. A., Jewitt, D. C., & Luu, J. X. 2001a, AJ, 122, 457
Trujillo, C. A., Luu, J. X., Bosh, A. S., and Elliot, J. L. 2001b, AJ 122, 2740.
Veillet, C., et al. 2002, Nature, 416, 711
Weidenschilling, S. J. 1977, Ap&SS, 51, 153
Weidenschilling, S. J. 2002, Icarus, 160, 212
Whipple, F. L. 1950, ApJ, 111, 375
van Woerkom, A. J. J. 1948, Bull. Astron. Inst. Netherlands, 10, 445
Yanagisawa, M. 2002, Icarus, 159, 300
CHAPTER
2
Detectability of
KBO lightcurves
ABSTRACT
We present a statistical study of the detectability of lightcurves of Kuiper
Belt objects (KBOs). Some Kuiper Belt objects display lightcurves that
appear ”flat”, i.e., there are no significant brightness variations within the
photometric uncertainties. Under the assumption that KBO lightcurves
are mainly due to shape, the lack of brightness variations may be due to
(1) the objects have very nearly spherical shapes, or (2) their rotation axes
coincide with the line of sight. We investigate the relative importance of
these two effects and relate it to the observed fraction of “flat” lightcurves.
This study suggests that the fraction of KBOs with detectable brightness
variations may provide clues about the shape distribution of these objects.
Although the current database of rotational properties of KBOs is still
insufficient to draw any statistically meaningful conclusions, we expect
that, with a larger dataset, this method will provide a useful test for
candidate KBO shape distributions.
Pedro Lacerda & Jane Luu
Icarus, 161, 174–180, (2003)
22
Detectability of KBO lightcurves
2.1
Introduction
T
he Kuiper Belt holds a large population of small objects which are thought
to be remnants of the protosolar nebula (Jewitt & Luu 1993). The Belt
is also the most likely origin of other outer solar system objects such as PlutoCharon, Triton, and the short-period comets; its study should therefore provide
clues to the understanding of the processes that shaped our solar system. More
than 650 Kuiper Belt objects (KBOs) are known to date and a total of about
105 objects larger than 50 km are thought to orbit the Sun beyond Neptune
(Jewitt & Luu 2000).
One of the most fundamental ways to study physical properties of KBOs is
through their lightcurves. Lightcurves show periodic brightness variations due
to rotation, since, as the KBO rotates in space, its cross-section as projected in
the plane of the sky will vary due to its non-spherical shape, resulting in periodic
brightness variations (see Fig. 2.1). A well-sampled lightcurve will thus yield the
rotation period of the KBO, and the lightcurve amplitude has information on the
KBO’s shape. This technique is commonly used in planetary astronomy, and has
been developed extensively for the purpose of determining the shapes, internal
density structures, rotational states, and surface properties of atmosphereless
bodies. These properties in turn provide clues to their formation and collisional
environment.
Although lightcurves studies have been carried out routinely for asteroids
and planetary satellites, the number of KBO lightcurves is still meager, with
few of sufficient quality for analysis (see Table 2.1). This is due to the fact that
most KBOs are faint objects, with apparent red magnitude of mR ∼ 23 (Trujillo
et al. 2001), rendering it very difficult to detect small amplitude changes in
their brightness. One of the few high quality lightcurves is that of (20000)
Varuna, which shows an amplitude of ∆m = 0.42 ± 0.02 mag and a period of
Prot = 6.3442±0.0002 hrs (Jewitt & Sheppard 2002). Only recently have surveys
started to yield significant numbers of KBOs bright enough for detailed studies
(Jewitt et al. 1998).
Another difficulty associated with the measurement of the amplitude of a
lightcurve is the one of determining the period of the variation. If no periodicity
is apparent in the data, any small variations in the brightness of an object must
be due to noise. Furthermore, a precise measurement of the amplitude of the
lightcurve requires a complete coverage of the rotational phase. Therefore, any
conclusion based on amplitudes of lightcurves must assume that their periods
have been determined and confirmed by well sampled phase plots of the data.
However, not all of the observed KBOs show detectable brightness variations
(the so-called “flat” lightcurves). The simplest explanations for this could be
due to (1) the object is axisymmetric (the two axes perpendicular to the spin
vector are equal), or (2) its rotation axis is nearly coincident with the line of
The Shapes and Spins of KBOs
23
∆m = 2.5log(πab/πcb) = 2.5log(a/b)
mag
c
c
a
b
∆m
time
Figure 2.1 – The lightcurve of an ellipsoidal KBO observed at aspect angle θ = π/2. Crosssections and lightcurve are represented for one full rotation of the KBO. The amplitude, ∆m,
of the lightcurve is determined for this particular case. See text for the general expression.
Table 2.1 – KBOs with measured lightcurves.
Name
1993 SC
1994 TB
1996 TL66
1996 TP66
1994 VK8
1996 TO66
(20000) Varuna
1995 QY9
1996 RQ20
1996 TS66
1996 TQ66
1997 CS29
1999 TD10
Classa
Hb
[mag]
∆mc
[mag]
C
P
S
P
C
C
C
P
C
C
C
C
S
6.9
7.1
5.4
6.8
7.0
4.5
3.7
7.5
7.0
6.4
7.0
5.2
8.8
<0.04
0.3
<0.06
<0.12
0.42
0.1
0.42
0.6
—
<0.16
<0.22
<0.2
0.68
Pd
[hrs]
6.5
9.0
6.25
6.34
7.0
5.8
Sourcee
RT99
RT99
RT99
RT99
RT99
Ha00
JS02
RT99
RT99
RT99
RT99
RT99
Co00
a dynamical
class (C=classical KBO, P=plutino, S=scattered KBO)
magnitude
c lightcurve amplitude
d spin period
e RT99=Romanishin & Tegler (1999), Ha00=Hainaut (2000),
Co00=Consolmagno et al. (2000), JS02=Jewitt & Sheppard (2002)
b absolute
sight (see Fig. 2.3). In other words, the undetectable variations are either a
consequence of the KBO’s shape, or of the observational geometry. By studying
the relative probabilities of these two causes, and relating them to the observed
fraction of ”flat”lightcurves, we might expect to improve our knowledge of the
intrinsic shape distribution of KBOs. In this paper we address the following
question: Can we learn something about the shape distribution of KBOs from
the fraction of “flat” lightcurves?
24
Detectability of KBO lightcurves
Symbol
Description
a≥b≥c
ã ≥ b̃ ≥ c̃
θ
∆mmin
θmin
K
axes of ellipsoidal KBO
normalized axes of KBO (b̃ = 1)
aspect angle
minimum detectable lightcurve amplitude
aspect angle at which ∆m = ∆mmin
100.8∆mmin
2.2
Table 2.2 – Used symbols
and notation.
Definitions and Assumptions
The observed brightness variations in KBO lightcurves can be due to:
· eclipsing binary KBOs
· surface albedo variations
· irregular shape
In general the brightness variations will arise from some combination of these
three factors, but the preponderance of each effect among KBOs is still not
known. In the following calculations we exclude the first two factors and assume
that shape is the sole origin of KBO brightness variations. We further assume
that KBO shapes can be approximated by triaxial ellipsoids, and thus expect a
typical KBO lightcurve to show a set of 2 maxima and 2 minima for each full
rotation (see Fig. 2.1). Table 2.2 summarizes the used symbols and notations.
The listed quantities are defined in the text.
The detailed assumptions of our model are as follows:
1. The KBO shape is a triaxial ellipsoid. This is the shape assumed by a
rotating body in hydrostatic equilibrium (Chandrasekhar 1969). There
are reasons to believe that KBOs might have a “rubble pile” structure
(Farinella et al. 1981), justifying the approximation even further.
2. The albedo is constant over surface. Although albedo variegation can in
principle explain any given lightcurve (Russell 1906), the large scale brightness variations are generally attributed to the object’s irregular shape
(Burns & Tedesco 1979).
3. All axis orientations are equally probable. Given that we have no knowledge
of preferred spin vector orientation, this is the most reasonable a priori
assumption.
4. The KBO is in a state of simple rotation around the shortest axis (the axis
of maximum moment of inertia). This is likely since the damping timescale
of a complex rotation (e.g., precession), ∼ 103 yr, (Burns & Safronov 1973),
(Harris 1994) is smaller than the estimated time between collisions (10 7 –
1011 yr) that would re-excite such a rotational state (Stern 1995; Davis &
Farinella 1997).
The Shapes and Spins of KBOs
25
5. The KBO is observed at zero phase angle (α = 0). It has been shown
from asteroid data that lightcurve amplitudes seem to increase linearly
with phase angle,
A(θ, 0) = A(θ, α)/(1 + mα) ,
where θ is the aspect angle, α is the phase angle and m is a coefficient
which depends on surface composition. The aspect angle is defined as the
angle between the line of sight and the spin axis of the KBO (see Fig. 2.2a),
and the phase angle is the Sun-object-Earth angle. The mean values of
m found for different asteroid classes are m(S) = 0.030, m(C) = 0.015,
m(M) = 0.013, where S, C, and M are asteroid classes (see (Michalowsky
1993). Since KBO are distant objects the phase angle will always be small.
Even allowing m to be one order of magnitude higher than that of asteroids
the increase in the lightcurve amplitude will not exceed 1%.
6. The brightness of the KBO is proportional to its cross-section area (geometric scattering law). This is a good approximation for KBOs because
(1) most KBOs are too small to hold an atmosphere, and (2) the fact
that they are observed at very small phase angles reduces the influence of
scattering on the lightcurve amplitude (Magnusson 1989).
The KBOs will be represented by triaxial ellipsoids of axes a ≥ b ≥ c rotating
around the short axis c (see Fig. 2.2b). In order to avoid any scaling factors we
normalize all axes by b, thus obtaining a new set of parameters ã, b̃ and c̃ given
by
ã = a/b ,
b̃ = 1 ,
c̃ = c/b .
(2.1)
As defined, ã and c̃ can assume values 1 ≤ ã < ∞ and 0 < c̃ ≤ 1. Note that the
parameters ã and c̃ are dimensionless.
The orientation of the spin axis of the KBO relative to the line of sight will be
defined in spherical coordinates (θ, φ), with the line of sight (oriented from the
object to the observer) being the z-axis, or polar axis, and the angle θ being the
polar angle (see Fig. 2.2a). The solution is independent of the azimuthal angle
φ, which would be measured in the plane perpendicular to the line of sight,
between an arbitrary direction and the projection of the spin axis on the same
plane. The observation geometry is parameterized by the aspect angle, which in
this coordinate system corresponds to θ.
As the object rotates, its cross-section area S will vary periodically between
Smax and Smin (see Fig. 2.1). These areas are simply a function of a, b, c and
the aspect angle θ. Given the assumption of geometric scattering, the ratio
between maximum and minimum flux of reflected sunlight will be equal to the
ratio between Smax and Smin . The lightcurve amplitude can then be calculated
26
Detectability of KBO lightcurves
a)
s
b)
s
φ
θ
c
θmin
b
a
line of sight
Figure 2.2 – a) A spherical coordinate system is used to represent the observing geometry.
The line of sight (oriented from the object to the observer) is the polar axis and the azimuthal
axis is arbitrary in the plane orthogonal to the polar axis. θ and φ are the spherical angular
coordinates of the spin axis ~s. In this coordinate system the aspect angle is given by θ.
The “non-detectability” cone, with semi-vertical angle θmin , is represented in grey. If the
spin axis lies within this cone the brightness variations due to changing cross-section will be
smaller than photometric errors, rendering it impossible to detect brightness variations. b)
The picture represents an ellipsoidal KBO with axes a ≥ b ≥ c.
from the quantities ã, c̃ and θ and is given by
s
ã2 cos2 θ + ã2 c̃2 sin2 θ
∆m = 2.5 log
.
ã2 cos2 θ + c̃2 sin2 θ
2.3
(2.2)
“Flat” Lightcurves
It is clear from Eqn. (2.2) that under certain conditions, ∆m will be zero, i.e., the
KBO will exhibit a flat lightcurve. These special conditions involve the shape of
the object and the observation geometry, and are described quantitatively below.
Taking into account photometric error bars will bring this “flatness” threshold to
a finite value, ∆mmin , a minimum detectable amplitude below which brightness
variation cannot be ascertained.
The two factors that influence the amplitude of a KBO lightcurve are:
1. Sphericity For a given ellipsoidal KBO of axes ratios ã and c̃ the
lightcurve amplitude will be largest when θ = π/2 and smallest when θ = 0
or π. At θ = π/2, Eqn. (2.2) becomes
∆m = 2.5 log ã .
(2.3)
Even at θ = π/2, having a minimum detectable amplitude, ∆mmin , puts constraints on ã since if ã is too small, the lightcurve amplitude will not be detected.
The Shapes and Spins of KBOs
27
Figure 2.3 – Illustration of a rotating ellipsoid at different aspect angles. A quarter of a
full rotation is represented. Rotational phase of ellipsoid increasing from top to bottom and θ
decreasing from left to right. T is the period of rotation. Axes ratios are ã = 1.2 and c̃ = 0.9.
This constraint is thus
ã < 100.4∆mmin ⇒ “flat” lightcurve .
(2.4)
2. Observation geometry If the rotation axis is nearly aligned with the
line of sight, i.e., if the aspect angle is sufficiently small, the object’s projected
cross-section will hardly change with rotation, yielding no detectable brightness
variations (see Fig. 2.3). The finite accuracy of the photometry defines a minimum aspect angle, θmin , within which the lightcurve will appear flat within
the uncertainties. This angle rotated around the line of sight generates the
“non-detectability cone” (see Fig. 2.2a), with the solid angle
Z 2π Z θmin
Ω(θmin ) =
sin θ dθ dφ .
(2.5)
0
0
28
Detectability of KBO lightcurves
Any aspect angle θ which satisfies θ < θmin falls within the “non-detectability
cone” and results in a non-detectable lightcurve amplitude. Therefore, the probability that the lightcurve will be flat due to observing geometry is
2 × Ω(θmin )
= 1 − cos θmin
4π
= cos θmin .
pã,c̃ (non-detection)
=
pã,c̃ (detection)
(2.6)
(2.7)
The factor of 2 accounts for the fact that the axis might be pointing towards
or away from the observer and still give rise to the same observations, and the
4π in the denominator represents all possible axis orientations.
From Eqn. (2.2) we can write cos θmin as a function of ã and c̃,
s
c̃2 (ã2 − K)
,
cos θmin = Ψ(ã, c̃) =
c̃2 (ã2 − K) + ã2 (K − 1)
(2.8)
where K = 100.8∆mmin . The function Ψ(ã, c̃), represented in Fig. 2.4, is the
probability of detecting brightness variation from a given
√ ellipsoid of axes ratios
(ã, c̃). It is a geometry weighting function. For ã in [1, K] we have Ψ(ã, c̃) = 0
by definition, since in this case the KBO satisfies Eqn. (2.4) and its lightcurve
amplitude will not be detected irrespective of the aspect angle. It is clear from
Fig. 2.4 that it is more likely to detect brightness variation from an elongated
body.
2.4
Detectability of Lightcurves
In order to generate a “non-flat” lightcurve, the KBO has to satisfy both the
shape and observation geometry conditions. Mathematically this means that
the probability of detecting brightness variation from a KBO is a function of the
probabilities of the KBO satisfying both the sphericity and observing geometry
conditions.
We will assume that it is possible to represent the shape distribution of KBOs
by two independent probability density functions, f (ã) and g(c̃), defined as
p(ã1 ≤ ã ≤ ã2 ) =
Z
ã2
f (ã) dã,
ã1
p(c̃1 ≤ c̃ ≤ c̃2 ) =
Z
c̃2
Z
∞
g(c̃) dc̃,
c̃1
f (ã) dã = 1 ,
(2.9)
1
Z
1
g(c̃) dc̃ = 1 ,
(2.10)
0
where the integrals on the left represent the fraction of KBOs in the given ranges
of axes ratios. This allows us to write the following expression for p(∆m >
The Shapes and Spins of KBOs
29
Figure 2.4 – The function Ψ(ã, c̃) (Eqn. 2.8). This plot assumes photometric errors
∆mmin = 0.15 mag. The detection probability is zero when ã < 100.4∆mmin ≈ 1.15.
∆mmin ), where both the shape and observation geometry constraints are taken
into account,
Z 1Z ∞
Ψ(ã, c̃)f (ã)g(c̃) dã dc̃.
(2.11)
p(∆m > ∆mmin ) =
0
1
The right hand side of this equation represents the probability of observing a
given KBO with axes ratios between (ã, c̃) and (ã + dã, c̃ + dc̃), at a large enough
aspect angle, integrated for all possible axes ratios. This is also the probability
of detecting brightness variation for an observed KBO.
√
The lower limit of integration for ã in Eqn. (2.11) can be replaced
by K,
√
with K defined as in Eqn. (2.8), since Ψ(ã, c̃) is zero for ã in [1, K]. In fact,
this is how the sphericity constraint is taken into account.
Provided that we know the value of p(∆m > ∆mmin ) Eqn. (2.11) can test
candidate distributions f (ã) and g(c̃) for the shape distribution of KBOs. The
best estimate for p(∆m > ∆mmin ) is given by the ratio of “non-flat” lightcurves
(ND ) to the total number of measured lightcurves (N ), i.e.,
p(∆m > ∆mmin ) ≈
ND
.
N
(2.12)
Because N is not the total number of KBOs there will be an error associated
with this estimate. Since we do not know the distributions f (ã) and g(c̃) we will
assume that the outcome of an observation can be described by a binomial distribution of probability p(∆m > ∆mmin ). This is a good approximation given that
N is very small compared with the total number of KBOs. Strictly speaking,
30
Detectability of KBO lightcurves
the hypergeometric distribution should be used since we will not unintentionally observe the same object more than once (sampling without replacement).
However, since the total number of KBOs (which is not known with certainty) is
much larger than any sample of lightcurves, any effects of repeated sampling will
be negligible, thereby justifying the binomial approximation. This simplification
allows us to calculate the upper (p+ ) and lower (p− ) limits for p(∆m > ∆mmin )
at any given confidence level, C. These values, known as the Clopper–Pearson
confidence limits, can be found solving the following equations by trial and error
(Barlow 1989),
N
X
r=ND +1
NX
D −1
r=0
¡
¢ C +1
P r; p+ (∆m > ∆mmin ), N =
2
¡
¢ C +1
,
P r; p− (∆m > ∆mmin ), N =
2
(2.13)
(2.14)
(see Table 2.2 for notation) where C is the desired confidence level and P (r; p, N )
is the binomial probability of detecting r lightcurves out of N observations, each
lightcurve having a detection probability p. Using the values in Table 2.1 and
∆mmin = 0.15 mag we have ND = 5 and N = 13 which yields
p(∆m > ∆mmin ) = 0.38+0.18
−0.15
at a C = 0.68 (1σ) confidence level. At C = 0.997 (3σ) we have
p(∆m > ∆mmin ) = 0.38+0.41
−0.31 .
The value of p(∆m > ∆mmin ) could be smaller since some of the flat lightcurves
might not have been published.
Note that for moderately elongated ellipsoids (small ã) the function Ψ(ã, c̃) is
almost insensitive to the parameter c̃ (see Fig. 2.4), in which case the axisymmetric approximation with respect to ã can be made yielding c̃ ≈ 1. Equation (2.11)
then has only one unknown parameter, f (ã).
p(∆m > ∆mmin ) ≈
Z
ãmax
√
K
Ψ(ã, 1)f (ã) dã ≈ 0.38+0.41
−0.31 .
(2.15)
If we assume the function f (ã) to be gaussian, we can use Eqn. (4.3) to determine its mean µ and standard deviation σ, after proper normalization to
satisfy Eqn. (2.9). The result is represented in Fig. 2.5, where we show all
possible pairs of (µ,σ) that would satisfy a given p(∆m > ∆mmin ). For example, the line labeled ”0.38”identifies all possible pairs of (µ,σ) that give rise to
p(∆m > ∆mmin ) = 0.38, the line labeled ”0.56”all possible pairs of (µ,σ) that
give rise to p(∆m > ∆mmin ) = 0.56, etc.
The Shapes and Spins of KBOs
31
Figure 2.5 – Contour plot of the theoretical probabilities of detecting brightness variation in
KBOs (assuming ∆mmin = 0.15 mag), drawn from gaussian shape distributions parameterized
by µ and σ (respectively the mean and spread of the distributions). The solid lines represent
the observed ratio of “non-flat” lightcurves (at 0.38) and 0.68 confidence limits (at 0.23 and
0.56 respectively).
Clearly, with the present number of lightcurves the uncertainties are too
large to draw any relevant conclusions on the shape distribution of KBOs. With
a larger dataset, this formulation will allow us to compare the distribution of
KBO shapes with that of the main belt asteroids. The latter has been shown
to resemble, to some extent, that of fragments of high-velocity impacts (Catullo
et al. 1984). It deviates at large asteroid sizes that have presumably relaxed to
equilibrium figures. A comparison of f (ã) with asteroidal shapes should tell us,
at the very least, whether KBO shapes are collisionally derived, as opposed to
being accretional products.
The usefulness of this method is that, with more data, it would allow us
to derive such quantitative parameters as the mean and standard deviation of
the KBO shape distribution, if we assume a priori some intrinsic form for this
distribution. The method’s strength is that it relies solely on the detectability of
lightcurve amplitudes, which is more robust than other lightcurve parameters.
This paper focuses on the influence of the observation geometry and KBO
shapes in the results of lightcurve measurements. In which direction would
our conclusions change with the inclusion of albedo variegation and/or binary
KBOs?
Non-uniform albedo would cause nearly spherical KBOs to generate detectable brightness variations, depending on the coordinates of the albedo patches
on the KBO’s surface. This means that our method would overestimate the num-
32
Detectability of KBO lightcurves
ber of elongated objects by attributing all brightness fluctuations to asphericity.
Binary KBOs would influence the results in different ways depending on
the orientation of the binary system’s orbital plane, on the size ratio of the
components, and on the individual shapes and spin axis orientations of the
primary and secondary. For example, an elongated KBO observed equator-on
would have its lightcurve flattened by a nearly spherical moon orbiting in the
plane of the sky, whereas two spherical KBOs orbiting each other would generate
a lightcurve if the binary would be observed edge-on.
These effects are not straightforward to quantify analytically and might require a different approach. We intend to incorporate them in a future study.
Also, with a larger sample of lightcurves it would be useful to apply this model
to subgroups of KBOs based on dynamics, size, etc.
2.5
Conclusions
We derived an expression for the probability of detecting brightness variations
from an ellipsoidal KBO, as a function of its shape and minimum detectable
amplitude. This expression takes into account the probability that a “flat”
lightcurve is caused by observing geometry.
Our model can yield such quantitative parameters as the mean and standard
deviation of the KBO shape distribution, if we assume a priori an intrinsic form
for this distribution. It concerns solely the statistical probability of detecting
brightness variation from objects drawn from these distributions, given a minimum detectable lightcurve amplitude. The method relies on the assumption
that albedo variegation and eclipsing binaries play a secondary role in the detection of KBO lightcurves. The effect of disregarding albedo variegation in our
model is that we might overestimate the fraction of elongated objects. Binaries
in turn could influence the result in both directions depending on the geometry
of the problem, and on the physical properties of the constituents. We intend
to incorporate these effects in a future, more detailed study.
Acknowledgments
We are grateful to Garrelt Mellema, Glenn van de Ven, and Prof. John Rice for
helpful discussion.
The Shapes and Spins of KBOs
33
References
Barlow, R. 1989. Statistics. A guide to the use of statistical methods in the physical
sciences. The Manchester Physics Series, New York: Wiley, 1989.
Burns, J. A. and V. S. Safronov 1973. Asteroid nutation angles. MNRAS 165, 403-411.
Burns, J. A. and E. F. Tedesco 1979. Asteroid lightcurves - Results for rotations and
shapes. Asteroids 494-527.
Catullo, V., V. Zappalà, P. Farinella, and P. Paolicchi 1984. Analysis of the shape
distribution of asteroids. A&A 138, 464-468.
Chandrasekhar, S. 1987. Ellipsoidal figures of equilibrium. New York : Dover, 1987. .
Consolmagno, G. J., S. C. Tegler, T. Rettig, and W. Romanishin 2000. Size, Shape,
Rotation, and Color of the Outer Solar System Object 1999 TD10. AAS/Division
for Planetary Sciences Meeting 32, 21.07.
Davis, D. R. and P. Farinella 1997. Collisional Evolution of Edgeworth-Kuiper Belt
Objects. Icarus 125, 50-60.
Farinella, P., P. Paolicchi, E. F. Tedesco, and V. Zappalà 1981. Triaxial equilibrium
ellipsoids among the asteroids. Icarus 46, 114-123.
Hainaut, O. R., C. E. Delahodde, H. Boehnhardt, E. Dotto, M. A. Barucci K. J. Meech,
J. M. Bauer, R. M. West, and A. Doressoundiram 2000. Physical properties of
TNO 1996 TO66 . Lightcurves and possible cometary activity. A&A 356, 10761088.
Harris, A. W. 1994. Tumbling asteroids. Icarus 107, 209.
Jewitt, D. and J. Luu 1993. Discovery of the candidate Kuiper belt object 1992 QB1.
Nature 362, 730-732.
Jewitt, D., J. Luu, and C. Trujillo 1998. Large Kuiper Belt Objects: The Mauna Kea
8K CCD Survey. AJ 115, 2125-2135.
Jewitt, D. C. and J. X. Luu 2000. Physical Nature of the Kuiper Belt. Protostars and
Planets IV 1201.
Jewitt, D. C. and S. S. Sheppard 2002. Physical Properties of Trans-Neptunian Object
(20000) Varuna. AJ 123, 2110-2120.
Magnusson, P. 1989. Pole determinations of asteroids. Asteroids II 1180-1190.
Michalowski, T. 1993. Poles, shapes, senses of rotation, and sidereal periods of asteroids. Icarus 106, 563.
Romanishin, W. and S. C. Tegler 1999. Rotation rates of Kuiper-belt objects from
their light curves. Nature398, 129-132.
Russell, H. N. 1906. On the light variations of asteroids and satellites. ApJ 24, 1-18.
Stern, S. A. 1995. Collisional Time Scales in the Kuiper Disk and Their Implications.
AJ 110, 856.
Trujillo, C. A., J. X. Luu, A. S. Bosh, and J. L. Elliot 2001. Large Bodies in the Kuiper
Belt. AJ 122, 2740-2748.
CHAPTER
3
The shape distribution
ABSTRACT
If we assume that the periodic brightness variations in a Kuiper Belt
lightcurve are determined only by their aspherical shapes and the observing geometry (no spin rate bias is considered), the fraction of detectable
Kuiper Belt lightcurves and the lightcurve amplitude distribution can be
used to constrain the shapes of Kuiper Belt objects. The results indicate
that most Kuiper Belt objects (∼ 85%) have shapes that are close to spherical (a/b ≤ 1.5), but there is a small but significant fraction (∼ 12%) possessing highly aspherical shapes (a/b ≥ 1.7). The distribution cannot be
well fitted by a gaussian and is much better approximated by a power law.
Jane Luu & Pedro Lacerda
Earth Moon and Planets, 92, 2007, (2003)
36
The shape distribution
3.1
Introduction
S
ince their discovery in 1992, the Kuiper Belt objects (KBOs) have attracted
a great deal of interest in planetary astronomy because of the information
they might contain. Thought to be a relic from the original protoplanetary
disk, they are expected to still bear signatures of their origin and evolution. In
particular, they are believed to be much less evolved than other known solar
system objects, and thus might show planetary formation at an early stage.
Although it has been a decade since their discovery, not much is known about
the KBOs physical properties, mainly because most are too faint (red magnitude mR ≥ 20) for detailed studies. Most of the existing data are broadband
photometry, with a few low-resolution optical and near-IR spectra. Broadband
photometry indicates that the KBOs possess diverse colors, ranging from neutral to very red (Luu & Jewitt 1996; Tegler & Romanishin 2000; Jewitt & Luu
2001). The low-resolution KBO spectra are usually featureless, although a few
show weak 2 µm water ice absorption (Brown, Cruikshank & Pendleton 1999;
Jewitt & Luu 2001). Some broadband photometric data have been obtained
for the purpose of studying KBO rotational properties, and although reliable
lightcurves are still sparse, the sample is sufficient for detailed analysis. In this
paper we collect data from reliable lightcurves and examine their implications
for the shape distribution of KBOs.
3.2
Observations
The largest and most systematic studies of KBO rotational properties were carried out by Sheppard & Jewitt (2002, hereafter SJ02) and Sheppard & Jewitt
(2003, hereafter SJ03), which together present optical lightcurves of 27 KBOs.
Their sample included most of the largest and brightest KBOs, with red absolute
magnitude HR in the range 6.0 – 7.5, corresponding to the diameter range 200
– 400 km. These large objects are unlikely to be collisional fragments. Interestingly, only 7 of the 27 yielded periodic lightcurves, which they defined as periodic
brightness variations of amplitude ∆m ≥ 0.15. Other lightcurves exist besides
those presented by SJ02 and SJ03, but they vary widely in quality and sometimes contradict each other. For example, Romanishin & Tegler (1999) report
a flat lightcurve for 1996 TO66, while Hainaut et al. (2000) and SJ03 detected
periodic lightcurve for the same object. Of the lightcurves in the literature that
were not measured by them, SJ02 deemed 6 to be reliable and included them in
their analysis. Of these 6, 2 showed periodic lightcurves. In this work we adopt
the same practice and make use of the 27 lightcurves from Sheppard & Jewitt,
plus the 6 reliable lightcurves from the literature. The KBO lightcurve statistics
are thus as follows: out of 33 reliable lightcurves, 9 showed periodic lightcurves
with amplitudes ∆m ≥ 0.15. The fraction of KBOs with detectable periodic
The Shapes and Spins of KBOs
37
Table 3.1 – KBO rotational properties
KBO
P [hr]
∆m
Reference
KBOs lightcurves considered to have ∆m < 0.15 mag
1993 SC
–
0.04
RT99, DMcG97
1994 TB
–
SJ02
1996 GQ21
–
< 0.10
SJ02
1996 TL66
–
0.06
RT99, LJ98
1996 TP66
–
0.12
RT99, CB99
1997 CS29
–
< 0.08
SJ02
1998 HK151
–
< 0.15
SJ02
1998 VG44
–
< 0.10
SJ02
(Chaos) 1998 WH24
–
< 0.10
SJ02
1999 DE9
–
< 0.10
SJ02
(47171) 1999 TC36
–
< 0.10
SJ03
(Huya) 2000 EB173
–
< 0.06
SJ02
2000 YW134
–
< 0.10
SJ03
2001 CZ31
–
< 0.20
SJ02
2001 FP185
–
< 0.10
SJ03
2001 FZ173
–
< 0.06
SJ02
2001 KD77
–
< 0.10
SJ03
(28978) Ixion 2001 KX76
–
< 0.10
SJ03,O01
2001 QF298
–
< 0.10
SJ03
(42301) 2001 UR163
–
< 0.10
SJ03
(42355) 2002 CR46
–
< 0.10
SJ03
(55636) 2002 TX300
16.24 ± 0.08
0.08 ± 0.02 SJ03
(55637) 2002 UX25
–
< 0.10
SJ03
(55638) 2002 VE95
–
< 0.10
SJ03
KBO lightcurves considered to have ∆m ≥ 0.15 mag
1995 QY9
0.60
SJ02, RT99
(24835) 1995 SM55
8.08 ± 0.03
0.19 ± 0.05 SJ03
1996 TO66
–
0.26 ± 0.03 SJ03, H00
1998 SM165
0.45
SJ02, R01
1998 BU48
9.8 ± 0.1
0.68 ± 0.04 SJ02
12.6 ± 0.1
1999 KR16
11.858 ± 0.002 0.18 ± 0.04 SJ02
11.680 ± 0.002
2000 GN171
8.329 ± 0.005
0.61 ± 0.03 SJ02
(Varuna) 2000 WR106
6.34
0.42 ± 0.03 SJ02
2003 AZ84
13.42 ± 0.05
0.14 ± 0.03 SJ03
CB99 = Collander-Brown et al. 1999, DMcG97 = Davies, McBride & Green 1997,
H00 = Hainaut et al. 2000, LJ98 = Luu & Jewitt 1998, O01 = Ortiz et al. 2001, R01
= Romanishin et al. 2001, RT99 = Romanishin & Tegler 1999, SJ02 = Sheppard &
Jewitt 2002, SJ03 = Sheppard & Jewitt 2003.
38
The shape distribution
lightcurves is then:
f (∆m ≥ 0.15) = 9/33 = 27%.
(3.1)
All 33 KBOs and their lightcurve parameters are listed in Table 1; the 9 KBOs
considered in this work as having periodic lightcurves are clustered at the bottom
of the Table. Assuming that the lightcurves are modulated by aspherical shapes
and are therefore double-peaked, the periods range from 6 to 12 hrs.
3.3
Discussion
Lacerda & Luu (2003) show that the fraction of detectable KBO lightcurves
can be used to infer these objects shape distribution, if certain (reasonable)
assumptions are made. The assumptions are
1. Asphericity. The lightcurve modulations are assumed to arise from an
aspherical shape, taken to be a triaxial ellipsoid with axes a ≥ b ≥ c. The
minimum detectable lightcurve amplitude ∆mmin must be as large as the
photometric error, or it will not be detected.
2. Observation geometry. The angle between the KBO spin axis and the line
of sight – the aspect angle θ – should also be large enough to give rise to
a detectable lightcurve amplitude, i.e., larger than ∆mmin .
Here we adopt SJ02’s photometric error of 0.15 mag, i.e., ∆mmin = 0.15. If a
lightcurve does not show periodic modulations, it is assumed that this is because
neither the asphericity nor the observation geometry criterion is satisfied. No
spin rate bias is considered here.
With these assumptions, the probability p of detecting a lightcurve can be
written as (Lacerda & Luu 2003)
Z 1Z ∞
Ψ(ã, c̃) f (ã) g(c̃) dã dc̃.
(3.2)
p(∆m > ∆mmin ) =
0
1
where, for the sake of being concise, we define ã = a/b, c̃ = c/b, and Ψ(ã, c̃) is
the probability of detecting brightness variations from a given ellipsoid of axis
ratio (ã, c̃). Ψ(ã, c̃) is given by
s
c̃2 (ã2 − K)
Ψ(ã, c̃) =
,
(3.3)
2
2
c̃ (ã − K) + ã2 (K − 1)
where K = 100.8∆mmin . The right hand side of Eqn. (3.2) represents the probability of observing a given KBO with axis ratios between (ã, c̃) and (ã+dã, c̃+dc̃)
at a large enough aspect angle, integrated over all possible axis ratios. For moderately elongated ellipsoids (small ã), the function Ψ(ã, c̃) is almost independent
of c̃. If we further assume c̃ ≈ 1, then g(c̃) is ≈ 1, and Eqn. (3.2) becomes
Z ∞
p(∆m > ∆mmin ) ≈
Ψ(ã, 1) f (ã) dã.
(3.4)
1
The Shapes and Spins of KBOs
39
1.4
mean
1.3
0.55
0.46
1.2
0.37
0.27
0.19
1.1
0.13
0.09
1
0
0.1
0.2
0.3
sigma
0.4
0.5
0.6
Figure 3.1 – The probability p as a function of the mean µ and standard deviation σ of a
gaussian ã distribution. The thick black line represents all µ−σ pairs that give rise to p = 0.27.
The shaded areas immediately adjacent to the line represent the µ and σ values within the 1σ
limits, the next shaded areas outward represent the 2σ limits, and the outermost shaded area
the 3σ limits. The number to the left of each boundary line indicates the p value corresponding
to that line. The circle with the cross marks the best-fit µ − σ: µ = 1.00, σ = 0.24.
From Eqn. (3.1), p(∆m > ∆mmin ) = 0.27. Since the data did not sample
the entire KBO population, there are necessarily errors associated with p. The
1-, 2- and 3σ error bars on p can be calculated based on the Clopper-Pearson
confidence limits (Lacerda & Luu 2003):
1σ
2σ
3σ
p = 0.27+0.10
−0.08
p = 0.27+0.19
−0.14
p=
(3.5)
0.27+0.28
−0.18
With p known, the problem then becomes inverting Eqn. (3.4) to determine the
shape distribution f (ã). This can be done if f (ã) is assumed to be a simple
analytical function.
3.3.1
Gaussian distribution
First we assume f (ã) to be a gaussian with a mean denoted by µ and a standard
deviation σ. All possible combinations of µ−σ that satisfy Eqn. (3.5) are plotted
in Fig. 3.1. The thick black curve represents all possible combinations of µ − σ
that give rise to p = 0.27, and the shaded regions represent the areas enclosed
40
The shape distribution
cumulative fraction
1
0.8
0.6
0.4
0.2
1
2
3
ab
4
5
Figure 3.2 – Cumulative fractions of KBOs as a function of a/b = ã. The solid line corresponds to the best-fit gaussian (µ = 1.00, σ = 0.24), the dashed line a gaussian with
µ = 1.22, σ = 0.24. The data are from Table 1, with vertical error bars calculated from
Poisson statistics. The horizontal error bars are calculated from 1σ deviation from the average
aspect angle of 60o , assuming that the aspect angle is uniformly distributed in sin θ. Note:
our x-axis is a/b, which is the inverse of SJ02’s x-axis, b/a.
by p’s 1-, 2-, and 3σ error bars. The entire shaded regions thus represent all
possible combinations of µ − σ that are consistent with p = 0.27+0.28
−0.18 .
We can constrain µ − σ further by making use of the observed axis ratios ã.
Using the data from Table 1, for each detected lightcurve, the observed ∆m is
converted to the axis ratio ã by using the relation (Lacerda & Luu 2003)
s
ã2 cos2 θ + ã2 c̃2 sin2 θ
,
(3.6)
∆m = 2.5 log
ã2 cos2 θ + c̃2 sin2 θ
and by assuming an average aspect angle θ = 60o . (θ = 60o is the average
angle if θ is distributed uniformly in sin θ). The observed cumulative fractions
of KBOs, as a function of ã, are plotted in Fig. 3.2. These can then be compared
with the cumulative fractions predicted by each allowed µ − σ pair to yield the
best-fit gaussian. A grid search is performed through all possible µ − σ pairs
allowed by Eqn. (3.5); using χ2 as the comparison criterion, the best-fit gaussian
is given by
µ = 1.00(+0.22), σ = 0.24+0.18
(3.7)
−0.13 .
The error bars in Eqn. (3.7) are 1σ error bars. There is no lower error bar to
The Shapes and Spins of KBOs
41
cumulative fraction
1
0.8
0.6
0.4
0.2
1
2
3
ab
4
5
Figure 3.3 – Cumulative fractions of KBOs as a function of a/b = ã. Same as Fig. 3.2, but
this time with the dotted line representing a gaussian with µ = 1.00, σ = 0.11, the dashed line
a gaussian with µ = 1.00, σ = 0.42.
µ since µ represents the mean ã, defined to be ≥ 1. The best-fit µ − σ is also
marked in Fig. 3.1.
The goodness of the fit is shown graphically in Fig. 3.2 where we plot the
observed cumulative fractions of KBOs, as a function of ã = a/b, with the
cumulative fractions predicted by the best-fit gaussian (Eqn. 3.7). The fit is
good for ã ≤ 1.5 but poor at larger ã’s. The Figure also shows that if we
increase µ to µ = 1.22 (1 standard deviation away from the best-fit µ), the
theoretical curve comes closer to fitting the larger ã’s, but misses practically all
the data points. [Note: our x-axis is ã = a/b, which is the inverse of SJ02’s
x-axis, b/a].
In Fig. 3.3 we try fitting the data with gaussians of different widths (σ = 0.11
and σ = 0.42, both being 1 standard deviation away from the best-fit σ), while
keeping µ fixed at µ = 1.00. The fit offered by σ = 0.11 is much poorer than
those seen in Fig. 3.2; σ = 0.42 comes closer to fitting all the data, but still
misses all the data points.
What can we infer from Figs. 3.2 and 3.3? The best-fit gaussian (solid line in
Fig. 3.2) is skewed toward small axis ratios and predicts that 95% of KBOs have
axis ratios a/b ≤ 1.48 (within 2σ from the mean). This agrees reasonably well
with the data which indicate that ∼ 85% have axis ratios a/b ≤ 1.48. However,
the best-fit gaussian fails badly at larger a/b’s: it predicts that only 0.3% of
42
The shape distribution
KBOs have axis ratios a/b ≥ 1.72 (larger than 3σ from the mean), while the
data indicate that ∼ 12% have axis ratios in this range. Increasing µ to µ = 1.22
(dashed line in Fig. 3.2) reduces some of this skewness but does not significantly
improve the fit at larger a/b’s.
Keeping µ fixed and decreasing σ to σ = 0.11 (dotted line in Fig. 3.3) worsens
the fit, as expected. Keeping µ fixed and increasing to σ = 0.42 (dashed line in
Fig. 3.3) arguably produces the best fit yet (as judged by eye) since it tries to
fit all the data points and does so equally well for all of them (or equally badly,
depending on one’s point of view). In short, none of the gaussians presented
in Figs. 3.2 and 3.3 offers a good fit to the data. The conclusion to draw from
the Figures is that the KBO shape distribution is not well approximated by a
gaussian.
This is because the KBO shape distribution has two characteristics that cannot be met simultaneously by a standard gaussian: (1) a large fraction (∼ 85%)
has shapes that are close to spherical (a/b ≤ 1.5), yet (2) there is a significant
tail to the distribution (∼ 12%) that has highly aspherical shapes (a/b ≥ 1.7).
In other words, most KBOs are nearly spherical, but a signicant fraction is not.
We note that, using a smaller data set, SJ02 came to the conclusion that a broad
gaussian was needed to fit their available data. This is roughly consistent with
our result here. With the benefit of a larger data sample, and the additional
constraint from the detection probability p, we are able to improve SJ02’s conclusion: the best description of the shape distribution is actually more like a
moderately narrow peak with a long tail.
3.3.2
Power law distribution
Considering how poorly a gaussian fits the shape distribution, we try approximating f (ã) with a power law, f (ã) ∝ ã−q . The solution is shown graphically in
Fig. 3.4, where the thick horizontal line represents p = 0.27, and the solid black
curve represents the detection probability p as a function of q:
p(ã)dã = ã−q dã.
(3.8)
In Eqn. (3.8), p(ã)dã is the fraction of a KBO with axis ratios in the range ã to
ã + dã. The probability is normalized so that the integral of p(ã) from ã = 1 to
ã = 5 is equal to 1. The shaded areas represent all possible values of p within its
1-, 2-, and 3σ error bars, so the allowed values of q are those that lie within these
shaded areas. We note that the horizontal line and the curved line intersect at
q = 6.7.
As was done in the previous section, we use the observed ã’s to constrain q
further. We compare the observed cumulative fractions of KBOs, as a function
of ã, with the cumulative fractions predicted by each possible q. A grid search is
The Shapes and Spins of KBOs
43
detection probability
0.8
0.6
0.4
0.2
0
-5
0
5
10
15
20
q
Figure 3.4 – The thick horizontal line represents all values of q that that give rise to p = 0.27.
The shaded areas immediately adjacent to the line represent the p’s within the 1σ limits, the
next shaded areas outward the 2σ limits, and the outermost shaded area the 3σ limits. The
solid curve represents p as a function of the exponent q (Eqn. 3.8). The intersections of the
shaded areas and the curve satisfy both Eqns. (3.2) and (3.8). The vertical line marks q = 6.4.
cumulative fraction
1
0.8
0.6
0.4
0.2
1
2
3
ab
4
5
Figure 3.5 – Cumulative fractions of KBOs as a function of ã. Same as Fig. 3.2, but this
time with the curves representing power laws. The dotted line represents the exponent q = 7.8,
solid line q = 6.4, and dashed line q = 5.0.
44
The shape distribution
4
3.5
3
fHabL
2.5
2
1.5
1
0.5
1
1.5
2
ab
2.5
3
Figure 3.6 – Distribution of axis ratio ã. The dashed line is the q = 6.4 power law (from Eqn.
3.9), the solid line a gaussian with µ = 1.00, σ = 0.24 (from Eqn. 3.7). f (ã) is normalized to
1 between ã = 1 and ã = 5.
performed through all possible values of q; using χ2 as the comparison criterion,
we obtain the best-fit q:
q = 6.4 ± 1.4 (1σ error bars).
(3.9)
It is reassuring that χ2 statistics yield q = 6.4 as the best fit, as this is very
close to the q = 6.7 found independently by the Lacerda & Luu method. The
fits offered by q = 5.0, 6.4 and 7.8 are shown in Fig. 3.5. It can be seen that
the power laws generally fit the data better than the gaussians. The shape
distribution f (ã) as a gaussian and a power law is shown in Fig. 3.6.
As Fig. 3.6 shows, the KBO shape distribution is characterized by a large
peak at small a/b’s, accompanied by a slow decline at larger a/b’s. The dominance of small a/b’s might be explained by (a) a preponderance of nearly spherical bodies, (b) a preponderance of very slow rotators whose lightcurve amplitudes
could not be determined from the limited data, or some combination of these two
factors. The likelihood of these scenarios will be evaluated in a future work. If
the observational bias against slow rotators can be ruled out, the challenge will
then be how to explain the dominance of nearly spherical bodies in the Kuiper
Belt. As for the (small) fraction of KBOs with larger a/b’s, Jewitt & Sheppard
(2002) and SJ02 have tentatively identified them as rotationally deformed ”rubble piles,”much like the ”rubble piles”that have been proposed to exist among
asteroids. If this hypothesis is correct, there should be a correlation between the
The Shapes and Spins of KBOs
45
KBO shapes and spin rates (high spin rate → large ∆m). The data sample is
as yet too small to confirm such a trend (e.g., see Fig. 13 of SJ02).
3.4
Summary
We have applied the method described in Lacerda & Luu (2003) to the available
KBO lightcurve data to constrain the KBO shape distribution. The method assumes that the detectability of KBO lightcurves depends only on the KBO shape
and the observing geometry; it does not take into account any spin frequency
effect (e.g., the bias against very slow spinners). The results can be summarized
as follows:
1. With 9 out of 33 reliable KBO lightcurves showing periodic brightness
variations, the fraction of detectable KBO lightcurves is f (∆m ≥ 0.15) =
0.27. This implies that the probability of detecting a KBO lightcurve is
p = 0.27+0.28
−0.18 (3σ error bars).
2. The KBO shape distribution has a steep peak at small axis ratios and
drops off quickly to form a long tail: most of the distribution (∼ 85%)
has shapes that are close to spherical (a/b ≤ 1.5), yet (2) there is also a
significant fraction (∼ 12%) that has highly aspherical shapes (a/b ≥ 1.7).
3. Fitting the KBO a/b distribution with a gaussian yields the best-fit mean
µ = 1.00(+0.22) and standard deviation σ = 0.24+0.18
−0.13 (1σ error bars).
However, this gaussian is strongly skewed toward small axis ratios (a/b ≤
1.5), and offers a bad fit for larger axis ratios. Increasing the standard
deviation reduces the skewness, but then all data points are fitted equally
poorly.
4. The KBO a/b distribution is better fitted with power law distributions
of the form f (a/b) ∼ (a/b)−q , with the best-fit exponent q = 6.4 ±
1.4 (1σ error bars).
Acknowledgments
We thank Ronnie Hoogerwerf and Dave Jewitt for helpful discussion; we especially appreciate RH’s expertise with LATEX and SM. PL also thanks Scott
Kenyon for a very fruitful and enjoyable visit to the Center for Astrophysics.
46
The shape distribution
References
Brown, R. H., Cruikshank, D. P., & Pendleton, Y. 1999, ApJ, 519, L101
Collander-Brown, S. J., Fitzsimmons, A., Fletcher, E., Irwin, M. J., & Williams, I. P.
1999, MNRAS, 308, 588
Davies, J. K., McBride, N., & Green, S. F. 1997, Icarus, 125, 61
Green, S. F., McBride, N., O Ceallaigh, D. P., Fitzsimmons, A., Williams, I. P., &
Irwin, M. J. 1997, MNRAS, 290, 186
Hainaut, O. R., et al. 2000, A&A, 356, 1076
Jewitt, D. C. & Luu, J. X. 2001, AJ, 122, 2099
Jewitt, D. C. & Sheppard, S. S. 2002, AJ, 123, 2110
Lacerda, P. & Luu, J. 2003, Icarus, 161, 174
Luu, J. & Jewitt, D. 1996, AJ, 112, 2310
Luu, J. X. & Jewitt, D. C. 1998, ApJ, 494, L117
Ortiz, J. L., Lopez-Moreno, J. J., Gutierrez, P. J., & Baumont, S. 2001, Bulletin of the
American Astronomical Society, 33, 1047
Romanishin, W. & Tegler, S. C. 1999, Nature, 398, 129
Romanishin, W., Tegler, S. C., Rettig, T. W., Consolmagno, G., & Botthof, B. 2001,
Bulletin of the American Astronomical Society, 33, 1031
Sheppard, S. S. & Jewitt, D. C. 2002, AJ, 124, 1757
Sheppard, S. S. & Jewitt, D. C. 2003, Earth Moon and Planets, 92, 207
Tegler, S. C. & Romanishin, W. 2000, Nature, 407, 979
CHAPTER
4
Analysis of
the rotational properties
ABSTRACT
We use optical data of 10 Kuiper Belt objects (KBOs) to investigate their
rotational properties. Of the 10, three (30%) exhibit light variations with
amplitude ∆m ≥ 0.15 mag, and 1 out of 10 (10%) has ∆m ≥ 0.40 mag,
which is in good agreement with previous surveys. These data, in combination with the existing database, are used to discuss the rotational
periods, shapes, and densities of Kuiper Belt objects. We find that, in the
sampled size range, Kuiper Belt objects have a higher fraction of low amplitude lightcurves and rotate slower than main belt asteroids. The data
also show that the rotational properties and the shapes of KBOs depend
on size. If we split the database of KBO rotational properties into two
size ranges with diameter larger and smaller than 400 km, we find that:
(1) the mean lightcurve amplitudes of the two groups are different with
98.5% confidence, (2) the corresponding power-law shape distributions are
different, and (3) the two groups occupy different regions on a spin period
vs. lightcurve amplitude diagram. These differences are interpreted in the
context of KBO collisional evolution.
Pedro Lacerda and Jane Luu
to be submitted to Icarus
48
Analysis of the rotational properties
4.1
Introduction
T
he Kuiper Belt (KB) is an assembly of small icy objects, orbiting the Sun
beyond Neptune. Kuiper Belt objects (KBOs) are likely to be remnants of
outer solar system planetesimals (Jewitt & Luu 1993). Their physical, chemical, and dynamical properties should therefore provide valuable information regarding both the environment and the physical processes responsible for planet
formation.
At the time of writing, 763 KBOs are known, with 363 of them having been
followed for more than one opposition. A total of ≈ 105 objects larger than
50 km are thought to orbit the Sun beyond Neptune (Jewitt & Luu 2000). Studies of KB orbits has revealed an intricate dynamical structure, with signatures
of interactions with Neptune (Malhotra 1995). The size distribution follows a
differential power-law of index q = 4 for bodies & 50 km (Trujillo et al. 2001a),
becoming slightly shallower at smaller sizes (Bernstein et al. 2004).
KBO colours show a large diversity, from slightly blue to very red (Luu & Jewitt 1996; Tegler & Romanishin 2000; Jewitt & Luu 2001), and seem to correlate
with inclination and perihelion distance (e.g., Jewitt & Luu 2001; Doressoundiram et al. 2002; Trujillo & Brown 2002). The few low-resolution optical and
near-IR KBO spectra are mostly featureless, with the exception of a weak 2 µm
water ice absorption line present in some of them (Brown et al. 1999; Jewitt &
Luu 2001).
About 4% of known KBOs are binaries with separations larger than 0.00 15
(Noll et al. 2002). All the observed binaries have primary-to-secondary mass
ratios ≈ 1. Two binary creation models have been proposed. Weidenschilling
(2002) favours the idea that binaries form in three-body encounters. This model
requires a 100 times denser Kuiper Belt at the epoch of binary formation, and
predicts a higher abundance of large separation binaries. The alternative scenario (Goldreich et al. 2002), in which the energy needed to bind the orbits of
two approaching bodies is drawn from the surrounding swarm of smaller objects,
also requires a much higher density of KBOs than the present, but it predicts
a larger fraction of close binaries. Recently, Sheppard & Jewitt (2004) have
shown evidence that 2001 QG298 could be a close or contact binary KBO, and
estimated the fraction of similar objects in the Belt to be ∼ 10%–20%.
Other physical properties of KBOs, such as their shapes, densities, and albedos, are still poorly constrained. This is mainly because KBOs are extremely
faint, with mean apparent red magnitude mR ∼23 (Trujillo et al. 2001b).
The study of KBO rotational properties through time-series broadband optical photometry has proved to be the most successful technique to date to
investigate some of these physical properties. Light variations of KBOs are believed to be caused mainly by their aspherical shape: as KBOs rotate in space,
their projected cross-sections change, resulting in periodic brightness variations.
The Shapes and Spins of KBOs
49
One of the best examples to date of a KBO lightcurve—and what can be
learned from it—is that of (20000) Varuna (Jewitt & Sheppard 2002). The authors explain the lightcurve of Varuna as a consequence of its elongated shape
(axes ratio, a/b ∼ 1.5). They further argue that the object is centripetally deformed by rotation because of its low density, “rubble pile” structure. The term
“rubble pile” is generally used to refer to gravitationally bound aggregates of
smaller fragments. The existence of rubble piles is thought to be due to continuing mutual collisions throughout the age of the solar system, which gradually
fracture the interiors of objects. Rotating rubble piles can adjust their shapes
to balance centripetal acceleration and self-gravity. The resulting equilibrium
shapes have been studied in the extreme case of fluid bodies, and depend on the
body’s density and spin rate (Chandrasekhar 1969).
Lacerda & Luu (2003, hereafter LL03a) showed that under reasonable assumptions the fraction of KBOs with detectable lightcurves can be used to constrain the shape distribution of these objects. A follow-up (Luu & Lacerda 2003,
hereafter LL03b) on this work, using a database of lightcurve properties of 33
KBOs (Sheppard & Jewitt 2002, 2003), shows that although most Kuiper Belt
objects (∼ 85%) have shapes that are close to spherical (a/b ≤ 1.5) there is a
significant fraction (∼ 12%) with highly aspherical shapes (a/b ≥ 1.7).
In this paper we use optical data of 10 KBOs to investigate the amplitudes
and periods of their lightcurves. These data are used in combination with the existing database to investigate the distributions of KBO spin periods and shapes.
We discuss their implications for the inner structure and collisional evolution of
objects in the Kuiper Belt.
4.2
Observations and Photometry
We collected time-series optical data of 10 KBOs at the Isaac Newton 2.5m
(INT) and William Herschel 4m (WHT) telescopes. The INT Wide Field Camera (WFC) is a mosaic of 4 EEV 2048×4096 CCDs, each with a pixel scale of
0.00 33/pixel and spanning approximately 11.0 3×22.0 5 in the plane of the sky. The
targets are imaged through a Johnson R filter. The WHT prime focus camera
consists of 2 EEV 2048×4096 CCDs with a pixel scale of 0.00 24/pixel, and covers
a sky-projected area of 2×8.0 2×16.0 4. With this camera we used a Harris R filter.
The seeing for the whole set of observations ranged from 1.0 to 1.900 FWHM.
We tracked both telescopes at sidereal rate and kept integration times for each
object sufficiently short to avoid errors in the photometry due to trailing effects
(see Table 4.1). No light travel time corrections have been made.
We reduced the data using standard techniques. The sky background in the
flat-fielded images shows variations of less than 1% across the chip. Background
variations between consecutive nights were less than 5% for most of the data.
Cosmic rays were removed with the package LA-Cosmic (van Dokkum 2001).
Analysis of the rotational properties
Table 4.1 – Observing Conditions and Geometry
Object Designation
ObsDatea
Tel.b
Seeingc
[00 ]
MvtRtd
[00 /hr]
ITimee
[sec]
(19308) 1996 TO66
1996 TS66
1996 TS66
1996 TS66
(35671) 1998 SN165
(35671) 1998 SN165
(19521) Chaos
(19521) Chaos
1999 DF9
1999 DF9
1999 DF9
2000 CM105
2000 CM105
2000 CM105
1999 RZ253
1999 RZ253
1999 RZ253
(47171) 1999 TC36
(47171) 1999 TC36
(47171) 1999 TC36
(38628) Huya
(38628) Huya
(38628) Huya
2001 CZ31
2001 CZ31
01-Oct-99
30-Sep-99
01-Oct-99
02-Oct-99
29-Sep-99
30-Sep-99
01-Oct-99
02-Oct-99
13-Feb-01
14-Feb-01
15-Feb-01
11-Feb-01
13-Feb-01
14-Feb-01
11-Sep-01
12-Sep-01
13-Sep-01
11-Sep-01
12-Sep-01
13-Sep-01
28-Feb-01
01-Mar-01
03-Mar-01
01-Mar-01
03-Mar-01
WHT
WHT
WHT
WHT
WHT
WHT
WHT
WHT
WHT
WHT
WHT
WHT
WHT
WHT
INT
INT
INT
INT
INT
INT
INT
INT
INT
INT
INT
1.8
1.3
1.1
1.5
1.5
1.4
1.0
1.5
1.7
1.6
1.4
1.5
1.4
1.5
1.9
1.4
1.8
1.9
1.4
1.8
1.5
1.8
1.5
1.3
1.4
2.89
2.62
2.67
2.70
3.24
3.22
1.75
1.79
3.19
3.21
3.22
3.14
3.12
3.11
2.86
2.84
2.82
3.85
3.86
3.88
2.91
2.97
3.08
2.72
2.65
500
400,600
600
600,900
360,400
360
360,400,600
400,600
900
900
900
600,900
900
900
600
600
600
600
900
900
600
360
360
600,900
600,900
a
23
02
02
02
23
23
03
03
10
10
10
09
09
09
22
22
22
00
00
00
13
13
13
09
08
59
26
26
25
32
32
44
44
27
26
26
18
18
18
02
02
02
16
16
16
31
31
31
00
59
46
06
02
58
46
41
37
34
04
50
46
48
39
34
57
53
49
49
45
39
13
09
01
03
54
decg
[◦000 ]
+03
+21
+21
+21
−01
−01
+21
+21
+09
+09
+09
+19
+19
+19
−12
−12
−12
−07
−07
−07
−00
−00
−00
+16
+16
36
41
40
40
18
18
30
30
45
46
46
41
42
43
31
31
31
34
35
36
39
38
36
29
30
42
03
50
35
15
47
58
54
16
25
50
59
40
02
06
26
49
59
33
13
04
23
59
23
04
Rh
[AU]
∆i
[AU]
αj
[deg]
45.950
38.778
38.778
38.778
38.202
38.202
42.399
42.399
39.782
39.783
39.783
41.753
41.753
41.753
40.963
40.963
40.963
31.416
31.416
31.416
29.769
29.768
29.767
41.394
41.394
44.958
37.957
37.948
37.939
37.226
37.230
41.766
41.755
38.818
38.808
38.806
40.774
40.778
40.781
40.021
40.026
40.033
30.440
30.437
30.434
29.021
29.009
28.987
40.522
40.539
0.1594
0.8619
0.8436
0.8225
0.3341
0.3594
1.0616
1.0484
0.3124
0.2436
0.2183
0.1687
0.2084
0.2303
0.4959
0.5156
0.5381
0.4605
0.4359
0.4122
1.2725
1.2479
1.1976
0.6525
0.6954
UT date of observation; b Telescope used for observations; c Average seeing of the data [00 ]; d Average rate of motion of KBO [00 /hr]; e
Integration times used; f Right ascension; g Declination; h KBO–Sun distance; i KBO–Earth distance; j Phase angle (Sun–Object–Earth
angle) of observation.
50
RAf
[hhmmss]
The Shapes and Spins of KBOs
Object Designation
Classa
Hb
[mag]
ic
[deg]
ed
ae
[AU]
(19308) 1996 TO66
1996 TS66
(35671) 1998 SN165
(19521) Chaos
1999 DF9
2000 CM105
1999 RZ253
(47171) 1999 TC36
(38628) Huya
2001 CZ31
C
C
C1
C
C
C
C
Pb
P
C
4.5
6.4
5.8
4.9
6.1
6.2
5.9
4.9
4.7
5.4
27.50
7.30
4.60
12.00
9.80
3.80
0.60
8.40
15.50
10.20
0.12
0.13
0.05
0.11
0.15
0.07
0.09
0.22
0.28
0.12
43.20
44.00
37.80
45.90
46.80
42.50
43.60
39.30
39.50
45.60
51
Table 4.2 – Properties of Observed KBOs. a Dynamical class (C = classical KBO, P =
Plutino, b = binary KBO); b Absolute magnitude; c Orbital inclination; d Orbital eccentricity;
e Semi-major axis.
We performed aperture photometry on all objects in the field using the SExtractor software package (Bertin & Arnouts 1996). This software performs circular aperture measurements on each object in a frame, and puts out a catalog
of both the magnitudes and the associated errors. Below we describe how we
obtained a better estimate of the errors. We used apertures ranging from 1.5 to
2.0 times the FWHM for each frame and selected the aperture that maximized
signal-to-noise. An extra aperture of 5 FWHMs was used to look for possible
seeing dependent trends in our photometry. The catalogs were matched by selecting only the objects that are present in all frames. The slow movement of
KBOs from night to night allows us to successfully match a large number of
stars in consecutive nights. We discarded all saturated objects as well as those
identified to be galaxies.
The KBO lightcurves were obtained from differential photometry with respect to the brightest non-variable field stars. An average of the magnitudes
of the brightest stars (the ”reference”stars) provides a reference for differential
photometry in each frame. This method allows for small amplitude brightness
variations to be detected even under non-photometric conditions.
The uncertainty in the relative photometry was calculated from the scatter in
the photometry of fainter field stars that are similar to the KBOs in brightness
(the ”comparison”stars, see Fig.4.1). This error estimate is more robust than the
errors provided by SExtractor (see below), and was used to verify the accuracy
of the latter. This procedure resulted in consistent time series brightness data
for ∼ 100 objects (KBO + field stars) in a time span of 2–3 consecutive nights.
We observed Landolt standard stars whenever conditions were photometric,
and used them to calibrate the zero point of the magnitude scale. The extinction
coefficient was obtained from the reference stars.
52
Analysis of the rotational properties
Variance H´10-3 L
5.
Figure 4.1 – Frame-to-frame
photometric variances of all
stars (gray circles and black
crosses) in the 1998 SN165 (a)
and Huya (b) fields, plotted
against their relative magnitude. The trend of increasing
photometric variability with increasing magnitude is clear.
The intrinsically variable stars
clearly do not follow this trend,
and are located towards the
upper left region of the plot.
The KBOs are shown as black
squares.
1998 SN165 , in the
top panel shows a much larger
variability than the comparison
stars (shown as crosses, see Section 4.3.1), while Huya is well
within the expected variance
range, given its magnitude.
aL
4.
3.
2.
1.
0.
Variance H´10-3 L
5.
bL
4.
3.
2.
1.
0.
0.
-1.
1.
2.
3.
Relative magnitude
UT Date
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
1.84831
1.85590
1.86352
1.87201
1.87867
1.88532
1.89302
1.90034
1.90730
1.91470
mR
[mag]
Julian Date
2451453.34831
2451453.35590
2451453.36352
2451453.37201
2451453.37867
2451453.38532
2451453.39302
2451453.40034
2451453.40730
2451453.41470
21.24
21.30
21.20
21.22
21.21
21.28
21.27
21.30
21.28
21.31
±
±
±
±
±
±
±
±
±
±
0.07
0.07
0.07
0.07
0.07
0.07
0.06
0.06
0.06
0.06
Table 4.3 – Photometric measurements of 1996 TO66 . Columns are UT
date at the start of the exposure, Julian date at the start of the exposure,
and apparent red magnitude. The errors include the uncertainties in relative and absolute photometry added
quadratically.
Since not all nights were photometric the lightcurves are presented as variations with respect to the mean brightness. These yield the correct amplitudes
and periods of the lightcurves but do not provide their absolute magnitudes.
The orbital parameters and other properties of the observed KBOs are given
in Table 4.2. Tables 4.3, 4.4, 4.5, and 4.6 list the absolute R-magnitude photometric measurements obtained for 1996 TO66 , 1996 TS66 , 1998 SN165 , and Chaos,
respectively.
4.3
Lightcurve Analysis
The results in this paper depend solely on the amplitude and period of the KBO
lightcurves. It is therefore important to accurately determine these parameters
and the associated uncertainties.
The Shapes and Spins of KBOs
Table 4.4 – Photometric measurements of 1996 TS66 . Columns are UT
date at the start of the exposure, Julian date at the start of the exposure,
and apparent red magnitude. The errors include the uncertainties in relative and absolute photometry added
quadratically.
4.3.1
UT Date
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
30.06087
30.06628
30.07979
30.08529
30.09068
30.09695
30.01250
30.10936
30.11705
30.12486
30.13798
30.14722
30.15524
30.16834
30.17680
30.18548
30.19429
30.20212
30.21010
30.21806
30.23528
30.24355
01.02002
01.02799
01.03648
01.04422
01.93113
01.94168
01.95331
01.97903
01.99177
02.00393
02.01588
02.02734
Julian Date
2451451.56087
2451451.56628
2451451.57979
2451451.58529
2451451.59068
2451451.59695
2451451.60250
2451451.60936
2451451.61705
2451451.62486
2451451.63798
2451451.64722
2451451.65524
2451451.66834
2451451.67680
2451451.68548
2451451.69429
2451451.70212
2451451.71010
2451451.71806
2451451.73528
2451451.74355
2451452.52002
2451452.52799
2451452.53648
2451452.54422
2451453.43113
2451453.44168
2451453.45331
2451453.47903
2451453.49177
2451453.50393
2451453.51588
2451453.52734
53
mR
[mag]
21.94
21.83
21.76
21.71
21.75
21.67
21.77
21.76
21.80
21.77
21.82
21.74
21.72
21.72
21.83
21.80
21.74
21.78
21.72
21.76
21.73
21.74
21.81
21.82
21.81
21.80
21.71
21.68
21.73
21.69
21.74
21.73
21.78
21.71
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.06
0.06
0.06
0.07
0.06
0.06
0.08
0.07
0.06
0.07
0.07
0.07
0.09
0.07
0.08
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.05
0.05
0.05
Can we detect the KBO brightness variation?
We begin by investigating if the observed brightness variations are intrinsic to
the KBO, i.e., if the KBOs intrinsic brightness variations are detectable given
our uncertainties. This was done by comparing the frame-to-frame scatter in
the KBO optical data with that of (∼ 10 − 20) comparison stars.
To visually compare the scatter in the magnitudes of the reference stars (see
Section 4.2), comparison stars, and KBOs, we plot a histogram of their frameto-frame variances (see Fig. 4.2). In general such a histogram should show the
reference stars clustered at the lowest variances, followed by the comparison stars
spread over larger variances. If the KBO appears isolated at much higher variances than both groups of stars (e.g., Fig. 4.2-j), then its brightness modulations
are significant. Conversely, if the KBO is clustered with the stars (e.g. Fig. 4.2f), any periodic brightness variations would be below the detection threshold.
Figure 4.1 shows the dependence of the uncertainties on magnitude. Objects
that do not fall on the rising curve traced out by the stars must have intrin-
54
Analysis of the rotational properties
UT Date
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Sep
Oct
Oct
29.87384
29.88050
29.88845
29.89811
29.90496
29.91060
29.91608
29.92439
29.93055
29.93712
29.94283
29.94821
29.96009
29.96640
29.97313
29.97850
29.98373
29.98897
29.99469
29.99997
30.00521
30.01144
30.02164
30.02692
30.84539
30.85033
30.85531
30.86029
30.86550
30.87098
30.87627
30.89202
30.89698
30.90608
30.91191
30.92029
30.92601
30.93110
30.93627
30.94858
30.95363
30.95852
30.96347
30.96850
30.97422
30.98431
30.98923
30.99444
30.99934
01.00424
01.00992
Julian Date
2451451.37384
2451451.38050
2451451.38845
2451451.39811
2451451.40496
2451451.41060
2451451.41608
2451451.42439
2451451.43055
2451451.43712
2451451.44283
2451451.44821
2451451.46009
2451451.46640
2451451.47313
2451451.47850
2451451.48373
2451451.48897
2451451.49469
2451451.49997
2451451.50521
2451451.51144
2451451.52164
2451451.52692
2451452.34539
2451452.35033
2451452.35531
2451452.36029
2451452.36550
2451452.37098
2451452.37627
2451452.39202
2451452.39698
2451452.40608
2451452.41191
2451452.42029
2451452.42601
2451452.43110
2451452.43627
2451452.44858
2451452.45363
2451452.45852
2451452.46347
2451452.46850
2451452.47422
2451452.48431
2451452.48923
2451452.49444
2451452.49934
2451452.50424
2451452.50992
mR
[mag]
21.20
21.19
21.18
21.17
21.21
21.24
21.18
21.25
21.24
21.26
21.25
21.28
21.25
21.21
21.17
21.14
21.12
21.15
21.15
21.16
21.12
21.09
21.18
21.17
21.32
21.30
21.28
21.31
21.21
21.26
21.28
21.23
21.30
21.20
21.26
21.15
21.19
21.14
21.16
21.18
21.16
21.13
21.17
21.16
21.18
21.18
21.17
21.16
21.20
21.16
21.18
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
0.06
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.06
0.06
0.06
0.06
0.06
0.05
0.05
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.05
0.06
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.06
0.05
0.05
0.05
0.06
Table 4.5 – Photometric measurements of 1998 SN165 . Columns are
UT date at the start of the exposure,
Julian date at the start of the exposure, and apparent red magnitude.
The errors include the uncertainties
in relative and absolute photometry
added quadratically.
The Shapes and Spins of KBOs
10.
10.
aL 1996 TO66
8.
8.
6.
6.
4.
4.
2.
2.
0.
55
bL 1996 TS66
0.
0.
0.002
0.004
Variance
12.
0.006
0.008
0.
0.002
0.004
Variance
10.
cL 1998 SN165
10.
8.
8.
6.
6.
0.006
0.008
dL Chaos
4.
4.
2.
2.
0.
0.
0.
0.001
0.002
Variance
10.
0.003
0.
0.004
0.0005 0.001 0.0015 0.002 0.0025 0.003
Variance
12.
eL 1999 DF9
fL 1999 RZ253
10.
8.
8.
6.
6.
4.
4.
2.
2.
0.
0.
0.
0.005
0.01
Variance
10.
0.015
0.
0.02
gL 1999 TC36
0.001 0.002 0.003 0.004 0.005 0.006
Variance
hL 2000 CM105
20.
8.
15.
6.
4.
10.
2.
5.
0.
0.
0.
0.0001
0.0002
Variance
10.
0.0003
0.0004
iL Huya
0.
0.001 0.002 0.003 0.004 0.005 0.006
Variance
15.
8.
12.5
6.
10.
jL 2001 CZ31
7.5
4.
5.
2.
2.5
0.
0.
0.
0.0001
0.0002
Variance
0.0003
0.0004
0.
0.002
0.004
Variance
0.006
0.008
Figure 4.2 – Stacked histograms of the frame-to-frame variance in the optical data of the
“reference” stars (in white), “comparison” stars (in gray), and the KBO (in black). In c), e),
and j) the KBO shows significantly more variability than the comparison stars, whereas in all
other cases it falls well within the range of photometric uncertainties of the stars of similar
brightness.
56
Analysis of the rotational properties
UT Date
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
1999
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
Oct
01.06329
01.06831
01.07324
01.07817
01.08311
01.08801
01.09370
01.14333
01.15073
01.15755
01.16543
01.17316
01.18112
01.18882
01.19652
01.20436
01.21326
01.21865
01.22402
01.22938
01.23478
01.24022
02.04310
02.04942
02.07568
02.08266
02.09188
02.10484
02.11386
02.12215
02.13063
02.13982
02.14929
Julian Date
2451452.56329
2451452.56831
2451452.57324
2451452.57817
2451452.58311
2451452.58801
2451452.59370
2451452.64333
2451452.65073
2451452.65755
2451452.66543
2451452.67316
2451452.68112
2451452.68882
2451452.69652
2451452.70436
2451452.71326
2451452.71865
2451452.72402
2451452.72938
2451452.73478
2451452.74022
2451453.54310
2451453.54942
2451453.57568
2451453.58266
2451453.59188
2451453.60484
2451453.61386
2451453.62215
2451453.63063
2451453.63982
2451453.64929
mR
[mag]
20.82
20.80
20.80
20.81
20.80
20.76
20.77
20.71
20.68
20.70
20.72
20.72
20.71
20.73
20.70
20.69
20.72
20.72
20.74
20.72
20.71
20.72
20.68
20.69
20.74
20.73
20.74
20.75
20.77
20.77
20.78
20.79
20.71
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.07
0.07
0.06
0.06
0.07
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.07
Table 4.6 –
Photometric measurements of Chaos. Columns are
UT date at the start of the exposure,
Julian date at the start of the exposure, and apparent red magnitude.
The errors include the uncertainties
in relative and absolute photometry
added quadratically.
sic brightness variations. By calculating the mean and spread of the variance
for the comparison stars (shown as crosses) we can calculate our photometric
uncertainties and thus determine whether the KBO brightness variations are
significant.
4.3.2
Period determination
In the cases where significant brightness variations were detected in the lightcurves, the phase dispersion minimization method was used (PDM, Stellingwerf
1978) to look for periodicities in the data. For each test period PDM computes
a statistical parameter θ that compares the spread of data points in phase bins
with the overall spread of the data. The period that best fits the data is the one
that minimizes θ. The advantages of PDM are that it is non-parametric, i.e., it
does not assume a shape for the lightcurve, and it can handle unevenly spaced
data. Each data set was tested for periods ranging from 2 to 24 hours, in steps
of 0.01 hr.
The Shapes and Spins of KBOs
57
Table 4.7 – Lightcurve Properties of Observed KBOs. a Mean red magnitude. Errors include
uncertainties in relative and absolute photometry added quadratically; b Number of nights with
useful data. Numbers in brackets indicate number of nights of data from other observers used
for period determination. Data for 1998 SN165 taken from Peixinho et al. (2002) and data for
2001 CZ31 taken from SJ02; c Lightcurve amplitude; d Lightcurve period.
mR a
[mag]
Ntsb
∆mR c
[mag]
Pd
[hr]
(35671) 1998 SN165
1999 DF9
2001 CZ31
21.20±0.05
–
2(1)
3
2(1)
0.16±0.01
0.40±0.02
0.21±0.02
8.84
6.65
4.71
(19308) 1996 TO66
1996 TS66
(19521) Chaos
2000 CM105
1999 RZ253
(47171) 1999 TC36
(38628) Huya
21.26±0.06
21.76±0.05
20.74±0.06
–
–
–
–
1
3
2
2
3
3
2
?
<0.15
<0.10
<0.14
<0.05
<0.07
<0.04
Object Designation
4.3.3
Amplitude determination
We used a Monte Carlo experiment to determine the amplitude of the lightcurves
for which a period was found. We generated several artificial data sets by randomizing each point within the error bar. Each artificial data set was fitted
with a Fourier series, using the best-fit period, and the mode and central 68%
of the distribution of amplitudes were taken as the lightcurve amplitude and 1σ
uncertainty, respectively.
For the ”null”lightcurves, i.e. those where no significant variation was detected, we subtracted the typical error bar size from the total amplitude of the
data to obtain an upper limit to the amplitude of the KBO photometric variation.
In this section we present the results of the lightcurve analysis for each of the
observed KBOs. We found significant brightness variations (∆m > 0.15 mag)
for 3 out of 10 KBOs (30%), and ∆m ≥ 0.40 mag for 1 out of 10 (10%). This
is consistent with previously published results: SJ02 found a fraction of 31%
with ∆m > 0.15 mag and 23% with ∆m > 0.40 mag, both consistent with our
results. The other 7 KBOs do not show detectable variations. The results are
summarized in Table 4.7.
58
Analysis of the rotational properties
4.4
Results
4.4.1
1998 SN165
The brightness of 1998 SN165 varies significantly (> 5σ) more than that of the
comparison stars (see Figs. 4.1 and 4.2-c). The periodogram for this KBO shows
a very broad minimum around P = 9 hr (Fig. 4.3a). The degeneracy implied
by the broad minimum would only be resolved with additional data. A slight
weaker minimum is seen at P = 6.5 hr, which is close to a 24hr alias of 9 hr.
Peixinho et al. (2002, hereafter PDR02) observed this object in September
2000, but having only one night’s worth of data, they did not succeed in determining this object’s rotational period unambiguously. We used their data to
solve the degeneracy in our PDM result. The PDR02 data have not been absolutely calibrated, and the magnitudes are given relative to a bright field star.
To be able to combine it with our own data we had to subtract the mean magnitudes. Our periodogram of 1998 SN165 (centered on the broad minimum) is
shown in Fig. 4.3b and can be compared with the revised periodogram obtained
with our data combined with the PDR02 data (Fig. 4.3c). The minima become
much clearer with the additional data, but because of the 1-year time difference
between the two observational campaigns, many close aliases appear in the periodogram. The absolute minimum, at P = 8.84 hr, corresponds to a double
peaked lightcurve (see Fig. 4.4). The second best fit, P = 8.7 hr, produces a
more scattered phase plot, in which the peak in the PDR02 data coincides with
our night 2. We will use P = 8.84 hr in the rest of the paper, as this corresponds
to the PDM absolute minimum.
The amplitude, obtained using the Monte Carlo method described in Section 4.3.3, is ∆m = 0.16 ± 0.01 mag. This value was calculated using only our
data, but it did not change when recalculated adding the PDR02 data.
4.4.2
1999 DF9
1999 DF9 shows large amplitude photometric variations (∆mR ∼ 0.4 mag). The
PDM periodogram for 1999 DF9 is shown in Fig. 4.5. The best-fit period is P =
6.65 hr, which corresponds to a double-peak lightcurve (Fig. 4.6). Other PDM
minima are found close to P/2 ≈ 3.3 hr and 9.2 hr, a 24 hr alias of the best period.
Phasing the data with P/2 results in a worse fit because the two minima of the
double peaked lightcurve exhibit significantly different morphologies (Fig. 4.6);
the peculiar shape of the brighter minimum, which is reproduced on two different
nights, may be caused by an effect other than shape, e.g., a darker (lower albedo)
region on the KBO’s surface.
The amplitude of the lightcurve, estimated as described in Section 4.3.3, is
∆mR = 0.4 ± 0.02 mag.
The Shapes and Spins of KBOs
1.
59
aL
Theta
0.8
0.6
0.4
0.2
4.
1.
6.
bL
8.
10.
Period HhrL
12.
14.
cL
Theta
0.8
0.6
0.4
0.2
8.5
9.0
9.5
10.0
8.5
Period HhrL
9.0
9.5
10.0
Figure 4.3 – Periodogram for the data of 1998 SN165 . The lower left panel (b) shows an
enlarged section near the minimum calculated using only the data published in this paper,
and the lower right panel (c) shows the same region recalculated after adding the PDR02 data.
60
Analysis of the rotational properties
Relative mag.
-0.2
-0.1
0
0.1
0.2
0
0.2
0.4
0.6
0.8
1
Phase
Figure 4.4 – Lightcurve of 1998 SN165 . The figure represents the data phased with the best
fit period P = 8.84 hr. Different symbols correspond to different nights of observation. The
gray line is a 2nd order Fourier series fit to the data. The PDR02 data are shown as crosses.
4.4.3
2001 CZ31
This object shows substantial brightness variations (4.5σ above the comparison
stars) in a systematic manner. The first night of data seems to sample nearly one
complete rotational phase. As for 1998 SN165 , the 2001 CZ31 data span only two
nights of observations. In this case, however, the PDM minima (see Figs. 4.7a
and 4.7b) are very narrow and only two correspond to independent periods,
P = 4.69 hr (the minimum at 5.82 is a 24 hr alias of 4.69 hr), and P = 5.23 hr.
2001 CZ31 has also been observed by Sheppard & Jewitt (2002, hereafter
SJ02) in February and April 2001, with inconclusive results. We used their data
to try to rule out one (or both) of the two periods we found. We subtracted the
SJ02 mean magnitudes in order to combine it with our uncalibrated observations.
Figure 4.7c shows the section of the periodogram around P = 5 hr, recalculated
using SJ02’s first night plus our own data. The aliases are due to the 1 month
time difference between the two observing runs. The new PDM minimum is at
P = 4.71 hr—very close to the P = 4.69 hr determined from our data alone.
Visual inspection of the combined data set phased with P = 4.71 hr shows
a very good match between SJ02’s first night (2001 Feb 20) and our own data
(see Fig. 4.8). SJ02’s second and third nights show very large scatter and were
not included in our analysis. Phasing the data with P = 5.23 hr yields a more
scattered lightcurve, which confirms the PDM result. We will use P = 4.71 hr
throughout the rest of the paper.
We measured a lightcurve amplitude of ∆m = 0.21 ± 0.02 mag. If we use
both ours and SJ02’s first night data, ∆m rises to 0.22 mag.
The Shapes and Spins of KBOs
61
1
Theta
0.8
0.6
0.4
0.2
5
10
15
Period HhrL
20
Figure 4.5 – Periodogram for the 1999 DF9 data. The minimum corresponds to a lightcurve
period P = 6.65 hr.
Relative mag.
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0
0.2
0.4
0.6
0.8
1
Phase
Figure 4.6 – Same as Figure 4.4 for KBO 1999 DF9 . The best fit period is P = 6.65 hr. The
lines represent a 2nd order (solid line) and 5th (dashed line) order Fourier series fits to the
data. Normalized χ2 values of the fits are 2.8 and 1.3 respectively.
62
Analysis of the rotational properties
1.
Theta
0.8
0.6
0.4
0.2
aL
0.
1.
5.
15.
Period HhrL
10.
bL
20.
cL
Theta
0.8
0.6
0.4
0.2
4.6
4.8
5.0
5.2
4.6
Period HhrL
4.8
5.0
5.2
Figure 4.7 – Periodogram for the 2001 CZ31 data. The lower left panel (b) shows an enlarged
section near the minimum calculated using only the data published in this paper, and the lower
right panel (c) shows the same region recalculated after adding the SJ02 data.
The Shapes and Spins of KBOs
63
Relative mag.
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0
0.2
0.4
0.6
0.8
1
Phase
Figure 4.8 – Same as Figure 4.4 for KBO 2001 CZ31 . The data are phased with period
P = 4.71 hr. The points represented by crosses are taken from SJ02.
4.4.4
Flat Lightcurves
The fluctuations detected in the optical data of KBOs 1996 TO66 , 1996 TS66 ,
1999 TC36 , 1999 RZ253 , 2000 CM105 , and Huya, are well within the uncertainties.
Chaos shows some variations with respect to the comparison stars but no period
was found to fit all the data. See Table 4.7 and Fig. 4.9 for a summary of the
results.
4.4.5
Other lightcurve measurements
The KBO lightcurve database has increased considerably in the last few years,
largely due to the observational campaign of SJ02, with a recent updates in
Sheppard & Jewitt (2003, hereafter SJ03) and Sheppard & Jewitt (2004). These
authors have published observations and rotational data for a total of 30 KBOs
(their SJ02 paper includes data for 3 other previously published lightcurves
in the analysis). Other recently published KBO lightcurves include those for
(50000) Quaoar (Ortiz et al. 2003) and the scattered KBO (29981) 1999 TD 10
(Rousselot et al. 2003). Of the 10 KBO lightcurves presented in this paper, 6
are new to the database, bringing the total to 41.
Table 4.8 lists the rotational data of the 41 KBOs that will be analyzed in
the rest of the paper.
4.5
Analysis
In this section we examine the lightcurve properties of KBOs and compare them
with those of main-belt asteroids (MBAs). The lightcurve data for these two
families of objects cover different size ranges. MBAs, being closer to Earth,
can be observed down to much smaller sizes than KBOs; in general it is very
64
Analysis of the rotational properties
-0.3
-0.2
-0.1
0.
0.1
0.2
0.3
1996 TO66
1999Oct01
19.
-0.2
-0.1
0.
0.1
0.2
-0.2
-0.1
0.
0.1
0.2
-0.2
-0.1
0.
0.1
0.2
-0.2
-0.1
0.
0.1
0.2
-0.2
-0.1
0.
0.1
0.2
-0.2
-0.1
0.
0.1
0.2
23.
24.
-0.2
-0.1
0.
0.1
0.2
-0.2
-0.1
0.
0.1
0.2
1996 TS66
1999Oct01
Chaos
1999Oct01
Chaos
1999Oct02
1.
1996 TS66
1999Oct02
0.
2.
UT hours
4.
6.
-0.2
-0.1
0.
0.1
0.2
-0.2
-0.1
0.
0.1
0.2
1999 RZ253
2001Sep12
-0.2
-0.1
0.
0.1
0.2
1999 RZ253
2001Sep13
3.
4.
UT hours
5.
6.
1999 TC36
2001Sep12
1999 TC36
2001Sep13
1.
2.
3.
UT hours
4.
5.
-0.1
1999 RZ253
2001Sep14
-2.
2.
1999 TC36
2001Sep11
0.
-3.
-0.2
-0.1
0.
0.1
0.2
21.
22.
UT hours
1996 TS66
1999Sep30
-2.
-0.2
-0.1
0.
0.1
0.2
20.
0.
-1.
UT hours
0.
1.
0.1
Huya
2001Feb28
-0.1
0.
2000 CM105
2001Feb13
0.1
Huya
2001Mar01
-0.1
0.
2000 CM105
2001Feb14
0.
0.1
1.
2.
UT hours
3.
Huya
2001Mar02
2.
3.
4.
5.
UT hours
6.
7.
Figure 4.9 – The “flat” lightcurves are shown. The respective amplitudes are within the
photometric uncertainties.
The Shapes and Spins of KBOs
65
Table 4.8 – Database of KBOs Lightcurve Properties.
Object Designation
Classa
Sizeb
[km]
Periodc
[hr]
∆mR d
[mag]
Sourcee
0.04
<0.04
<0.10
0.06
0.12
<0.08
<0.15
<0.10
<0.10
<0.10
<0.07
<0.04
<0.10
<0.10
<0.06
<0.10
<0.10
<0.10
<0.10
<0.10
0.08±0.02
<0.10
<0.10
<0.14
<0.05
<0.14
RT99, DMcG97
SJ02
SJ02
RT99, LJ98
RT99, CB99
SJ02
SJ02
SJ02
LL, SJ02
SJ02
LL, SJ03
LL, SJ03
SJ03
SJ03
SJ02
SJ03
SJ03, O01
SJ03
SJ03
SJ03
SJ03
SJ03
SJ03
LL
LL
LL
0.60±0.04
0.19±0.05
0.26±0.03
0.45±0.03
0.68±0.04
0.18±0.04
0.61±0.03
0.42±0.03
0.14±0.03
1.14±0.04
0.17±0.02
0.53±0.03
0.16±0.01
0.40±0.02
0.21±0.06
SJ02, RT99
SJ03
SJ03, H00
SJ02, R01
SJ02, R01
SJ02
SJ02
SJ02
SJ03
SJ04
O03
Rou03
LL
LL
LL
KBOs considered to have ∆m < 0.15 mag
15789 1993 SC
15820 1994 TB
1996 GQ21
1996 TL66
15875 1996 TP66
1997 CS29
1998 HK151
33340 1998 VG44
19521 Chaos
1999 DE9
47171 1999 TC36
38628 Huya
2000 YW134
2001 FP185
2001 FZ173
2001 KD77
28978 Ixion
2001 QF298
42301 2001 UR163
42355 2002 CR46
55636 2002 TX300
55637 2002 UX25
55638 2002 VE95
2000 CM105
1999 RZ253
1996 TS66
P
P
S
S
P
C
P
P
C
S
Pb
P
S
S
S
P
P
P
S
S
C
C
P
C
Cb
C
240
220
730
480
250
630
170
280
600
700
600
740
790
400
430
430
1310
580
1020
210
1250
1090
500
330
380
300
16.24
KBOs considered to have ∆m ≥ 0.15 mag
32929
24835
19308
26308
33128
40314
47932
20000
1995 QY 9
1995 SM55
1996 TO66
1998 SM165
1998 BU48
1999 KR16
2000 GN171
Varuna
2003 AZ84
2001 QG298
50000 Quaoar
29981 1999 TD10
35671 1998 SN165
1999 DF9
2001 CZ31
a Dynamical
P
C
C
R
S
C
P
C
P
Pcb
C
S
C
C
C
180
630
720
400
210
400
360
980
900
240
1300
100
400
340
440
7.3
8.08
7.9
7.1
9.8
11.858
8.329
6.34
13.44
13.7744
17.6788
15.3833
8.84
6.65
4.71
class (C = classical KBO, P = Plutino, b = binary KBO).
in km assuming an albedo of 0.04, except for (20000) Varuna which has a known
albedo of 0.07.
c Period of the lightcurve in hours. For KBOs with both single and double peaked possible
lightcurves the double peaked period is listed.
d Lightcurve amplitude in magnitudes.
e DMcG97 = Davies et al. (1997), CB99 = Collander-Brown et al. (1999), H00 = Hainaut
et al. (2000), LJ98 = Luu & Jewitt (1998), O01 = Ortiz et al. (2001), O03 = Ortiz et al. (2003),
RT99 = Romanishin & Tegler (1999), R01 = Romanishin et al. (2001), Rou03 = Rousselot
et al. (2003), SJ02 = Sheppard & Jewitt (2002), SJ03 = Sheppard & Jewitt (2003), SJ04 =
Sheppard & Jewitt (2004), LL = this work.
b Diameter
66
Analysis of the rotational properties
KBOs
6.
4.
2.
0.
Asteroids
6.
4.
Figure 4.10
– Histograms of the spin
periods of KBOs (upper
panel) and main belt
asteroids (lower panel)
satisfying D > 200 km,
∆m
≥
0.15 mag,
P < 20 hr. The mean rotational periods of KBOs
and MBAs are 9.23 hr
and 6.48 hr, respectively.
The y-axis in both cases
indicates the number of
objects in each range of
spin periods.
2.
0.
4.
7.
10.
13.
Spin period HhrL
16.
19.
difficult to obtain good quality lightcurves for KBOs with diameters D < 50 km.
Furthermore, some KBOs surpass the 1000 km barrier—assuming a 0.04 albedo,
5 objects in the current database have diameters in excess of 1000 km—whereas
the largest asteroid, Ceres, does not reach 900 km. This will be taken into
account in the analysis.
The lightcurve data for asteroids was taken from the Harris Lightcurve Catalog2 , Version 5, while the diameter data were obtained from the Lowell Observatory database of asteroids orbital elements3 . The sizes of KBOs were calculated
from their absolute magnitude assuming an albedo of 0.04, except for Varuna
for which simultaneous thermal and optical observations have yielded a red geometric albedo of 0.07+0.30
−0.17 (Jewitt et al. 2001).
4.5.1
Spin period statistics
As Fig. 4.10 shows, the spin period distributions of KBOs and MBAs are significantly different. Because the sample of KBOs of small size or large periods is
poor, to avoid bias in our comparison we consider only KBOs and MBAs larger
than 200 km and with periods below 20 hr. In this range the mean rotational
periods of KBOs and MBAs are 9.23 hr and 6.48 hr, respectively, and the two
are different with a 98.5% confidence according to Student’s t-test. However,
the different means do not rule that the underlying distributions are the same,
and could simply mean than the two sets of data sample the same distribution
differently. This is not the case, however, according to the Kolmogorov-Smirnov
2 http://pdssbn.astro.umd.edu/sbnhtml/asteroids/colors
3 ftp://ftp.lowell.edu/pub/elgb/astorb.html
lc.html
The Shapes and Spins of KBOs
67
(K-S) test, which gives a probability that the periods of KBOs and MBAs are
drawn from the same parent distribution of 0.7%.
Although it is clear that KBOs spin slower than asteroids, it is not clear
why this is so. If collisions have contributed as significantly to the angular
momentum of KBOs as they did for MBAs (Farinella et al. 1982; Catullo et al.
1984), then the observed difference should be related to how these two families
react to collision events.
We will address the question of the collisional evolution of KBO spin rates
in Chapter 5.
4.5.2
Lightcurve amplitudes and the shapes of KBOs
The cumulative distribution of KBO lightcurve amplitudes is shown in Fig. 4.11.
It rises very steeply in the low amplitude range, and then becomes shallower
reaching large amplitudes. In quantitative terms, ∼ 70% of the KBOs possess
∆m < 0.15 mag, while ∼ 12% possess ∆m ≥ 0.40 mag, with the maximum value
being ∆m = 0.68 mag. [Note: Fig. 4.11 does not include the KBO 2001 QG298
which has a lightcurve amplitude ∆m = 1.14 ± 0.04 mag, and would further
extend the range of amplitudes. We do not include 2001 QG298 in our analysis because it is thought to be a contact binary (Sheppard & Jewitt 2004)].
Figure 4.11 also compares the KBO distribution with that of MBAs. The distributions of the two populations are clearly distinct: in the low amplitude range
(∆m < 0.15 mag) the KBO distribution is steeper than the MBA distribution
and extends to larger values of ∆m.
Figure 4.12 shows the lightcurve amplitude of KBOs and MBAs plotted
against size. KBOs with diameters larger than D = 400 km seem to have lower
lightcurve amplitudes than KBOs with diameters smaller than D = 400 km. Student’s t-test confirms that the mean amplitudes in each of these two size ranges
are different at the 98.5% confidence level. For MBAs the transition is less sharp
and seems to occur at a smaller size (D ∼ 200 km). In the case of asteroids,
the accepted explanation is that small bodies (D . 100 km) are fragments of
high-velocity impacts (Catullo et al. 1984), whereas their larger counterparts
(D > 200 km) are not. The lightcurve data of small KBOs are still too sparse
to permit a similar analysis.
The distribution of lightcurve amplitudes can be used to infer the shapes of
KBOs, if certain reasonable assumptions are made (see, e.g., LL03a). Generally,
objects with elongated shapes produce large brightness variations due to their
changing projected cross-section as they rotate. Conversely, round objects, or
those with the spin axis aligned with the line of sight, produce little or no
brightness variations, resulting in ”flat”lightcurves. Figure 4.12 shows that the
lightcurve amplitudes of KBOs with diameters smaller and larger than D =
400 km are significantly different. Does this mean that the shapes of KBOs
68
Analysis of the rotational properties
Cumulative fraction
1
0.8
0.6
0.4
0.2
0
KBOs
Asteroids
0
0.2
0.4
0.6
Lightcurve amplitude HmagL
0.8
Lightcurve amplitude HmagL
Figure 4.11 – Cumulative distribution of lightcurve amplitude for KBOs (circles) and asteroids (crosses) larger than 200 km. We plot only KBOs for which a period has been determined.
KBO 2001 QG298 , thought to be a contact binary (Sheppard & Jewitt 2004), is not plotted
although it may be considered an extreme case of elongation.
KBOs
Asteroids
1
0.8
0.6
0.4
0.2
0
200
400
600
800
1000
Diameter HkmL
1200
1400
Figure 4.12 – Lightcurve amplitudes of KBOs (black circles) and main belt asteroids (gray
crosses) plotted against object size. All sizes assume albedo of 0.04 except for Varuna, which
has a known albedo of 0.07.
The Shapes and Spins of KBOs
69
are also different in these two size ranges? To investigate this possibility of
a size dependence among KBO shapes we will consider KBOs with diameter
smaller and larger than 400 km separately. We shall loosely refer to objects with
diameter D > 400 km and D ≤ 400 km as larger and smaller KBOs, respectively.
We approximate the shapes of KBOs by triaxial ellipsoids with semi-axes
a > b > c. For simplicity we consider the case where b = c and use the axis ratio
ã = a/b to characterize the shape of an object. The orientation of the spin axis
is parameterized by the aspect angle θ, defined as the smallest angular distance
between the line of sight and the spin vector. On this basis the lightcurve
amplitude ∆m is related to ã and θ via the relation (Eqn. 2.2 with c̄ = 1)
s
2 ã2
.
(4.1)
∆m = 2.5 log
2
1 + ã + (ã2 − 1) cos(2 θ)
Following LL03b we model the shape distribution by a power-law of the form
f (ã) dã ∝ ã−q dã
(4.2)
where f (ã) dã represents the fraction of objects with shapes between ã and
ã + dã. We use the measured lightcurve amplitudes to estimate the value of q by
employing both the method described in LL03a, and by Monte Carlo fitting the
observed amplitude distribution (SJ02; LL03b). The latter consists of generating
artificial distributions of ∆m (Eqn. 4.1) with values of ã drawn from distributions
characterized by different q’s (Eqn. 4.2), and selecting the one that best fits the
observed cumulative amplitude distribution (Fig. 4.11). The values of θ are
generated assuming random spin axis orientations. We use the K-S test to
compare the different fits. The errors are derived by bootstrap resampling the
original data set (Efron 1979), and measuring the dispersion in the distribution
of best-fit power-law indexes, qi , found for each bootstrap replication.
Following the LL03a method we calculate the probability of finding a KBO
with ∆m ≥ 0.15 mag:
s
Z ãmax
ã2 − K
f (ã)
p(∆m ≥ 0.15) ≈ √
dã.
(4.3)
(ã2 − 1)K
K
where K = 100.8×0.15 , f (ã) = C ã−q , and C is a normalization constant. This
probability is calculated for a range of q’s to determine the one that best
matches the observed fraction of lightcurves with amplitude larger than 0.15 mag.
These fractions are f (∆m ≥ 0.15 mag; D ≤ 400 km) = 8/18, and f (∆m ≥
0.15 mag; D > 400 km) = 6/22, and f (∆m ≥ 0.15 mag) = 14/40 for the complete set of data. The results are summarized in Table 4.9 and shown in Fig. 4.13.
Both methods yield steeper shape distributions for larger KBOs, implying
more spherical shapes in this size range (Table 4.9). A distribution with q = 9.8
70
Analysis of the rotational properties
Cumulative fraction
1.2
1.
0.8
0.6
0.4
0.2
D < 400km
Cumulative fraction
0.
1.
0.8
0.6
0.4
0.2
D > 400km
All sizes
0.
0.
0.2
0.4 0.6 0.8 1.
Amplitude (mag)
0.
0.2
0.4 0.6 0.8 1.
Amplitude (mag)
Figure 4.13 – Observed cumulative lightcurve amplitude distributions of KBOs (black circles)
with diameter smaller than 400 km (upper left panel), larger than 400 km (lower left panel),
and of all the sample (lower right panel) are shown as black circles. Error bars are poissonian.
The best fit power-law shape distributions of the form f (ã) dã = ã −q dã were converted to
amplitude distributions using a Monte Carlo technique (see text for details), and are show as
solid lines. The best fit q’s are listed in Table 4.9. The gray crosses show the distributions for
asteroids in the same size ranges.
Method
Size range
LL03
D ≤ 400 km
q=
D > 400 km
q=
All sizes
q=
+2.0
4.0−1.5
+2.7
6.7−2.1
+1.4
5.8−1.3
MC
q = 4.5 ± 2.3
q = 9.8 ± 2.6
q = 6.1 ± 1.7
Table 4.9 – Best fit parameter to the
KBO shape distribution. Column 1 is
the range of KBO diameters, in km, considered in each case. Columns 2 and
3 are the results from the method described in LL03, and from a Monte Carlo
fit of the lightcurve amplitude distribution, respectively. See text for detailed
discussion.
predicts that ∼80% of the large KBOs have a/b < 1.2. For the smaller objects
we find a shallower distribution, q ∼ 4 − 4.5, which implies a significant fraction
of very elongated objects: ∼ 15% have a/b > 1.7. Although based on small
The Shapes and Spins of KBOs
71
numbers, the shape distribution of large KBOs is well fit by a simple power-law
(the K-S rejection probability is 0.3%). This is not the case for smaller KBOs
for which the fit is poorer (K-S rejection probability is 61%, see Fig. 4.13).
Our results are in agreement with previous studies of the overall KBO shape
distribution, which had already shown that a simple power-law does not explain
the shapes of KBOs as a whole (LL03b; SJ02).
The results presented in this section suggest that the shape distributions of
smaller and larger KBOs are different. However, given the size of our sample, we
cannot conclusively state that this difference is significant. Better observational
contraints, particularly of smaller KBOs, are necessary to understand the origin
of the KBO shape distribution. A comparison with asteroids suggests that the
collisional evolution may have played an important role. Collisions in the asteroid belt have left different marks in smaller and larger bodies. The shapes of
smaller asteroids (D ≤ 100 km) are consistent with collisional fragments (Catullo
et al. 1984), indicating that they are most likely by-products of disruptive collisions. Larger asteroids have in principle survived collisional destruction for the
age of the solar system, but may nonetheless have been converted to rubble piles
by repeated impacts. As a result of multiple collisions, the “loose” pieces of the
larger asteroids may have reassembled into shapes close to triaxial equilibrium
ellipsoids (Farinella et al. 1981).
4.5.3
The inner structure of KBOs
In this section we wish to investigate if the rotational properties of KBOs show
any evidence that they have a rubble pile structure; a possible dependence on
object size is also investigated.
Figure 4.14 plots the lightcurve amplitudes versus spin periods for the 15
KBOs whose lightcurve amplitudes and spin period are known. Open and filled
symbols indicate the KBOs with diameter smaller and larger than D = 400 km,
respectively. Clearly, the smaller and larger KBOs occupy different regions of
the diagram. For the larger KBOs (black filled circles) the (small) lightcurve
amplitudes are almost independent of the objects’ spin periods. In contrast,
smaller KBOs span a much broader range of lightcurve amplitudes. Two object
objects have very low amplitudes: 1998 SN165 and 1999 KR16 , which have diameters D ∼ 400 km and fall precisely on the boundary of the two size ranges. The
remaining objects hint at a trend of increasing ∆m with lower spin rates. The
one exception is 1999 TD10 , a Scattered Disk Object (e = 0.872, a = 95.703 AU)
that spends most of its orbit in rather empty regions of space and most likely
has a different collisional history.
For comparison, Fig. 4.14 also shows results of N-body simulations of collisions between “ideal” rubble piles (gray filled circles; Leinhardt et al. 2000), and
the lightcurve amplitude-spin period relation predicted by ellipsoidal figures of
72
Analysis of the rotational properties
hydrostatic equilibrium (dashed and dotted lines; Chandrasekhar 1969; Holsapple 2001). The latter is calculated from the equilibrium shapes that rotating
uniform fluid bodies assume by balancing gravitational and centrifugal acceleration. The spin rate-shape relation in the case of uniform fluids depends solely
on the density of the body. Although fluid bodies behave in many respects differently from rubble piles, they may, as an extreme case, provide insight on the
equilibrium shapes of gravitationally bound agglomerates.
The simulations of Leinhardt et al. (2000, hereafter LRQ2000) consist of
collisions between agglomerates of small spheres meant to simulate collisions between rubble piles. Each agglomerate consists of ∼ 1000 spheres, held together
by their mutual gravity, and has no initial spin. The spheres are indestructible,
have no sliding friction, and have coefficients of restitution of ∼ 0.8. The bulk
density of the agglomerates is 2000 kg m−3 . The authors collide two equal-size
rubble piles for a range of impact velocities and impact parameters. The impact
velocities range from ∼ zero at infinity to a few times the critical velocity for
which the impact energy would exceed the mutual gravitational binding energy
of the two rubble piles. The impact geometries range from head-on collisions
to grazing impacts. The mass, final spin period, and shape of the largest remnant resulting from each collision are registered and presented (see Table 1 of
LRQ2000). From their results, we selected the outcomes for which the mass of
the largest remnant is equal or larger than the mass of one of the colliding rubble
piles, i.e., we selected only accreting outcomes. The spin periods and lightcurve
amplitudes that would be generated by such remnants (assuming they are observed equator-on) are plotted in Fig. 4.14 as gray circles. Nine points, resulting
from head-on collisions, are not visible because the periods are larger than 19 hr,
up to ∼ 650 hr. Note that the simulated rubble piles have radii of 1 km—much
smaller than the sizes of the KBOs in our database. However, since the effects of
the collision scale with the ratio of impact energy to gravitational binding energy
of the colliding bodies (Benz & Asphaug 1999), the model results should apply
to other sizes. Clearly, the LRQ2000 model makes several specific assumptions,
and represents one possible idealization of what is usually referred to as “rubble
pile”. Nevertheless, the results are illustrative of how collisions may affect this
type of structures, and are useful for comparison with the KBO data.
The lightcurve amplitudes resulting from the LRQ2000 experiment are relatively small (∆m < 0.25 mag) for spin periods larger than P ∼ 5.5 hr (see
Fig. 4.14). Objects spinning faster than P = 5.5 hr have more elongated shapes,
resulting in larger lightcurve amplitudes, up to 0.65 magnitudes. The latter are
the result of collisions with higher angular momentum transfer than the former
(see Table 1 of LRQ2000). The maximum spin rate attained by the rubble piles,
as a result of the collision, is ∼ 4.5 hr. This is consistent with the maximum
spin expected for bodies in hydrostatic equilibrium with the same density as the
rubble piles (ρ = 2000 kg m−3 ; see long-dashed line in Fig. 4.14). The results of
LRQ2000 show that collisions between ideal rubble piles can produce elongated
The Shapes and Spins of KBOs
73
Lightcurve amplitude (mag)
2001QG 298
1.
0.8
1998BU48
0.6
2000GN171
1999TD10
Varuna
0.4
1999KR16
0.2
1998SN165
2001CZ31
0.
4.
6.
8.
10.
12.
14.
16.
18.
Spin period (hr)
Figure 4.14 – Lightcurve amplitudes versus spin periods of KBOs. The black filled and open
circles represent objects larger and smaller than 400 km, respectively. The smaller gray circles
show the results of numerical simulations of “rubble-pile” collisions (Leinhardt et al. 2000). The
lines represent the locus of rotating ellipsoidal figures of hydrostatic equilibrium with densities
ρ = 500 kg m−3 (dotted line), ρ = 1000 kg m−3 (shorter dashes) and ρ = 2000 kg m−3 (longer
dashes). The point towards the upper right corner of the plot corresponds to 2001 QG 298
(P = 13.77 hr, ∆m = 1.14 mag), a likely contact binary KBO (Sheppard & Jewitt 2004).
remnants (when the projectile brings significant angular momentum into the
target), and that the spin rates of the collisional remnants do not extend much
beyond the maximum spin permitted to fluid uniform bodies with the same bulk
density.
The distribution of KBOs in Fig. 4.14 is less clear. Indirect estimates of
KBO bulk densities indicate values ρ ∼ 1000 kg m−3 (Luu & Jewitt 2002). If
KBOs are strengthless rubble piles with such low densities we would not expect
to find objects with spin periods lower than P ∼ 6 hr (as indicated by the
dashed line in Fig. 4.14). However, one object (2001 CZ31 ), is found to have
a spin period below 5 hr. If this object has a rubble pile structure, its density
must be at least ∼ 2000 kg m−3 (see Fig. 4.14). The remaining 14 objects have
spin periods below the expected upper limit, given their estimated density. Of
the 14, 4 objects lie close to the line corresponding to equilibrium ellipsoids of
density ρ = 1000 kg m−3 . One of these objects, (20000) Varuna, has been studied
in detail by Sheppard & Jewitt (2002). The authors conclude that Varuna is
best interpreted as a rotationally deformed rubble pile with ρ ≤ 1000 kg m −3 .
One object, 2001 QG298 , has an exceptionally large lightcurve amplitude (∆m =
1.14 mag), indicative of a very elongated shape (axes ratio a/b > 2.85). Given
74
Analysis of the rotational properties
its modest spin rate (P = 13.8 hr) and approximate size (D ∼ 240 km) it is
unlikely that it would be able to keep such an elongated shape against the crush
of gravity. Analysis of the lightcurve of this object (Sheppard & Jewitt 2004)
suggests it is a contact (or very close) binary KBO. The same applies to two
other KBOs, 2000 GN171 and (33128) 1998 BU48 , also very likely to be contact
binaries.
To summarize, based on the available rotational properties of KBOs, it is
not clear if they have a rubble pile structure. Comparing the KBO data with
results from computer simulations of rubble pile collisions (LRQ2000) we find
that 11 out of 15 (∼ 73%) KBOs have rotational data that follows essentially the
same trends as the LRQ2000 results. Of the 11 KBOs, 5 (45%) have diameters
D ≤ 400 km, and 6 (55%) have diameters D > 400 km. If these KBOs are rubble
piles then their spin rates set a lower limit to their bulk density: one object
(2001 CZ31 ) spins fast enough that its density must be at least ρ ∼ 2000 kg m−3 ,
while 4 other KBOs (including (20000) Varuna) must have densities larger than
ρ ∼ 1000 kg m−3 . A better assessment of the inner structure of KBOs will
require more observations, and detailed modelling of the collisional evolution of
rubble-piles.
4.6
Conclusions
We have collected and analyzed R-band photometric data for 10 Kuiper Belt
objects, 5 of which have not been studied before. No significant brightness variations were detected from KBOs 2000 CM105 , 1999 RZ253 , 1996 TS66 . Previously
observed KBOs Chaos, 1999 TC36 , and Huya were confirmed to have very low
amplitude lightcurves (∆m ≤ 0.1 mag). 1998 SN165 , 1999 DF9 , and 2001 CZ31
were shown to have periodic brightness variations. Our lightcurve amplitude
statistics are thus: 3 out of 10 (30%) observed KBOs have ∆m ≥ 0.15 mag,
and 1 out of 10 (10%) has ∆m ≥ 0.40 mag. This is consistent with previously
published results.
The rotational properties that we obtained were combined with existing data
in the literature and the total data set was used to investigate the distribution of
spin period and shapes of KBOs. Our conclusions can be summarized as follows:
1. KBOs with diameters D > 200 km have a mean spin period of 9.23 hr,
and thus rotate slower on average than main belt asteroids of similar size
(hP iMBAs = 6.48 hr). The probability that the two distributions are drawn
from the same parent distribution is 0.7%.
2. 26 of 37 KBOs (70%, D > 200 km) have lightcurve amplitudes below
0.15 mag. In the asteroid belt only 10 of the 27 (37%) asteroids in the
same size range have such low amplitude lightcurves.
3. KBOs with diameters D > 400 km have lightcurves with significantly
The Shapes and Spins of KBOs
75
(98.5% confidence) smaller amplitudes (h∆mi = 0.13 mag, D > 400 km)
than KBOs with diameters D ≤ 400 km (h∆mi = 0.26 mag, D ≤ 400 km).
4. The shape distributions are different for these two size ranges, and predict
a larger fraction of round objects in the D > 400 km size range (f (a/b >
1.2) ∼ 80%) than in the group of smaller objects (f (a/b > 1.2) ∼ 50%).
5. The rotational properties of KBOs suggest that some of these objects may
have a rubble pile structure: 73% of KBOs with measured spin periods
have lightcurve amplitudes and spin periods consistent with rubble piles
of density ρ & 1000 kg m−3 . However, the data are too sparse to allow a
conclusive assessment of the inner structure of KBOs.
6. KBO 2001 CZ31 has a spin period of P = 4.71 hr. If this object has a
rubble pile structure then its density must be ρ & 2000 kg m−3 . If the
object has a lower density then it must have internal strength.
Acknowledgments
This work was supported by grants from the Netherlands Foundation for Research (NWO), and the Leids Kerkhoven-Bosscha Fonds (LKBF). We are grateful to Scott Kenyon and Ivo Labbé for helpful discussion.
76
Analysis of the rotational properties
References
Benz, W. & Asphaug, E. 1999, Icarus, 142, 5
Bernstein, G. M., Trilling, D. E., Allen, R. L., Brown, M. E., Holman, M., & Malhotra,
R. 2004, AJ, 128, 1364
Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393
Brown, R. H., Cruikshank, D. P., & Pendleton, Y. 1999, ApJ, 519, L101
Catullo, V., Zappala, V., Farinella, P., & Paolicchi, P. 1984, A&A, 138, 464
Chandrasekhar, S. 1969, Ellipsoidal figures of equilibrium (The Silliman Foundation
Lectures, New Haven: Yale University Press, 1969)
Collander-Brown, S. J., Fitzsimmons, A., Fletcher, E., Irwin, M. J., & Williams, I. P.
1999, MNRAS, 308, 588
Davies, J. K., McBride, N., & Green, S. F. 1997, Icarus, 125, 61
Dobrovolskis, A. R. & Burns, J. A. 1984, Icarus, 57, 464
Doressoundiram, A., Peixinho, N., de Bergh, C., Fornasier, S., Thébault, P., Barucci,
M. A., & Veillet, C. 2002, AJ, 124, 2279
Efron, B. 1979, Annals of Statistics, 7, 1
Farinella, P., Paolicchi, P., Tedesco, E. F., & Zappala, V. 1981, Icarus, 46, 114
Farinella, P., Paolicchi, P., & Zappala, V. 1982, Icarus, 52, 409
Goldreich, P., Lithwick, Y., & Sari, R. 2002, Nature, 420, 643
Hainaut, O. R., Delahodde, C. E., Boehnhardt, H., Dotto, E., Barucci, M. A., Meech,
K. J., Bauer, J. M., West, R. M., & Doressoundiram, A. 2000, A&A, 356, 1076
Holsapple, K. A. 2001, Icarus, 154, 432
Jewitt, D., Aussel, H., & Evans, A. 2001, Nature, 411, 446
Jewitt, D. & Luu, J. 1993, Nature, 362, 730
Jewitt, D. C. & Luu, J. X. 2000, Protostars and Planets IV, 1201
—. 2001, AJ, 122, 2099
Jewitt, D. C. & Sheppard, S. S. 2002, AJ, 123, 2110
Lacerda, P. & Luu, J. 2003, Icarus, 161, 174
Leinhardt, Z. M., Richardson, D. C., & Quinn, T. 2000, Icarus, 146, 133
Luu, J. & Jewitt, D. 1996, AJ, 112, 2310
Luu, J. X. & Jewitt, D. C. 2002, ARA&A, 40, 63
Luu, J. & Lacerda, P. 2003, Earth Moon and Planets, 92, 221
Luu, J. X. & Jewitt, D. C. 1998, ApJ, 494, L117+
Noll, K. S., Stephens, D. C., Grundy, W. M., Millis, R. L., Spencer, J., Buie, M. W.,
Tegler, S. C., Romanishin, W., & Cruikshank, D. P. 2002, AJ, 124, 3424
Ortiz, J. L., Gutiérrez, P. J., Sota, A., Casanova, V., & Teixeira, V. R. 2003, A&A,
409, L13
Ortiz, J. L., Lopez-Moreno, J. J., Gutierrez, P. J., & Baumont, S. 2001, Bulletin of the
American Astronomical Society, 33, 1047
Peixinho, N., Doressoundiram, A., & Romon-Martin, J. 2002, New Astronomy, 7, 359
Romanishin, W. & Tegler, S. C. 1999, Nature, 398, 129
Romanishin, W., Tegler, S. C., Rettig, T. W., Consolmagno, G., & Botthof, B. 2001,
Bulletin of the American Astronomical Society, 33, 1031
Rousselot, P., Petit, J.-M., Poulet, F., Lacerda, P., & Ortiz, J. 2003, A&A, 407, 1139
Sheppard, S. S. & Jewitt, D. C. 2002, AJ, 124, 1757
—. 2003, Earth Moon and Planets, 92, 207
Sheppard, S. S. & Jewitt, D. 2004, AJ, 127, 3023
The Shapes and Spins of KBOs
Stellingwerf, R. F. 1978, ApJ, 224, 953
Tegler, S. C. & Romanishin, W. 2000, Nature, 407, 979
Trujillo, C. A., Jewitt, D. C., & Luu, J. X. 2001a, AJ, 122, 457
Trujillo, C. A., Luu, J. X., Bosh, A. S., & Elliot, J. L. 2001b, AJ, 122, 2740
Trujillo, C. A. & Brown, M. E. 2002, ApJ, 566, L125
van Dokkum, P. G. 2001, PASP, 113, 1420
Weidenschilling, S. J. 2002, Icarus, 160, 212
77
CHAPTER
5
Origin and evolution of
KBO spins
ABSTRACT
We present a numerical study of the collisional evolution of the masses
and spins of Kuiper Belt objects (KBOs). Our model follows one KBO
at a time (the target), as it collides with the surrounding bodies. The
collisional environment, described by the total mass, size and velocity
distributions of KBOs, determines the total number, and the character
of individual collisions. Changes in the target’s spin rate and mass are
calculated for each collision, as a function of a several of parameters describing individual objects and the environment. We find that the spins
of KBOs do not depend strongly on their bulk properties. Furthermore,
the observed spins of KBOs larger than ∼ 200 km cannot be explained
by collisions, if the objects had no spin at the end of the primary growth
phase. This suggests that the large KBOs must have attained their spin
rates very early in their evolution. We investigate the possibility that the
accretion process was not entirely isotropic, and contributed angular momentum to the growing KBOs. We find that a ∼ 10% asymmetry in the
net angular momentum of accreted particles would explain the observations. However, if the accreted particles were comparable in size to the
growing body, no anisotropy is required because the accretion of individual particles can produce significant spin changes. These two scenarios
make different predictions about the distribution of KBO spin rates and
spin axis orientations: (1) Anisotropic accretion favours low scatter in the
spin rates; (2) Isotropic accretion of larger building blocks predicts a large
scatter in KBO spin rates and random spin axis orientations. The existing
data is insufficient to discern between the two possibilities.
Pedro Lacerda, Carsten Dominik, Jane Luu & Scott Kenyon
To be submitted to Icarus
80
Origin and evolution of KBO spins
5.1
Motivation
C
ollisions are the cause for many of the observed properties of our solar
system, and of solar system objects. The craters on the Moon (Proctor
1873; Wegener 1975), the inclination of planetary spin axes (Safronov 1969), the
zodiacal light and the meteoritic flux (Whipple 1967), and the very origin of the
Earth-Moon system (Cameron 1997), to cite a few examples, are all thought to
be related to impact events.
The asteroid belt is an example of a system shaped by collisions. This is
apparent, for example, from the asteroid spin rate distribution (Harris 1979;
Farinella et al. 1981), and from the distribution of shapes of asteroids smaller
than r ∼ 100 km, which is consistent with that of fragments of high-velocity
impact experiments carried out in the laboratory (Catullo et al. 1984).
Kuiper Belt objects (KBOs, Luu & Jewitt 2002) collide on timescales comparable to those of asteroids of the same size (Davis & Farinella 1997). Their
physical and dynamical properties should therefore show signatures of such encounters. The importance of collisions to the distribution of spins and shapes
of KBOs is not known. The lack of significant amounts of rotational data has
discouraged investigation of the collisional evolution of KBO spins. However,
several recent surveys (Sheppard & Jewitt 2002, 2003; Lacerda and Luu 2005,
references therein) have provided the community with a database of lightcurves
of 41 KBOs, from which 15 spin periods have been derived. Analysis of the
amplitudes of these lightcurves (Luu & Lacerda 2003; Lacerda and Luu 2005)
has revealed a broad variety of KBO shapes, from round to very elongated.
Probably the most striking example is (20000) Varuna (Jewitt & Sheppard
2002). Nearly 500 km in radius, and with an axis ratio a/b ∼ 1.5, it completes
one full rotation every 6.3 hours. The authors interpret this object’s high spin
angular momentum as the result of a collision. In the current low density Kuiper
Belt the chance that (20000) Varuna could hit a body large enough to significantly alter its spin and shape is very low. This collision must have happened
soon after the largest bodies formed.
KBOs grew by accretion of dust condensates. Numerical models yield a
formation timescale of the largest KBOs of ∼ 100 Myr (Kenyon & Luu 1999).
The bulk of the population was essentially formed by that time. If accretion was
isotropic, i.e., if the accreted material supplied zero average torque to the growing
body, angular momentum conservation should lead to these objects having little
spin by the end of the growth phase. This is very different from the what is
observed today, where 10 of the 15 KBOs have spin periods below 10 hr.
In this paper we wish to investigate how collisions between Kuiper Belt objects may have changed their spin rates, in the last ∼ 4 Gyr. Attempting to fit
the observed distribution of KBO spins is not realistic at this point. The number
of measured periods is small, and many aspects of KBOs are poorly understood.
The Shapes and Spins of KBOs
81
Our goal is to explore the role of several parameters in the collisional evolution
of KBO spins, and to gauge the overall importance of collisions in the KBO spin
distribution.
5.2
Spin evolution model
We start by giving a general, qualitative explanation of how the model works, and
below we describe in more detail what our assumptions are, and how individual
steps are calculated.
5.2.1
General description
We want to investigate how collisions might contribute to the observed spins of
KBOs. Our computation starts when the KBO population is already formed. We
place the transition between the epoch of formation and the epoch of collisions
at ≈ 4 Gyr ago. The particular value 4 Gyr is a safe order-of-magnitude estimate
based on the best current models of KBO formation, which claim the population
is essentially formed in . 100 Myr (Kenyon & Luu 1999). We assume, as a first
order approximation, that the population of KBOs has a total mass and size
distribution that do not change throughout the collisional evolution. This is
clearly a simplified version of what happened since the Kuiper Belt has changed
much over the last 4 Gyr. We adopt this simplification because we want to focus
on the effects of collisions; the bulk properties of the Belt, such as the total
mass, distribution of sizes, number of collisions, etc., are treated in a statistical,
time-averaged way.
The KBO population is in Keplerian orbit around the Sun. For KBOs to
collide, they must have random velocities relative to circular Keplerian orbits.
Hereafter, when we refer to the velocities of the KBOs, we mean the random
velocities in a Keplerian reference frame.
To simulate each collision, our model follows one individual KBO (the target)
as it collides with other KBOs (the projectiles), for 4 Gyr time. The population
of projectiles is binned according to size. By making assumptions about the
total mass and the size distribution of the projectiles, we estimate how many
projectiles fall in each size bin. Assuming KBOs to be spherical, with a given
density, we can translate the size bins into mass bins. In general, and as a
consequence of energy equipartition, which tries to make each object contribute
a similar fraction to the total kinetic energy of the system, KBO velocities are
a function of size: the smaller the mass, the higher the velocity.
First we calculate the total number of collisions experienced by the target
KBO. These collisions occur with projectiles from different bins, so we need to
determine how many collisions will come from a given bin. This depends on
how many projectiles there are in the bin, as well as the size and velocity of
82
Origin and evolution of KBO spins
these projectiles. Then we need to determine in what sequence the collisions
will occur, i.e., which projectile from which bin collides with the target first,
which collides second, and so on. This collisional sequence is randomly selected.
Once the collisional sequence is established we are ready to initiate the spin
evolution. Starting with an initial size and spin rate for the target body, for each
collision (in the predetermined sequence), we calculate the change in the target’s
spin rate and mass (size) resulting from the collision. In each collision a projectile
hits the surface of the target at a random location, with a random orientation.
Depending on the impact energy, the target may lose or gain mass. An important
assumption in how we compute each collision is that all the angular momentum 1
transferred by the projectile to the target stays in the target, i.e., that the ejecta
leaves the target body with no angular momentum. This assumption maximizes
the spin rate change of the target per collision.
A catastrophic collision can have two possible outcomes: (1) the ejected
mass may be equal to the total mass of projectile plus target or (2) the target
may reach a critical spin rate where centrifugal acceleration is equal to the
gravitational acceleration at the surface. If outcome 1 occurs the calculation
stops. If outcome 2 occurs the calculation is not stopped—since an object can
spin faster than the critical spin rate without flying apart, depending on its
material properties—but the target is flagged. Since the exact critical spin is
uncertain we choose to let the calculation continue and simply register the fact.
After all collisions have been accounted for and if the target still survives, we
register the final spin and mass of the target, and can start the process all over
again. By running the model several times with the same initial conditions, we
obtain a Monte Carlo estimate of the distribution of spins that a particular set of
initial set of parameters generates. Randomness is introduced in the collisional
sequence and the individual collisional geometries.
5.2.2
Model Details
The input parameters of the model are listed in Table 5.1; these parameters
are used by default, unless mentioned otherwise. The collisional environment
is characterized by (1) the total mass in the projectiles, (2) the total volume
accessible to all projectiles, (3) the size distribution of the projectiles, and (4)
the velocity distribution of the projectiles. As a first order approximation we
keep the total mass, and the size and velocity distributions constant throughout
the ∼ 4 Gyr of collisional evolution. The particular choice of 4 Gyr was made
based on models of KBO accretion (Kenyon & Luu 1998, 1999) which estimate
that Pluto-size objects finished forming by τP ≈ 100 Myr. By choosing a starting
1 The projectile’s angular momentum has an orbital component and a spin component.
Since the latter is much smaller than the former, we assume that the projectiles have no spin
angular momentum.
The Shapes and Spins of KBOs
Parameter
Symbol
General simulation and Belt parameters
Total mass in KBOs
MKBO
Distance Sun–Center of Kuiper Belt
a
Width of Kuiper Belt
2 × ∆a
Simulation timescale
∆t
Size distribution of KBOs
Number of size bins
Nb
Small body power-law exponent
qS
Large body power-law exponent
qL
Intermediate body power-law exponent
qI
Small body size interval
Rmin · · · R1
Large body size interval
R2 · · · Rmax
Individual KBO bulk properties
Density
ρ
Qb
Disruption energy parameters
Qg
βg
83
Value
1 M⊕
40 AU
20 AU
4 Gyr
64
2.5
3.5
0.0
100 · · · 102 m
103.5 · · · 106 m
1000 kg m−3
0
1.5 × (105 )1.25−βg
1.25
Table 5.1 – Model input parameters. These parameters are used by default, unless mentioned
otherwise.
time ti = 5 τP , we place our model safely beyond the formation epoch. Kenyon
& Bromley (2004a) have modelled the evolution of the size distribution of KBOs.
Based on their estimate of the mass loss rate from the Kuiper Belt in the last
≈ 4 Gyr we assign a total mass of 1 M⊕ to all KBOs, and use a set of power laws
to parameterize the size distribution:

r ≤ R1 ,
 nS r−qS
R1 < r ≤ R 2 ,
nI r−qI
n(r) =
(5.1)

nL r−qL
r > R2 ,
where n(r) dr is the number of objects with radius between r and r + dr. The
power law slopes are qS = 2.5, qI = 0, and qL = 3.5, and break radii R1 = 102 m
and R2 = 103.5 m.
The KBOs occupy a belt of width 2∆a = 20 AU, whose center lies at a
distance a = 40 AU from the Sun. The vertical extent of the belt is determined
by the velocities of the projectiles and is specified below. The projectiles are
binned according to their radius in Nb intervals whose centers form a geometric
series defined as
r0 = Rmax ,
ri+1 = (Rmin /Rmax )
1/Nb
ri ,
i = 1, . . . , Nb − 1
(5.2)
84
Origin and evolution of KBO spins
Figure 5.1 – The cumulative size distribution and the distribution of velocities for parameters,
MKBO = 1 M⊕ , qS = 2.5, qI = 0, qL = 3.5, break radii, R1 = 102 m and R2 = 103.5 m, and
ρ = 1000 kg m−3 .
where Rmin and Rmax are the minimum and maximum radius of the bodies. The
number of objects in each bin, ni , is determined from Eqn. (5.1).
We calculate the distribution of random velocities (measured relative to a reference frame in Keplerian rotation around the Sun) using the “two-group”approximation (Wetherill & Stewart 1989; Goldreich et al. 2004). This simplification
considers the velocity evolution of only two groups of bodies, the small and the
large, and provides an expression for the velocities of bodies of intermediate sizes.
The approximation is in agreement with numerical simulations of the velocity
evolution of a disk of planetesimals (Kenyon & Bromley 2004b). The velocity v
of a body of radius r is then given by (Goldreich et al. 2004)
(
1/2
1/3
(Σ/σ) Ve
r < (Σ/σ) Rmax
v(r) =
(5.3)
3/4
−3/4
(Σ/σ) (r/Rmax )
Ve
otherwise
where Σ and σ are the surface mass densities of the large and small bodies,
respectively, Rmax is the radius of the large bodies, and Ve is the escape velocity
from the surface of the large bodies.
The size and velocity distributions (Fig. 5.1), together with the total mass of
the Belt, MKBO , and the minimum and maximum radii of the bodies, Rmin and
The Shapes and Spins of KBOs
85
Figure 5.2 – The average number of collisions per target of a given size (solid contours) occurring in 4 Gyr with projectiles of different sizes. The dotted diagonal line indicates collisions
between equal sized bodies. The dashed line confines the region where collisions are erosive.
The figure was calculated using parameters listed in Table 5.1.
Rmax , completely define the collisional environment. Since the KBOs are binned
according to their radii, their masses (calculated from the radii assuming spheres
of density ρ) and the random velocities (Eqn. 5.3) are discretized accordingly.
The number of collisions between the target body and projectiles from the
ith bin that occur in the time interval ∆t, is estimated using a particle-in-a-box
approach (Safronov 1969), and is given by
·
¸
2 G(Mt + mi )
ni
2
π(Rt + ri ) 1 +
vrel ∆t,
Ci =
2
4π a ∆a hi
(Rt + ri )vrel
(5.4)
where G is the universal gravitational constant, a and ∆a are the radius and
half-width of the Kuiper Belt, Rt and Mt are the mass and radius of the target
body, ni , mi , and ri are the number, mass, and radius of the projectiles in the ith
bin, and vrel is the relative velocity at infinity. The relative velocity is obtained
from
2
vrel
= vt2 + vi2
86
Origin and evolution of KBO spins
with vt and vi are the target and projectile velocities, respectively, given by
Eqn. (5.3). The vertical extent of the orbits of the ith bin is given by (Kenyon
& Luu 1998)
vi
hi ≈ (a − ∆a) √
,
3 vKep
where vKep is the Keplerian velocity, and assumes the eccentricities and inclinations of the orbits are related by e ≈ 2 i. Figure 5.2 shows the number of
collisions in a 4 Gyr interval, as a function of target and projectile radius. Once
the total number of collisions per size bin is known, the collisional sequence is
randomized.
The collisional evolution is then initiated, with each collision being evaluated
in turn. For each collision, the model computes the change in target mass and
spin rate. The post-collision target mass depends on the ratio QI /Qd , where QI
is the center-of-mass impact energy per total (target+projectile) mass, and Q d
(the disruption energy per total mass) is defined as the energy needed to disperse
50% of the combined mass of the two bodies to infinity (see, e.g, Melosh & Ryan
1997; Benz & Asphaug 1999, and references therein). QI is given by (Wetherill
& Stewart 1993)
µ vI2
,
(5.5)
QI =
4(Mt + mi )
where Mt is the target mass, µ = Mt mi /(Mt + mi ) is the reduced mass, and vI
is the impact velocity, given by
2
vI2 = vt2 + vi2 + vesc
(5.6)
where vesc is the mutual escape speed. Qd has a bulk strength component and
a gravitational component (Benz & Asphaug 1999):
Qd = Q b r β b + ρ Q g r β g .
(5.7)
The value of r in the equation above is the radius of a sphere of mass Mt + mi .
This expression ignores a weak dependence of Qd on the impact velocity (see,
e.g., Benz & Asphaug 1999). We will consider two extreme cases for the bulk
strength component, Qb = 0 and Qb = 108 erg/g, both size independent (βb = 0),
and show below that, in the target size range we choose to study, this property
does not significantly influence the results. As for the gravity component, we
follow Kenyon & Bromley (2004a) and normalize it at r = 1 km by making
Qg = 1.5 × (105 )1.25−βg ,
(5.8)
and make use of scaling laws corresponding to icy targets (βg = 1.25), rocky
targets (βg = 1.40) (Benz & Asphaug 1999), and asteroids (βg = 2) (Durda
et al. 1998). The mass remaining in the target after the impact is given by
(Davis et al. 1985; Benz & Asphaug 1999)
¤
£
(5.9)
Mt0 = (Mt + mi ) 1 − 0.5(QI /Qd )βe
The Shapes and Spins of KBOs
87
where βe is a constant of order unity. The dashed line in Fig. 5.2 indicates
collisions for which Mt0 = Mt . Below the line the target’s post-impact mass is
smaller than its pre-impact mass, whereas for pairs target-projectile above the
line the target mass increases.
The post-impact spin vector, ω
~ t0 , is calculated by solving
8π
8π
5 0
~ t + mi ~r × ~vI ,
ρ Rt0 ω
~t =
ρ Rt5 ω
15
15
(5.10)
where ρ is the target and projectile density, Rt and Rt0 are the target radii before
and after the collision, ω
~ is the target spin frequency vector before the collision, ~r
is the radius vector of the impact point on the surface of the target, in the target’s
reference frame, and ~vI is the impact velocity vector. The post-impact target
radius, Rt0 , is calculated from Eqn. (5.9), assuming a spherical shape and density
ρ. The impact point is chosen randomly on the surface of the target (entry
point), and the impact velocity direction is determined by randomly picking
another point on the surface (exit point). This process generates an isotropic
distribution of collision geometries, from head-on to grazing. Equation (5.10)
assumes that the ejected mass carries no angular momentum. This assumption
maximizes the change in target spin per collision.
The change in mass and spin rate is computed for each collision. Catastrophic
impacts can have two outcomes: (1) disintegration of the target, i.e., Q I > 2Qd ,
or (2) centripetal disruption of the target, i.e., if the spin frequency exceeds the
critical spin frequency, given by
ωcrit =
p
(4/3)πGρ.
(5.11)
If the target is completely destroyed by a collision, the calculation stops. If the
target exceeds the critical spin rate the calculation is not stopped but the object
is flagged. As mentioned in Section 5.2.1, we choose not to stop the calculation
at the critical spin rate because the latter, as given by Eqn. (5.11), is only a
lower limit for the range of spin rate that can cause an object to blow itself
apart. We thus choose to let the calculation continue and simply register the
fact for statistical purposes.
If the target survives all collisions, we register its final spin and mass. The
simulation can then begin all over, to yield a statistically significant distribution. Although the model follows all three components of ω
~ , at the end of the
calculation only the norm of the spin vector is registered. Only one parameter
is changed at a time, and the model is run several times using each value of that
particular parameter; in this way we obtain a Monte Carlo estimate of how the
distribution of spins depends on a given parameter.
88
Origin and evolution of KBO spins
Figure 5.3
– Collisional
evolution results for the parameters listed in Table 5.1.
(a) The change in target radius, after 4 Gyr of collisional
evolution, as a function of
initial size. (b) The mean
final spin rate of the target,
as a function of initial size.
Each point represents the
mean of 50 model runs. 1σ
dispersion bars are shown.
Plus (square) symbols indicate that a fraction (all) of
the 50 bodies have exceeded
the critical spin (Eqn. 5.11).
5.3
Results and Discussion
The results yielded by the simulations are described below. A few are nearly
independent of the choice of parameters:
1. Collisions do not change the spins of bodies with radii larger than roughly
200 km. In ∼4 Gyr, these objects do not collide with enough large projectiles to alter their mass or spin angular momentum (see Fig. 5.2). Farinella
et al. (1992) derived a similar result for the largest asteroids (r > 100 km).
2. The relative velocities of KBOs, ∼ a few × 100 m s−1 , are low enough to
allow bodies & 100 km to grow via collisions, albeit very slowly.
3. Most bodies with initial radii of 50 km or smaller do not survive 4 Gyr of
collisional evolution. Catastrophic collisions destroy many of these bodies.
Most of the rest reach spin rates larger than the critical spin rate and must
lose mass to remain stable.
Figure 5.3 summarizes the results of our simulations for the parameters listed in
Table 5.1. Figure 5.3a shows the change in size of KBOs due to 4 Gyr of collisional evolution, as a function of initial size. The horizontal gray line corresponds
to the case where initial and final size are the same. With our standard parameters, the transition radius from net accretion to net erosion is rtr ∼ 100 km.
Each point in Fig. 5.3a is the mean change obtained from 50 simulations. The
bottom panel (5.3b) shows the final spin period of KBOs as a function of their
initial size and spin. The horizontal gray line shows the initial spin period of the
objects in the simulation. Here we use P = 9.23 hr, which is the observed mean
spin period of KBOs with r > 100 km (Lacerda and Luu 2005, see Chapter 4).
As in Fig. 5.3a, each point represents the mean final spin period, obtained from
50 simulations. The error bars show the 1σ dispersion. In the following sections
we investigate how different parameters affect the trends shown in Fig. 5.3.
The Shapes and Spins of KBOs
5.3.1
89
Effect of disruption energy scaling laws
We varied parameters Qb and βg to simulate different material properties. Given
the large size of our targets (r & 50 km), the bulk strength component of the
disruption energy must play a less significant role than the gravitational component. Nevertheless we explored two extreme values, Qb = 0 and Qb = 108 , with
no dependence on target size (βb = 0), to test how our result might depend on
material strength. For the gravity component we considered values representative of different materials: βg = 1.25 and βg = 1.40, for ice and rock, respectively
(Benz & Asphaug 1999), and βg = 2.00 for asteroids (Durda et al. 1998).
Figures 5.4 and 5.5 show the change in radius and spin rate, respectively, as a
function of initial size. Each point on the figures corresponds to the mean value
obtained from 50 model runs. Parameter Qb has little impact on the results,
aside from rendering objects . 40 km more resistant to destruction. As for β g ,
only βg = 2 results in net accretion for all simulated initial radii, from 20 km
to 1000 km. For βg = 1.25 and βg = 1.40 the transition radii from net erosion
to net accretion are approximately rtr = 100 km and rtr = 70 km, respectively,
and in both cases all bodies initially smaller than 40 km have been centrifugally
disrupted. The values obtained for rtr are consistent with previous simulations
of KBO collisional evolution (Davis & Farinella 1997; Kenyon & Bromley 2004a).
The spin properties of KBOs do not seem to depend significantly on how
strong they are, especially for bodies & 100 km. Smaller, weaker objects (smaller
βg ) are spun up more easily by collisions, mainly because of mass loss. Being
weaker, these objects are easily eroded by the average collision, and thus spin up
to conserve angular momentum, because the ejected material carries none (see
Section 5.2.2).
5.3.2
Effect of density
Figures 5.6 and 5.7 summarize the results for three different body densities. In
all three cases we obtain rtr ≈ 100 km. Changing the density has an opposite
effect on bodies above and below rtr : smaller bodies are more easily eroded if
their density is lower, whereas for bodies & 100 km, the lower the density the
larger the growth. The reason is that, since all realizations of the model have the
same total mass and same size distribution, lower bulk densities imply a larger
number of bodies, and thus more collisions (see Fig. 5.8). For large targets
(higher disruption energies), the extra collisions result in accretion, while at
small sizes they result in erosion.
The spin properties as a function of size follow a similar trend. At sizes
r & 100 km, where collisions are accreting, lower density objects end up with
slower spins, while the contrary is true for sizes where collisions are erosive
. 100 km. This raises the question whether the change in spin might simply
be due to conservation of spin angular momentum. To verify this we recompute
90
Origin and evolution of KBO spins
Figure 5.4 – The change in target radius, after 4 Gyr of collisional evolution, as a function
of initial size, for different combinations of material properties. The lines simply connect the
points. The horizontal gray line indicates no size change.
Figure 5.5 – The mean final spin rate of the target, after 4 Gyr of collisional evolution, as
a function of initial size. Different lines indicate different combinations of material strength
properties. Points represent the mean of 50 model runs. 1σ dispersion bars are shown at the
top for each curve. Plus (square) symbols indicate that a fraction (all) of the 50 bodies have
exceeded the critical spin (Eqn. 5.11).
The Shapes and Spins of KBOs
91
Figure 5.6 – Same as Fig. 5.4 but for different body densities, ρ = 250 kg m −3 (dotted
line), ρ = 1000 kg m−3 (dashed line), and ρ = 4000 kg m−3 (solid line). The remaining body
parameters are Qb = 0 and βg = 1.25.
Figure 5.7 – The same as Fig. 5.5, but for different bulk densities.
92
Origin and evolution of KBO spins
Figure 5.8 – The same as Fig. 5.2 but for a
bulk density of bodies ρ = 250 kg m−3 .
Figure 5.9 – The same as Fig. 5.7, but with every collision made head-on collisions, i.e., bringing
no spin angular momentum to the target.
Fig. 5.7 but making every impact head-on. In this way collisions add (or remove)
mass to the target, but no spin angular momentum. The result is shown in
Fig. 5.9. Indeed, angular momentum conservation alone is enough to explain
the spin-down for bodies larger than 100 km. This is not the case for smaller
objects, where the contribution from collisions is more noticeable.
5.3.3
High angular momentum collisions
Figures 5.5 and 5.7 show that in the size range r = 50 − 120 km, for some parameters, a fraction of the targets does not survive the high spin rates attained as a
result of high angular momentum collisions (generally with projectiles of comparable size). The plus (+) signs in Figs. 5.5 and 5.7 mark sizes for which a fraction of the model runs resulted in bodies exceeding the critical spin (Eqn. 5.11).
Square symbols indicate sizes for which all of the bodies exceed this limit.
These high angular momentum collisions could be responsible for the formation of fast spinning KBOs with elongated shapes (e.g., (33128) 1998 BU 48 ).
Table 5.2 lists the KBOs from the current lightcurve database (Lacerda and Luu
2005, see Chapter 4) that meet this description. (Note: the listed sizes are based
on the assumption that the albedo of KBOs is 0.04—assuming
an albedo twice
√
as large would result in sizes smaller by a factor 2). Based on our results, only
The Shapes and Spins of KBOs
Object Designation
Radiusa
[km]
Pb
[hr]
∆mc
[mag]
32929 1995 QY9
33128 1998 BU48
1999 DF9
47932 2000 GN171
26308 1998 SM165
20000 Varuna
90
105
170
180
200
490
7.3
9.8
6.65
8.33
7.1
6.34
0.60±0.04
0.68±0.04
0.40±0.02
0.61±0.03
0.45±0.03
0.42±0.03
a : bd
1.7 : 1
1.9 : 1
1.5 : 1
1.8 : 1
1.5 : 1
1.5 : 1
93
Ref.e
1, 2
1, 3
4
1
1, 3
1
Table 5.2 – KBOs with high amplitude lightcurve and high spin rate. a The radii assume
an albedo of 0.04, except for Varuna which has a known albedo of 0.07(Jewitt et al. 2001);
b Spin period in hours; c Lightcurve amplitude in magnitudes; d Axis ratio calculated from ∆m
assuming an aspect angle θ = π/2; e References: (1) Sheppard & Jewitt (2002), (2) Romanishin
& Tegler (1999), (3) Romanishin et al. (2001), (4) Lacerda and Luu (2005).
Figure 5.10 – The spin distribution of KBOs initially 100 km (top), and 500 km in radius
(bottom). All bodies start with no spin. Different densities are shown as different fill patterns.
Histograms are based on 250 runs of the model.
Varuna is too large to have suffered a high angular momentum collision in the
last 4 Gyr of collisional evolution. If Varuna’s spin rate is due to a collision with
a similar sized body, then it must have happened at a very early stage, just after
it formed, when the Kuiper Belt was more massive (Jewitt & Sheppard 2002).
94
Origin and evolution of KBO spins
Figure 5.11 – The final mean spin period after 4 Gyr of collisional evolution, as a function of
initial size, for bodies with no initial spin (dotted lines), and with initial spin period P = 9.23 hr
(dashed lines). The effect of the total mass in the KBO region is also shown: thick lines are
for MKBO = 10 M⊕ , and thin lines are for MKBO = 1 M⊕ . The result for each size represents
the mean of 50 runs.
5.4
The origin of KBO spin rates
Can collisions in the last 4 Gyrs explain the observed spins of KBOs? For bodies
in the size range r = 50 − 100 km they can. Figure 5.10 shows the distribution
of spin rates for 100 km- and 500 km-radius KBOs predicted by our simulations.
Note that these KBOs have no spin angular momentum initially. The Figure
shows that 100 km bodies can be spun to P ∼ 10 hr by collisions, but this is not
the case for 500 km bodies. If these bodies have negligible spin by the end of the
formation epoch (when our simulations start), collisions would only spin them
up to P ∼ 500 hr after 4 Gyr.
To test the robustness of this result we repeated the simulations, this time
using a total KBO mass of 10 M⊕ throughout the 4 Gyr of collisions, i.e., we assumed that the Kuiper Belt lost no mass since the formation epoch, 4 − 4.5 Gyr
ago. This increases the total number of collisions, including those between large
KBOs. The results for two distinct initial conditions, zero spin angular momentum and P = 9.23 hr (Note: P = 9.23 hr is the mean spin period of KBOs with
r > 100 km (Lacerda and Luu 2005, see Chapter 4), are shown in Fig. 5.11. As
expected, a more massive Kuiper Belt spins up KBOs more efficiently on average,
resulting in mean periods of order P ≈ 100 hr for r = 500 km bodies. But the
consequences of a more intense collisional environment extend to all sizes: KBOs
with r = 100 km are centripetally disrupted in such a massive environment (see
The Shapes and Spins of KBOs
95
Figure 5.12 – The distribution of spins for 1000 bodies of initial radius R = 500 km. The
histogram was calculated for MKBO = 10 M⊕ .
Fig. 5.11), which is not consistent with the observations. Besides, mean periods
of ∼ 100 hr still do not explain the spins of the largest KBOs (see Table 5.3).
To assess the statistical significance of our result, we repeat the simulations for
r = 500 km KBOs 1000 times, assuming a 10 M⊕ Kuiper Belt. As before we consider KBOs with no initial spin. The result is shown in Fig. 5.12. None of the
1000 objects attained spin rates P < 25 hr. This implies that the probability of
finding an object spinning that fast (with no initial spin) is p < 0.001. However,
existing KBO data show that 4/7 (57%) of KBOs ∼ 500 km in radius have spin
periods P < 20 hr. The remaining 3/7 have “flat” lightcurves, which makes it
impossible to measure their spins. Note that flat lightcurves may indicate very
low spin periods, round shapes, or unfavourable observing geometry (Lacerda &
Luu 2003, see Chapter 2).
As our simulations fail to reproduce the observed spin rates, it is therefore
logical to conclude that, if a large fraction of 500 km KBOs spin today with a
∼ 10 hr period, they must have had similar spins at the start of the collisional
evolution. But if KBOs grew from isotropic accretion, angular momentum conservation would argue in favour of them having very little spin by the time they
reached several hundreds of kilometers in size. It has been suggested that the
high angular momentum of (20000) Varuna was the result of a collision with
a similar size body, occurring very shortly after the largest KBOs had formed,
while the KB was still massive (Jewitt & Sheppard 2002). The high fraction of
96
Origin and evolution of KBO spins
Object Designation
20000
42301
55637
55636
50000
28978
2003 AZ84
Varuna
2001 UR163
2002 UX25
2002 TX300
Quaoar
Ixion
Radius
[km]
P
[hr]
Ref.
450
490
510
545
625
650
655
13.44
6.34
–
–
16.24
17.69
–
1
2
1
1
1
3
1, 4
Table 5.3 – Spin rates of large
KBOs. The radii were calculated
assuming an albedo of 0.04, except for Varuna which has a known
albedo of 0.07(Jewitt et al. 2001);
References: (1) Sheppard & Jewitt (2003), (2) Sheppard & Jewitt
(2002), (3) Ortiz et al. (2003), (4)
Chapter 4 of this thesis.
large KBOs with spin periods P ∼ 15 hr would require this type of collisions to
be very common.
5.4.1
Anisotropic accretion
An alternative explanation for the high fraction of large KBOs with high angular
momentum is that accretion was not entirely isotropic. How anisotropic would
accretion need to be to explain the spins of the largest KBOs? To try to answer
this question we devised the following experiment: a body initially 5 km in radius
is grown by accretion of smaller projectiles, until it reaches a radius of 500 km.
The mass of each particle is set to be equal to a constant fraction k of the
instantaneous mass of the growing body. The projectiles impact with a velocity
equal to the escape velocity of the target, and always adhere to the target,
i.e., no mass is ejected in the impacts. This is appropriate for simulating the
runaway accretion phase, where relative velocities are small. The impact point
and velocity are determined by randomly selecting two points on the surface of
the target. The first (entry point) is the impact point, and the second (exit
point) defines the direction of the impact velocity vector.
To parameterize anisotropy during accretion we define a parameter α, 0 ≤
α ≤ 1, where α = 1 corresponds to completely isotropic accretion and α = 0
completely anisotropic accretion. The regions allowed for the “entry” and “exit”
points2 of each projectile depend on α, and are given by
θentry (x) = arccos(2x α − 1),
ψentry (y) = πα(2y − 1) + π/2,
θexit (z) = arccos(2z α − 1),
(5.12)
ψexit (w) = πα(2w − 1) + 3π/2,
where x, y, z, and w are random real numbers selected in the [0, 1] interval. These
regions are illustrated in Fig. 5.13.
2 The exit point defines, together with the entry point, the direction of the impact velocity.
See Section 5.2.2.
The Shapes and Spins of KBOs
Figure 5.13 – Allowed impact
geometries for different values of
the parameter α. The transparent
mesh represents the entry region,
i.e., the fraction of the target’s surface allowed to entry points, and
the opaque grayscale section represents the exit region, i.e., fraction of the target’s surface allowed
to exit points. When α > 0.5 the
exit region is not shown because it
partly overlaps with the entry region.
®
®
z
z
Α=0.50
Α=0.25
®
®
z
z
Α=0.60
Figure 5.14 – The mean zprojection of the torque due
to accreting projectiles, in arbitrary units, as a function of the
anisotropy parameter α. The solid
line, hzi = −α, is shown for reference. See text and Fig. 5.13.
97
Α=0.80
98
Origin and evolution of KBO spins
Figure 5.15 – The mean final spin of full grown (r = 500 km) bodies, as a function of α (left)
and hzi (right). Different lines indicate different ratios of projectile mass to target mass, k.
The thick gray line is a fit “by eye” of the form P ∝ 1/hzi.
Figure 5.14 illustrates how α translates into a more physically intuitive,
torque-like, quantity. The vertical axis shows the mean z-projected (see Fig. 5.13)
angular momentum brought into the target by each collision, in arbitrary units.
For each different value of α (each point in Fig. 5.14) we generated 5000 pairs
of vectors, each pair consisting of (1) the position vector of the impact point,
connecting the origin with a randomly generated point on the surface of the
target (entry point, Eqn. 5.12), and (2) the impact velocity vector, connecting
the impact point with a second randomly chosen on the surface of the target
(exit point, Eqn. 5.12, see above). Then we calculated the cross product of each
pair of vectors3 . The mean of the z-axis projections, hzi, of all 5000 cross products is plotted. A value hzi ≈ 0 corresponds to nearly isotropic accretion where
projectiles bring no preferential spin direction. If all projectiles tend to spin the
target in the same direction then hzi ≈ 1.
In Fig. 5.15 we show the final spin period of the fully grown 500 km KBO
as a function of α and as a function of hzi. Is is clear that the closer α is to 0,
the faster the final spin. The Figure points out that accretion does not need to
3 To normalize the result, we used the corresponding unit vectors instead of the vectors
themselves.
The Shapes and Spins of KBOs
99
be very anisotropic to make a 500 km body spin with a ∼ 10 − 20 hr period by
the time it reached full size. Values α = 0.7 − 0.8 (corresponding to hzi ≈ 0.1)
would explain the spin rates shown in Table 5.3. The ratio of projectile mass
to target mass, k, also has an influence in the final spin of the growing KBO,
but only if the accretion is nearly isotropic, i.e., if α ≈ 1 (see Fig. 5.15a). For
values α < 0.8 the final spin period is independent of how massive the accreted
projectiles are with respect to the target. If accretion is isotropic, however, the
final spin of the target depends considerably on k. Ratios of projectile mass to
target mass of k & 0.01 could explain the measured spin periods, even under
completely isotropic growth. This corresponds to a ratio of projectile to target
radius of k 1/3 ≈ 0.2.
Figures 5.16 and 5.17 show the evolution of the spin rate as the KBO grows.
An equilibrium spin rate is attained very quickly in all cases, and it does not
depend on the initial spin rate. The fluctuations in the spin rate due to individual
projectiles are considerably smaller both with decreasing α (more anisotropic
accretion), and with decreasing mass ratio of projectile to target. If the spin
rate fluctuates the object tends to spend more the time at the higher spin rate.
The reason for this is geometrical. Only collisions with a very specific impact
geometry can slow down the spin rate; most impact geometries either change
the spin direction or contribute to increase (or maintain) it. Moreover, if a
collision happens to have the right geometry to slow down the spin, then any
next collision will most likely increase the spin rate again. This is why the slower
spin states are not lasting.
These results have the following implications:
1. Slight anisotropies in the accretion process can result in considerable spin
angular momentum for the full grown bodies. A ∼ 10% asymmetry in the
angular momentum brought by accreted particles is enough to explain the
observed spin rates. Anisotropic accretion also implies that the scatter in
the final spin distribution of large KBOs should be small.
2. Isotropic accretion can explain the observed spin rates if the accreted particles are comparable in size to the growing object. If the particles are
small, isotropic growth can only produce very slowly spinning objects
(P ∼ 1000 hr). Another consequence of isotropic accretion is that the
scatter in the final spin distribution should be large.
3. Isotropic and anisotropic accretion should produce very different distribution of spin axis orientations. Isotropic accretion produces random axis
orientations (hzi ≈ 0). Anisotropic accretion aligns the spin axes of KBOs,
in the z-direction (see Fig. 5.13). Since the material from which KBOs accreted had cold, low inclination orbits (Malhotra 1995), it is reasonable
to assume that any asymmetry originating in such a flat disk will favour
an alignment of KBO spin axes perpendicular to the ecliptic. This is also
what is observed for the majority of planets.
100
Origin and evolution of KBO spins
Figure 5.16 – The evolution of the target’s spin period as it grows for different values of
α. Solid lines mean that the target is initially spining with the critical spin period for its
density, P ≈ 3.3 hr (ρ = 1000 kg m−3 ), the dotted lines correspond to an initial spin period
P = 9.23 hr, and dashed lines correspond to no initial spin. Parameter k was set to 0.5 × 10 −4 .
Figure 5.17 – The same as Fig. 5.16 but fot k = 10−2 .
The Shapes and Spins of KBOs
101
Speculating on the origin of anisotropies in the process of accretion is beyond
the scope of this work. However, the explanation may lie in the dynamics of
accreted particles close to and inside the region of gravitational influence of the
growing bodies. We plan to investigate this in the future.
5.5
Limitations and future improvements
One of the main assumptions of the model is that each collision leaves all the
angular momentum in the target. If this assumption is dropped, the conclusion
that large KBOs have not had their spins significantly altered by 4 Gyr of collisions is only reinforced. However, the sizes for centrifugal disruption may change
under more a detailed treatment of angular momentum partition in an impact
event.
Another assumption is that we adopt a constant total mass and size distribution for KBOs. In reality the KB has probably lost ∼ 99% of its original mass in
the last 4 Gyr, and the size distribution has changed as a result of the collisional
cascade (Kenyon & Bromley 2004a). An obvious extension to the model is to
parameterize the time evolution of the KBO size distribution. Our results are
a good first-order approximation, but collisions in a very early phase are not
well accounted for. The results of Section 5.4.1 of this chapter give a hint of the
effect of collisions with similar size bodies in an early KB.
Finally, the total number of collisions is calculated in a very deterministic
way, i.e., a body of a given size always experiences the same number of collisions. KBOs exist in different dynamical classes, with different orbital properties.
Bodies from different classes should thus have different collisional probabilities,
and this is not accounted for. The data, however, are not numerous enough to
distinguish between the spin properties of bodies in different dynamical classes.
5.6
Summary and Conclusions
We presented results of numerical simulations of KBO collisions. The time scale
of our calculation starts when the bulk of the Kuiper Belt population is formed,
4 Gyr ago, and ends in the current epoch. In this period collisions are assumed to
be the main type of interaction between KBOs. Each simulation follows a single
target body as it collides with the surrounding bodies. Changes in the target’s
spin rate and mass are calculated for each collision. We studied the influence of
various properties of the Kuiper Belt on the final distribution of spin rates. Our
conclusions are as follows:
1. The spins of KBOs do not depend strongly on their bulk strength parameters.
2. KBOs with initial radius < 50 km do not survive the collisional evolution.
They are disrupted into smaller pieces either by high energy collisions, or
102
Origin and evolution of KBO spins
by centripetal disruption due to the high spin rates attained. This is true
for low tensile strength, icy material, as well as for moderately strong rocky
composition. Therefore, most KBOs with r < 50 km are in principle not
primordial, and should be by-products of collisions between larger bodies.
3. Collisions are slightly accreting for bodies with radius r > 100 − 200 km,
resulting in a size growth of a few percent in the last 4 Gyr.
4. KBO with initial radii ∼ 50 − 120 km lose 25 − 85% of their mass (10 − 50%
decrease in size) as a result of collisions. Although not disrupted, these objects have suffered high angular momentum collisions capable of producing
fast spin rates and rather elongated shapes. The current database of KBO
spin properties indicates that, out of the 7 KBOs with fast spin rates and
elongated shapes, 6 have sizes in this range.
5. The spins of KBOs larger than ∼ 200 km cannot be explained by collisions
if the objects had no spin angular momentum at the end of accretion. This
suggests that the large KBOs must have attained their spin rates during,
or very shortly after the accretion period.
The last point led us to the investigation of anisotropic accretion, as an explanation of the observed spins. We found that a ∼ 10% asymmetry in the
net angular momentum of accreted particles is enough to explain the observed
mean spin rate. However, if the accreted particles were comparable in size to the
growing body, no anisotropy is required. These two scenarios, anisotropic accretion of small particles, and isotropic accretion of large particles, make different
predictions about the distribution of KBO spin rates and spin axis orientations:
(1) Anisotropic accretion favours low scatter in the spin rates; (2) Isotropic accretion of larger building blocks predicts a large scatter in KBO spin rates and
random spin axis orientations. The existing data are not large enough to discern
between the two possibilities.
Acknowledgments
We would like to thank Ronnie Hoogerwerf for helpful discussion, and assistance
with LATEX.
The Shapes and Spins of KBOs
103
References
Benz, W. & Asphaug, E. 1999, Icarus, 142, 5
Cameron, A. G. W. 1997, Icarus, 126, 126
Catullo, V., Zappala, V., Farinella, P., & Paolicchi, P. 1984, A&A, 138, 464
Davis, D. R. & Farinella, P. 1997, Icarus, 125, 50
Durda, D. D., Greenberg, R., & Jedicke, R. 1998, Icarus, 135, 431
Davis, D. R., Chapman, C. R., Weidenschilling, S. J., & Greenberg, R. 1985, Icarus,
62, 30
Farinella, P., Davis, D. R., Paolicchi, P., Cellino, A., & Zappala, V. 1992, A&A, 253,
604
Farinella, P., Paolicchi, P., & Zappala, V. 1981, A&A, 104, 159
Goldreich, P., Lithwick, Y., & Sari, R. 2004, ARA&A, 42, 549
Harris, A. W. 1979, Icarus, 40, 145
Jewitt, D., Aussel, H., & Evans, A. 2001, Nature, 411, 446
Jewitt, D. C. & Sheppard, S. S. 2002, AJ, 123, 2110
Kenyon, S. J. & Luu, J. X. 1998, AJ, 115, 2136
Kenyon, S. J. & Luu, J. X. 1999, AJ, 118, 1101
Kenyon, S. J. & Bromley, B. C. 2004a, AJ, 128, 1916
Kenyon, S. J. & Bromley, B. C. 2004b, AJ, 127, 513
Lacerda, P. & Luu, J. 2003, Icarus, 161, 174
Lacerda, P. and Luu, J. 2005, in preparation.
Luu, J. X. & Jewitt, D. C. 2002, ARA&A, 40, 63
Luu, J. & Lacerda, P. 2003, Earth Moon and Planets, 92, 221
Malhotra, R. 1995, AJ, 110, 420
Melosh, H. J. & Ryan, E. V. 1997, Icarus, 129, 562
Ortiz, J. L., Gutiérrez, P. J., Sota, A., Casanova, V., & Teixeira, V. R. 2003, A&A,
409, L13
Proctor, R. A. 1873, London, Longmans, Green, and co., 1873.
Safronov, V. S. 1969, Evoliutsiia doplanetnogo oblaka.
Romanishin, W. & Tegler, S. C. 1999, Nature, 398, 129
Romanishin, W., Tegler, S. C., Rettig, T. W., Consolmagno, G., & Botthof, B. 2001,
Bulletin of the American Astronomical Society, 33, 1031
Sheppard, S. S. & Jewitt, D. C. 2003, Earth Moon and Planets, 92, 207
Sheppard, S. S. & Jewitt, D. C. 2002, AJ, 124, 1757
Wegener, A. 1975, Moon, 14, 211
Wetherill, G. W. & Stewart, G. R. 1989, Icarus, 77, 330
Wetherill, G. W. & Stewart, G. R. 1993, Icarus, 106, 190
Whipple, F. L., Southworth, R. B., & Nilsson, C. S. 1967, SAO Special Report, 239
Nederlandse samenvatting
Het zonnestelsel
N
egen planeten, Mercurius, Venus, Aarde, Mars, Jupiter, Saturnus, Uranus,
Neptunus en Pluto, samen met de ster die wij de Zon noemen, vormen
ons zonnestelsel. Tenminste, dat is wat we leren op school. De bovengenoemde
planeten zijn geordend volgens hun afstand tot de Zon: van de dichtstbijzijnde
tot de meest verafstaande. De eerste vier — Mercurius, Venus, Aarde en Mars
— bestaan uit vaste materie, voornamelijk rotsgesteente, en zijn relatief klein,
met diameters variërend van 4 900 km (Mercurius) tot 12 800 km (Aarde); zij
zijn de zogenaamde “rotsachtige planeten”. Als we ons wegbewegen van de Zon
zijn de vier die daarop volgen Jupiter, Saturnus, Uranus en Neptunus. Zij zijn
enorme gasbollen, zonder een begaanbaar oppervlak. Deze “gasreuzen” zijn veel
groter dan de rotsachtige planeten. Neptunus, de kleinste van de giganten, meet
een diameter van bijna vier “Aardes”, en Jupiter, de grootste van alle planeten,
heeft een diameter die elf keer groter is dan die van de Aarde. Als Jupiter de
grootte had van een voetbal, dan zou Neptunus die van een tennisbal hebben
en de Aarde die van een knikker. Op deze schaal zou de Zon een diameter van
meer dan twee meter hebben.
De eerste acht planeten lijken een patroon te volgen — vier kleine rotsachtige
in de binnenste regio en vier gigantische gasachtige in de buitenste regio. Bovendien zijn er grote overeenkomsten tussen de banen die planeten volgen: alle
banen liggen in bijna hetzelfde vlak, de ecliptica, ze zijn bijna cirkelvormig en
de banen worden allemaal in dezelfde richting doorlopen. De zes planeten die
het dichtst bij de Zon staan, van Mercurius tot Saturnus, zijn met het blote
oog waarneembaar en waren derhalve al bekend bij de eerste mensen die naar
de hemel begonnen te turen. De twee verste planeten, Uranus en Neptunus,
werden ontdekt met behulp van telescopen in respectievelijk 1781 en 1846. De
ontdekking van Pluto, in 1930, compliceerde enigszins de ogenschijnlijke eenvoud
106
Nederlandse samenvatting
van het zonnestelsel. Pluto draait op hetzelfde vlak als de andere planeten rond
de Zon, zijn baanvlak is nogal geheld ten opzichte van de ecliptica en de baan is
zoveel langgerekter dat Pluto’s baan gedeeltelijk ligt binnen die van Neptunus.
Verder bestaat Pluto voornamelijk uit ijs en hij is heel klein. Was de Aarde
zo groot als een knikker, dan zou Pluto iets kleiner zijn dan een peperkorrel.
Deze gegevens tonen duidelijk aan dat Pluto verschilt van de andere planeten.
Verderop in de tekst zullen we nog op deze planeet terugkomen.
Het zonnestelsel kent naast de Zon en de planeten ook nog andere families
van objecten. Tussen de banen van Mars en Jupiter bestaat een ring vol kleine
rotsachtige lichamen die asteroı̈den worden genoemd. Deze “kleine planeten”
vormen de zogenaamde “asteroı̈dengordel”. Ceres, de grootste van de asteroı̈den,
meet een diameter van ongeveer 900 km en werd als eerste ontdekt, in 1801.
Daarna werden nog enkele honderdduizenden asteroı̈den ontdekt, waarvan er
slechts 26 een diameter hebben van meer dan 200 km. Het merendeel van de
asteroı̈den is heel klein, en hoe kleiner, hoe talrijker ze zijn. Stel je voor dat
alle asteroı̈den samengeperst zijn in een enorme bal van fimoklei; deze bal zou
kleiner zijn dan de Maan. Als we deze bal verdelen in vier ballen van gelijke
grootte, dan zou een van hen de grootte van Ceres hebben. Vervolgens maken
we van een andere bal nog 25 asteroı̈den, degenen die samen met Ceres groter
dan 200 km in doorsnee zijn. Uiteindelijk kunnen we met de laatste twee ballen
alle andere, kleinere asteroı̈den maken.
Kometen
Een andere belangrijke familie van objecten in het zonnestelsel is die van de
kometen. Vanwege hun spookachtige verschijning hebben deze hemellichamen altijd bewondering en zelfs angst opgeroepen. Zoals doorgaans gebeurt met wetenschappelijk onverklaarbare natuurlijke fenomenen, werden kometen beschouwd
als gevreesde boodschappers van de goden en als voortekenen van catastrofes.
Het was de invloedrijke Griekse filosoof Aristoteles die ze rond het jaar 340 v.C.
voor het eerst als iets fysisch probeerde te duiden. Hij dacht dat het om lichtgevende wolken ging die, gegeven hun veranderlijke gedrag, geen deel uit konden
maken van het firmament, maar thuis hoorden in het ondermaanse. Aristoteles doopte hen “kometes” (“harig” in het Oud-Grieks). Eeuwen later, in 1577,
gebruikte de Deense astronoom Tycho Brahe waarnemingen van een komeet,
die op verschillende plaatsen in Europa waren gedaan, om aan te tonen dat er
geen sprake kon zijn van een wolk in de dampkring. De komeet verscheen steeds
in dezelfde positie in relatie tot de sterren, waar de waarnemingen ook waren
gedaan. Als het een wolk in de dampkring was geweest en deze zou zich bijvoorbeeld boven Parijs bevinden, dan zou hij vanuit Lissabon in het noordoosten te
zien zijn en vanuit Amsterdam in het zuiden. Brahe concludeerde dat de komeet
zich verder van de Aarde moest bevinden dan de Maan.
107
In de 17de eeuw ontwikkelde de Engelse wis- en natuurkundige Isaac Newton
een theorie die fundamenteel is gebleken voor de studie van het zonnestelsel en
de gehele Kosmos. Hij kwam erachter dat alle objecten elkaar aantrekken met
een kracht die proportioneel is aan hun massa. Op onze planeet geeft deze kracht
betekenis aan de richtingen “naar boven” en “naar beneden”. De theorie van
Newton, die de Universele Wet van de Zwaartekracht heet, verklaart waarom
dingen vallen, waarom er getijden zijn, waarom de Maan om de Aarde draait,
waarom de Aarde om de Zon draait, etcetera. Circa 20 jaar nadat Newton zijn
theorie publiceerde, gebruikte Edmund Halley haar om de banen van kometen te
berekenen. Deze banen zijn ellipsvormig. Een ellips1 is een soort cirkel met twee
middelpunten, die “brandpunten” worden genoemd. Hoe verder deze uit elkaar
liggen, hoe langwerpiger de ellips is. In de banen van kometen en planeten 2 is de
Zon een van die brandpunten. De berekeningen van Halley brachten hem tot de
voorspelling dat een bepaalde komeet, die in 1456, 1531 en 1607 was verschenen,
opnieuw in 1758 te zien zou moeten zijn. De komeet verscheen inderdaad met
Kerst van dat jaar en staat sindsdien bekend als de komeet van Halley — Halley
zelf was toen al 12 jaar dood. De laatste verschijning van de komeet van Halley
was in 1986.
Men wist echter nog steeds niet waar de kometen vandaan kwamen of waaruit
ze bestonden. Aan het eind van de 18de eeuw formuleerden de Duitse filosoof
Immanuel Kant en de Franse wetenschapper Pierre Laplace de hypothese dat
alle lichamen van het zonnestelsel afkomstig waren van een enorme draaiende
wolk van gas en stof. Deze wolk zou zich door de zwaartekracht hebben samengetrokken en de Zon in het middelpunt hebben gecreëerd. Maar een deel
van de materie kwam terecht in een sneldraaiende schijf om de Zon en uit de
materie daarvan ontstonden de planeten. Dit idee, dat 250 jaar geleden voor het
eerst onder woorden werd gebracht, wordt tegenwoordig ondersteund door tal
van waarnemingen en is een algemeen aanvaarde theorie geworden. Dit scenario
zou suggereren dat kometen op dezelfde manier als planeten worden gevormd,
namelijk vanuit die opeenhopingen. Maar waarom zijn ze dan geen planeten
geworden? Meer aanwijzingen voor het doorgronden van deze mysteries werden
ontdekt rond 1950 door Fred Whipple en Jan Hendrik Oort.
De eerste, een Amerikaanse astronoom, kwam tot de conclusie dat kometen
een soort vuile, stoffige sneeuwballen zijn. De uitdrukking “dirty snowball” is
van Whipple zelf afkomstig. Dankzij hun sterk elliptische banen bevinden deze
objecten van ijs zich voor een groot deel van de tijd ver van de Zon, daar waar
de temperaturen het absolute nulpunt, −273◦ C, naderen. Wanneer een komeet
richting het centrum van het zonnestelsel beweegt, dan stijgt de temperatuur
door de nabijheid van de Zon en gaat zijn bevroren oppervlak direct over van
vast naar gasvormig. Een deel van dit gas omgeeft de bevroren kern van de
1 De
figuur op de voorkant van dit boek bestaat uit zeven, binnen elkaar liggende ellipsen.
het geval van de planeten liggen de brandpunten zo dicht bij elkaar dat de ellips haast
een cirkel is.
2 In
108
Nederlandse samenvatting
komeet in de vorm van een lichtgevende halo, die “coma” wordt genoemd. De
rest wordt door het zonlicht weggeduwd, waardoor de komeet een staart krijgt
die altijd van de Zon af wijst. Het stof dat inmiddels is vrijgekomen uit het ijs bij
diens overgang van vast naar gas, vormt een tweede staart die bijna samenvalt
met de eerste, maar die vanwege zijn “zwaardere gewicht” enigszins naar achteren
helt.
In het jaar 1950 gebruikte de Nederlandse astronoom Jan Oort twee observationele gegevens om de oorsprong van kometen te verklaren. Terwijl de
astronomen de banen bepaalden van meer en meer van deze objecten, werd
duidelijk dat er twee verschillende subgroepen bestonden: kortperiodieke kometen, die iedere 5 tot 200 jaar terugkeren naar het centrum van het zonnestelsel
(hierbij hoort de komeet van Halley, met een periode van ±75 jaar), en langperiodieke kometen die er meer dan 200 jaar over doen om terug te keren. Bij
deze laatste groep horen ook kometen die sinds het ontstaan van de mensheid
slechts één keer de Zon genaderd zijn — kometen die maar eens in de 10 miljoen
jaar terugkeren. Deze kometen hebben extreem langgerekte banen, wier uiterste
punten meer dan 100 000 astronomische eenheden3 van de Zon verwijderd zijn.
Oort merkte op dat: (1) een grote hoeveelheid kometen van dergelijke afstanden
kwamen, dus van meer dan 100 000 AU, en (2) deze kometen in de nabijheid van
de Zon vanuit alle richtingen kwamen. Deze constateringen brachten Oort ertoe
het bestaan voor te stellen van een grote bol om de schijf van het zonnestelsel heen, van waaruit de kometen naar de kern, dus naar de Zon vallen. Deze
grote bol, gelegen op een afstand van meer dan 100 000 AU van de Zon, zou de
bron zijn van de kometen die ons in de kern van het zonnestelsel bezoeken. Als
eerbetoon aan de man die dit alles bedacht had, werd deze bron de “Oortwolk”
gedoopt. Het woord “wolk” werd gekozen als een verwijzing naar een bolvormige
stofwolk, waarin de kometen de stofdeeltjes zijn.
Toch is het vreemd dat kometen zo ver van de Zon ontstaan. De wolk van
gas en stof die door Kant en Laplace voorzien was, van waaruit zich het zonnestelsel gevormd zou hebben, zou mogelijk een te lage dichtheid hebben op die
afstand voor de vorming van de opeenhopingen. Een Nederlandse astronoom,
die genaturaliseerd was tot Amerikaan, Gerard Kuiper, was zich ook bewust van
dit probleem en had een ander idee. Als kometen uit ijs bestaan worden ze in
principe ver van de warmte van de Zon gevormd — maar er niet zo ver vandaan
als de Oortwolk. Kuiper bedacht het volgende: misschien werden kometen dichtbij de grens van ons planetaire stelsel “geboren”, voorbij Neptunus, en werden
ze na verloop van tijd de Oortwolk in geslingerd door de gigantische gasplaneten. Als de ruimte voorbij Neptunus inderdaad de wieg van de kometen was,
dan zouden er zich daar veel meer moeten bevinden en die zouden dan Pluto
in zijn baan rond de Zon gezelschap houden. Deze potentiële kometen waren
waarschijnlijk bevroren en wachtten het moment af waarop ze de warmte van
de Zon konden opzoeken, of gelanceerd werden richting de Oortwolk. Er is in3 Een
astronomische eenheid (AU) komt overeen met de afstand van de Aarde naar de Zon.
109
derdaad gezocht naar deze “trans-Neptuniaanse” kometen, maar dit bracht geen
succes. Op zo’n afstand van de Zon, ongeveer 40 AU, gaf alleen Pluto een teken
van leven. De mislukte pogingen leidden ertoe dat de hypothese van Kuiper aan
kracht verloor.
In 1988 zorgde een andere kwestie, die van de banen van de kortperiodieke
kometen, ervoor dat drie onderzoekers, Martin Duncan, Thomas Quinn en Scott
Tremaine weer in het idee van Kuiper gingen geloven. De kortperiodieke kometen hebben banen die bijna liggen in het baanvlak van de planeten. Daarnaast
zijn er de langperiodieke kometen die, zoals reeds is gezegd, komen van de verre
Oortwolk en de Zon vanuit allerlei richtingen naderen. Met deze twee groepen
van kometen voor zich werd de vraag — zoals de kwestie van de kip en het ei
— welke van de twee de eerstgeborene was. Kon het zo zijn dat de langperiodieke kometen, vanuit alle richtingen afkomstig en met een afwijkende baan,
“getemd” konden worden, om zo kortperiodieke kometen te worden, met een
weinig afwijkende baan? Of waren het toch de brave kortperiodieke kometen,
die zo nu en dan richting de Oortwolk gelanceerd werden in langgerekte en afwijkende banen? Duncan, Quinn en Tremaine testten deze eerste hypothese met
een computersimulatie. De conclusie was dat je de langperiodieke kometen niet
kunt temmen; bovendien concludeerden de drie dat de kortperiodieke kometen
vanuit een “donut”vormig gebied moesten komen, die aan de rand van het zonnestelsel gelegen was, precies zoals Kuiper het had bedacht. In het artikel dat
zij publiceerden, doopten Duncan en zijn collega’s deze hypothetische gordel van
kometen de “Kuiper Belt”.
De technologische revolutie aan het eind van de 20ste eeuw bracht grote
voordelen voor de astronomie met zich mee. Telescopen die telkens groter werden
en beter uitgerust waren, lieten ons tot in detail het Heelal zien, iets wat tot dan
toe onmogelijk was geweest. Dave Jewitt en Jane Luu, twee astronomen van de
Universiteit van Hawaii, waren zich bewust van dit feit. Zij besloten de jacht
te openen op deze kometengordel van Kuiper met behulp van de telescopen op
de top van de vulkaan Mauna Kea, op 4 000 m hoogte. Geduldig zochten zij
nacht na nacht een lichtpunt waarvan de beweging zou aangeven dat het om een
trans-Neptuniaans object ging. Maar de jaren gingen voorbij en Jewitt en Luu
zagen niets. Tot in 1992, vijf jaar nadat ze aan hun zoektocht waren begonnen,
zij een klein object zagen dat een diameter had van ongeveer 200 km. Dit object
bewoog zich in een bijna perfect circulaire baan, niet ver van die van Pluto.
Zes maanden later vonden ze een ander object in een soortgelijke baan, aan de
andere zijde van het zonnestelsel.
Sindsdien zijn er bijna duizend “Kuiper Belt Objects” ontdekt. Deze nieuwe
familie van trans-Neptuniaanse objecten veranderde de status van Pluto in de
hiërarchie van het zonnestelsel. De vreemde, kleine en bevroren planeet blijkt
de grootste van de KBO’s te zijn, zoals Ceres de grootste van de asteroı̈den
is. De ontdekking van de Kuiper Belt riep een nieuw onderzoeksgebied in het
leven. Hoewel de trans-Neptuniaanse objecten de oplossing lijken te zijn van het
110
Nederlandse samenvatting
probleem van de oorsprong van kometen, is wat men over ze weet eigenlijk bijna
niets. Opdat we een van die objecten vanaf de Aarde kunnen zien, moet het
zonlicht, in de vorm van minuscule deeltjes genaamd fotonen, meer dan vijf uur
reizen, weerkaatsen op het oppervlak van het object en terugkeren om ons te
bereiken. Het is niet zo dat de fotonen hier moe aankomen. Het probleem is dat
er maar weinig ons bereiken — het overgrote deel gaat tijdens de tocht verloren.
In de astronomie zijn de fotonen de boodschappers van alle informatie die ons
bereikt. Hoe meer fotonen we ontvangen van een bepaald object, des te meer
informatie we kunnen bemachtigen. Doordat er maar weinig arriveren van de
trans-Neptuniaanse objecten weten we zo weinig over hen. Waarschijnlijk zullen
we op een dag sondes kunnen zenden, zoals die welke naar Mars gaan, opdat we
meer over deze kleine bevroren werelden te weten kunnen komen.
Dit proefschrift
In de wetenschappelijke literatuur worden de Kuiper Belt objecten aangeduid
met KBO’s. Soms worden ze ook wel TNO’s genoemd, “Trans-Neptunian Objects”. KBO’s lijken een soort babyplaneten te zijn. In het al genoemde model
van Kant-Laplace klonteren de opeenhopingen die rond de Zon achterblijven
samen en groeien ze tot ze planeten worden. Astronomen noemen deze opeenhopingen in de groeifase “planetesimals”. De regio van de KBO’s bevindt zich
ver van de Zon, waar minder materiaal aanwezig was om planetesimals te vormen. Daarom groeiden de KBO’s langzamer. Toen het materiaal op was, waren
de KBO’s nog in hun kindertijd, dus nog lang geen planeten. En zo zijn ze
gebleven, bevroren. Om deze reden is de bestudering van de KBO’s van groot
belang voor meer inzicht in het formatieproces van planeten.
Dit proefschrift presenteert een studie van de vormen en rotaties van de
KBO’s. Om uit te leggen waarin het belang ligt de vormen van de KBO’s te
kennen, kunnen we het voorbeeld van de asteroı̈den gebruiken. De vorm van
de kleinste asteroı̈den komt overeen met die van stukken die erafvliegen wanneer je een steen breekt. Dit geeft aan dat deze objecten inderdaad dergelijke
dingen zijn: brokstukken als gevolg van zeer krachtige botsingen tussen grote
asteroı̈den. De hoofdstukken 3 en 4 van dit proefschrift laten zien dat een groot
deel van de KBO’s relatief rond is, maar dat een aanzienlijk percentage ovale
of langwerpige vormen heeft. Waarom dit zo is, is nog niet bekend. Wel geeft
de snelheid waarmee een KBO om zijn as draait ons informatie over diens interne structuur, vooral als we de vorm van de KBO weten. Bijvoorbeeld: als
een object massief is, kan het vrij snel draaien, zonder dat zijn vorm verandert.
Maar als een KBO een opeenhoping van kleine fragmenten is (een “afvalhoop”
of “rubble pile”), dan maken de rotaties het object ovaler. Als het te snel draait
kan het zelfs uiteenspatten. In hoofdstuk 4 worden de vormen en rotaties van de
KBO’s samen geanalyseerd en die lijken aan te tonen dat de KBO’s inderdaad
opeenhopingen van veel kleinere brokken zijn die alleen bij elkaar blijven door
111
de zwaartekracht en niet doordat ze een samenhangende substantie zijn. Hoofdstuk 5 bestudeert de botsingen tussen de KBO’s. Het doel is om te verifiëren of
de draaiing van de KBO’s wordt veroorzaakt door botsingen met andere KBO’s
vanaf het moment van hun geboorte. De conclusie luidt dat de grootste KBO’s
dezelfde rotatiesnelheid hebben als toen ze ontstonden, maar dat de rotatiesnelheid van de kleinste compleet veranderd is door de botsingen. De botsingen
lijken overigens de reden te zijn van het feit dat de KBO’s opeenhopingen van
fragmenten zijn: voortdurende collisies breken en vergruizen geleidelijk aan deze
objecten, die door de zwaartekracht bijeengehouden worden.
Aan het eind van hoofdstuk 1 staat een meer gedetailleerde samenvatting van
dit proefschrift.
Resumo em Português
O sistema solar
N
ove planetas, Mercúrio, Vénus, Terra, Marte, Júpiter, Saturno, Urano,
Neptuno, e Plutão, juntamente com a estrela a que chamamos Sol, formam
o nosso sistema solar. Pelo menos é isto que aprendemos na escola. Os planetas
acima citados estão ordenados do mais próximo para o mais distante do Sol.
Os primeiros quatro — Mercúrio, Vénus, Terra e Marte — são sólidos, constituı́dos maioritariamente por rocha, e relativamente pequenos, com diâmetros
entre 4 900 km (Mercúrio) e 12 800 km (Terra); são os chamados “planetas rochosos”. Os quatro que se seguem, à medida que nos afastamos do Sol, —
Júpiter, Saturno, Urano e Neptuno — são enormes esferas de gás, sem uma superfı́cie onde se possa caminhar. Estes “gigantes gasosos” são muito maiores do
que os planetas rochosos. Neptuno, o mais pequeno dos gigantes, mede quase 4
“Terras” em diâmetro, e Júpiter, o maior de todos os planetas, tem um diâmetro
11 vezes superior ao da Terra. Se Júpiter fosse do tamanho de uma bola de futebol, Neptuno seria do tamanho de uma bola de ténis e a Terra seria do tamanho
de um berlinde. Nesta escala, o Sol teria mais de dois metros de diâmetro.
Os primeiros oito planetas parecem seguir um padrão — quatro pequenos
rochosos na região interior, e quatro gigantes gasosos na região exterior. Além
disso, movem-se de forma bastante regular — todos circulam o Sol no mesmo
sentido, em órbitas heliocêntricas1 , quase circulares e coplanares2 . Os seis planetas mais próximos do Sol, Mercúrio a Saturno, são visı́veis a olho nú e, portanto,
do conhecimento dos homens desde que estes olharam para os céus com atenção.
Os dois mais distantes, Urano e Neptuno, foram descobertos com o auxı́lio de
telescópios em 1781 e 1846, respectivamente. A descoberta de Plutão, em 1930,
veio complicar ligeiramente a aparente simplicidade do sistema solar, e aguçar a
curiosidade dos cientistas. Plutão orbita o Sol no mesmo sentido dos restantes
1 Centradas
2 Todas
no Sol.
assentes no mesmo plano, como as faixas de um disco de vinil imaginário.
114
Resumo em Português
planetas, mas fá-lo de uma forma mais irregular. A sua órbita é inclinada em
relação ao plano da eclı́ptica3 , e é alongada, de tal forma que Plutão umas vezes
está mais perto do Sol do que Neptuno, outras vezes está mais longe. Para além
disso, Plutão é composto principalmente de gelo, e é muito pequeno. Na nossa
escala em que a Terra tem o tamanho de um berlinde, Plutão seria ligeiramente
mais pequeno do que um grão de pimenta. Estes factos mostram claramente que
Plutão é diferente dos outros planetas. Voltaremos a Plutão mais adiante.
O sistema solar tem outras famı́lias de objectos, para além do Sol e dos
planetas. Entre as órbitas de Marte e Júpiter existe uma região repleta de pequenos corpos rochosos, chamados asteróides. Estes “planetas menores” formam
a chamada “cintura de asteróides”. Ceres, o maior dos asteróides, tem cerca de
900 km de diâmetro, e foi o primeiro a ser descoberto, em 1801. Desde então,
várias centenas de milhares de asteróides foram descobertos, dos quais apenas
26 têm mais de 200 km em diâmetro. Os asteróides são, na sua grande maioria,
muito pequenos, e quanto mais pequenos mais numerosos. Imaginem todos os
asteróides “amassados” numa única e gigante bola de plasticina: essa bola seria
menor do que a Lua. Se a dividı́ssemos em quatro bolas iguais, uma delas seria do
tamanho de Ceres. Depois, usarı́amos outra bola para fazer mais 25 asteróides,
os tais que, juntamente com Ceres, são maiores que 200 km. Finalmente, as
últimas duas bolas seriam suficientes para fazer todos os outros asteróides.
Cometas
Outra importante famı́lia de objectos do sistema solar é a dos cometas. Por causa
da sua aparência fantasmagórica, estes astros sempre suscitaram admiração, e
até medo. Como geralmente acontece com fenómenos naturais para os quais não
existe explicação cientı́fica, os cometas foram considerados temı́veis mensageiros
dos deuses, e prenúncio de catástrofes. Foi o influente filósofo grego Aristóteles
que cerca do ano 340 a.C. tentou, pela primeira vez, explicar os cometas como
algo de fı́sico. Ele julgava tratarem-se de nuvens luminosas que, dado o seu
comportamento errático, não poderiam fazer parte do firmamento. Aristóteles
baptizou-os de kometes (“cabeludos”, do Grego antigo). Séculos mais tarde,
em 1577, o astrónomo dinamarquês Tycho Brahe recorreu a observações de um
cometa feitas de diversos locais na Europa para mostrar que este não podia ser
uma nuvem na atmosfera: o cometa aparecia na mesma posição no céu, em
relação às estrelas, de onde quer que as observações fossem feitas. Se fosse uma
nuvem, e a nuvem estivesse por exemplo sobre Paris, quando vista de Lisboa esta
apareceria a nordeste, mas vista de Amesterdão apareceria a sul. Brahe concluiu
que o cometa teria que estar longe da Terra, mais longe do que a própria Lua.
No século XVII o fı́sico e matemático inglês Isaac Newton desenvolveu uma
teoria que viria a ser fundamental para o estudo do sistema solar, e do Cosmos.
3 O plano da órbita da Terra em torno do Sol, ou o tal “disco invisı́vel” onde assentam as
órbitas dos planetas.
115
Newton percebeu que todos os objectos se atraem mutuamente, com uma força
que é proporcional à sua massa. No nosso planeta essa força dá significado às
direcções “para cima” e “para baixo”. A teoria de Newton, chamada Lei da
Gravitação Universal, explica porque é que as coisas caem para a Terra, porque
é que há marés, porque é que a Lua anda à volta da Terra, porque é que a
Terra anda à volta do Sol, etc. Cerca de 20 anos depois de Newton publicar a
sua teoria, Edmund Halley usou-a para calcular as órbitas dos cometas. Estas
órbitas são elipses. Uma elipse4 é uma espécie de circunferência com dois centros,
chamados “focos”. Quanto mais afastados são os focos mais alongada é a elipse.
Nas órbitas de cometas e planetas5 , o Sol ocupa um dos focos. Os cálculos de
Halley levaram-no a propôr que um determinado cometa, que tinha aparecido
em 1456, 1531 e 1607, deveria reaparecer em 1758. De facto o cometa apareceu,
no Natal desse ano, e ficou desde então conhecido como o cometa de Halley. A
última aparição do cometa de Halley foi em 1986.
Mas os cientistas continuavam sem saber de onde vinham, ou do que eram
feitos, os cometas. No final do século XVIII, o filósofo alemão Immanuel Kant e
o cientista francês Pierre Laplace puseram a hipótese de que todos os corpos do
sistema solar provinham de uma enorme nuvem de gás e poeira. Essa nuvem,
rodando no espaço, ter-se-ia contraı́do sobre si própria, por acção da força da
gravidade, e formado o Sol no centro. As regiões exteriores formariam aglomerados que, devido à rotação da nuvem, ficariam em órbita em torno do Sol, dando
origem aos planetas. Esta ideia, formulada há 250 anos, veio a ser reforçada
por inúmeras observações e é hoje a explicação aceite para a formação do nosso
(e outros) sistema planetário. Neste cenário, faz sentido pensar que os cometas
se formaram da mesma maneira que os planetas, a partir dos tais aglomerados.
Mas então porque é que não são como os planetas? Mais pistas para entender
estes mistérios foram descobertas por volta de 1950 por Fred Whipple e Jan
Hendrik Oort.
O primeiro, astrónomo americano, chegou à conclusão que os cometas são
uma espécie de bolas de neve sujas de poeira. A expressão “dirty snowball” é
da autoria do próprio Whipple. Graças às suas órbitas elı́pticas, estes objectos
passam grande parte do tempo congelados longe do Sol, onde as temperaturas se
aproximam do zero absoluto, −273◦ C. Quando um cometa se aproxima do centro
do sistema solar, o aumento de temperatura resultante da proximidade do Sol faz
com que a sua superfı́cie gelada passe directamente do estado sólido ao estado
gasoso. Uma parte deste gás envolve o núcleo gelado do cometa sob a forma
um halo luminoso, chamado “coma”, e o restante é literalmente empurrado pela
radiação solar, formando a cauda do cometa. Por isso as caudas dos cometas
apontam na direcção oposta à do Sol. A poeira entretanto libertada do gelo
quando este passa a gás forma uma segunda cauda que quase coincide com a
primeira, mas por ser mais “pesada” fica ligeiramente para trás.
4A
figura na capa deste livro é formada por sete elipses, umas dentro das outras.
caso dos planetas os focos estão tão próximos que a órbita é quase uma circunferência.
5 No
116
Resumo em Português
No ano de 1950, o astrónomo holandês Jan Oort usou dois dados observacionais para propôr uma origem para os cometas. À medida que os astrónomos
determinavam as órbitas de mais e mais destes objectos, tornou-se claro que
existiam dois subgrupos distintos: os cometas de curto perı́odo, que voltam ao
centro do sistema solar cada 5 a 200 anos (onde se inclui o cometa de Halley,
cujo perı́odo órbital são ±75 anos), e os cometas de longo perı́odo que demoram
mais de 200 anos a voltar. Este último grupo inclui cometas que só se aproximaram do Sol uma vez desde o aparecimento dos primeiros homens — cometas
que só voltam uma vez em cada 10 milhões de anos. Estes cometas têm órbitas
extremamente alongadas, que os transportam a distâncias de mais de 100 000
unidades astronómicas6 do Sol. Oort reparou que: (1) uma grande quantidade
de cometas vinha de distâncias dessa ordem, de mais de 100 000 AU, e (2) esses
cometas chegavam às proximidades do Sol vindos de todas as direcções. Estas
constatações levaram Oort a propôr a existência de uma região esférica, envolvendo o disco do sistema solar como uma gigante bola de vidro, de onde os
cometas caem para o centro, onde se encontra o Sol. Esta região esférica, situada a mais de 100 000 AU do Sol, funcionaria assim como fonte para os cometas
que nos visitam no centro do sistema solar. Em homenagem ao homem que a
idealizou, esta fonte foi baptizada de “nuvem de Oort”. A palavra “nuvem” foi
escolhida em alusão a uma nuvem esférica de poeira, em que os cometas são os
grãos de pó.
É estranho, no entanto, que os cometas se tenham formado tão longe do Sol.
A nuvem de gás e poeira idealizada por Kant e Laplace, de onde se formou o
sistema solar, seria demasiado ténue a essa distância para permitir a formação
de aglomerados. Ciente deste problema, um astrónomo holandês naturalizado
americano, Gerard Kuiper, teve outra ideia. Se os cometas são feitos de gelo, em
princı́pio formaram-se longe do calor do Sol. Mas não tão longe quanto a nuvem
de Oort. Kuiper imaginou o seguinte: talvez os cometas tenham “nascido” perto
do limite do nosso sistema planetário, na região além-Neptuno, e com o passar do
tempo alguns deles foram sendo “lançados” para a nuvem de Oort pelos planetas
gigantes gasosos7 . Se de facto a região trans-Neptuniana é o berço dos cometas,
então ainda lá deverão residir vários, fazendo companhia a Plutão na sua órbita
em torno do Sol. Estes candidatos a cometas estarão congelados, aguardando a
sua vez de se aproximarem do calor do Sol, ou de serem lançados em direcção
à nuvem de Oort. Os astrónomos apontaram os seus telescópios para o céu, em
busca dos tais potenciais cometas mas não tiveram sucesso. De tal distância do
Sol, cerca de 40 AU, só Plutão dava sinais de existência. As tentativas falhadas
fizeram com que a hipótese de Kuiper fosse perdendo força.
Em 1988, uma outra questão, a das órbitas dos cometas de curto perı́odo,
fez com que três cientistas, Martin Duncan, Thomas Quinn e Scott Tremaine,
6 Uma
unidade astronónomica é a distância entre a Terra e o Sol, e designa-se por AU.
um modo semelhante ao usado pelos atletas olı́mpicos quando lançam uma bola de
metal presa na ponta de um cabo, rodando sobre si próprios e largando o cabo.
7 De
117
voltassem a acreditar na ideia de Kuiper. Os cometas de curto perı́odo têm
órbitas pouco inclinadas, ou seja, que não se afastam muito do disco onde os
planetas se movem. Por outro lado, e como foi referido atrás, os cometas de longo
perı́odo vêm da distante nuvem de Oort, e aproximam-se do Sol vindos de todas
as direcções. Face a estes dois grupos de cometas os cientistas interrogaram-se —
ao jeito da história do ovo e da galinha — qual teria “nascido” primeiro. Será que
os cometas de longo perı́odo, vindos de todas as direcções e inclinações, podem
ser “domados”, passando a ter órbitas de curto perı́odo, e pouco inclinadas? Ou
serão, pelo contrário, os bem comportados cometas de curto perı́odo que, de
vez em quando, são lançados para órbitas alongadas e inclinadas, em direcção
à nuvem de Oort? Duncan, Quinn e Tremaine decidiram fazer uma simulação
usando um computador para testar a primeira hipótese. A conclusão foi que
não é possı́vel domar os cometas de longo perı́odo. Além disso, os três cientistas
concluı́ram que os cometas de curto perı́odo têm necessáriamente de vir de uma
cintura em forma de bolo-rei, em torno do sistema solar, exactamente como
Kuiper imaginara. No artigo que publicaram, Duncan e colegas baptizaram a
hipotética cintura de cometas de “cintura de Kuiper”.
A revolução tecnológica dos finais do século XX trouxe grandes vantagens
para a astronomia. Telescópios cada vez maiores e melhor equipados mostravam-nos o Universo com um detalhe até então impossı́vel. Dave Jewitt e Jane Luu,
dois astrónomos da Universidade do Havai, aperceberam-se desse facto e decidiram voltar à caça da tal cintura de cometas de Kuiper, usando telescópios
situados no topo do vulcão Mauna Kea, a 4 000 metros de altitude. Pacientemente, noite após noite, procuraram um ponto luminoso cujo movimento indicasse tratar-se de um objecto trans-Neptuniano. Mas os anos passavam e Jewitt
e Luu não viam nada. Até que em 1992, cinco anos depois de terem iniciado a
busca, eles avistaram um pequeno objecto, com cerca de 200 km de diâmetro.
Este objecto movia-se numa órbita quase perfeitamente circular, ligeiramente
mais distante do que a de Plutão. Seis meses mais tarde encontraram outro
objecto, numa órbita semelhante, do lado oposto do sistema solar.
Desde então quase mil “objectos de Kuiper” foram descobertos. Esta nova
famı́lia de objectos trans-Neptunianos veio alterar o estatuto de Plutão na hierarquia do sistema solar. O estranho, pequeno, e gelado planeta é afinal o maior
dos objectos de Kuiper, assim como Ceres é o maior dos asteróides. A descoberta
da “cintura de Kuiper” entusiasmou a comunidade cientı́fica e fez nascer uma
nova área de investigação. Embora os objectos trans-Neptunianos pareçam ser
a solução do problema da origem dos cometas, o que se sabe sobre eles é quase
nada. Para que consigamos ver um destes objectos a partir da Terra, é necessário
que a luz do Sol, sob a forma de minúsculas partı́culas chamadas fotões, viage
durante mais de cinco horas, seja reflectida pela superfı́cie do objecto, e volte a
viajar outro tanto de volta até nós. Não é que os fotões cheguem cá cansados.
O problema é que chegam poucos — a maior parte perde-se pelo caminho. Em
astronomia, os fotões são os mensageiros de toda a informação que nos chega.
118
Resumo em Português
Quanto mais fotões recebemos de um determinado objecto, mais informação
podemos obter. É por chegarem poucos vindos dos objectos trans-Neptunianos
que nós sabemos tão pouco acerca deles. Provavelmente, um dia teremos de
mandar sondas, semelhantes às que vão a Marte, para aprendermos mais sobre
estes pequenos mundos gelados.
Esta tese
Nos textos cientı́ficos, os objectos de Kuiper são vulgarmente designados por
KBOs (do inglês, Kuiper Belt objects). Por vezes, também se usa a designação
TNOs, de Trans-Neptunian objects. Julga-se que os KBOs sejam uma espécie
de planetas bebés. No modelo de Kant-Laplace referido atrás, os aglomerados
que ficam em órbita em torno do Sol vão-se juntando e crescendo até formar os
planetas. Os astrónomos chamam a estes aglomerados em fase de crescimento,
“planetesimais”. Na região dos KBOs, por ser distante do Sol, havia menos
material para formar planetesimais. Por isso os KBOs cresceram mais devagar.
Quando o material se esgotou os KBOs estavam ainda na fase da infância, longe
de se tornarem planetas. E assim ficaram, congelados. Por essa razão, o estudo
dos KBOs é vital para perceber o processo da formação dos planetas.
Esta tese apresenta um estudo das formas e rotações dos KBOs. Para explicar a importância de conhecer as formas dos KBOs, podemos usar o exemplo
dos asteróides. Os mais pequenos tem formas equivalentes às dos estilhaços que
saltam ao partir uma pedra. Isto indica que estes objectos são isso mesmo: estilhaços resultantes de violentas colisões entre asteróides maiores. Os Capı́tulos
3 e 4 desta tese mostram que grande parte dos KBOs são relativamente redondos, mas que existe uma percentagem considerável com formas mais ovóides, ou
alongadas. Não é possı́vel para já saber porquê. Por outro lado, a velocidade de
rotação de um KBO dá-nos informação acerca da sua estrutura interna, especialmente se soubermos a forma do KBO. Por exemplo, se o objecto for maciço,
pode rodar bastante depressa, e a sua forma não se alterará significativamente.
Mas se o KBO for um amontoado de pequenos fragmentos, a rotação deformará
o objecto tornando-o mais ovóide. Se rodar demasiado depressa pode mesmo
desintegrar-se. No Capı́tulo 4, as formas e as rotações dos KBOs são analizadas
em conjunto, e parecem indicar que os KBOs são mesmo amontoados de blocos
mais pequenos, que se mantém juntos graças à força da gravidade. O Capı́tulo 5
estuda as colisões entre KBOs. O objectivo é verificar se a rotação dos KBOs foi
provocada por colisões entre eles, desde que se formaram até aos nossos dias. A
conclusão é que os maiores KBOs têm a mesma rotação que tinham quando se
formaram, mas a rotação dos mais pequenos foi completamente alterada pelas
colisões. Por outro lado as colisões parecem ser a razão pela qual os KBOs são
amontoados de fragmentos: impactos sucessivos foram rachando e partindo o
interior destes objectos transformando-os lentamente em blocos de entulho.
No final do Capı́tulo 1 é apresentado um resumo mais detalhado desta tese.
Curriculum vitae
M
arch 27th, 1975. That was my birthday. I was born in Lisbon, Portugal.
I learned to speak relatively early, and as soon as I could say a few words,
I started asking questions. I had no particular preference on the subject, which
later proved to be one of my good (and bad) characteristics. I went to school
at the Escola do Magistério Primário, Escola Preparatória Delfim Santos, and
finally Escola Secundária de Benfica. I liked school a lot. Mathematics was
my favourite subject. Teachers say I was always agitated, and distracting my
colleagues. Besides school I did swimming, tennis, and judo. And I have always
liked music, which I often played with friends. When I was about ten years old
my uncle gave me his old computer, a ZX Sinclair 48K. In hindsight I think
this was very good for me. I lived most of my youth in a neighbourhood called
Benfica. Most of my friends are there.
After highschool I went to the Instituto Superior Técnico in Lisbon, to become an engineer. A little over 3 years later I decided that I wanted to study
theoretical Physics instead, and moved to the Faculty of Sciences of the University of Lisbon. In the meantime I worked part-time at an internet service
provider. Another 3 years later, when I was in my last year of Physics, I worked
at the Center for Astronomy and Astrophysics of the University of Lisbon. During that time I joined observing runs with Prof. Yun at La Silla, and Dr.
Roos-Serote at Calar Alto. In 1999 I attended a summer school on solar system
science (EVISS, Coimbra), organized by Dr. Roos-Serote, where I met Prof.
Jewitt. Shortly after, I applied, and was accepted, to a PhD position in Leiden,
to work with Dr. Jane Luu on Kuiper Belt research.
I was very fortunate to be able to live and study in Leiden. During my PhD
research I carried out several observing runs at the William Hershel and Isaac
Newton telescopes on La Palma, and at the Dutch 0.9m, Danish 1.5m, ESO
2.2m telescopes at La Silla. I participated in the NOVA fall school in Dwingeloo
(2000), and attended conferences at Meudon (2001), Penn State (2001), the
NAC in 2002, Berlin (2002), Porto (2002), and the IAU General Assembly in
Sydney (2003). I assisted Dr. Luu with the teaching of the Leiden-Dwingeloo
Summerschool in Astrophysics (2001). I enjoyed a very fruitful visit to the
Harvard-Smithsonian Center for Astrophysics (2002), where I worked with Dr.
Kenyon. At the CfA, I assisted Profs. Donnelly and Stanek with the course
Astronomy 1.
In the coming 3 years, I will continue my studies of Kuiper Belt objects at
Coimbra, with Prof. João Fernandes, and at Hawaii, with Prof. Dave Jewitt.
Nawoord / Acknowledgments
W
ritten words take space whereas feelings and memories don’t. Therefore
I will only be able to mention a few of the many people that helped in the
making of this thesis. Nevertheless, I would like to thank everyone I have had
contact with in the past, for I have learned from all of you.
To everybody at the Sterrewacht, my deepest thanks for making my stay
such a pleasant one. I leave with the wish that I may come back. Marja Zaal,
Janet Soulsby, Kirsten Groen, and Jeanne Drost, thank you for being always
so helpful. The Sterrewacht computer group has been outstanding everytime
I needed their support. I thank Garrelt for being the best office mate I could
possibly have. I am grateful to Glenn, Michele, and others for many fruitful
discussions, in the spirit of how I believe that science should be pursued. I
thank the SocCom and the Big Dipper for organizing many entertaining events,
and for the Friday borrels. I have enjoyed the hospitality of everyone at the
Harvard-Smithsonian Center for Astrophysics, where I conducted a part of the
research presented here. I thank Mango and her housemates for all their support,
patience, and friendship, in a crucial step during the preparation of this thesis.
Antonietta, Chris (moran taing!), Gaetano e Luca, vi ringrazio per tanti
giorni carini, e perché siete delle persone molto belle. Mariska, thank you for
not allowing any day at the Sterrewacht to be boring. I thank Maaike for always
smiling, and adding a bit of poetry to many days. Davor, thanks for being a
good friend. As my allochtoon buddy, you have a unique perception of how much
I will miss Leiden. Ivo, thank you for your friendship, which I deeply treasure.
I am greatly indebted to all my past teachers for my academic education. In
connection with this thesis, I thank Prof. Yun and Prof. Agostinho for motivating
me to pursue my studies in astronomy. I acknowledge Dr. Roos-Serote for his
contribution to Portuguese astronomy, and for organizing a great summer school
in Coimbra, which eventually triggered my coming to Leiden. Catarina, obrigado
por me teres encorarajado. Obrigado a todos os meus amigos em Portugal, por
serem sempre uma boa razão para voltar.
Lieve Maaike, dankjewel voor de mooiste vertaling van mijn samenvatting,
voor jouw geduld met me, en voor 10000 andere dingen.
Por último, quero agradecer a toda a minha famı́lia porque foi nela, e dela,
que formei a base de quem eu sou.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement

Languages