THE SIZE DISTRIBUTION OF TRANS-NEPTUNIAN BODIES
1
G. M. Bernstein and D. E. Trilling
Department of Physics and Astronomy, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street,
Philadelphia, PA 19104; garyb@physics.upenn.edu, trilling@astro.upenn.edu
R. L. Allen
Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada;
lallen@astro.ubc.ca
M. E. Brown
Division of Geological and Planetary Sciences, Mail Code 150-21, California Institute of Technology, Pasadena, CA 91125;
mbrown@gps.caltech.edu
M. Holman
Harvard-Smithsonian Center for Astrophysics, Mail Stop 51, 60 Garden Street, Cambridge, MA 02138;
mholman@cfa.harvard.edu
and
R. Malhotra
Department of Planetary Sciences, University of Arizona, 1629 East University Boulevard,
Tucson, AZ 85721; renu@lpl.arizona.edu
Receivved 2003 August 26; accepted 2004 May 25
ABSTRACT
We search 0.02 deg
2
of the invariable pla ne for trans-Neptunian objects (TN Os) 25 AU or more distant using
the Advanced Camera for Surveys (ACS) aboard the Hu bble Spac e Telescope. With 22 ks per pointing, the
search is more than 50% complete for m
606W
29:2. Three new objects are discovered, the faintest with mean
magnitude m ¼ 28:3 (diameter 25 km), which is 3 mag fainter than any previously well-measured solar system
body. Each new discovery is verified with a f ollow-up 18 ks observation with the ACS, and the detection
efficiency is verified with implanted objects. The three detections are a factor of 25 le ss than would be expected
under extrapolation of the power-law differential sky density for brighter o bjects, (m) dN=dm d / 10
m
with 0 :63. Analysis of the ACS data and recent TNO surveys from the literature reveals departures from this
power law a t both the bright and faint ends. Division of the TNO sample by distance and inclination into
‘‘classical Ku iper belt’’ (CKB) and ‘‘Exc ited’’ samples reveals that (m) differs for the two populations at 96%
confidence, and both samples show departures from power-l aw behavior. A d ouble power-law (m) adequately
fits all data. Implications of these departures include the f ollowing: (1) The total mass of the ‘‘classical’’ Kuiper
belt is 0.010 M
, only a few times Pluto’s mass, and is predominantly in the form of 100 km bodies (barring a
secondary pe ak in the mass distribution at sub–10 km sizes). The mass of Excited objects is perhaps a few times
larger. (2) The Excited class has a shallower bright-end magnitude (and, presumably, size) distribu tion; the
largest objects, including Pluto, make up ten s of percent of the total mass whereas the largest CKB objects are
only 2% of its mass. (3) The derived size distributions predict that the largest Excited body should be roughly
the mass of Pluto, and the largest CKB body should have m
R
20—hence, Pluto is feas ibly considere d to have
originated from the same physical pro cess as the Excited T NOs. (4) The observed deficit of small TNOs occurs in
the size regime where present-day collisions are expected to be disruptive, suggesting extensive depletion by
collisions. The Excited and CKB size distributions are qualitatively similar to some numerical models of growth
and erosion, with both a ccretion and eros ion appearing to have proce eded to more advanced stag es in the Excited
class tha n in the CKB. (5) The lack of detections of d istant TNOs implies that if a mass of TNOs co mparab le to
the CKB is present near the invariable plane beyond 5 0 AU, that distant population must be composed primarily
of bodies smaller than 40 km. (6) There are too few small C KB objects for this populatio n to be the reservoir of
Jupiter-family comet precursors with out a significant upturn in the population at diameters unde r 20 km. With
optimistic model parameters and extrapola tions, the Excited population could be the source reservoir. Implica-
tions of these discoveries for the formation and evolution of the outer solar system are disc ussed.
Key words: Kuiper belt — solar system: formation
1. MOTIVATION
The nebular hypothesis for the formation of planeta ry sys-
tems is nearly 250 years old (Kant 1755), and yet observa-
tional support f or the model is relatively recent. In the stan-
dard scenario, solids in the disk surrounding t he protostar
begin to coagulate into macroscopic objec ts, which accrete to
kilometer sizes. When the planetesimals become massive
enough for gravitatio nal focusing, runaway accre tion begins.
In the oligarchic growth phase, accretion is limited by exci-
tations in the population induced by the largest few objects. In
a protoplanetary disk, these largest planete simals ca n reach a
1
Based on observations made with the NASA / ESA Hubble Space Telescope,
obtained at the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA con-
tract NAS 5-26555. These observations are associated with program GO-9433.
1364
The Astronomical Journal, 128:1364–1390, 2004 September
# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
few Earth masses, sufficient to tr ap the nebular gas, and rapid
growth of gas giants can ensue. The nebular gas is cleared by
the stellar wind, and the remaining planetesimals are scattered
away by the giant planets.
Today we have many observations of dust and gas disks
around young stars (O’Dell & Beck with 1997; Beckwith et a l.
2000), evidence that supports the nebular hypothesis. In ad-
dition, observations of dust disks around somewhat older stars
suggest the presence of a population of dust-produc ing plan-
etesimals in those systems (e.g., Smith & Terrile 1984; Greaves
et al. 199 8; Koerner et al. 2001). Some of these dust disks
exhibit structures that can perhaps be ascribed to embedded
planetary s ystems (Kuchner & Holman 2003). There is also
now abundant e vidence for the final stag e of accretion—planet
formation—as extrasolar giant planets have been de tected b y
radial velocity and transit observations (Marcy et al. 2000).
Though the basic idea o f the nebular hypothesis remains in-
tact, each new round of observations has led to fundamental
changes in our v iew of planet formation. The presence of gas
giants at less than 1 AU, for example, was not well anticipated
by the ory, and migration is now recognized a s an importa nt
process.
It is unfortunate that direct observation of planetesimals
smaller t han 1000 km in extrasolar systems i s currently in-
feasible and likely to remain so for many decades. Such obser-
vations would likely r eveal further failures of imagination
in our m odeling of the planetesimal phase. Fortunate ly, a
portion of the Sun ’s planetesimal population is preserved for
our examination in the region be yond Neptu ne, where g rowth
timescales are longer, the accr etion process apparently did n ot
proceed to formation of planets, and the influence of the giant
planets was not sufficient to remove all small bodies. Study of
trans-Neptunian objects (TNOs) provides ‘‘ground truth’’ for
models of the accretion, collisional erosion, and dynamical
evolution of planetes imal populations. True to fo rm, the TNO
population only vaguely resembles the prec onception of a
dynamically pristine planetesimal disk. With over 800 TNOs
discovered between 1992 and the pr esent, it is cle ar that th e
TNO population has several distinct dyna mical components,
all of which appear to have eccentricity and inclination dis-
tributions that are too broad to be the undisturbed remnants o f
the primord ial population. The TNO population contains un-
mistakable signatures of interactions with Neptune and per-
haps other massive bodies. With further study, we can hop e to
understand the dynamical history of this region.
The physical properties of the TNOs, particularly the size
distribution, are indicative of the accretion proce ss. O bserva-
tions to date are consistent with a distribution of diameters D
that is a power law, dN=dD / D
q
with q ¼ 4:0 0:5 (Trujillo
et al. 2001). This distribution must fail at some D > 0to
avoid a divergence in the mass or reflected surface bright-
ness of the trans-Neptunian cloud, but the scale of the
breakdown in the power law is not usefully bounded by these
constraints (Kenyon & Windhorst 2001). In the current dy-
namical environment, TNO collisions are erosive for objects
with diameters P100 km, so that small objects have been re-
moved from the population since the events or processes t hat
excited the TNO dynamics (Stern 1996). Rather so on after the
discovery of the Kuiper belt, there was s pecu lation tha t t he
size distribution might break at 50 km sizes (Weissman &
Levison 1997), but observa tions to date have n ot evidenced
this phenomenon. A generic prediction of accretion/erosion
models is a break to a shallower size distribution b elow some
size, but the size break is dependent upon factors such as the
duration of the accretion epoch (Farinella et al. 2000). Th e
mass in the trans-Neptunian region must have been substan-
tially larger in th e past in order to support the migration of
Neptune ( Hahn & Malhotra 1999) and accretion of the present
TNO pop ulation (Stern 1996), but the relative importance
of scattering and collisional grinding in mass removal is
unknown.
Extending our knowledge to the faintest (and hence
smallest) possible TNOs is clearly desirable, as there may be
signatures of the collisional evolution or processes unantici-
pated by present theory. It is of further interest to see if the size
distribution has a dependence upon dynamical properties, as
this can provide fur ther i nsight in to the dependence of the
accretion/collision process upon the dynamics of the parent
population.
The Hubble Space Telescope (HST ) is currently the obser-
vatory of choice for detection of the faintest possible point
sources. A detection of a ve ry high density of m
V
> 27:8
TNOs using the Wide Field Planetary Ca mera 2 ( WFPC2) on
HST is reported by Cochran et al. (199 5, hereafter CLSD). For
various reasons, it is likely that these detections were merely
noise (Brown et al. 1997; Gladman et al. 1998; see Bernstein
et al. 2004a for further analysis of the WFPC2 results). The
installation of the Advanced Camera for Surveys (ACS) on
HST subtanstially improved the field of view, efficiency, and
sampling. This paper describes the results of a large invest-
ment of HST time (125 orbits) into a search for TNOs using
the ACS.
Detection of faint objects requires long integration times,
but a typical TNO moves the width of the HST point-sprea d
function (PSF) in only a few minutes. The ACS survey
therefore uses a tec hnique we call ‘‘digital tracking,’’ in which
a long series of exposures is acquired, with each individu al
exposure short enough to avoid trailing losses. The short
exposures are shifted to follow a candidate TNO orbit and
then summed, y ielding an image with long exposure time that
will detect TNOs on the chosen orbit with no trailing. The
summation must be repeated for all plausible TNO orbits that
diverge by more than the PSF over the time span of the ob-
servations. This computationally intensive technique has been
used successfully for several ground-based faint-TNO searches
(Tyson et al. 1992; Allen et al. 2001; Gladman et al. 1998;
Chiang & B rown 1999), and a varia nt was used by CLSD
with WFPC2 data. We are able to detect TNOs to the funda-
mental limits set by photon noise in the 22 k s total ex posure
time of each ACS sear ch field. The survey is ov er 50%
complete for m
606W
< 29:2 mag, which is 2 mag fainter than
any succ essful published TNO surv ey and 1.5 mag deeper
than the onset of false positives in the CLSD data. The area
covered by t he search is 0.02 de g
2
, 13 times the area of the
CLSD search. The lessons learned f rom the ground-based and
CLSD digital-tracking survey s have helped us to produce
results that we believe are optimal and reliable.
The concepts and fundamental limits of digital tracking in
this and other applica tions are detailed in Bernstein et al.
(2004a). This paper summa rizes the me thodology of th e ACS
search, presents the detections and efficiencies, derives bounds
on the apparent magnitude distribution of the TNOs and some
dynamical subsamples, and discusses the implications for the
evolution of the TNO system. Trilling & Bernstein (2004) pre-
sent the variability data for the objects detected in the ACS
survey. Bernstein et al. (2004b) examin es the current state of
the art in astrometry for moving objects and the utility of
high-precision as trometry for orbit determinatio n.
SIZE DISTRIBUTION OF TNO
s 1365
2. DETECTION TECHNIQUES
The search for moving objects to the photon-no ise limit of a
22,000 s ACS integration requires a sop histicated analy sis,
attention to detail, approximately 30,000 lines of code, and
several CPU-year s’ worth of computation on 2.4 GHz Pentium
processors. The unique tools of this data reduction are de-
scribed in detail i n B ernstein et al. (2004a), but we summarize
here the aspects that are important for understanding the results.
2.1. Obs erv vations
The survey covers six slightly overlapping fie lds of view of
the A CS. The spacecraft is oriented so that detector rows and
columns are aligned to the lo cal ecliptic cardinal directions.
The six pointings are arranged in a 2 ; 3 mosaic, with the long
axis in the ecliptic north-south direction. The southern two
pointings are labeled A and B, the central two C and D, and
the northern two E and F. The ACS pixel scale is nominally
0B050, and nominal coverage of the full mosaic field of view
(FOV) is 400
00
; 600
00
¼ 0:01 9 deg
2
. The exposures at a
given pointing are dithered by noninteger pixel steps, up to
a few pixels, in order to improve the sampling of the static
sky objec ts. The imaged field is not contiguous, because our
dithers do not span the gap between the two ACS CCDs.
The field location was chosen subject to a number of cri-
teria. The field center, 14
h
07
m
53
s
:
3, 11
21
0
38
00
(J2000), is
only 3
0
from the invariable plane. The field trails Neptune by
99
, within the libration region for perihelia of TNOs in 2:1
and 3:2 resonance with Neptune (Malhotra 1996; Ch iang &
Jordan 2002). A known T NO, 2000 FV
53
, is within pointing A
for the full observing period, allowing us to verify our navi-
gation and orb ital calculations. The field is place d a nd th e
observations timed to minimize the loss of observing time
to moonlight and South Atlantic Anomaly crossings, and to
place the fi eld 88
from opposition at the start of the observing
sequence (see below).
All exposures wer e taken through the F606W filter of the
ACS using the Wide Field Channel (WFC). In the period UT
2003 January 26. 014–31.341, which we call the ‘‘discovery
epoch,’’ 55 ; 40 0 s exposures were taken at each of the six
pointings.
2
During 2003 February 5.835–9.703, the ‘‘ re-
covery epoch,’’ a n additional 40 ; 400 s exposures were taken
at each pointing. The two sets of obse rvations, 88
–83
and
77
–73
from opposition, were chosen to straddle th e tra nsi-
tion from prograde to retrograde motion for most TNOs.
Hence any discovered objects have a maximal chance of
remaining in the mosaic FOV for the full 15 day duration of
the HS T observations, and the image trailing due to apparent
motion is minimized.
Individual exposures are 340–410 s long, averaging 400 s.
Five exposures fit into a typical HST orbit, with fewer during
radiation-impacted orbits. A set of 10 or 1 5 exposures is taken
during each HST visit to a given pointing. The pointings are vis-
ited in the pattern ABAB-CDCD-EFEF-ABAB-CDCD-EFEF
during each of the two epochs of observation. So pointing
C, for example, is sampled sporadically, at intervals as clo se
as 8 minutes, over a time span of ap proximately 24 hr, during
the first CDCD set of visits. Approximately 2 days later, the
CDCD set of visits are repeated . The n, 7 days later, the cy cle
repeats for the recovery epoch .
A few shorter ex posures of the six pointings and of the
outskirts of 47 Tucanae were taken in order to map the WFC
point-spread function and provide astrometric tie-ins. The
performance of HST and ACS during the observations was
nearly flawless. Comparison of the 47 Tuc images before and
after the TNO observing cycle showed negligible change in
the PSF, so we use a time-invariant (but spatially dependent)
PSF map.
2.2. Preprocessinggand ‘‘Brigght’’ Object Detection
Once the data are placed in the HST archive, they are pre-
pared for the moving-object search as follows:
1. Bias removal, fla t-fielding, and bad-pixel flagging are
done by the HST ‘‘on the fly’’ processing. Engineering key-
words are checked for guiding errors or other problems. The
uncertainty images are corrected for some errors in the STScI
pipeline, and we creat e a weight image with the value w at each
pixel being 1/
2
(where is the pixel’s flux uncertainty). The
weight is zeroed for defective and saturated pixels, and the data
and weight images are changed into flux units.
2. Objects in individual exposures are cataloged using
SExtractor (Bertin & Arnouts 199 6).
3. The exposures of 47 Tuc are used to produce a map of
the PSF for the WFC (Bernstein et al. 2004a).
4. WFC distortion maps from STScI or from Anderson
(2003) a re used to transform pixel positions into a local tangent
plane for each exposure, to accuracy 10 mas; a translation
and linear transformation are derived for each exposure to
register all the cataloged objects onto a global tangent-plane
coordinate system centered on the mosaic center.
5. All exposures from the discovery epoch are combined
into a deep image of the fixed sky. This template image has
0B025 pixels that are square (no distortions) on the global
tangent plane, so that the PSF is now sampled near the Nyquist
density. Because there are 55 exposures per pointing in the
discovery epoch, each template pixel has 10 or more contrib-
uting images, and the template noise level is well below the
individual exposures’. Sigma-clipping eliminates cosmic rays
and bright moving objects from the template images.
6. We interpolate the template image to the location of each
pixel of each individual exposure. The interpolated template is
then subtracted from each exposure. At pixels with very high
flux (centers of bright stars and galaxies), we zero the weight
image because the residuals to the template subtraction will rise
above the noise. Note that the individual exposures have not
been resampled in producing these ‘‘subtracted images.’’
7. Artificial TNOs are added to the subtracted images. One
of us (M. H.) produces a list of objects with orbital elements
and light-curve parameters selected at random from a chosen
range. The positions, magnitudes, and motions of these objects
are calculated for each exposure. The position-dependent PSF
is trailed for the motion and each artificial object added into
the subtracted images, with appropriate Poisson noise in each
pixel. One-third of the objects on the list are later revealed to
the searchers (G. M. B. and D. E. T.) for use in tuning the
search algorithms. The searchers remain blind to the other two-
thirds of the artificial objects until after a final TNO candidate
list is produced.
8. The subtracted images are searched for potential bright
TNOs as follows. A PSF-matched, compensated filter is
scanned a cross each subtracted image. Using the weight image,
we can calculate t he significance (i.e., the signal-to-noise
ratio) of each candidate point-source peak in the subtracted
image. All peaks with j j3:5 are noted and the
2
of a fit to
the PSF is calculated. Those that sufficiently resemble the PSF
2
Exposure times varied slightly because of spacecraft constraints.
BERNSTEIN ET AL.1366 Vol. 128
are recorded to a file of bright-TNO candidates, to be examined
later. Note that real TNOs will not fit the PSF precisely, be-
cause of trailin g, so our criterion for matching the PSF is kept
loose, and the vast majority of candidates are cosmic rays.
9. The subtracted images are ‘‘cleaned’’ in preparation for
the faint-object search as follows. Every pixel in the subtracted
image that deviates by more than 5 from the mean sky level
is flagged. All weights are set to zero within a 2 pixel radius of
each flagged pixel. This effectively masks all cosmic rays and
non-Gaussian noise in the subtracted images, which is ex-
tremely important for avoiding fals e positive detections in the
faint-object search. This process also masks bright TNOs and
asteroids; the former have already been detected, however, in
the previous step.
10. A ‘‘flux image’’ is now created for each exposure. The
flux image is created on a regular grid in the global tangent-
plane coordinates. Each such grid point is mappe d back to a
pixel position on the masked subtracted image, and we record
the best-fit PSF flux and its uncertainty for a point source at
that location. Hence the ‘‘flux image’’ is a map of the bright-
ness of a potential point source at any location in that exposure,
and a weight (uncertainty) image is propaga ted as well. These
flux images are the raw material for the faint-object search.
Any pote ntial bright moving objects must now be found on
lists produced in step 8, because the masking in step 9 may
preclude their later detection. ‘‘Bright’’ in this context means
detectable at 3.5 in a single 400 s HST ex posu re, which in
practice corresponds to m 27:6. We us e h ere and henceforth
the HST F606W magnitude system unless otherwise noted.
The filter passband is roughly the union of V and R passbands,
and the AB zero point is similar to a V zero point.
Over 900,000 flux peaks trigger the 3:5 threshold in the
discovery epoch. To fish the real (and implanted) TNOs from
this sea of cosmic rays, we first re quire that a flux peak repeat
in the same sky location (to 0B2) on successive exposures in
one orbit, leaving 7700 pairs of detection s. We rej ect linked
detections that occur on the same detector pixels to avoid
CCD defec ts. We next require two pairs of detections to exist
within the same visit and be within 2
00
hr
1
of each other,
leaving 1300 candidate q uadr uples of detections. Next a pre -
liminary orbit is fitted to each quadruple, and the methods of
x 2.3.2 are used to check whe ther the subtracted images are
consistent with a point source moving on the putative orbit.
This reduces the candidate list to 49 objects, of which 46 are
then re vealed to be on the a rtificial-object list. The detectio n
efficiency of the bright search for artificial objects is found to
be 100% for m P 27:6.
The three remaining objects are r eal: one is 2000 FV
53
,the
previously known object, which at m ¼ 23:4 is blindingly
bright here, appearing at 80 in each of the 55 disc over y-
epoch exposures and 40 recovery-epoch exposures. The second
bright detection is a new object, now given the preliminary
designation 200 3 BG
91
, with time- averaged magnitude hmi¼
26:95 0:02. The third d etection from the bright search , 2003
BF
91
,hashmi¼28:15 0:04 but is hig hly va riable a nd ris es
above the ¼ 3:5 single-exposure threshold several times.
The bright-object search is executed independently on the
recovery-epoch observation s, revea ling the sa me th ree objec ts,
which are thus undoubtedly real.
2.3. Faint-Object Search
The search for moving objects t hat are below the single-
exposure detection threshold is much more computationally
intensive. We must sum the available exposures along any
potential TNO path through the discovery-epoch exposures
and then ask whether the best-fit flux for this path is safely
above the expected noise level.
2.3.1. The Search Space
The space of TNO orbits is six-dimensional, with one
possible parameterization being {, , d,
˙
,
˙
,
˙
d}, where
and are the angular position relative to th e center of the
mosaic at some reference time T
0
, d is the geocentric distance
at T
0
, and the overdots deno te the TNO’s space velocity in the
same basis (cf. Bernstein & Khushalani 2000) . Th e line- of-
sight motion
˙
d has negligible observable effect over the course
of the 15 day HST observation, so we may set it to zero in our
searches. T his means we have five dimensions of TNO orbit
space to search. We search on a grid of points i n this space.
The grid spacing in an d is the pixel scale P of the flux
images discussed above . The grid spacing v in the velocity
space (
˙
,
˙
) should be fin e enough that tracking errors are
held to less than 1 pixel: v P=T,whereT is the time
span of the observations being combined. Finally we must
choose a grid in distanc e d. The primary eff ect of d upon the
apparent motion of the TNO is fro m the re flex of Earth’s orbit
around the Sun (and HST ’s orbit around Earth). The reflex
motions scale as 1/d, so we choose a grid that is uniform in
1=d. We also note that the nonlinear components of the
TNO apparent motion all depend solely upon —primarily the
reflex of Earth’s orbital acceleration, but also the Newtonian
gravitational acceleration of the TNO itself. The spacing
must be fine enough that errors in these nonlinear motion
components are held toTP.
The numb er of grid points tha t must be searched then scales
roughly as P
5
T
3
. We conduct o ur faint-object search in
two passes: first with P ¼ 0B050, and then a finer pass with
P ¼ 0B030. The first pass ru ns quickly enough to have been
completed between receipt of the HST data in mid-Fe bruary
and scheduled follow-up observations at the Keck an d
Magellan telescope s i n late April (see x 2.4). But the PSF of
the WFC is only 0B05 across, so mistracking by 0.5P at
P ¼ 0B05 causes significant blurring of the PSF in digitally
tracked images, degrading our magnitude limit by 0.2 mag.
Hence we later run the finer grid search to reach the ultimate
limit of the WFC data.
The bounds of the s earch space are determined as follows:
a) We search 25 AU < d < 1. Even objects at d
1000 AU would move several ACS pixels over the course of
our visits.
b) The perihelion of the orbit is constrained to be 10 AU.
This places a lower limit on the transverse motion at a given d.
c) The orbit is assumed to be bound. This places an upper
limit on the transverse motion at a given d.
d) The inclination of the orbit is assumed to be i < 45
.
This bounds the vertical component of the apparent motion.
Note that we search only prograde orbits.
2.3.2. Steps for the Faint Search
The faint search pro ceeds after ste p 10 abov e as follo ws for
each of the coarse P ¼ 0B05 (discovery and r ecovery epochs)
and the fine P ¼ 0B03 (discovery only) searches:
1. The flux images produced for this P in the search are
split into six sets of visits. Set 1 contains the first ABAB
sequence, set 2 the first CDCD sequence, etc. The digital-
tracking sums wi ll be accumulated over a set’s worth of
SIZE DISTRIBUTION OF TNO
s 1367No. 3, 2004
images, with time span T P 24 hr. Digital tracking over the
full 5 day time span of the discovery epoch would be com-
putationally infeasible.
2. For each set, the outermost loop is over the distance grid.
The next inner loop is over the velocities
˙
and
˙
. At a given
distance and velocity, we calculate an orbital shift for each
exposure relative to the first exposure. The inner loops consist
of summing the individual flux images at each pixel, with
integer pixel shifts defined by the velocity and distance.
In the fine search, there are 13 distance grid points and a total
of 7 ; 10
5
velocity grid points in the f
˙
;
˙
; dg-space. For each
set there are two pointings spanning 1 ; 10
8
pixels in the flux
images, with 25–30 exposures per pixel per set. In total, the fine
search tests 10
14
points in the TNO phase space, requiring
10
16
pixel additions to do so. This takes several CPU-years
for 2.4 GHz Pentium 4 proces sors, but a cluster of 10 CPUs at
Penn and eight at Arizona reduces the required real time.
3. At each grid point of the TNO search, the point-source
fluxes along the track of the putative TNO from all exposures
in the set are summed, as weighted by their inverse uncertainties,
to form a total best-fit flux and uncertainty. If the significance
f =
f
exceeds a threshold of 4.0, the grid point is saved.
4. Above-threshold grid points that abut in phase space are
aggregated, and the most significant is saved. The output of the
fine search is a list of 1:5 ; 10
8
significance peaks in the TNO
phase space.
5. For each detected peak a ‘‘tune-up’’ program is run,
which fits a model moving point source to the pixel values in
postage stamps from all subtracted images. The gridded peak is
the starting point, and , ,
˙
, and
˙
are allowed to vary. The
significance of real (or impl anted) objects typically rises after
tune-up, since the optimized orbit is a bett er fit than the nearest
grid point, and the position and velocity estimates become
more accurate. Significance peaks that are noise tend, however,
to become less significant, to have poor
2
, and /or to fail to
converge. The tune-up step reduces the number of 4:0
peaks in the fine search to 6 ; 10
7
.
6. The tuned-up peak catalog from one set is now compared
with all other sets of the epoch; any pair of peaks that might
correspond to a common orbit are linked and passed to the next
step. Note that TNOs that cross the boundaries of the ACS
pointings are found as efficiently as those that do not. There are
3 ; 10
5
(nonunique) linked peak pairs in the fine search, of
which 3 ; 10
4
have total significance 7.
7. All the linked pairs with 7 are again run through the
tune-up program, but this time all of the exposures from the
entire epoch are used. The arc is now sufficiently long (typi-
cally 3 days) that we can allow the distance d to vary without
fear of degeneracy. A few detection candidates with
2
dof
>150 in the fit of the moving-source model to the data are
rejected; inspection shows these to be spurious detections near
the residuals of diffraction spikes of bright stars. We apply
a threshold of 8:2( 10 for the coarse search) to ob-
tain the TNO candidate list of 100 objects.
The histogram of detections versu s significance rises very
rapidly below ¼ 8:2, wh ich is to be e xpected from Gaussian
noise in a search of 10
14
or so p hase- space locations (Bern stein
et al. 2004a). The t hresho ld is p laced at the tail of this fal se-
positive distribution. Detection candidates above this threshold
are inspected by e ye, with two to three being c learly associated
with subtraction residuals and other data flaws.
In the coarse search, there were 92 detections with 10.
The blind list of artificial TNOs was then revealed, and 89
of the 92 were f ound to be implanted objects. Two of the
remaining detections, 2000 FV
53
and 2003 BF
91
, coincide
with bright detections . The last is a new object, 2003 BH
91
,
discovered with significance ¼ 16:7 and mean magnitude
m ¼ 28:35.
For t he fine search (which had an inde pendent set of
implanted objects), there were 67 detections, of which 64 were
found to be on the list of impla nted TNOs. The three re-
maining d etections are again 2000 FV
53
, 2003 BF
91
, and 2003
BH
91
.
The fa int-sea rch techniqu e was also applied to the re covery
epoch with a coarse (P ¼ 0B05) grid. The same candidates
were independe ntly detected above the ¼10 threshold.
Figure 1 shows postage-stamp images of 2000 FV
53
and the
three new detections, as we improve the depth of images by
summing more exposures.
2.4. Orbit Determination and Recovvery
Each of the TNOs detected in the discovery epoch is also
clearly detected in the recovery e poch. We now combine the
information from all exposures in the en tire A CS campaign
to get the be st po ssible constraint on each ob ject’s o rbit.
We again invoke the tune-up program, whereby the orbital
parameters are varied to maximize the significance of the de-
tection of the moving p oint source. More specifically, the
orbital p arameters determine the location of the PSF and the
degree of trailing in each individual exposure. The two end-
points of the trail are converted into pixel coordinates using
the registration information and the distortion maps. We cal-
culate the PSF at the TNO location using the spatially varying
PSF ma ps from 47 Tuc, and we smear this PSF to the requ ired
trail length. The flux of th e TNO is allowed to vary in a
stepwise fashion from orbit to orbit (or from exposure to ex-
posure for the high signal-to-no ise ratio 2000 FV
53
). A model
with constant flux for a give n TNO would be a poor fit, as
all the detected objects have significant flux v ariations . The
moving, variable-flux model is then fitted to the subtrac ted
images, with all orbital elements and fluxes being optimized.
A by-product of this orbital optimization is an optimally
measured light curve f or each object. Analysis and interpre-
tation of these light curves is presented in Trilling & Bernstein
(2004).
For the final orbit determination, all six orbital parame-
tersareallowedtovary.The2000FV
53
data are of such
high quality—po sitional accuracy of 1masforeachof
the 95 exp osures—that the line-of-sight velocity, and hence
a and e, are significantly constra ined with only a 13 day a rc.
Bernstein et al. (20 04b) will consider in d etail the techniques,
limitations, and benefits of such high-precision astrometry for
the determination of solar system orbits.
For the three newly detected objects, the line-of-sight mo-
tion is still poorly determined over the 13 day arc. In the final
orbit fit to the HST data, we include a prior constraint on the
kinetic and potential energies that weak ly pushes the orbit to
circularity:
2
prior
¼ 4(2 KE=PE þ 1)
2
: ð1Þ
An unbound or plunging orbit is thus penalized as a 2
deviation. The results of the fitting process are best-fitting or-
bital parameters (in the {, , ... } ba sis) for each object and
covariance matrices for each, whic h can be used as described
BERNSTEIN ET AL.1368 Vol. 128
