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arXiv:0912.2996v1 [astro-ph.EP] 15 Dec 2009
A single sub-km Kuiper Belt object from a stellar
Occultation in archival data
H. E. Schlichting1,2, E. O. Ofek1,3, M. Wenz4, R. Sari1,5,
A. Gal-Yam6, M. Livio7, E. Nelan7, S. Zucker8
1Department of Astronomy, 249-17, California Institute of Technology, Pasadena, CA 91125, USA
2CITA, University of Toronto, 60 St. George St., ON, M5S 3H8, Canada
3Einstein Fellow
4Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA
5Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel
6Faculty of Physics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel
7Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
8Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv 69978, Israel
The Kuiper belt is a remnant of the primordial Solar System. Measurements of
its size distribution constrain its accretion and collisional history, and the importance
of material strength of Kuiper belt objects (KBOs)(1; 2; 3; 4). Small, sub-km sized,
KBOs elude direct detection, but the signature of their occultations of background
stars should be detectable(5; 6; 7; 8; 9). Observations at both optical(10) and X-
ray(11) wavelengths claim to have detected such occultations, but their implied KBO
abundances are inconsistent with each other and far exceed theoretical expectations.
Here, we report an analysis of archival data that reveals an occultation by a body with
a ∼500m radius at a distance of 45AU. The probability of this event to occur due to
random statistical fluctuations within our data set is about 2%. Our survey yields a
surface density of KBOs with radii larger than 250m of 2.1+4.8
inferred surface densities from previous claimed detections by more than 5σ. The
fact that we detected only one event, firmly shows a deficit of sub-km sized KBOs
compared to a population extrapolated from objects with r > 50 km. This implies that
sub-km-sized KBOs are undergoing collisional erosion, just like debris disks observed
around other stars.
−1.7× 107deg−2, ruling out
A small KBO crossing the line of sight to a star will partially obscure the stellar light,
an event which can be detected in the star’s light curve. For visible light, the characteristic
scale of diffraction effects, known as the Fresnel scale, is given by (λa/2)1/2∼ 1.3km, where
a ∼ 40AU is the distance to the Kuiper belt and λ ∼ 600nm is the wavelength of our
observations.
Diffraction effects will be apparent in the star’s light curve due to occulting KBOs
provided that both star and the occulting object are smaller than the Fresnel scale (12; 13).
Occultations by objects smaller than the Fresnel scale are in the Fraunhofer regime. In
this regime the diffraction pattern is determined by the size of the KBO and its distance
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to the observer, the angular size of the star, the wavelength range of the observations and
the impact parameter between the star and the KBO (see Supplementary Information for
details). The duration of the occultation is approximately given by the ratio of the Fresnel
scale to the relative velocity perpendicular to the line of sight between the observer and the
KBO. Since the relative velocity is usually dominated by the Earth’s velocity around the
Sun, which is 30kms−1, typical occultations only last of order of a tenth of a second.
Extensive ground based efforts have been conducted to look for optical occultations
(10; 9; 14; 15). To date, these visible searches have announced no detections in the region
of the Kuiper belt (30-60AU), but one of these quests claims to have detected some events
beyond 100AU and at about 15AU (10). Unfortunately, ground based surveys may suffer
from a high rate of false-positives due to atmospheric scintillation, and lack the stability
of space based platforms. The ground breaking idea to search for occultations in archival
RXTE X-ray data resulted in several claimed occultation events (11). Later, revised analysis
of the X-ray data (16; 17; 18; 19) conclude that the majority of the originally reported events
are most likely due to instrumental dead time effects. Thus, previous reports of optical and
X-ray events remain dubious (14) and their inferred KBO abundance is inconsistent with the
observed break in the KBO size distribution, which has been obtained from direct detections
of large KBOs (20; 21; 22). Furthermore, they are also difficult to reconcile with theoretical
expectations, which predict collisional evolution for KBOs smaller than a few km in size
(23; 4) and hence a lower KBO abundance than inferred from extrapolation from KBOs
with r > 50km.
For the past 14 years, the Fine Guidance Sensors (FGS) on board of Hubble Space
Telescope (HST) have been collecting photometric measurements of stars with 40Hz time
resolution, allowing for the detection of the occultation diffraction pattern rather than a
simple decrease in the photon count. We examined four and a half years of archival FGS
data, which contain ∼ 12,000 star hours of low ecliptic latitude (|b| < 20◦) observations.
Our survey is most likely to detect occultations by KBOs that are 200-500m in radius
given the signal-to-noise of our data (Supplementary Figure 3) and a power-law size distri-
bution with power-law index between 3 and 4.5. Occultation events in this size range are
in the Fraunhofer regime where the depth of the diffraction pattern varies linearly with the
area of the occulting object and is independent of its shape. The theoretical light curves for
our search algorithm were therefore calculated in this regime. We fitted these theoretical
occultation templates to the FGS data and performed χ2analysis to identify occultation
candidates (see Supplementary Information). We detected one occultation candidate, at
ecliptic latitude 14◦, that significantly exceeds our detection criterion (Figure 1). The best
fit parameters yield a KBO size of r = 520 ± 60m and a distance of 45+5
assumed a circular KBO orbit and an inclination of 14◦. Using bootstrap simulations, we
estimate a probability of ∼ 2% that such an event is caused by statistical fluctuations over
the whole analyzed FGS data set (Supplementary Figure 7). We note that for objects on
circular orbits around the sun two solutions can fit the duration of the event. However, the
−4AU where we
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other solution is at a distance of 0.07 AU from the Earth, and is therefore unlikely. It is
also unlikely that the occulting object was located in the Asteroid belt, since the expected
occultation rate from Asteroids is about two orders of magnitude less than our implied rate.
Furthermore, an Asteroid would have to have an eccentricity of order unity to be able to
explain the duration of the observed occultation event.
Using the KBO ecliptic latitude distribution from Elliot et. al (2005) (24), our detection
efficency, and our single detection, we constrain the surface density around the ecliptic
(averaged over −5◦< b < 5◦) of KBOs with radii larger than 250m to 2.1+4.8
(see Supplementary Information Sections 5 and 6). This surface density is about three
times the implied surface density at 5.5◦ecliptic latitude and about five times the surface
density at 8 − 20◦ecliptic latitude. This is the first measurement of the surface density of
hecto-meter-sized KBOs and it improves previous upper limits by more than an order of
magnitude (9; 15). Figure 2 displays our measurement for the sub-km KBO surface density
and summarizes published upper limits from various surveys. Our original data analysis
focused on the detection of KBOs located at the distance of the Kuiper belt between 30AU
and 60AU. In order to compare our results with previously reported ground-based detections
beyond 100AU (10), we performed a second search of the FGS data that was sensitive to
objects located beyond the classical Kuiper belt. Our results challenge the reported ground-
based detections of two 300m-sized objects beyond 100AU (10). Given our total number of
star hours and a detection efficency of 3% for 300m-sized objects at ∼ 100AU we should have
detected more than twenty occultations. We therefore rule out the previously claimed optical
detections (10) by more than 5σ. This result accounts for the broad latitude distribution of
our observations (i.e., |b| < 20◦) and the quoted detection efficency of our survey includes
the effect of the finite angular radii of the guide stars at 100AU.
−1.7× 107deg−2
The KBO cumulative size distribution is parameterized by N(> r) ∝ r1−q, where N(>
r) is the number of objects with radii greater than r, and q is the power-law index. The
power-law index for KBOs with radii above ∼45km is ∼ 4.5 (21; 22) and there is evidence
for a break in the size distribution at about rbreak∼ 45km (20; 21; 22). We hence use this
break radius and assume a surface density for KBOs larger than rbreak (25) of 5.4 deg−2
around the ecliptic. Accounting for our detection efficency, the velocity distribution of the
HST observations, and assuming a single power-law for objects with radii less than 45km
in size, we find q = 3.9+0.3,+0.4
deficit of sub-km sized KBOs compared to large objects. This confirms the existence of the
previously reported break and establishes a shallower size distribution extending two orders
of magnitude in size down to sub-km sized objects. This suggests that sub-km sized KBOs
underwent collisional evolution, eroding the smaller KBOs. This collisional grinding in the
Kuiper belt provides the missing link between large KBOs and dust producing debris disks
around other stars. Currently our results are consistent with a power-law index of strength
dominated collisional cascade (23), q = 3.5, within 1.3σ and with predictions for strengthless
rubble piles (4), q = 3.0, within 2.4σ. An intermediate value of 3 < q < 3.5 implies that
−0.3,−0.7(1 and 2σ errors) below the break. Our results firmly show a
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KBOs are strengthless rubble piles above some critical size, rc< r < 45km, and strength
dominated below it, r < rc. Our observations constrain for the first time rc. At the 2σ level
we find rc> 3km.
Using our estimate for the size distribution power-law index (q = 3.9) and our KBO
surface density for 250m sized KBOs at an ecliptic latitude of b = 5.5◦, which is the ecliptic
latitude of the RXTE observations of Scorpius X-1, we predict that there should be ∼
3.6 × 10930m-radius objects per square degree. This is about 150 times less than the
original estimate from X-ray observations of Scorpius X-1 that reported 58 events (11), and
it is about 30 times less than the revised estimate from the same X-ray observations, which
concludes that up to 12 events might be actual KBO occultations (16). Our results rule out
the implied surface density from these 12 events at 7σ confidence level. One can reconcile
our results and the claimed X-ray detections only by invoking a power-law index of q ∼ 5.5
between 250m and 30m. More recent X-ray work reports no new detections in the region of
the Kuiper belt but places an upper limit of 1.7×1011deg−2for objects of 50m in radius and
larger (18). This is consistent with the KBO surface density of N(> 50m) = 8.2×108deg−2
that we derive by extrapolating from our detection in the hecto-meter size range.
The statistical confidence level on our detection is 98%. However, our conclusions that
there is a significant break in the size distribution and that collisional erosion is taking place
and the significant discrepancy with previously claimed occultation detections rely on the low
number of events we discovered. These conclusions would only be strengthened if this event
was caused by an unlikely statistical fluctuation or a yet unknown instrumental artifact.
Ongoing analysis of the remaining FGS data, which will triple the number of star hours,
together with further development of our detection algorithm (i.e., including a larger number
of light curve templates) holds the promise for additional detections of occultation events
and will allow us to constrain the power-law index of the size distribution further.
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Acknowledgments We thank Dr. H. K. Chang for valuable comments that helped to im-
prove this manuscript. Some of the numerical calculations presented here were performed on
Caltech’s Division of Geological and Planetary Sciences Dell cluster. Partial support for this
research was provided by NASA through a grant from the Space Telescope Science Institute.
R. S. acknowledges support from the ERC and the Packard Foundation. A. G. is supported
by the Israeli Science Foundation, an EU Seventh Framework Programme Marie Curie IRG
fellowship and the Benoziyo Center for Astrophysics, a research grant from the Peter and
Patricia Gruber Awards, and the William Z. and Eda Bess Novick New Scientists Fund at
the Weizmann Institute. S. Z. acknowledges support from the Israel Science Foundation –
Adler Foundation for Space Research.
Author Contributions H. E. S. wrote the detection algorithm, analyzed the FGS data for
occultation events, calculated the detection efficency of the survey, preformed the bootstrap
analysis and wrote the paper. E. O. O. calculated the stellar angular radii, the velocity
information of the observations, the correlated noise and other statistical properties of the
data. R. S. guided this work and helped with the scientific interpretation of the results. A.G.
proposed using HST FGS data for occultation studies and helped to make the data avail-
able for analysis. M. W. extracted the FGS photometry streams and provided coordinates
and magnitudes of the guide stars. M. L. helped in gaining access to the FGS data and
provided insights into the operation and noise properties of the FGS . E. N. provided expert
interpretation of the FGS photometric characteristics in the HST operational environment.
S. Z. took part in the statistical analysis of the data. All authors discussed the results and
commented on the manuscript.
Author Information Reprints and permissions information is available at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to H. E. S. (hes@astro.caltech.edu)
or E. O. O. (eran@astro.caltech.edu).
This preprint was prepared with the AAS LATEX macros v5.2.
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80
100
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140
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180
200
220
-0.3 -0.2-0.1 0 0.1 0.2 0.3
photon count
time relative to mid eclipse (s)
b
80
100
120
140
160
180
200
220
-2 -1.5-1
time relative to mid eclipse (s)
-0.5 0 0.5 1 1.5 2
photon count
a
Fig. 1.— Photon counts as a function of time of the candidate occultation event observed by FGS2. Part
a) shows the photon count spanning ±2seconds around the occultation event. Part b) displays the event
in detail. The red crosses and error bars are the FGS data points with Poisson error bars, the dashed,
blue line is the theoretical diffraction pattern (calculated for the 400-700nm wavelength range of the FGS
observations), and the pink squares correspond to the theoretical light curve integrated over 40Hz intervals.
Note, the actual noise for this observation is about 4% larger than Poisson noise due to additional noise
sources such as dark counts (about 3 to 6 counts in a 40Hz interval), and jitter due to the displacement of
the guide star (by up to 10mas) from its null position. The mean signal-to-noise ratio in a 40Hz interval
for the roughly half an hour of observations is ∼ 12. The event occurred at UTC 05:17:49 2007, Mar 24.
The best fit χ2/dof is 20.1/21. The star has an ecliptic latitude of +14. Its angular radius and effective
temperature are ≈ 0.3 of the Fresnel scale and ≈ 4460K, respectively. These values were derived by fitting
the 2MASS (26) JHK and USNO-B1 BR (27) photometry with a black-body spectrum. The position of
the star is R.A.=186.87872◦, Dec=12.72469◦(J2000) and its estimated V-magnitude is 13.4. The auto-
correlation function (excluding lag zero) of the photometric time series of this event is consistent with zero
within the statistical uncertainty. Each FGS provides two independent PMT readings and we confirmed that
the occultation signature is present in both of these independent photon counts. We examined the photon
counts of the other guide star that was observed by FGS1 at the time of the occultation and confirmed that
the occultation signal is only present in the observations recorded by FGS2. We examined the engineering
telemetry for HST around the time of the event and verified that the guiding performance of HST was
normal. We therefore conclude that the above occultation pattern is not caused by any known instrumental
artifacts.
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0
0.01 0.01
2 2
4 4
6 6
8 8
10 10
12 12
0.1 0.1 1 1 10 10 100 100
log N(>r) (deg-2)
radius (km)radius (km)
FGS FGS
RXTE Jones et al. (2008)
RXTE Liu et al. (2008)
RXTE Chang et al. (2007)
TAOS Zhang et al. (2008)
MMT Bianco et al. (2009)
Bickerton et al. (2008)
Roques et al. (2008)
0
log N(>r) (deg-2)
Fig. 2.— Cumulative KBO size distribution as a function of KBO radius for objects located between 30 and
60AU. The results from our FGS survey are shown in red and are presented in three different ways: (i) The
cross is derived from our detection and represents the KBO surface density around the ecliptic (averaged over
−5◦< b < 5◦) and is shown with 1σ error bars. The cross is plotted at r = 250m, which is roughly the peak
of our detection probability (see Supplementary Information Section 6 for details). (ii) The upper and lower
red curves correspond to our upper and lower 95% confidence level which were derived without assuming any
size distribution. (iii) The region bounded by the two straight red lines falls within 1σ of our best estimate
for the power-law size distribution index, i.e. q = 3.9 ± 0.3, which was calculated for low ecliptic latitudes
(|b| < 5◦). These lines are anchored to the observed surface density at r = 45km. For comparison, the
green (long-dashed) line is the observed size distribution of large KBOs (i.e., r > 45km), which has q = 4.5,
extrapolated as a single power-law to small sizes. The blue (short-dashed) line is a double power-law with
q = 3.5 (collisional cascade of strength dominated bodies) for KBOs with radii less than 45km and q = 4.5
above. The cyan (dot-dashed) line corresponds to q = 3.0 (collisional cascade of strengthless rubble piles)
for KBOs below 45km in size. All distributions are normalized to N(> r) = 5.4deg−2at a radius of 45km
(25). In addition, 95% upper limits from various surveys are shown in black. Note, a power-law index of 3.9
was used for calculating the cumulative KBO number density from the RXTE observations.
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Supplementary Information
1 The FGS data set
There are three FGS on board of HST . Each FGS consists of four photomultipliers (PMTs).
Nominal HST operation uses two FGS for guiding, with each FGS observing its own guide
star. The photon counts recorded by each FGS are therefore different, but global instru-
mental artifacts and Observatory level transients will display in both FGS and can therefore
be identified and removed.
Observations of the inclination distribution of large KBOs find that about 75% have an
inclination angle |i| ? 20◦(29; 30; 31). We therefore divide the FGS observations into a low
ecliptic latitude (|b| < 20◦) and a high ecliptic latitude (|b| > 20◦) sample. The high-ecliptic
latitude observations (|b| > 20◦) provide an excellent control sample.
2 FGS Guide Stars
The FGS guide stars span a broad range of magnitudes and spectral types. The signal-to-
noise ratio, S/N, in a 1/40s data bin depends on the magnitude of the star. Its distribution
is shown in Supplementary Figure 3.
The angular sizes of guide stars were derived by fitting the 2MASS (32) JHK and USNO-
B1 BR (33) photometry with a black-body spectrum. Supplementary Figure 4 shows the
angular radii distribution of the guide stars. About 66% of the stars in our data set subtend
angular sizes less than 0.5 of the Fresnel scale at a distance of 40AU. The diffraction pattern
that is produced by a sub-km sized KBO occulting an extended background star is smoothed
over the finite stellar disk. This effect becomes clearly noticeable for stars that subtend sizes
larger than about 0.5 of a Fresnel scales (34; 35) and it reduces the detectability of occul-
tation events around such stars. The effect of finite angular radii of the guide stars on the
detection efficiency of our survey is taken into account (see Detection Efficency section 5 for
details).
3 Detection Algorithm
Our detection algorithm performs a template search with theoretical light curves and uses a
χ2fitting procedure to identify occultation candidates. Our survey is most likely to detect
KBO occultation events caused by objects that are 200-500m in radius given the signal-
to-noise of our data and for a power-law index of the KBO size distribution, q, between 3
and 4.5. Occultation events in this size range are in the Fraunhofer regime. The theoretical
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light curves for our search algorithm are therefore calculated in the Fraunhofer regime. Our
templates are calculated for various impact parameters assuming a point source background
star and are integrated over the 400-700nm wavelength range of the FGS observations. For
a given impact parameter between the KBO and the star, our theoretical light curves have
three free parameters that we fit for. The first is the mean number of photon counts, which
is the normalization of the light curve. The second is the amplitude of the occultation, which
is proportional to the size of the KBO, and the third is the width of the occultation, which
is independent of the object size, and is determined by the ratio of the Fresnel scale to the
relative speed between HST and the KBO perpendicular to the line of sight. This relative
speed is determined by the combination of HST ’s velocity around the Earth, Earth’s veloc-
ity around the Sun and the velocity of the KBO itself. We use this information to restrict the
parameter space for the template widths in our search such that we are sensitive to KBOs
located at the distance of the Kuiper belt between 30AU and 60AU.
4 Detection Criterion and Significance Estimates
The significance of occultation candidates can be measured by their ∆χ2which is defined
here as the difference between the χ2calculated for the best fit of a flat light curve, which
corresponds to no event, and the χ2of the best fit template. Occultation events have large
∆χ2, since they are poorly fit by a constant. Cosmic ray events, which give rise to one very
large photon count reading in a 40Hz interval, can also result in a large ∆χ2but the fit of
the occultation template is very poor. We examined all flagged events for which the tem-
plate fit of the diffraction pattern was better than 15σ. About a handful of false-positives
where flagged by our detection algorithm that have a value of ∆χ2comparable to or larger
than the occultation event. However, in all cases these false-positives were caused by a 1 Hz
jitter due to the displacement of the guide star from its null position. The occultation event
itself did not show any such jitter. To determine the ∆χ2detection criterion for our search
algorithm and to estimate the probability that detected events are due to random noise we
use the bootstrap technique (36). Specifically, from a given FGS time series of length N we
randomly selected N points with repetitions and created ‘artificial’ time series from it. We
analyzed these ‘artificial’ data sets using the same search algorithm that we applied to the
actual FGS data. This technique creates random time series with noise properties identical
to those of the actual data, but it will lose any correlated noise. Therefore, this technique
is justified if there is no correlated noise in the data sets. To look for correlated noise we
calculated the auto-correlation function, with lags between 0 to 1s. Most of the data sets
are free of statistical significant correlated noise. The ∼ 12% of the data sets that did show
correlated noise exceeding 4σ, which was often due to slopes (e.g., long-term variability) in
the data sets, were excluded from the bootstrap analysis.
The FGS data set consists of observations of many different stars with magnitudes rang-
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ing from 9 to 14. The number of photon counts and signal-to-noise properties vary therefore
from observation to observation (see Supplementary Figure 3 for the signal-to-noise ratio
distribution of the FGS observations). Our ∆χ2calculation accounts for the Poisson noise
of the data. Therefore, the probability that occultation candidates are due to random noise
can be characterized by a single value of ∆χ2for all observations, irrespective of the mean
photon count of a given observation provided that the noise properties across all observa-
tions are well characterized by a Poisson distribution. In reality, the noise properties are
different from observation to observation; especially non-Poisson tails in the photon counts
distribution will give rise to slightly different ∆χ2distributions. Therefore, ideally, we would
determine a unique detection criterion for each individual data set. However, this would re-
quire to simulate each data set, which contains about an hour of observations in a single HST
orbit, over the entire length of our survey (∼ 12,000 star hours). This is not feasible due to
the enormous computational resources that would be required, i.e. simulating a single one
hour data set over the entire survey length requires about 5CPU days, which corresponds
to ∼ 60,000CPU days for the entire FGS survey. Instead, we perform the bootstrap sim-
ulation over all the FGS data sets together, where each individual data set was simulated
about a 100 times, which required about ∼ 500CPU days in total. This way we estimate the
typical ∆χ2value that corresponds to having less than one false-positive detection over the
∼ 12,000 star hours of low ecliptic observations. For all occultation candidates that exceed
this detection threshold, we determined their statistical significance, i.e. the probability that
they are due to random noise, by extensive bootstrap simulations of the individual data sets
(Supplementary Figure 7).
5 Detection Efficency
The ability to detect an occultation event of a given size KBO depends on the impact
parameter of the KBO, the duration of the event, the angular size of the star and the signal-
to-noise ratio of the data. We determined the detection efficiency of our survey by recovering
synthetic events that we planted into the observed photometric time series by multiplying
the actual FGS data with theoretical light curves of KBO occultation events. The synthetic
events correspond to KBO sizes ranging from 130m < r < 650m, they have impact pa-
rameters from 0 to 5.5 Fresnel scales and a relative velocity distribution that is identical to
that of the actual FGS observations. To account for the finite angular sizes of the stars we
generated light curve templates with stellar angular radii of 0.1, 0.2, 0.3, 0.4, 0.6, 0.8 and
1 Fresnel scales distributed as shown in Supplementary Figure 4. The modified light curves
with the synthetic events were analyzed using the same search algorithm that we used to
analyze the FGS data. The detection efficency of our survey was calculated using the angu-
lar size distribution of the FGS guide stars assuming a distance of 40AU. We normalize our
detection efficency for a given size KBO, η(r) , to 1 for an effective detection cross section
with a radius of one Fresnel scale.
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The detection efficiency of our survey is ∼ 0.05 (∼ 0.6) for objects with r = 200m
(r = 500m) located at 40AU. Note that this value for the detection efficency already ac-
counts for the angular radii distribution of the guide stars (e.g., for comparison, stars that
subtend angular radii less than 0.5 of the Fresnel scale result in a detection efficency of
∼ 0.08 [∼ 0.8] for objects with r = 200m [r = 500m].).
6 Calculating the KBO Surface Density
The number of occultation events is given by
Nevents≃ −2vrelF
?rmax
rmin
?b
−b
η(r)∆t
∆b
dN(r,b)
dr
dbdr (1)
where vrel = 23km/s is the typical relative velocity between the KBO and the observer,
b is the ecliptic latitude, ∆t/∆b is the time observed per degree in ecliptic latitude (see
Supplementary Figure 5) and F = 1.3km is the Fresnel scale. The number density of KBOs
is both a function of ecliptic latitude and the KBO radius, r. Here we assume that the KBO
latitude distribution, f(b), is independent of size and we take the distribution provided in
Elliot et al. (2005) (31). We further assume that the KBO size distribution follows a power
law. It can therefore be written as N(r,b) = n0×r−q+1×f(b) where n0is the normalization
factor for the cumulative surface density of KBOs. Substituting for dN(r,b)/dr in equation
1 and solving for n0we have
n0≃
Nevents
2vrelF(q − 1)?rmax
rminη(r)r−qdr?b
−bf(b)∆t
∆bdb. (2)
Evaluating equation 2 yields a cumulative KBO surface density averaged over the ecliptic
(|b| < 5◦) of
N(r > 250m) ≃ 2.1 × 107deg−2
(3)
We assumed q = 4 when evaluating the integral over r. We note however that the value for
the cumulative KBO surface density at r = 250m only depends weakly on the exact choice
for q [e.g. N(r > 250m) only ranges from 2.3 × 107deg−2to 2.1 × 107deg−2for values of
q between 3 and 4.5]. We quote our results as the KBO surface density of objects larger
than 250m in radius since this is roughly the size of KBOs, which our survey is most likely
to detect given our detection efficency and a power-law size distribution with q = 3 − 4.5.
The implied surface density for KBOs with radii larger than 250m is 7.7 × 106deg−2at
b = 5.5◦, which is the ecliptic latitude of the RXTE observations of Scorpius X-1, and it is
4.4 × 106deg−2for 8◦< |b| < 20◦.
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0
500
1000
1500
2000
2500
3000
3500
4000
0 20 40 60 80 100
star hours
S/N
Fig. 3.— Distribution of star hours as a function of the mean signal-to-noise ratio, S/N, in a 40Hz bin for
the 12,000 hours of low ecliptic latitude observations (|b| < 20◦) in the analyzed FGS data set.
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0
500
1000
1500
2000
2500
3000
3500
4000
0 0.5 1 1.5 2
star hours
angular radius (Fresnel scale)
Fig. 4.— Distribution of star hours as a function of angular radii of the guide stars. The angular radii are
given as fraction of the Fresnel scale both which are calculated at 40AU.
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