Detecting Earth’s Temporarily Captured
Natural Irregular Satellites
Bryce Bolin1 ([email protected]), Robert Jedicke1 , Mikael Granvik2 , Peter Brown3 ,
Monique Chyba4 , Ellen Howell5 , Mike Nolan5 , Geoff Patterson3 , Richard Wainscoat1
Received
;
accepted
N Pages, N Figures, N Table
1
University of Hawaii, Institute for Astronomy, 2680 Woodlawn Dr, Honolulu, HI, 96822
2
Department of Physics, P.O. BOX 64, 00014 University of Helsinki, Finland
3
University of Western Ontario, Physics & Astronomy Department, London,
Ontario, Canada
4
University of Hawaii, Department of Mathematics, Honolulu, HI, 96822
5
Arecibo Observatory, Arecibo, Puerto Rico
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ABSTRACT
JEDICKE WILL WRITE ONCE REMAINDER OF PAPER IS
COMPLETE.
Subject headings: Near-Earth Objects; Asteroids, Dynamics
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Proposed Running Head: Detecting Earth’s Natural Irregular Satellites
Editorial correspondence to:
Bryce Bolin
Institute for Astronomy
University of Hawaii
2680 Woodlawn Drive
Honolulu, HI 96822
Phone: +1 808 294 6299
Fax: +1 808 988 2790
E-mail: [email protected]
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1.
Introduction
Granvik et al. (2012) introduced the idea that there exists a steady state population of
temporarily captured orbiters (TCO) that are natural irregular satellites of the Earth. They
are captured from a dynamically suitable subset of the near Earth object (NEO) population
— those that are on Earth-like orbits with semi-major axis a ∼ 1.0 AU, eccentricity e ∼ 0.0
and inclinations i ∼ 0 deg — and complete an average of 2.88 ± 0.82(rms) revolutions
around Earth during their average capture duration of 286 ± 18(rms) days. In this work
we evaluate options for detecting the TCOs as they are being captured, while they are on
their geocentric trajectories, and in their meteor phase (about 1% of TCOs enter Earth’s
atmosphere).
There is almost no previous work on TCOs and what little exists is reviewed in Granvik
et al. (2012). Twenty years ago the Spacewatch project (Scotti et al. 1991) regularly
identified geocentric objects but was told that they must be artificial satellites because
there are no natural Earth satellites other than the Moon. Those objects were often on
high eccentricity orbits similar to the TCOs in Granvik et al. (2012)’s model. Then the
first confirmed TCO, 2006 RH120 , was discovered by the Catalina Sky Survey late in 2006
(Larson et al. 1998) and 2006 RH120 citepKwiatkowski2008. It’s pre- and post-capture
orbit, geocentric trajectory, size, and TCO lifetime were consistent with the Granvik et al.
(2012) model.
2006 RH120 had an absolute magnitude of H ∼ 29.5 corresponding to a diameter in the
range ∼3-7 meters if we assume S- and C-class albedos pv of 0.26 and 0.064 respectively
(Mainzer et al. 2012). We note that these sizes are considerably larger than the unofficially
measured ∼1-2 meter diameter from the Goldstone radar facility1 . The radar measurement
1
L. Benner, personal communication.
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implies that the 2006 RH120 must have a much higher albedo than even the S-class asteroids
For comparison, Granvik et al. (2012) suggest that the largest TCO in the steady state
population is in the 1-2 m diameter range, that objects comparable to 2006 RH120 are
captured every decade, and a 100 m diameter TCO is captured about every 100,000 years.
Thus, the predicted flux of objects in 2006 RH120 ’s size range is in rough agreement with
the length of time that optical telescopic surveys have been in regular operation and their
time-averaged sensitivity to objects like 2006 RH120 .
However, there is no a priori reason to expect that the actual TCO orbit and sizefrequency distribution (SFD) should match the Granvik et al. (2012) model because they
used a NEO orbit model strictly applicable only to much larger objects & 100 m in diameter
and only accounted for gravitational dynamics with no allowance for non-gravitational
forces like the Yarkovsky effect (e.g. Morbidelli and Vokrouhlický 2003). Their favored
TCO SFD from Brown et al. (2002) for bolides is appropriate for TCOs in the 0.1 cm to
1 m diameter range but the TCO orbit distribution should be considered suspect without
accounting for the Yarkovsky effect.
Furthermore, Granvik et al. (2012) used the orbit distribution from the Bottke et al.
(2002) NEO model that has several known problems including but not limited to 1) being
applicable only to the larger NEOs, 2) not including the Yarkovsky effect, 3) underestimating
the fraction of the NEO population with perihelion p < 1.0 AU, 4) underestimating the
number of objects on low inclincation orbits (Greenstreet and Gladman 2012) and 5) having
coarse resolution in (a, e, i)-space. The last three issues are particularly important because
they make it difficult to estimate the population of dynamically capturable NEOs. For
instance, the entire set of pre-capture TCO orbits is contained within just 8 bins in the
Bottke et al. (2002) NEO model.
Thus, measuring the TCO size and orbit distribution provides a sensitive means of
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testing current NEO and meteoroid SFD models in a size range that transitions between
the two regimes. The smallest known NEO with H ∼ 33.2 (2008 TS26 Boattini et al. 2008)
corresponds to a sub-meter diameter under the standard albedo assumption that makes
H = 18 correspond to D = 1 km. The smallest NEO with a measured period is 2006 RH120 ,
the only known TCO (Kwiatkowski et al. 2008), and the smallest NEO with measured
colors/spectrum is 2008 TC3 (Jenniskens et al. 2009), the only object that was discovered
prior to impact. The problem with discovering and characterizing the smallest NEOs is
that they are discovered close to Earth and moving so fast that there is almost no time to
coordinate followup during the time when they are brightest. On the other hand, some of
the TCOs will be visible for days and their location can be predicted accurately. Measuring
their spin-rates will have implications for studies of the YORP effect (e.g. Bottke et al.
2006) while their taxonomic classification will have implications for the relative delivery
rates of objects in this size range from the main belt (e.g. Bottke et al. 2002).
NEED PETER BROWN’S HELP HERE... REFERENCES, CONTEXT,
CONTENT, CORRECTIONS. The field of meteor physics has made tremendous
advances in the past decades with the advent of radar detection systems BAGGALEY,
BROWN, infrasound networks BROWN, +??, and high-speed all-sky optical CCD
surveys CAMPBELL, CZECHS, ETC.. They have had incredible success in modeling
the ‘dark flight’ of the meteor fragments through the atmosphere once they cease being
luminous and have reached terminal velocity. One critical element for meteor studies is the
absence of calibrating standards — objects for which the pre-meteor phase size, rotation
period, taxonomy, etc. are all known before impact. TCOs could provide the calibration
standards if enough of them can be discovered because 1% of them eventually become
low-speed meteors.
Finally, TCOs provide an opportunity for the lowest ∆v targets for spacecraft missions
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(Granvik 2013; Elvis et al. 2011). The opportunity to retrieve kilograms or even hundreds
of kilograms of pristine asteroidal material that unaffected by passage through Earth’s
atmosphere and uncontaminated by exposure to elements on Earth’s surface would be a
tremendous boon to solar system studies. We can even imagine multiple retrieval missions
to TCOs that have been taxonomically pre-classified as interesting targets — essentially
sampling the range of the main belt in our own backyard.
2.
Observable characteristics of the steady state TCO population
Granvik et al. (2012) provide trajectories (orbits) for about 17,000 TCOs that are
designed to be a representative, unbiased, population. i.e. they represent a steady state
set of TCOs. Their sky plane distribution without any constraints on e.g. their apparent
brightness, distance, rate of motion, etc., shows a strong enhancement at quadrature
(fig. 1). Their trajectories2 are such that they tend to be furthest from Earth at quadrature
and therefore moving the slowest in that area of the sky. Since they move slowly they
spend more time in the area, are more likely to be found in that region, and therefore have
the highest sky plane density. Other than the enhancements at quadrature the sky plane
distribution is distributed along the ecliptic and drops off rapidly with ecliptic latitude as
for most classes of objects in the solar system as viewed from Earth.
The TCO geocentric distribution illustrated in fig. 2 is critical to assessing opportunities
for detecting and even discovering TCOs at radar facilities (see §5). There is no significant
difference between the distributions in the L1 and L2 directions or in the east and west
quadratures but there is a clear difference between the two sets. The average±rms
2
we often refer to TCO motion around Earth as a trajectory because their motion does
not typically follow a nearly-closed elliptical path like the conventional notion of an ‘orbit’.
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geocentric distance is 4.3 ± 2.2 LD for the L1/L2 regions, 4.1 ± 1.63 LD for the entire TCO
population, and 5.7 ± 2.9 LD for the region near quadrature. The difference works in favor
of optical detection of the TCOs since they are closer to Earth near opposition but a 1 m
diameter object with absolute magnitude H ∼ 333 will have an apparent magnitude of
V = 23.1 at the average opposition distance.
In the remainder of this paper we will often refer to residence-time distributions —
the total time spent by the ensemble of particles (TCOs) that meet some constraints (e.g.
apparent brightness, rate of motion) as a function of, usually, their position on the sky.
Our sky plane distributions are invariably with respect to an opposition-centric ecliptic
coordinate system.
Figure 3 shows the residence time sky plane distribution for TCOs of >10 cm diameter
subject to the constraint of a modest limiting magnitude of V < 20 and maximum rate of
motion of 15 deg/day. The constraints are roughly consistent with the expected performance
characteristics of the ATLAS system described in §3.1 below but are generally applicable
to any telescopic TCO survey except for (roughly) a normalization constant. The sky
plane residency with magnitude and rate constraints is dramatically different from the
unconstrained sky plane distribution shown in fig. 1. Application of the standard asteroid
H and G photometric model (e.g. Bowell et al. 1988, we use G = 0.15 throughout this
work) eliminates the enhancements at quadrature observed in fig. 1 in favor of a strong
enhancement at opposition due to the photometric surge for very small phase angles.
The enhancement along the ecliptic is still present but decreases with increasing angular
separation from opposition. Thus, it is clear that a telescopic optical TCO survey is most
3
For convenience, unless stated otherwise, we use an albedo of pv ∼ 0.11 throughout this
work that results in a 1 km diameter object yielding an absolute magnitude of H = 18. A
1 m diameter object than has H = 33 and a 10 cm diameter objects H = 38.
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effective towards opposition and along the ecliptic — exactly the kind of surveying typically
employed by contemporary NEO surveys (e.g. Jedicke 1996; Jedicke et al. 2002). So we
expect that the NEO survey strategy should be finding TCOs or that the next generation of
sky surveys will do so as a matter of course in their regular surveying (e.g. Tokunaga 2007).
All our results that require calculating the TCOs’ apparent brightness account for
Earth’s umbral and penumbral shadowing (see §C). Earth’s umbral shadow is about 1◦ in
diameter at 1 LD (lunar distance, about 0.00275 AU) and it might be expected that most of
the faint TCOs can only be detected close to Earth, near opposition, when they are entirely
in the umbra. We assume that within the umbral shadow there is zero sunlight and if the
TCO is in the penumbral shadow we reduce its apparent brightness by 2.5 log f⊙ magnitudes
where f⊙ is the fraction of the Sun’s ‘surface’ visible at the TCO. We found that Earth’s
shadow makes only a 4%difference in the total sky plane residence time for the detection
parameters in fig. 3 and a 5% decrease in the bin at opposition. The modest shadowing
effect even for the relatively bright limiting magnitude is a consequence of requiring that
the rate of motion be <15 deg/day. The Moon’s rate of motion is about 12 deg/day so most
of the TCOs in the figure must be near or beyond the Moon and mostly unaffected by
Earth’s shadow. If an optical survey was capable of detecting faster rates of motion then
the Earth shadowing would have a larger effect on the TCO detection capability. In any
event, all the figures and values presented in the remainder of this work incorporate the
affect of Earth’s shadow.
In designing a TCO survey strategy (see §3) it is important to understand the TCOs’
rates of motion. Objects that move by more than about 2 PSF-widths during the course
of an exposure spread their total flux in a ‘trail’ that reduces the peak S/N in any pixel
along the trail compared to stationary objects of the same intrinsic brightness. Peak pixel
detection algorithms are therefore less likely to identify the trailed source. Even algorithms
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that can identify trailed detections, e.g. by summing flux along a line in the image, have
reduced sensitivity because the trail’s total S/N is also less than the overall S/N for a
stationary object of the same intrinsic brightness. In this work we partly ignore trailing
effects because they are detection-system and software-algorithm dependent. We simply
assume that the system will be able to detect objects that are not moving too fast as long as
their apparent magnitude is brighter than the system’s limiting value.4 While faster moving
objects can be identified simply by limiting the exposure time we restrict the problem by
placing a reasonable limit on the TCO rate of motion of (typically) 15 deg/day.
The average TCO with V < 20 and rate of motion < 15 deg/day has a rate of motion
of about 10 deg/day within about 30◦ of opposition as shown in fig. 3. Typical exposure
times for contemporary sky surveys are on the order of 60 s so the average TCO would
move about 25′′ during the exposure. This could hamper TCO detection by survey systems
with small PSFs and pixels because the TCO flux will be distributed over many pixels and
create a long trail. Conversely, survey systems with larger PSFs and pixels will be less
affected by the TCOs’ motion.
Figure 12 of Granvik et al. (2012) provides an idea for how to design a targeted TCO
survey (see §3.1). The figure shows that at the moment of capture, the time at which their
total energy becomes negative with respect to the Earth-Moon barycenter, most TCOs
are 1) in retrograde geocentric orbits 2) near L1 and L2 3) moving in roughly the same
direction 4) at similar rates of motion 5) just outside the Earth’s Hill sphere. Figure 4
shows that at the time of capture the TCOs are concentrated near the ecliptic in a ∼ 20◦
wide band centered at opposition (the L2 direction). Indeed, there are 6.1 TCOs larger
than 10 cm diameter captured every day in both L1 and L2 (half at each). The objects’
4
For more information on trail fitting see e.g. Vereš et al. (2012) who provide a technique
for fitting trailed source detections but not how to identify them.
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motion vectors bring them near the opposition point before or after their moment of
capture so that there is a steady stream of small objects in roughly the same direction
moving at roughly the same rate of motion. The rates of motion are relatively modest by
NEO standards at 1.8 ± 0.7(rms) deg/day with the objects’ motion at an average position
angle of 94◦ ± 38◦ (rms). The trailing rate corresponds to ∼ 0.075′′/sec so the average TCO
trails by about 1′′ in ∼ 13 s. However, if the telescope is tracked at the mean TCO rate
then all the objects with rates within ±1 rms of the mean will trail by less than 1′′ in about
34 s. A large telescope with a wide passband filter can reach V ∼ 24.5 at a good site with
1′′ seeing so that trailing losses can be dramatically reduced for the TCOs in exposures of
. 60 s. Furthermore, other TCOs that have already been captured may pass through the
same region and contribute more detections to this type of targeted TCO survey. Thus,
a targeted deep survey towards the opposition point might detect TCOs near the time of
capture and can take advantage of the relatively smooth flow of objects through the region.
It may also be possible to detect TCOs in the infrared from space-based platforms or
perhaps detections already exist in the WISE spacecraft images (e.g. Mainzer et al. 2011,
§4). If we assume that the TCOs in the size range from 10 cm to 1 m are small enough
and/or rotating rapidly enough to be in thermal equilibrium then there is an advantage to
detecting the TCOs in the infrared because they are not affected by phase angle effects.
The peak of the thermal radiation for a 1 m diameter blackbody object occurs at about
16 µm and the peak signal occurs in the 12 µm W3 passband of the 4 WISE filters. (We
used a wavelength-independent emissivity of 0.9 Harris et al. (2009)) Thus, the sky plane
TCO number distribution in the IR does not have any strong enhancements along the
ecliptic as shown in fig. 5. An IR TCO survey would be best served by surveying along the
ecliptic as much as possible and then expanding the search to higher latitudes.
Finally, we will consider detecting TCOs in the final moments of their geocentric
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trajectory as they enter Earth’s atmosphere (see §3.2 and §5.2). Granvik et al. (2012)
showed that about 1% of all TCOs become meteors and, interestingly, none of them do
so if the Moon does not exist. These TCO meteors strike the Earth’s atmosphere in a
narrow range of speeds with an average of 11.19 ± 0.03 km/s — nearly equal to the Earth’s
escape speed because the TCOs have essentially fallen to Earth from a large distance with
v∞ ∼ 0 km/s. The impacting TCOs have a flat distribution in sin θi where θi is the impact
angle, the acute angle between the TCO’s trajectory and the perpendicular to the Earth’s
surface. Thus, the mean impact angle for the TCOs in Granvik et al. (2012)’s sample is
44.6◦ ± 21.4◦.
3.
Optical Detection
In this section we consider different instances of multiple methods of detecting TCOs in
optical light: 1) all-sky, wide area and targeted telescopic surveys and 2) wide area meteor
surveys. We make the distinction between all-sky and wide-area because within the next
few years the ATLAS survey (Tonry 2011) will truly survey the entire night sky multiple
times each night. Other contemporary and anticipated surveys only cover wide areas of the
sky each night. The meteor surveys monitor the entire sky visible from their locations each
night but necessarily to a relatively bright limiting magnitude.
Since TCOs are strongly enhanced in the optical on the ecliptic and towards opposition
any optical survey with limited coverage should first target the region near opposition, then
expand to cover the ecliptic on either side of opposition, and only then survey at higher
ecliptic latitudes. Surveys with faint limiting magnitudes will have higher TCO discovery
rates and those with software capable of identifying the trailed TCO detections will also
have an advantage over those surveys that can only identify detections with stellar PSFs.
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3.1.
Telescopic Detection
We considered four different instances of optical telescopic surveys with which we have
some familiarity. Given the TCOs’ steady state sky-plane distribution as discussed in §2 we
focussed on next-generation surveys with wide field coverage or a targeted survey to a faint
limiting magnitude with a next-generation camera. They are listed along with their survey
characteristics and TCO performance in table 1.
The Pan-STARRS1 survey has been in operation since the spring of 2010. The
1.8 m telescope with an ∼ 7 deg2 field of view discovers most of its moving objects in 45 s
exposures obtained with their wide-band wP1 filter. Since the onset of operations they
have surveyed about 7700 deg2 /month in modes suitable for TCO detection to a limiting
magnitude of V ∼ 21.7. Pan-STARRS1 has not yet implemented trail detection and
currently has essentially zero efficiency for identifying objects moving faster than about
3 deg/day (e.g. Denneau and Jedicke 2013). Our estimated Pan-STARRS1 TCO discovery
statistics in table 1 of ∼ 8.1 × 10−3/month thus assumes Vlim = 21.7, ωlim = 3.0 deg/day
and are averaged over the entire night sky because Pan-STARRS1 does not concentrate its
survey pattern on the eclipitc or at opposition. Given Pan-STARRS1’s current operations
it is unsurprising that it has not yet reported a TCO discovery. However, it is likely that
they have already imaged trailed TCOs that were not identified by their image processing
pipeline or linked by the moving object processing system. Furthermore, Pan-STARRS1
typically requires 12-18 hours to process moving objects so that even if a TCO is detected
and reported to the MPC it is unlikely that it can be reacquired by any followup observatory
because of the TCOs’ relatively high rates of motion and GCRs.
The ATLAS survey is expected to begin regular operations in early 2016 (Tonry 2011).
They plan to survey the entire sky 4× each night to V ∼ 20 with relatively small but
very wide field telescopes. The system has one major limitation from the perspective of
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detecting TCOs — its relatively bright limiting magnitude — but this is compensated
somewhat by its ∼ 4′′ pixels that dramatically reduce trailing losses. The large pixel scale
limits their ability to detect all but the largest GCRs but they expect to deliver moving
objects to the MPC within minutes of the final image acquired at a specific boresite so
that followup activity can begin almost immediately. There are about 2.5 × 10−3 TCOs at
any time on the night sky with V < 20.0 and ω < 15 deg/day, a rate of motion that will
leave just 5 pixel long trails on the ATLAS images. We thus expect that ATLAS should
detect about 0.02 TCOs/lunation or, equivalently, about a 52% chance of detecting a TCO
in its first two years of operation. One advantage to the bright limiting magnitude is that
almost all moving objects with V < 20 will already exist in the MPC catalog so that any
new moving object that appears in the ATLAS survey must be interesting — e.g. an NEO,
asteroid cratering or disruption event, or TCO. While the ATLAS TCO detection rate
might be relatively low it is guaranteed that the discovered objects will be bright, large,
and relatively easy to track over many nights.
With its ∼9.6 deg2 FOV, V ∼ 24.7 limiting magnitude, and 15 s exposures, the LSST is
nearly the ultimate TCO detection machine but it ‘only’ surveys about 20-25% of the sky
each night (Ivezic et al. 2008). Its short and back-to-back pairs of exposures should make
identifying fast moving objects easy and eliminate much of the confusion from systematics
and noise in the image plane. Their 0.2′′ pixel scale should make it straightforward to
detect the GCR residual within a set of linked TCO detections but, in combination with
their good site on Cerro Pachòn, will mean that TCO detections will be trailed. Our LSST
TCO detection estimate in table 1 used a peak rate of detectable motion of 10 deg/day for
which a TCO would leave a 33 pixel long trail in the system’s focal plane. We expect that
LSST could detect about 2.2 TCOs/lunation but a major problem with this success rate is
that most will be too faint for followup by other observatories.
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We also considered a targeted survey with Hyper Suprime-Cam (HSC; Takada 2010)
on the Subaru telescope located on Mauna Kea, Hawaii, as described in §2. The 870 Mpix
HSC with a ∼ 1.5◦ wide FOV mounted on a 8.2 m telescope at one of the best sites in the
world could be an impressive TCO survey instrument if dedicated to the task. For our
study we assumed that the HSC can reach V ∼ 25 in a 39 s exposure while non-sidereal
tracking at 3.3 deg/day on the TCOs being captured moving in position angle of θ = 94 deg.
This system should allow detection of TCOs of only about 65 cm in diameter since the mean
geocentric distance of the objects at capture is 6.3 ± 1.4(rms) LD. Given the 39 s exposure
time, HSC’s FOV, and 20 s readout time we expect that we could survey ∼ 300 deg2 /night
centered on opposition with this system. While there are ∼4.5 TCOs visible in the night
sky with V < 25 the restrictions on the position angle, rate of motion and the survey region
suggest that the targeted HSC survey could detect about 0.2 TCOs/night or, equivalently,
have a 89% chance of detecting a TCO in a 10-night observing run.
The four surveys discussed above are sensitive to TCOs of different sizes due to their
different survey capabilities (see fig. 6). The ATLAS survey has a peak sensitivity to TCOs
with H ∼ 30 or about 2 m in diameter. The problem is that there are only 1-2 objects of
this diameter in orbit at any time, and the probability that they pass near opposition is
relatively small (where the phase angle is close to zero at a close enough distance to be
detected by ATLAS). Pan-STARRS1 could do much better than ATLAS but is limited
by the relatively small amount of time devoted to surveying for moving objects and the
inability to identify trails in the images. The wide area LSST and targeted HSC surveys
both have good sensitivity to TCOs of sub-meter diameter. The HSC survey discovers
larger TCOs because it is targeting objects as they are captured at typically more than 5 LD
— the objects simply must be large in order to detect the TCOs at many lunar distances.
While this survey mode requires dedicated time on a large telescope in a mode that is
unlikely to be useful for other purposes it does offer the opportunity of 1) detecting larger
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TCOs 2) as they are being captured. Both of these advantages make the objects easier to
detect in dedicated followup efforts, with radar, and maximizes the time the objects can be
studied while captured. These benefits are particularly interesting when considering TCOs
as spacecraft targets (see §6 and Chyba and Patterson 2013).
3.2.
Meteor Survey Detection
The Granvik et al. (2012) Earth-Moon captured NEO simulation yielded 14,909 TCOs
of which 189 or ∼1% eventually struck Earth. They normalize their results to the Brown
et al. (2002) bolide data and argue that about 0.1% of meteors are TCOs prior to striking
the atmosphere. Considering that all-sky meteor surveys have existed for decades and
that modern networks like the CAMO (Weryk et al. 2008) and CAMS (Jenniskens et al.
2011) systems have the capability of measuring the pre-meteor phase orbit it might seem
surprising that their have as yet been no reports that 1 in a 1,000 meteors were originally
in geocentric orbit.
The solution to the puzzle is that not all meteors are equally visible. The apparent
brightness of a meteor ∝ m−2.02±0.15 s−7.17±0.41 where m is the meteoroid’s mass and s is
its speed (Sarma and Jones 1985; Campbell et al. 2000) (there is also an insignificant
dependence on the zenith angle that is consistent with zero that we ignore in our analysis).
Thus, a meteor’s apparent brightness is exceedingly sensitive to its rate of motion. Since
the CAMS system has a limiting apparent magnitude of ∼ 4.8 at the TCOs’ sluggish
∼ 11.19 ± 0.03(rms) km/s impact speed the meteoroid has to have a mass of & 0.06 grams
to be detected. This suggests that TCOs must have a diameter of more than about
3.7 mm to be detected assuming a typical meteorite grain density of 3.0 gm/cm3 (Britt and
Consolmagno 2004). The lower diameter limit corresponds roughly to the diameter of a
small pea.
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We calculate that CAMS observed about 3.3 meteors/year with speeds in the narrow
range of TCO impact speeds within the set of all objects reported by CAMS from Oct
2010 through December 2011. i.e. We assumed a Gaussian probability density in the speed
of each object, integrate the total number probability of detected objects over the range
[11.16, 11.22] km/s, and then normalize to the annual rate. Extrapolating from the Brown
et al. (2002) meteoroid SFD down to 3.7 mm diameter suggests that there are 162 ± 12 × 103
TCOs of this size or larger striking Earth every year corresponding to about 3.2 ± 0.3 × 10−4
TCOs/km2 /year. The CAMS system monitors the sky above an altitude of about 30◦ and
at a typical meteor phase onset altitude of about 100 km the system therefore monitors
about 2 × 105 km2 . Accounting for the fact that the survey can only operate at night (about
8 hours/day) and for weather losses (we assume 1/3) we estimate that CAMs should detect
about 14 ± 1 TCO meteors/year.
The agreement to within a factor of about 4 between the predicted and observed
rate of meteors with TCO-like speeds is incredibly good considering all the unknowns and
uncertainties. We think that the agreement should not be overinterpreted as any value
within a couple orders of magnitude could have been argued as being due to detection
efficiency within the CAMS system, errors in the TCO model at the smallest sizes due to
Yarkovsky, contamination of the CAMS TCO-like speeds by underestimated uncertainties
or artificial satellite debris, etc.
The CAMO system has been in operation for about six years and recently detected
its first TCO-like meteor with an atmospheric entry speed of 11.2 ± 0.8 km/s and mass of
∼100 g (about 44 mm diameter). SHOULD WE INCLUDE A FOOTNOTE LINK
TO THE METEOR VIDEO? The measured meteor speed is consistent with the
expected TCO meteor impact speed but at the 3-σ level in the range of possible TCO
speeds there is only a 10% probability that it is a TCO-like speed assuming a Gaussian
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distribution in the uncertainty. We estimate that CAMO should see about 0.01 TCOs/year
of this size or larger implying that its detection of a TCO-like meteor in only six years
would be unusual — perhaps ∼15× higher than what we predict from the TCO model.
On the other hand, with only a 10% chance that the CAMO object has a TCO-like speed
perhaps the numbers are in excellent agreement. Once again, we do not want to speculate
too much on a disagreement between the predicted and observed TCO meteor rate for the
same reasons as discussed above for CAMS.
4.
Infrared Detection
Figure 5 hints that an infrared ecliptic survey could be effective at detecting TCOs.
Ground-based infrared facilities do not currently have the sensitivity and wide-fields
necessary for the TCO survey but the successful WISE spacecraft mission (e.g. Wright
et al. 2010) and its asteroid detecting NEOWISE sub-component (e.g. Mainzer et al. 2011)
shows that space-based IR surveys can be very effective. Thus, we use the NEOWISE
mission as a baseline for an IR TCO survey even though future IR spacecraft surveys such
as the proposed NEOCAM mission (Mainzer 2006) will certainly be even more effective at
asteroid discovery.
Thus, to characterize the relative utility of a space-based TCO survey we estimate the
number of TCOs that might have been detected by NEOWISE in their cryogenic 12µm W3
band images, the band most sensitive to meter-scale TCOs as described in §2. WISE was
sensitive to sources with flux ΦW 3 > 0.65 mJy in the W3 band (Wright et al. 2010) and the
fastest solar system object they reported to the MPC was moving at 3.22 deg/day. The
WISE FOV was 47′ with a 90 minute re-visit time at each sky location. In the worst-case
scenario a TCO moving at up to 6 deg/day should still be detected 3× but in practice
NEOWISE required that the object be detected > 3× to reduce the false detection rate.
– 19 –
Figure 5 shows the sky plane residency time distribution for TCOs with a slightly more
conservative ΦW 3 > 0.65 mJy moving at < 3 deg/day. At any time there are ∼ 1 TCOs on
the entire sky plane that meet these requirements. This corresponds to about 0.005 TCOs
in a 47′ wide strip extending 360◦ around the sky at quadrature — the region surveyed
by WISE in a 90 minute time interval. With a ∼ 2 day refresh time for TCOs with these
properties we calculate that WISE had about a 7% probability of detecting one in each
of the 10 months of the survey or about a 54% chance of detecting one TCO during the
spacecraft’s W3 band’s operational lifetime.
Figure 7 shows that WISE’s peak sensitivity for TCOs occurs for objects in the 1 m size
range — well-tuned to the expected size of the largest TCOs expected in the steady-state.
Since the WISE mission’s cryogenic lifetime of about 10 months is well matched to the
average TCO lifetime it is likely that a WISE-like survey would detect one of the 1-2
one-meter scale TCOs in orbit around Earth at any time. Larger objects are unlikely to be
detected simply because they are unlikely to be captured and the smaller objects are not
detected due to the flux limitations. Thus, a NEOWISE-like spacecraft mission could be an
effective technique for detecting the largest, and therefore most interesting, TCOs available
in the steady state.
Of course, despite its high detection efficiency, NEOWISE is not known to have
reported a TCO to the MPC. Short of arguing that the TCO model is invalid we think the
most likely explanation is that only XXX% GET THIS NUMBER FROM MASIERO
OR MPC of reported NEOWISE candidates had ground-based followup confirmation
turning them into NEO discoveries. Ground-based followup typically assumes that the
objects are on heliocentric orbits when creating ephemeris predictions so that recovery of
objects that are actually on geocentric orbits will be unlikely if not effectively impossible.
Thus, we conclude that space-based IR surveys like WISE could be very effective for
– 20 –
discoverying TCOs if the analysis pipeline and ground-based recrovey efforts were sensitive
to the possibility of geocentric orbits.
5.
RADAR Detection
There are multiple radar systems capable of detecting TCOs either while in orbit
around Earth or as they plummet through the atmosphere. One option that we will not
discuss in detail here is the capability of the Air Force Space Surveillance Systems. There
is understandably little verifiable public information on the system’s performance but it ‘is
claimed to be able to detect objects as small as 10 cm diameter (four inches) at heights up
to 30,000 km5 as they pass through a ‘space fence”. This system might be very powerful at
detecting TCOs as they pass quickly through perigee. Instead, we focus on detecting TCOs
with the more traditional monostatic and bistatic radar facilities at Goldstone and Arecibo
and meteor radar facilities.
5.1.
Meteoroid RADAR Detection
REALLY NEED NOLAN & HOWELL TO GIVE REALITY CHECK ON
THIS SECTION.
Detecting asteroids by their reflected radar signals is now routine operations at the
Arecibo and Greenbank facilities where almost 500 objects have been detected as of October
2012.6 They now bounce radar off asteroids on a WEEKLY? basis and the smallest
5
http://en.wikipedia.org/wiki/Air Force Space Surveillance System;
2011
6
http://echo.jpl.nasa.gov/asteroids/
5
Aug
– 21 –
detected object7 was 2006 RH120 the only known TCO, with H ∼ 29.6 corresponding to a
few meters in diameter. However, they have not discovered new asteroids other than the
orbiting companions of the targeted objects because the radar beam has limited angular
coverage and sensitivity that drops off like ∆4 where ∆ is the geocentric distance.
Figure 8 shows that the TCOs range and range-rate is within Arecibo’s radar detection
capabilities. A 100 cm/25 cm diameter non-rotating TCO is detectable if it is within
∼11/∼6 LD (we assume a minimum S/N=0.5 per run). NEED NOLAN & HOWELL
TO CHECK/CONFIRM/REVISE. SHOULD WE USE A LIMIT ON THE
SN/RUN OF 0.5 OR SHOULD WE USE A LIMIT ON THE S/N OF 8
INSTEAD? The problem is that the reflected signal is doppler spread by the object’s
rotation and at some size- and range-dependent rotation rate the received S/N will be
below the system’s detection limit.
The rotation rate distribution of meter-scale meteoroids is addressed in §B where we
argue that we have a technique for calculating the distribution for TCOs in our size range of
interest. While most TCOs are likely spinning too fast to be radar-detected the long tail of
slower rotators makes their detection possible and we proceed using our normalized rotation
period (p) distribution given by a Maxwellian spin-rate (ω = p−1 ) probability distribution
f (ω; d) =
2
p 2 ω2
− 21 [ ω ω(d) ]
0
exp
( )
π ω0 (d)3
(1)
where d is the object’s diameter in meters, ω is the rotation rate in rev/hour, and ω0 (d) is a
parameter selected so that the median rotation period of the meteoroids at that diameter is
given by p(d) = 0.005d (Farinella et al. 1998). The median rotation rate for the distribution
is given by ω̂ ≈ 1.54 ω0 .
Our calculations suggest that detecting the TCOs is essentially impossible with
7
http://echo.jpl.nasa.gov/∼lance/small.neas.html
– 22 –
monostatic operations at Arecibo in which the site both transmits and receives the radar
signal. The minimum geocentric distance of detectable objects of 4-5 LD is then fixed by
the time to switch between transmitting and receiving and there are simply not enough
TCOs rotating slow enough beyond that distance to make the technique viable (see fig. 8).
There may be opportunities to discover TCOs using bistatic operations where the
radar signal is transmitted by Arecibo and received at Greenbank NEED CITATION(S).
We assumed a transmitting power of 0.9 MW, calculated the returned S/N accounting
for the TCOs rotation rate according to (Renzetti et al. 1988), and required a minimum
S/N=0.5 per run HOW MANY RUNS FOR A DETECTION? THE TOTAL
NUMBER OF RUNS IS DETERMINED BY A RELATIONSHIP BETWEEN
NUMBER OF RUNS AND GEODISTINACE. THE S/N SCALES AS THE
S/N PER RUN TIMES THE SQUARE ROOT OF THE NUMBER OF RUNS
(SEE FIG. ??) on reception. We find that there are ∼ 3 × 10−6 radar detectable TCOs at
any time in a cone with 1◦ opening angle centered at opposition and about 4 × 10−6 at the
quadratures. Since the cone has an area of ∼ 0.79 deg2 these values correspond to about
4 × 10−6 and 5 × 10−6 detectable TCOs/deg2 at the two locations respectively.
The detectable TCOs must be close to Earth and typically are well within 1 LD as
shown in fig. 8. They are dominated by objects . 0.5 LD with range-rate speeds in the
range [−0.5 : 0.5] km/s and, as described above, must be objects in the tail of the rotation
distribution with relatively long spin rates. Since the objects are so close to Earth their
apparent transverse speeds are high and the refresh rate of objects in the cone is about
3 minutes. The competing effects of the different distance distributions for TCOs at
opposition and at quadrature with their different refresh rates in the two directions leads
to a roughly equal probability of detecting TCOs. While the probability of detecting an
object is small in a single pass over the area of the cone if the search could be accomplished
– 23 –
in less than the refresh time and if it could be maintained continuously for 10 days there
is a better than 1 in 3 chance of detecting a TCO. BRYCE, DO YOU HAVE THE
DETECTED SIZE DISTRIBUTION AVAILABLE? YES SEE FIG. ??.
5.2.
Meteor RADAR Detection
NEED BROWN TO WRITE THIS SECTION.
I THINK THAT REFLECTED SIGNAL GOES SOMETHING LIKE V 4
SO NO CHANCE OF SEEING OBJECTS AT ESCAPE SPEED. I THINK WE
CHECKED WITH BROWN LONG AGO THAT THE METEOR RADARS
CAN’T SEE OBJECTS AT 11.2 KM/S. NEED TO CHECK WITH BROWN.
IT WOULD BE BEST IF BROWN COULD WRITE THIS SECTION...
DISCUSS SENSITIVITY VS. MASS AND SPEED AND ARGUE THAT
TCOS MOST LIKELY CAN’T BE DETECTED THIS WAY?
6.
Discussion
We have shown above while it will be challenging to discover TCOs on a regular basis
it is not out of the question. Since the scientific opportunities are substantial the TCOs
might generate enough interest to dedicate resources to the task.
One TCO discovery opportunity that has not been explored is opportunities for data
mining of the Minor Planet Center’s (MPC) so called one-night-stand (ONS) file. This list
of detections that have never been linked to known heliocentric objects with longer arcs
could be searched specifically for geocentric objects. As mentioned in the introduction, we
have anecdotal evidence suggesting that unknown geocentric objects were/are discovered
– 24 –
and followed by asteroid surveys but they are not flagged as interesting by the MPC because
it is difficult or impossible for them to distinguish these objects from operational spacecraft
or space junk. However, the work of Granvik et al. (2012) provides a means of comparing
the derived geocentric orbit to those expected for TCOs that are typically very different
from artificial satellite orbits. Given the short arc lengths and lack of spectroscopic or
even colors of these historical ONS detections it is unlikely that any TCO could ever be
confirmed but if there were enough detected objects it might be possible to compare the
distribution of their geocentric orbits with the expected TCO distribution.
Of course, there will always remain the possibility that the detected objects were
classified satellites because the types of orbits occupied by TCOs are also the types of
orbits that might be desirable for ‘hiding’ spacecraft — objects on these orbits would spend
a great of time far from Earth where they are very faint and when they are brighter and
closer to Earth they would be moving extremely fast.
This work has addressed only the issue of TCO discoverability and ignored issues
related to observation cadence, followup, and time of discovery relative to time of capture
— all important issues for determining the viability of observation programs that would
target the TCO for physical characterization or spacecraft missions that might attempt to
intercept or even retrieve the object (Chyba and Patterson 2013). This kind of work would
require a high fidelity simulation of an asteroid survey system capable of integrating the
TCOs within the Earth-Moon system.
Even if TCOs are discovered by operational surveys it is important to understand
whether their orbits can be determined quickly and well enough to assure followup or even a
spacecraft mission. As is well known from NEO followup efforts, short-arc extrapolation of
discovery tracklets for the ‘simple’ heliocentric NEO motion can quickly lead to ephemeris
sky plane uncertainties of many deg2 making recovery impossible. We have begun to
– 25 –
study the evolution of the orbital uncertainties and error with TCO observational arc and
find that they converge rapidly, probably due to their close proximity and the advantage
afforded the orbit determination by the associated topocentric parallax (Granvik 2013).
7.
Conclusions
The prospects for discovering TCOs are challenging
JEDICKE WILL WRITE ONCE REMAINDER OF PAPER IS COMPLETE.
Acknowledgments
This work was supported by NASA NEOO grant NNXO8AR22G. MG was funded by
grants #125335 and #137853 from the Academy of Finland. XXX
ANYBODY ELSE NEED SOMETHING HERE?
– 26 –
A.
Appendix
JEDICKE ARGUES THAT MOST OF THIS APPENDIX SHOULD
*NOT* BE INCLUDED IN THE PAPER. IF PORTIONS OF THE PAPER
ARE UNCLEAR WITHOUT IT WE SHOULD CUT & PASTE FROM IT
INTO THE TEXT. I AM HAPPY TO BE CONVINCED OTHERWISE.
We assume that there exists an ensemble of particles whose trajectories and observable
properties from Earth are calculable.
A.1.
Sky plane residence time density
We define the time that a particle spends in the interval [λ0 − dλ0 /2, λ0 + dλ0 /2] and
[β0 − dβ0 /2, β0 + dβ0 /2] as its sky-plane residence time where λ0 and β0 represent ecliptic
longitude and latitude respectively. The sky-plane residence time density for an individual
particle j at location (λ0 , β0 ) is
Z+∞
ρj (λ0 , β0 ) =
dt δ(λj (t) − λ0 ) δ(βj (t) − β0 )
(A1)
−∞
where we introduce the Dirac δ-function and (λj (t), βj (t)) represent the ecliptic longitude
and latitude of the particle at time t respectively. Therefore, the residence time of the
particle within (dλ0 /2, dβ0/2) of (λ0 , β0 ) is ρj (λ0 , β0 ) dλ0 dβ0 , and the residence time for
the particle within the extended range λ1 ≤ λ < λ2 and β1 ≤ β < β2 is:
Tj (∆λ0 , ∆β0 ) =
Zλ2
dλ
λ1
Zβ2
dβ ρj (λ, β).
(A2)
β1
The sky-plane residence time density for a population of particles is the sum of the
individual sky-plane residence time densities
ρ(λ0 , β0 ) =
X
j
ρj (λ0 , β0 )
(A3)
– 27 –
Letting the normalization constant
C=
Z
dλ
Z
dβ ρ(λ, β)
(A4)
i.e. the cumulative time spent over the entire sky by all particles, the normalized residence
time density is
ρN (λ, β) =
1
ρ(λ, β).
C
(A5)
Thus, ρN (λ, β)∆λ∆β is the fraction of time that all the particles spend within (∆λ/2, ∆β/2)
of (λ, β).
A.2.
Sky-plane number density
Let the particles’ cumulative H-frequency distribution (HFD) be
NHF D (H) = N0 10α(H−H0 )
(A6)
i.e. NHF D (H) is the number of particles with absolute magnitude < H.
The differential HFD is then
nHF D (H) dH = N0 α ln 10 10α(H−H0 ) dH
(A7)
i.e. there are nHF D (H) dH objects in the interval dH at magnitude H.
Since ρN (λ, β) is the normalized sky-plane residence time density, the differential
sky-plane number density of objects at absolute magnitude H is
n(λ, β, H) = nHF D (H) ρN (λ, β).
(A8)
i.e. the number of objects in an absolute magnitude interval of width ∆H around H and in
longitude and latitude intervals of widths (∆λ, ∆β) around (λ, β) is n(λ, β, H) ∆λ ∆β ∆H.
– 28 –
The cumulative sky-plane number density of objects brighter than H0 at (λ0 , β0 ) is
N(λ0 , β0 , H0 ) =
Z
H0
dH n(λ0 , β0 , H)
= NHF D (H0 ) ρN (λ0 , β0 )
(A9)
so that the number of particles with H < H0 in longitude and latitude intervals of widths
(∆λ0 , ∆β0 ) around (λ0 , β0 ) is N(λ0 , β0 , H0 ) ∆λ0 ∆β0 .
Thus, the number of particles with absolute magnitude < H in the range λ1 ≤ λ < λ2
and β1 ≤ β < β2 is
Zλ2
dλ
λ1
A.3.
Zβ2
dβ N(λ, β, H).
(A10)
β1
Sky-plane residence times and number densities with constraints
Following the nomenclature of §A.1 and §A.2 we develop the sky-plane residence and
number densities subject to a set of constraints suitable for visible light surveys (§A.3.1),
infrared surveys (§A.3.2), and radar surveys (§A.4).
A.3.1.
Visible light surveys
In this section we impose constraints on the particles’ apparent magnitude V and rate
of motion ω.
The sky-plane residence time density for an individual particle j with absolute
magnitude H0 while it has apparent magnitude V (H0 ) < V0 and rate of motion ω < ω0 is
Z+∞
vis
ρj (λ0 , β0 , V0 , ω0 , H0 ) =
dt δ(λj (t) − λ0 ) δ(βj (t) − β0 )
−∞
(A11)
V (H0 ) < V0 , ω < ω0
– 29 –
This equation is identical to eq. A1 except for the delimiter that constrains the rate of
motion and apparent magnitude.
If all the particles have absolute magnitude H0 their sky-plane residence time
distribution density while they have V (H0 ) < V0 and ω < ω0 is
ρvis (λ0 , β0 , V0 , ω0 , H0 ) =
X
ρvis
j (λ0 , β0 , V0 , ω0 , H0 ),
(A12)
j
and their normalized sky-plane residence time distribution density is:
ρvis
N (λ0 , β0 , V0 , ω0 , H0 ) =
1 vis
ρ (λ0 , β0 , V0 , ω0 , H0 )
C
(A13)
where the normalization constant C is from eq. A4. This normalization constant ensures
that the normalized residence time in eq. A13 is the fraction of all possible particles rather
than the fraction of particles that satisfy the constraints so that we can determine the
number density of particles below.
Following the arguments in §A.2 the number density of particles with H < H0 ,
V (H0 ) < V0 and ω < ω0 as a function of sky-plane location is
N
vis
(λ0 , β0 , V0 , ω0 , H0 )
H0
=
dH n(λ0 , β0 , V0 , ω0 , H)
V (H0 ) < V0 , ω < ω0
Z H0
= ρN (λ0 , β0 , V0 , ω0 , H0 )
dH nHF D (H)
Z
(A14)
i.e. There are N vis (λ0 , β0 , V0 , ω0 , H0 ) ∆λ0 ∆β0 particles in the range [λ0 −∆λ0 /2, λ0 +∆λ0 /2]
and with [β0 − ∆β0 /2, β0 + ∆β0 /2] that also have H < H0 , V (H0 ) < V0 and ω < ω0 .
The total number of particles in the sky that have V (H0 ) < V0 , ω < ω0 and absolute
magnitude brighter than H0 is than
=
Z
dλ
Z
dβ N vis (λ, β, V0, ω0 , H0 )
(A15)
and the total number of particles in the range [λ1 , λ2 ] and [β1 , β2 ] with the same constraints
– 30 –
on V , ω and H is
=
Zλ2
dλ
λ1
Zβ2
dβ N vis (λ, β, V0, ω0 , H0 ).
(A16)
β1
A.3.2.
Infrared Surveys
In this section we impose constraints on the particles’ IR flux (F ), diameter (D), and
rate of motion. We convert between diameter and absolute magnitude using Fowler and
Chillemi (1992):
1.329 × 106 −H/5
D
=
10
√
meters
pv
(A17)
where pv is the albedo. We used pv = 0.11 for the albedo throughout this work to make the
conversion from absolute magnitude and diameter simple — H = 18 corresponds to 1 km
diameter and factors of 10 changes in the diameter cause a change of 5 units in absolute
magnitude.
Following the discussion of §A.3.1 the sky-plane residence time distribution density for
an individual particle j with diameter D0 while it has thermal infrared flux F (D0 ) > F0
and rate of motion ω < ω0 is
Z+∞
ρIR
dt δ(λj (t) − λ′ ) δ(βj (t) − β ′ )
j (λ0 , β0 , F0 , ω0 , H) =
.
(A18)
F (D0 ) > F0 , ω < ω0
−∞
The number density of particles in with D < D0 , F < F0 and ω < ω0 as a function of
sky-plane location is
N IR (λ0 , β0 , F0 , ω0 , D0 )
=
ZD0
dD n(λ0 , β0 , F0 , ω0 , D)
= ρIR
N (λ0 , β0 , F0 , ω0 , D0 )
F (D0 ) > F0 , ω < ω0
ZD0
dD nHF D (D)
(A19)
– 31 –
The total number of particles in the sky that have F (D0 ) > F0 , ω < ω0 and D > D0 is
Z
=
dλ
Z
dβ N IR (λ, β, F0 , ω0 , D0 )
(A20)
and the total number of particles in the range [λ1 , λ2 ] and [β1 , β2 ] with the same constraints
on flux, rate and diameter is
=
Zβ2
dβ
Zλ2
dλ N IR (λ, β, F0 , ω0 , D0 ).
(A21)
λ1
β1
Infrared flux
The observed IR flux for a particle at geocentric distance ∆ and heliocentric distance r was
calculated following Harris et al. (2009) and Mainzer et al. (2011):
2
ǫD
F =
4∆2
Z2π
0
π
dθ
Z2
0
dφ
Zλ2
dλ B(λ, Tss ) ǫb (λ) sin θ cos θ
(A22)
λ1
where ǫ is a particle’s thermal emissivity, ǫb is the passband efficiency as a function of
wavelength λ (not ecliptic longitude), B is the blackbody flux for an object of temperature
Tss — the temperature of the sub-solar point on the particle —
S(r) (1 − A)
Tss =
ηǫσ
14
(A23)
where S(r) is the solar flux at heliocentric distance r, η is the beaming parameter, σ is the
Stefan-Boltzmann constant and A is the bond albedo.
Our interest in small natural satellites of Earth allows us to assume that r ∼ 1 AU and,
since the detectable objects are typically in the range from ∼0.1 m to ∼1.0 m in diameter,
are in orbit around Earth for an average of ∼9 months, and rotate relatively quickly (see
§B below), we assume that the entire object is in thermal equilibrium. i.e. there are no
longitudinal or latitudinal variations in the blackbody flux emitted from the particle.
– 32 –
A.4.
Radar surveys
We now extend the discussion from the previous two sub-sections to a 4-dimensional
residence time distribution suitable for radar surveys in which a ground-based radar facility
beams a radar signal towards a particle and then detects the reflected signal. The radar
survey residence time distribution is in ecliptic longitude and latitude as derived in the last
two sub-sections but also in terms of the geocentric distance (range, ∆) and geocentric
˙ In this section we provide the 4-d residence time distribution with
speed (range-rate, ∆).
constraints on the particles’ reflected radar S/N (Φ) which is a function of the detected
particle’s diameter and rotation period (T ).
Following the developments in §A.1 the 4-d sky-plane/range/range-rate residence time
density distribution for an individual particle equivalent to eq. A1 is
˙ 0)
ρradar
(λ0 , β0 , ∆0 , ∆
j
(A24)
Z+∞
˙ j (t) − ∆
˙ 0)
dt δ(λj (t) − λ′ ) δ(βj (t) − β ′ )δ(∆j (t) − ∆0 )δ(∆
=
(A25)
−∞
Summing over all the particles and normalizing yields the radar equivalent to eq. A5:
˙ = 1 ρradar (λ, β, ∆, ∆),
˙
ρradar
(λ, β, ∆, ∆)
N
C
(A26)
the normalized residence time in the 4-d space for the particle ensemble. i.e. the particles
˙
˙ within (dλ/2, dβ/2, d∆/2, d∆/2)
˙
spend a fraction of time ρradar
(λ, β, ∆, ∆)dλdβd∆d
∆
of
N
˙
(λ, β, ∆, ∆).
Analagous to eqs. A6 and A7 we let the particles’ cumulative and differential
size-frequency distributions (SFD) be represented by NSF D (D) and nSF D (D) respectively.
i.e. There are NSF D (D) particles with diameter > D and nSF D (D)dD particles in an
interval of width dD centered at diameter D.
– 33 –
Furthermore, we assume that the rotation period distribution is a function of both
the particles’ diameters and the rotation period and the fractional differential period
distribution can be represented by p(D, T ). i.e. the fraction of the population with D in the
range [D − ∆D/2, D + ∆D/2] and T in the range [T − ∆T /2, T + ∆T /2] is p(D, T )∆D∆T .
Let P (D, T ) represent the cumulative fraction of particles with diameter > D and period
< T.
The number density of objects in the 4-d residency space with diameter D and rotation
period T is then represented by
˙ D, T ) = nSF D (D) P (D, T ) ρradar (λ, β, ∆, ∆)
˙
nradar (λ, β, ∆, ∆,
N
(A27)
˙ D, T ) dλ dβ d∆ d∆
˙ dD dT within (dλ/2, dβ/2, d∆/2, d∆/2,
˙
i.e. there are nradar (λ, β, ∆, ∆,
dD/2, dT /2)
˙ D, T ).
of (λ, β, ∆, ∆,
Similarly, the cumulative number of objects in the 4-d residency space with diameters
> D and rotation period < T is then represented by
˙ D, T ) = NSF D (D) P (D, T ) ρradar (λ, β, ∆, ∆)
˙
N radar (λ, β, ∆, ∆,
N
(A28)
˙ D, T ) dλ dβ d∆ d∆
˙ dD dT within (dλ/2, dβ/2, d∆/2, d∆/2,
˙
i.e. there are nradar (λ, β, ∆, ∆,
dD/2, dT /2)
˙ D, T ).
of (λ, β, ∆, ∆,
Finally, we can calculate the number of detectable objects by radar subject to
observational constraints. For instance, the detected S/N ≡ Φ(∆, D, T ) is a function of
the particles’ geocentric distance, diameter and rotation period and must be larger than a
specified threshold Φ0 to be detected. Thus, the number density of detectable objects with
˙ is
D, T and Φ > Φ0 at (λ, β, ∆, ∆)
radar
n
radar
˙
˙
(λ, β, ∆, ∆, D, T, Φ0 ) = nSF D (D) P (D, T ) ρN (λ, β, ∆, ∆)
φ(∆,D,T )>Φ0
.
(A29)
The number of particles detectable by radar with D > D0 and T < T0 within an
– 34 –
interval of width (dλ0 , dβ0 ) centered at (λ0 , β0 ) is then
= dλ0 dβ0
Z∞
D0
dD
ZT0
∞
dT
∆Zm ax
d∆
∆m in
˙ m ax
∆
Z
˙ m in
∆
˙ SF D (D) P (D, T ) ρradar (λ0 , β0 , ∆, ∆)
˙ d∆n
N
φ(∆,D,T )>Φ0
(A30)
where the limits on the range and range-rate integrals are defined by the radar operational
characteristics (e.g. the minimum time to switch between transmit and receive).
The reflected signal power received at Earth is
Prec =
Ptran ǫR AR
∆2 r 2
(A31)
where Ptran is the transmitted radar power, ǫR is the radar facility’s detection efficiency,
and AR is the particle’s radar albedo (the fraction of the signal at the particle reflected
back to the transmitter). The received power is detectable by the facility if it exceeds the
intrinsic detector and background noise. However, the received power is spread over a
range of wavelengths if the particle is rotating, further degrading the signal. The detected
signal-to-noise ratio is then (Renzetti et al. 1988):
SNR =
GT G2A λ2 σ(NL)0.5
∆4 kb T ( 2ΩD
)0.5
λ
(A32)
where GT is the peak transmitter gain, GA Is the antenna gain, λ is the radar wavelength
(in meters), σ is the target’s radar cross section (in m2 ), N is the number of observations,
∆ is the target’s distance (in meters), kb is Boltzmann’s constant, T is the system noise
temperature(degrees Kelvin), B the receiver bandwidth (s−1 ), and L the post-detection
integration time (seconds).
A.5.
Fluxes and lifetimes of particles with constraints
In the steady state the number of particles in a population that satisfy some constraints
NC is related to the mean lifetime LC of particles in the population and the flux of particles
– 35 –
FC into or, equivalently, out of the population:
NC = FC LC .
(A33)
‘The population’ may be any set of particles that satisfy the constraints. e.g. particles
brighter than V0 within 10 deg of opposition, e.g. particles with diameter larger than d0 in
a shell of geocentric range [∆1 , ∆2 ].
The mean lifetime of particles that satisfy the constraints is given by the total amount
of time all the particles spend satisfying the constraints divided by the total number of
particles, Ntot :
LC
Ntot Z
1 X
dt δ[Cj (t)],
=
Ntot j=1
(A34)
where Cj (t) = 0 only when the constraint is satisfied by particle j at time t.
If we let ρ(x̄, t) represent the number density as a function of time of particles with
respect to the set of parameters x̄ upon which the constraints are defined then the number
of particles that meet the constraints at any time is:
NC =
Z
dt
Z
dx̄ δ[C(x̄, t)] ρ(x̄, t)
(A35)
where C(x̄, t) = 0 only when the constraint is satisfied.
The steady state flux of particles into or out of the population can then be determined
from the mean lifetime and number of particles.
A.6.
Mean Values on the Skyplane
The mean value of a quantity Q(t) (e.g. rate of motion, position angle or geocentric
distance) for a set of particles j over a time interval [t1 , t2 ] at the sky plane location (λ0 , β0 )
– 36 –
under a set of constraints C is:
Q̄total (λ0 , β0 ) =
P Rt2
j t1
dt Q[λj (t), βj (t)] δ(λj (t) − λ0 ) δ(βj (t) − β0 )
C
.
t
2
R
P
dt δ(λj (t) − λ0 ) δ(βj (t) − β0 )
j t1
B.
(A36)
C
Meteoroid rotation rates
Rotation rates of meteoroid-scale objects (∼0.1 to 1 m diameter) are essentially
unknown. At the current time there is only one known NEO with H > 33 (about 1 m
diameter) and the smallest object in the Light Curve Database (LCDB, Warner et al.
(2009)) has H ∼29.5 (the only known TCO, 2006 RH120 ). The observational selection effects
against identifying the fastest and slowest rotators must be tremendous for objects with
H . 33 i.e. the objects are so small that they are faint and require relatively long exposure
times so that it is impossible to resolve fast rotation rates. Observations of fireballs add
the complication of their interaction with the atmosphere. Periodic variations in the flux
along a meteor’s trail on an image might represent the object’s underlying rotation period
but these observations will specifically select those objects with non-spherical shapes and
fast rotation periods. Thus, there is not much that can be done to constrain the objects’
rotation rates except to use the available limited and biased observations.
Farinella et al. (1998) adopted a spin-period diameter relationship of P = 5(D/km) hours
for kilometer-scale asteroids (P = 0.005(D/meters) hours). Extrapolating to the meteoroid
size range yields periods of 18 s at 1 m diameter and 2 s at 10 cm diameter. While these
rates seem fast Beech and Brown (2000) point out that they are much slower than those
observed for meteors in that size range. The meteor observations suggest a spin-period
diameter relationship of P = 0.5(D/meters) seconds or P ∼ 0.0001(D/meters) hours — 50×
faster than that suggested by Farinella et al. (1998). Their results imply rotation periods of
– 37 –
0.5 s at 1 m diameter and 0.05 seconds at 10 cm diameter.
The Beech and Brown (2000) and Farinella et al. (1998) spin-period vs. diameter
relationships are shown in fig. 9 that also includes the median values from the LCDB
in bins corresponding to 10 m wide intervals in meteoroid diameter. The agreement
between the Farinella et al. (1998) results and the LCDB is excellent in the range from
[0, 100] m diameter. CAN BROWN DISCUSS WHY THERE COULD BE SUCH
A DISCREPANCY BETWEEN EXTRAPOLATING UP IN SIZE FROM
BEECH AND DOWN IN SIZE FROM FARINELLA?
We fit the spin rates (ω = P −1) of the LCDB asteroids in the same size intervals to
Maxwellian distributions of the form:
f (ω) =
r
2
2 ω2
− 21 ( ωω )
0
exp
.
π ω03
(B1)
The Maxwellian distribution is the expected form of the rotation frequency distribution for
a collisionally-evolved population of asteroids (?).
The cumulative distribution is given by
F (x) =
Zx
f (y) dy
0
1
=
ω03
2
= √
π
r
2
π
Zx
2
2
y exp
− 21 [ y 2 ]
ω0
dy
0
x2
2α2
Z
√
z exp−z dz
0
3 x2 3
= γ , 2 Γ( )−1
2 2ω0
2
3 x2 = P , 2 .
2 2ω0
where P (a, x) is the incomplete gamma function.
(B2)
– 38 –
We compute the median by setting eq. B2 to
1
2
and solving numerically for x in terms
of ω0 which results in the following calculation for the median
r
3 1
≈ 1.54 α
P −1 ,
x̂ =
2 2
(B3)
as shown on fig. 9. We note that the medians of our fits are typically below both the
predicted and actual medians but that our calculated median approaches the predicted
value as the diameter decreases. Taken at face value it implies that the smaller objects
(near 10 m diameter) have a more Maxwellian spin rate distribution but the distribution
becomes less Maxwellian as the objects’ diameters increase (to 100 m diameter). While
there are tremendous biases and incompleteness in the known distribution of the small
objects’ rotation rates this result is surprising given that objects that are 3 orders of
magnitude larger in diameter already show non-Maxwellian rotation rate distributions (?).
For the purpose of this work it is not important because we are concerned with
modeling the spin-rate distribution of the smallest meteoroids where our fit agrees well with
the data and with Farinella et al. (1998).
Thus, we proceed under the assumption that 1) the median rotation rate of meteoroids
in the [0.1, 10] m diameter size range is given by Farinella et al. (1998) and 2) eq. B1
represents their spin-rate distribution.
The smallest object for which a measured rotation period exists is the TCO 2006 RH120
with a period of 165 s (Kwiatkowski et al. 2008). The median rotation rate of objects of
2 m diameter corresponding to the size of 2006 RH120 is about 1 s and 36 s using Beech and
Brown (2000) and Farinella et al. (1998) respectively. With these medians the probability
that a 2 m diameter object has a rotation period of ≥ 165 s in the two models is 1%
and ∼ 2.2 × 10−5 % respectively CALCULATED USING THE CUMULATIVE
DISTRIBUTION OF THE MAXWELL DISTRIBUTION. Thus, 2006 RH120 ’s
rotation period would be a common < 2.0 − σ event in our analysis but a & 3.6 − σ event if
– 39 –
the asteroids in that size range had a median given by Beech and Brown (2000).
C.
Earth shadowing
The typical formulae for calculating the apparent magnitude of an asteroid do not
account for shadowing by Earth since most asteroids are too far away for Earth’s disk to
make an observable decrease in the asteroid’s brightness. Even if Earth’s disk is entirely
superimposed on the Sun’s disk it reduces the apparent brightness of the Sun at the asteroid
by 10% at a geocentric distance of ∼0.03 AU and 1% at ∼0.1 AU. The TCOs considered
here are typically within 0.01 AU of the Earth so the effect is small to negligible for the
TCOs but we still account for the shadowing.
Referring to Fig. 10 and letting R1 and R2 be the actual radius of object 1 and 2
respectively, and letting d1 and d2 be the distance between the observer and the two
objects, the apparent angular radii of the objects are simply r1 = arcsin(R1 /d1 ) and
r2 = arcsin(R2 /d2 ) respectively.
If we let s represent the apparent angular separation between their centers there are
three possible scenarios related to their relative sizes:
1. s < r2 − r1 and r2 ≥ r1 : Earth’s disk entirely covers Sun’s disk
2. s ≥ r2 − r1 and r2 < r1 : Earth’s disk smaller than but entirely overlapping Sun’s disk
3. s < r2 + r1 : Partial overlap of Earth’s and Sun’s disk
(a) r1 ≥ r2 : Earth’s disk smaller than Sun’s disk
(b) r1 < r2 : Earth’s disk larger than Sun’s disk
– 40 –
In the first case, when disk 1 completely covers disk 2 the visible fraction of disk 2 is
zero: fvis = 0.
In the second case the visible fraction of disk 2 is simply the ratio of the area of the
two disks
fvis =
r2
r1
2
(C1)
Finally, for the third case, consider the definition of terms illustrating the relative
orientation between two overlapping disks shown in Fig. 10. The angles, α1 and α2 are
given by
ssec
α1
= arccos
2
r1
(C2)
α2
2
−ssec + s
.
= arccos
r1
where ssec is given by
ssec =
r12 − r22 + y22
.
2s
(C3)
If the apparent width of the Earth is greater than that of the Sun then all variables with
subscript 1 become subscript 2.
Finally, the area of the hatched region between the secant and the outer circumference
of each disk is
Ai =
1 2
r (αi − sin αi )
2 i
(C4)
with αi in radians. The fraction of the disk 1 (the Sun) that is visible is then
fvis =
A1 + A2
.
πr12
(C5)
– 41 –
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– 47 –
Survey
PS11
Type
FOR
Texp
(deg2 )
s
Vlim
ωlim
Nsky
deg/day
NT CO
TCO/day
days
wide-area
1,000
40 21.7
all-sky
20,000
30 20.0
15 0.0025
0.0016
LSST3
wide-area
7,000
15 24.7
10 1.7
0.1
Subaru-HSC4
targeted
300
40 25.0
3.3 0.2
0.011
7.5
ATLAS2
3 0.00075 0.00027
τref
0.4
2.5
15.9
CAMO5
meteor
20,000
N/A
7.5
N/A ∼ 0
N/A
N/A
CAMS6
meteor
20,000 0.017
4.8
N/A ∼ 0
0.04
26.1
Table 1: COMPLETE THIS TABLE. TCO detection performance for six optical surveys.
We consider only TCOs with absolute magnitude H < 38 corresponding roughly to those
>10 cm diameter. FOR is the Field-of-Regard — the total average amount of sky surveyed
each night in a mode suitable for identifying TCOs. Vlim and ωlim are the survey’s limiting
V -band magnitude and approximate rate of motion. Nsky and NT CO are the number of
TCOs on the sky that are detectable by the survey at any instant and the number of TCOs
detectable by the survey in a lunation. τref is the refresh rate for the observable TCOs,
equivalent to their average observable lifetime.
1
Denneau and Jedicke (2013)
2
Tonry (2011)
3
Ivezic et al. (2008)
4
Takada (2010)
5
Weryk et al. (2008)
6
Jenniskens et al. (2011)
– 48 –
Fig. 1.— TCO normalized sky-plane residence distribution with no restrictions on apparent
magnitude or rate of motion in an opposition-centric ecliptic reference system. Negative
opposition-centric longitudes are west of opposition. The values are the fraction of the
population in 3◦ × 3◦ bins.
– 49 –
Fig. 2.— Geocentric distance distribution in lunar distances (0.000257 AU) for all TCOs,
near the L2 and L1 Lagrange points, and near the two quadratures. ‘Near’ means within 15◦
in longitude and within 10◦ in latitude. The fraction density ρf (∆) provides the fraction of
the population ρf (∆)d∆ at geocentric distance ∆ in a bin of width d∆.
– 50 –
Fig. 3.— Sky plane distribution of TCOs with H < 38, apparent magnitude V < 20 and rate
of motion < 15 deg/day. The small dot in the center represents the size of Earth’s umbra
and penumbra at one lunar distance. The values are the number of TCOs in 3◦ × 3◦ bins.
– 51 –
Fig. 4.— Sky-plane distribution of TCOs at the moment of capture near L2 without constraints on the apparent magnitude or rate of motion. The values are the fraction of the
population at capture in the 3◦ × 3◦ bins. The vectors represent the average rate of motion
and direction of the TCOs in the bins at the time of capture.
– 52 –
Fig. 5.— Skyplane time residency distribution in WISE’s 12 micron band for TCOs with
flux > 0.65 mJy and rate of motion < 3 deg/day. The values are the steady state instantaneous number of TCOs in 3◦ × 3◦ bins.
– 53 –
Fig. 6.— Size distribution of detectable TCOs for four different optical surveys. The number
density gives the number of TCOs per unit absolute magnitude on the entire sky detectable
by the survey. It is not corrected for the survey’s sky coverage or cadence. ATLAS surveys
the entire night sky each night to V < 20.0 and ω < 15 deg/day; PS1 5% of the night sky to
V < 21.7 and ω ∼ 3 deg/day; LSST 35% of the night sky to V < 24.7 and ω = 10 deg/day.
The Subaru HSC survey is targeted and assumes a 300 deg2 field imaged to V < 25.0 with
non-sidereal tracking at 3.3 deg/day in position angle 94◦ .
– 54 –
Fig. 7.— Size distribution for TCOs detectable in WISE’s 12µm W3 band.
– 55 –
Fig. 8.— (Top 2 figs) Range-rate vs. range distribution for TCOs in a 1◦ opening angle
cone centered on L1 or L2 (i.e. opposition or the direction towards the Sun) and (bottom 2
figs) the same distribution towards the east or west quadratures. The top and third figures
are without any constraints on the objects’ detectability. The second and fourth figures
incorporate the effects of bi-static radar detectability including the TCOs’ rotation rates.
– 56 –
Fig. 9.— Median rotation period of small asteroids as a function of diameter as predicted by
(red)Farinella et al. (1998) and (black) Beech and Brown (2000). The blue squares joined by
straight dashed blue lines represent the median of the data in 10 meter diameter bins from
the Light Curve Database Warner et al. (LCDB 2009). The data points with uncertainties
show our calculated values from our fits to the LCDB.
– 57 –
Fig. 10.— A representation of the Sun (yellow circle, disk 1) and Earth (blue circle, disk 2)
as observed from a location relatively close to Earth when a small portion of the Earth’s disk
overlaps a portion of the Sun’s disk. In this geometry the observer is in Earth’s penumbra.
The observer is in Earth’s umbra when Earth’s disk entirely covers the Sun’s disk.