Detecting ’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

1University of Hawaii, Institute for Astronomy, 2680 Woodlawn Dr, Honolulu, HI, 96822 2Department of Physics, P.O. BOX 64, 00014 University of Helsinki, Finland 3University of Western Ontario, Physics & Astronomy Department, London, Ontario, Canada 4University of Hawaii, Department of Mathematics, Honolulu, HI, 96822 5Arecibo Observatory, Arecibo, Puerto Rico –2–

ABSTRACT

JEDICKE WILL WRITE ONCE REMAINDER OF PAPER IS COMPLETE.

Subject headings: Near-Earth Objects; , Dynamics –3–

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] –4–

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 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 RH had an absolute magnitude of H 29.5 corresponding to a diameter in the 120 ∼ range 3-7 meters if we assume S- and C-class albedos p of 0.26 and 0.064 respectively ∼ v (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 ∼

1L. Benner, personal communication. –5–

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 size- frequency 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 Vokrouhlick´y2003). Their favored TCO SFD from Brown et al. (2002) for bolides is appropriate for TCOs in the 0.1cm 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 –6– testing current NEO and SFD models in a size range that transitions between the two regimes. The smallest known NEO with H 33.2 (2008 TS Boattini et al. 2008) ∼ 26 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, , 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 –7–

(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 ±

2we 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’. –8– 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 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 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

3For convenience, unless stated otherwise, we use an albedo of p 0.11 throughout this v ∼ 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. –9– 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 2.5log 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 – 10 – 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 < 15deg/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’

4For more information on trail fitting see e.g. Vereˇs et al. (2012) who provide a technique for fitting trailed source detections but not how to identify them. – 11 – 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 – 12 – 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 θ where θ is the impact ∼ i i 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. – 13 –

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 V = 21.7, ω = 3.0 deg/day ∼ × lim lim 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 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 – 14 – 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 Pach`on, will mean that TCO detections will be trailed. Our LSST TCO detection estimate in table 1 used a peak rate of detectable motion of 10deg/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. – 15 –

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 – 16 –

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±0.15s−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. – 17 –

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 – 18 – 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 Φ > 0.65 mJy moving at < 3 deg/day. At any time there are 1 TCOs on W 3 ∼ 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

5http://en.wikipedia.org/wiki/Air Force Space Surveillance System; 5 Aug 2011 6http://echo.jpl.nasa.gov/asteroids/ – 21 – detected object7 was 2006 RH the only known TCO, with H 29.6 corresponding to a 120 ∼ 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 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

2 2 2 ω − 1 [ ω ] f(ω; d)= ( ) exp 2 ω0(d) (1) π ω (d)3 p 0

where d is the object’s diameter in meters, ω is the rotation rate in rev/, 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 byω ˆ 1.54 ω . ≈ 0 Our calculations suggest that detecting the TCOs is essentially impossible with

7http://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 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 COM- PLETE.

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 [λ dλ /2,λ + dλ /2] and 0 − 0 0 0 [β dβ /2, β + dβ /2] as its sky-plane residence time where λ and β represent ecliptic 0 − 0 0 0 0 0 longitude and latitude respectively. The sky-plane residence time density for an individual particle j at location (λ0, β0) is +∞

ρj(λ0, β0)= dt δ(λj(t) λ0) δ(βj(t) β0) (A1) Z − − −∞ 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 λ λ<λ and β β < β is: 1 ≤ 2 1 ≤ 2 λ2 β2

Tj(∆λ0, ∆β0)= dλ dβ ρj(λ, β). (A2) Z Z λ1 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)= ρj(λ0, β0) (A3) Xj – 27 –

Letting the normalization constant

C = dλ dβ ρ(λ, β) (A4) Z Z i.e. the cumulative time spent over the entire sky by all particles, the normalized residence time density is 1 ρ (λ, β)= ρ(λ, β). (A5) N C

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

α(H−H0) NHFD(H)= N0 10 (A6)

i.e. NHFD(H) is the number of particles with absolute magnitude

The differential HFD is then

α(H−H0) nHFD(H) dH = N0 α ln10 10 dH (A7)

i.e. there are nHFD(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) = nHFD(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

H0 N(λ0, β0,H0) = dH n(λ0, β0,H) Z

= NHFD(H0) ρN (λ0, β0) (A9)

so that the number of particles with H

(∆λ0, ∆β0) around (λ0, β0) is N(λ0, β0,H0) ∆λ0 ∆β0.

Thus, the number of particles with absolute magnitude

A.3. 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)

+∞ vis ρ (λ0, β0,V0,ω0,H0) = dt δ(λj(t) λ0) δ(βj(t) β0) (A11) j Z − − 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)

vis vis ρ (λ0, β0,V0,ω0,H0)= ρj (λ0, β0,V0,ω0,H0), (A12) Xj and their normalized sky-plane residence time distribution density is:

1 ρvis(λ , β ,V ,ω ,H )= ρvis(λ , β ,V ,ω ,H ) (A13) N 0 0 0 0 0 C 0 0 0 0 0 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 vis N (λ0, β0,V0,ω0,H0) = dH n(λ0, β0,V0,ω0,H) Z V (H0) < V0, ω<ω0

H0 = ρN (λ0, β0,V0,ω0,H0) dH nHFD(H) (A14) Z

i.e. There are N vis(λ , β ,V ,ω ,H ) ∆λ ∆β particles in the range [λ ∆λ /2,λ +∆λ /2] 0 0 0 0 0 0 0 0 − 0 0 0 and with [β ∆β /2, β + ∆β /2] that also have H

The total number of particles in the sky that have V (H0) < V0, ω<ω0 and absolute

magnitude brighter than H0 is than

vis = dλ dβ N (λ,β,V0,ω0,H0) (A15) Z Z

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 λ2 β2 vis = dλ dβ N (λ,β,V0,ω0,H0). (A16) Z Z λ1 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): D 1.329 106 = × 10−H/5 (A17) meters √pv 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

+∞ IR ′ ′ ρ (λ0, β0, F0,ω0,H)= dt δ(λj(t) λ ) δ(βj(t) β ) . (A18) j Z − − 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

D0 IR N (λ0, β0, F0,ω0,D0) = dD n(λ0, β0, F0,ω0,D) Z F (D0) >F0, ω<ω0

D0 IR = ρ (λ0, β0, F0,ω0,D0) dD nHFD(D) (A19) N Z – 31 –

The total number of particles in the sky that have F (D0) > F0, ω<ω0 and D > D0 is

IR = dλ dβ N (λ,β,F0,ω0,D0) (A20) Z Z

and the total number of particles in the range [λ1,λ2] and [β1, β2] with the same constraints on flux, rate and diameter is

β2 λ2 IR = dβ dλ N (λ,β,F0,ω0,D0). (A21) Z Z β1 λ1

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):

π 2π 2 λ2 ǫD2 F = dθ dφ dλ B(λ, Tss) ǫb(λ) sin θ cos θ (A22) 4∆2 Z Z Z 0 0 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 —

1 S(r) (1 A) 4 T = − (A23) ss  ηǫσ 

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 0.1m 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 speed (range-rate, ∆).˙ In this section we provide the 4-d residence time distribution with 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

radar ˙ ρj (λ0, β0, ∆0, ∆0) (A24)

+∞ ′ ′ = dt δ(λj(t) λ ) δ(βj(t) β )δ(∆j(t) ∆0)δ(∆˙ j(t) ∆˙ 0) (A25) Z − − − − −∞

Summing over all the particles and normalizing yields the radar equivalent to eq. A5:

1 ρradar(λ,β, ∆, ∆)˙ = ρradar(λ,β, ∆, ∆)˙ , (A26) N C

the normalized residence time in the 4-d space for the particle ensemble. i.e. the particles radar ˙ ˙ ˙ spend a fraction of time ρN (λ,β, ∆, ∆)dλdβd∆d∆ within (dλ/2,dβ/2,d∆/2,d∆/2) of (λ,β, ∆, ∆).˙

Analagous to eqs. A6 and A7 we let the particles’ cumulative and differential size-frequency distributions (SFD) be represented by NSFD(D) and nSFD(D) respectively.

i.e. There are NSFD(D) particles with diameter > D and nSFD(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

radar ˙ radar ˙ n (λ,β, ∆, ∆,D,T )= nSFD(D) P (D, T ) ρN (λ,β, ∆, ∆) (A27) i.e. there are nradar(λ,β, ∆, ∆˙ ,D,T ) dλ dβ d∆ d∆˙ dD dT within (dλ/2,dβ/2,d∆/2,d∆˙ /2,dD/2,dT/2) of (λ,β, ∆, ∆˙ ,D,T ).

Similarly, the cumulative number of objects in the 4-d residency space with diameters > D and rotation period < T is then represented by

radar ˙ radar ˙ N (λ,β, ∆, ∆,D,T )= NSFD(D) P (D, T ) ρN (λ,β, ∆, ∆) (A28) i.e. there are nradar(λ,β, ∆, ∆˙ ,D,T ) dλ dβ d∆ d∆˙ dD dT within (dλ/2,dβ/2,d∆/2,d∆˙ /2,dD/2,dT/2) of (λ,β, ∆, ∆˙ ,D,T ).

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

D, T and Φ > Φ0 at (λ,β, ∆, ∆)˙ is

radar ˙ radar ˙ n (λ,β, ∆, ∆,D,T, Φ0)= nSFD(D) P (D, T ) ρN (λ,β, ∆, ∆) . (A29) φ(∆,D,T )>Φ0

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

∞ T0 ∆max ∆˙ max ˙ radar ˙ = dλ0 dβ0 dD dT d∆ d∆nSFD(D) P (D, T ) ρN (λ0, β0, ∆, ∆) Z Z Z Z φ(∆,D,T )>Φ0 D0 ∞ ∆min ˙ ∆min (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 P ǫ A P = tran R R (A31) rec ∆2r2 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): G G2 λ2 σ(NL)0.5 SNR = T A (A32) 4 2ΩD 0.5 ∆ kb T ( λ ) 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: 1 Ntot LC = dt δ[Cj(t)], (A34) N Z tot Xj=1

where Cj(t) = 0 only when the constraint is satisfied by particle j at time t.

If we let ρ(¯x, t) represent the number density as a function of time of particles with respect to the set of parametersx ¯ upon which the constraints are defined then the number of particles that meet the constraints at any time is:

NC = dt dx¯ δ[C(¯x, t)] ρ(¯x, t) (A35) Z Z

where C(¯x, 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:

t2 dt Q[λj(t), βj(t)] δ(λj(t) λ0) δ(βj(t) β0) j t1 − − C Q¯total(λ0, β0)= P R . (A36) t2 dt δ(λ (t) λ ) δ(β (t) β ) j − 0 j − 0 Pj tR1 C

B. 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 1m diameter) and the smallest object in the Light Curve Database (LCDB, Warner et al. (2009)) has H 29.5 (the only known TCO, 2006 RH ). The observational selection effects ∼ 120 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:

2 2 2 ω − 1 ( ω ) 2 ω0 f(ω)= 3 exp . (B1) rπ ω0

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

x F (x) = f(y) dy Z 0 x 2 − 1 y 1 2 2 2 [ ω2 ] 0 = 3 y exp dy ω0 rπ Z 0 x2 2α2 2 = √z exp−z dz √π Z 0 2 3 x 3 −1 = γ , 2 Γ( ) 2 2ω0  2 3 x2 = P , 2 . (B2) 2 2ω0  where P (a, x) is the incomplete gamma function. – 38 –

1 We compute the median by setting eq. B2 to 2 and solving numerically for x in terms of ω0 which results in the following calculation for the median 3 1 xˆ = P −1 , 1.54 α (B3) r 2 2 ≈ 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 2m 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 0.03AU 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

2. s r r and r

3. s

(a) r r : Earth’s disk smaller than Sun’s disk 1 ≥ 2

(b) r1

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 2 r2 fvis = (C1) r1 

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

α s 1 = arccos sec 2  r1  (C2) α s + s 2 = arccos − sec . 2  r1  where ssec is given by r2 r2 + y2 s = 1 − 2 2 . (C3) sec 2s 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 1 A = r2 (α sin α ) (C4) i 2 i i − i with αi in radians. The fraction of the disk 1 (the Sun) that is visible is then

A1 + A2 fvis = 2 . (C5) πr1 – 41 –

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Survey Type FOR Texp Vlim ωlim Nsky NTCO τref (deg2) s deg/day TCO/day days

PS11 wide-area 1,000 40 21.7 3 0.00075 0.00027 0.4 ATLAS2 all-sky 20,000 30 20.0 15 0.0025 0.0016 2.5 LSST3 wide-area 7,000 15 24.7 10 1.7 0.1 15.9 Subaru-HSC4 targeted 300 40 25.0 3.3 0.2 0.011 7.5 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 NTCO 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.

1Denneau and Jedicke (2013)

2Tonry (2011)

3Ivezic et al. (2008)

4Takada (2010)

5Weryk et al. (2008)

6Jenniskens 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 con- straints 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 > 0.65mJy and rate of motion < 3 deg/day. The values are the steady state instanta- neous 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.