Albedos of Small Jovian Trojans
aa r X i v : . [ a s t r o - ph . E P ] J un appearing in Astron. J., July 2009, vol. 138, pp. 240-250 Albedos of Small Jovian Trojans
Yanga R. Fern´andez
Department of Physics, University of Central Florida,4000 Central Florida Blvd, Orlando, FL 32816-2385
David Jewitt
Institute for Astronomy, University of Hawaii,2680 Woodlawn Dr, Honolulu, HI 96822 andJulie E. Ziffer
Department of Physics, University of Southern Maine,96 Falmouth St, Portland, ME 04104-9300
ABSTRACT
We present thermal observations of 44 Jovian Trojan asteroids with diame-ters D ranging from 5 to 24 km. All objects were observed at a wavelength of24 µ m with the Spitzer Space Telescope. Measurements of the thermal emissionand of scattered optical light, mostly from the University of Hawaii 2.2-metertelescope, together allow us to constrain the diameter and geometric albedo ofeach body. We find that the median R-band albedo of these small Jovian Tro-jans is about 0.12, much higher than that of “large” Trojans with D >
57 km(0.04). Also the range of albedos among the small Trojans is wider. The smallTrojans’ higher albedos are also glaringly different from those of cometary nuclei,which match our sample Trojans in diameter, however they roughly match thespread of albedos among (much larger) Centaurs and trans-Neptunian objects.We attribute the Trojan albedos to an evolutionary effect: the small Trojans aremore likely to be collisional fragments and so their surfaces would be younger. Ayounger surface means less cumulative exposure to the space environment, whichsuggests that their surfaces would not be as dark as those of the large, primordialTrojans. In support of this hypothesis is a statistically significant correlation ofhigher albedo with smaller diameter in our sample alone and in a sample that 2 –includes the larger Trojans. This correlation of albedo and radius implies thatthe true size distribution of small Trojans is shallower than the visible magni-tude distribution alone would suggest, and that there are approximately half theTrojans with
D >
Subject headings: minor planets — infrared: solar system
1. Introduction
Jupiter’s Trojan asteroids inhabit two swarms centered on the L4 and L5 Lagrangianpoints located 5.2 AU from the Sun and from the planet. More than 2700 Trojans are knownat the time of writing. Based on optical studies, the total population larger than 1 km inradius has been estimated by various workers: Jewitt et al. (2000) estimated ∼ × suchobjects in the L4 swarm; Szab´o et al. (2007) estimated ∼ × in both swarms combined;Yoshida & Nakamura (2005) estimated ∼ × in the L4 swarm; and Nakamura & Yoshida(2008) estimated ∼ × in the L4 swarm and ∼ × in the L5. The magnitude-derived size distribution resembles a broken power law (Jewitt et al. 2000), and is such thatthe bulk of the mass (approximately 10 − M ⊕ , where M ⊕ = 6 × kg is the mass of theEarth) is contained within the largest objects. By number and by mass, the Trojan pop-ulation is only slightly inferior to the population of the main-belt asteroids. However, theobservational attention given to the Trojans so far is miniscule compared to that lavished onthe main-belt objects, and many of the basic properties of Jupiter’s Trojans remain poorlyknown. The Trojans have been reviewed alongside the irregular satellites of Jupiter, to whichthey may be closely related, by Jewitt et al. (2004) and separately by Dotto et al. (2008).Scientific interest in the Trojans focuses both on their origin and on their composition.How and when they were trapped in 1:1 mean-motion resonance with Jupiter remains un-known. Capture at a very early epoch in association with planet formation and capture muchlater, in a dynamical clearing phase in the Solar system, are both under current considera-tion (Morbidelli et al. 2005; Marzari & Scholl 2007). The snow-line in the Solar system wasmost likely inside the orbit of Jupiter (Garaud & Lin 2007), so if they formed in-situ or at amore distant location in the Sun’s protoplanetary disk, the Trojans could have incorporatedwater as bulk ice. In this sense, the Trojans might share as much in common with thenuclei of comets as with the classical, rocky asteroids. Observationally, the measured Tro-jans resemble the nuclei of short-period comets in their optical colors (Jewitt & Luu 1990;Fornasier et al. 2007) and albedos (Fern´andez et al. 2003, Paper I), tending to reinforce byassociation the possibility that they might be comet-like, ice-rich bodies. On the other hand,low-resolution spectral observations in the near infrared have uniformly failed to reveal ab- 3 –sorption bands that could be attributed to water ice or, indeed, to show any absorption bandsat all (Luu et al. 1994; Dumas et al. 1998; Emery & Brown 2003; Yang & Jewitt 2007). Thelow albedos, neutral to reddish optical colors and featureless near infrared spectra are com-patible with, but not uniquely diagnostic of, irradiated, complex organics (Cruikshank et al.2001).The absence of water ice is easily understood as a consequence of sublimation, even atJupiter’s distance. For example, dirty (absorbing) water ice exposed at the sub-solar pointon a Trojan at 5.2 AU sublimates in equilibrium at a rate of ˙ m ∼ × − kg m − s − ,corresponding to recession of the sublimating surface at speed ˙ m/ρ ∼ − , where ρ ∼ kg m − is the bulk density. In a few years, any exposed dirty water ice on a Jovian Trojanwould recede into the surface by a depth greater than the diurnal thermal skin depth (i.e.,approximately 5 to 10 cm on a body rotating with a period ∼ κ ∼ − m s − ). Clean (i.e. pure) surface ice could survive much longer, by virtue of itshigher albedo and lower temperature, but sustaining clean surface ice will be difficult in theface of micrometeorite gardening and contamination. Just as with the nuclei of comets, then,the Trojans could have ice-rich interiors but relatively (or even, completely) ice-free surfacescomposed of refractory, particulate matter (“mantles”). In this case, it is possible that colli-sions within each swarm (Marzari et al. 2002) occasionally cause previously-embedded andrelatively pristine material to be exposed to space. While no water ice has been definitivelydetected spectroscopically on the surfaces of larger Trojans (as mentioned above), smallerbodies, currently just beyond the range of ground-based spectroscopic observation, may holdsome remnant near-surface ice.In order to address these topics, we are investigating some of the physical propertiesof the known Trojans. In earlier work (Paper I), we found that the geometric albedos ofTrojans larger than ∼
60 km in diameter (“large” Trojans) are uniform. The mean R-bandgeometric albedo of such objects is 0 . ± .
002 and the standard deviation is just 0.008.(These are transformed from the paper’s V-band results using the average color derived byFornasier et al. (2007) of V − R = 0 . ± § § §
2. Observations
We have two datasets, one obtained with the Spitzer Space Telescope (SST, Werner et al.2004) that provided us with mid-infrared imaging, and another with the University of Hawaii2.2-meter Telescope that provided us with visible-wavelength imaging. Table 1 provides alist of our targets and the circumstances of the observations. The targets were chosen tohave excellent ephemerides so that there would be no doubt about the success of the SSTobservations. At the time we prepared the project, our targets were among the smallestnumbered Trojans known (as judged by H , the absolute magnitude). We used the Multiband Imaging Photometry for Spitzer (MIPS, Rieke et al. 2004)aboard SST to observe all 44 small Trojans during Cycle 1. Each Trojan was observedin “photometry” mode using the 24- µ m imager (effective wavelength λ = 23 . µ m), a 128-by-128 array of Si:As impurity band conduction detectors. The scale is 2.55 arcseconds perpixel, and the spatial resolution is diffraction-limited (Rayleigh criterion of 7.1 arcsec). Theintegration time was 132 seconds, using 3-second exposure times and 3 cycles, resulting in44 individual raw exposures. Each visit to a Trojan lasted 6.7 minutes, including observingoverheads. Raw data was processed by the SST pipeline version 14.4.0 to produce flux-calibrated “BCD” (basic calibrated data) images. A discussion of the pipeline processingis given by Gordon et al. (2005). In general, the data quality was high and the Trojansprovided good signal-to-noise ratios in the individual frames. No latents or streaks wereseen.To measure the flux density of each Trojan, we used two independent methods. First,we used MOPEX (Makovoz & Marleau 2005) to obtain photometry of an object using itsindividual BCD images. The targets were bright enough that stacking to boost the signalwas not necessary. This gave us 44 separate samples of a Trojan’s brightness, with which wecould calculate an appropriate mean and error, and also readily identify bad frames. Second,we used Interactive Data Language (IDL) software to analyze post-BCD mosaics providedby the SST pipeline. These post-BCD data are combinations of the BCD images and have 5 –had array distortions rectified (Spitzer Science Center 2007).In both methods, aperture photometry was performed usually using an aperture ofradius 3.0 BCD pixels (7.65 arcsec), though reduced to 2.0 or 2.5 BCD pixels when the Trojanwas near a background object. The results were compared and in all cases the differenceswere at the few percent level. Averages and propagated errors were then calculated.This photometry was then corrected for aperture loss and for color to produce a finalmeasurement of the monochromatic flux density. All Trojans appeared as point sources inall images, facilitating an aperture correction. Color corrections were calculated from theshape of the expected spectral energy distribution that results from the thermal model (see §
3) and the known Trojan-Spitzer-Sun angles and distances.Our photometry is listed in Table 2, with 1 σ error bars. Errors in the photometry resultfrom uncertainty in the photon counting, in measuring an appropriate sky background, andin the repeatability of the photometry from BCD to BCD. Optical photometry was obtained on the nights of UT 2005 April 7, April 8, June 28,June 29, and June 30 using the University of Hawaii 2.2-meter Telescope located atop MaunaKea, Hawaii. We used a Tektronix charge-coupled device (CCD) camera located at the f/10Cassegrain focus to image the Trojans through an R-band filter approximating the Kron-Cousins photometric system. Image scale with this set-up was 0.219 arcseconds per pixel.The image quality delivered by the telescope, including the effects of the atmosphere andwind-shake of the telescope, was typically 0.8 to 1.0 arcseconds full width at half maximum(FWHM).Photometric calibration was obtained using observations of standard stars from thelist by Landolt (1992), giving us effectively Cousins R-band magnitudes. We selected thefaintest standard stars and those having broadband colors most similar to the Sun in orderto minimize photometric uncertainties owing to the shutter and to color terms introducedby the use of broadband filters. We also observed the standards at airmasses similar to theairmasses of the Trojans, to minimize atmospheric extinction corrections. The sky on allnights was photometric except for part of the night of April 7, as judged by the real-timedata from the “Skyprobe” instrument on the Canada-France-Hawaii Telescope. Data takenthrough thin clouds on April 7 were calibrated using the photometry of the same field starsobserved on April 8. 6 –Photometry was performed using concentric, circular projected apertures, typically from4 to 7 pixels (0.9 to 1.5 arcseconds) in radius. Several of our targets were observed at lowGalactic latitude and so we took care to select an aperture size and sky location so as toexclude flux from background stars. Integration times employed were short enough thattrailing of the Trojans relative to the fixed stars was comparable to, or less than, the imageFWHM, so resulting in no photometric consequence.Aperture and color corrections were applied to our photometry and the resulting finalCousins R-band magnitudes are listed in Table 2, with 1 σ error bars. Note that for 12 ofour 44 objects, optical data were not obtained or were unusable due to stellar crowding.Error in the photometry results mainly from uncertainty in the aperture correction and inthe determination of an appropriate sky background.
3. Physical Parameters3.1. Thermal Model
The basic radiometric method to obtain an effective diameter D and geometric albedo p is to solve two equations with these two unknowns, first done many years ago (Allen 1970;Matson 1971; Morrison 1973) and described in detail by (e.g.) Lebofsky & Spencer (1989).One must observe the reflected sunlight (usually in visible wavelengths) and the thermalemission (usually in mid-infrared wavelengths); the former is proportional to D p , whilethe latter is proportional to D (1 − pq ), where q is the phase integral. In our study, weobserved Trojans only in Cousins R-band, so the geometric albedo is specific to that bandand represented by p R .The method requires knowing the distribution of temperature across the object’s surface,which itself depends on many parameters including the orientation and magnitude of thespin vector and the thermal diffusivity/thermal inertia of the surface materials. The spinvectors and thermal properties of the sample Trojans are unfortunately unknown. Themedian rotation period for Main-Belt asteroids of the appropriate diameter scale is about 6hours (Pravec et al. 2004). Thermal inertias of primitive asteroids are less well studied, butrecent work on cometary nuclei and Centaurs suggest that their thermal inertias are roughly ∼
10 J/m /K/ √ s (e.g. Fern´andez et al. 2006; Groussin et al. 2007; Li et al. 2007; Lamy et al.2008; Groussin et al. 2009). These parameters, if applicable to small Trojans, indicate thatat 5 AU the Trojans would lie in the “slow rotator” regime (cf. Spencer et al. 1989).The thermal model that we have employed to interpret our data is the “NEA ThermalModel” (NEATM) devised by Harris (1998), a simple and widely-used modification to the 7 –older “standard thermal model” (STM; Lebofsky & Spencer 1989). The STM and NEATMgenerally apply if the rotation is so slow or the thermal inertia so low that every point onthe surface is near instantaneous equilibrium with the impinging solar radiation. In the caseof zero thermal inertia, the temperature is a maximum at the subsolar point and decreasesas (cos ϑ ) / , where ϑ is the local solar zenith angle.To use NEATM we must make some assumptions. We assume that emissivity ǫ =0.9 and the phase slope parameter G = 0.05. We also assume a value for the beamingparameter, η , which is a rough proxy for thermal inertia and the effects of surface roughness,night-side emission, and beaming from e.g. craters. In Paper I we found that η = 0 . η , and that the beaming parameter is oftenstrongly dependent on the phase angle. Fortunately, all of our sample objects were observedat similar low phase angles. We address in § η would have on ourresults. Technically the derivation of diameter and albedo from this method requires that theobservations be done simultaneously, or at least while knowing the rotational context of theobservations. Neither condition was satisfied by our datasets, since it is difficult to scheduleground-based observations to match SST observations. This means that the diameters andalbedos that we derive may not be exactly correct for a specific object. Depending onthe different rotational phases at which the thermal and reflected signals are measured, thederived diameters and albedos could be either too high or too low by an amount that dependson the deviation of the shape from spheroidal.Fortunately, this effect should average out. Our sample size is large enough, and wehave detected all of our targets at significant signal-to-noise ratio so that we are not missingthe faint end of the sample. Thus, we have an approximately equal number of Trojans withboth too high and too low albedos. While the albedos for individual objects may be off fromtheir true values, the average and median of the ensemble of apparent albedos should beclose to the true average and median. The spread of the distribution will be wider than itreally is, but the extent of this spread can be estimated (see § r , geo/Spitzercentric distance 8 –∆, and geo/Spitzercentric phase angle α . In other words, we needed to estimate what eachTrojan’s magnitude would be had it been observed by the UH 2.2-meter Telescope at thesame geometry at which it was observed by SST. The correction to the measured magnitudeis 5 log( r i /r v ) + 5 log(∆ i / ∆ v ) + Φ( G, α i ) − Φ( G, α v ) , where subscripts “v” and “i” refer tothe visible and infrared observations, Φ is the phase function, and G is the phase slopeparameter.For the 12 objects with no visible-wavelength data, we have used the absolute magni-tude H (as given by the Minor Planet Center ) and the average Trojan V − R color (0.45,Fornasier et al. 2007) to predict what the visible magnitude would be. We assumed anuncertainty of ± H . Since there are two measurements and two parameters to be fit, there are no degreesof freedom with which to use, say, a χ -statistic. Therefore we employed a Monte Carlomethod with which to estimate the uncertainties of D and p R based on the uncertaintiesin the photometry. For each pair of photometric points – one mid-IR and one visible –we created 500 pairs of hypothetical measurements distributed normally about the actualmeasured values and with sigmas equal to the actual error bars. We then derived theappropriate D and p R that fits each pair, giving us 500 pairs of D and p R . The meansand standard deviations of these distributions of D and p R essentially became our “best-fitvalues” and “error bars.”For the 12 objects with no visible-wavelength data, we effectively have only the one vis-ible data point derived from H . For the other 32 objects, however, there are multiple visibledata points. For a Trojan with N such visible measurements, we paired each measurementin turn with the single mid-IR measurement to create N estimates for both D and p R usingthe Monte Carlo idea described above. We then averaged together all the estimates to createa single overall estimate of diameter and albedo. We also propagated the errors except incases where the variance of either D or p R among the N estimates was significantly largerthan the nominal error estimate, in which case we simply used the standard deviation itself.Our final values of diameter and albedo are given in Table 3. The table includes all 32objects with multiwavelength data, as well as the 12 with only infrared data. It is importantto note however that the table’s values do not account for the non-simultaneity of the IR URL
4. Discussion4.1. Ensemble Properties and Correlations
A plot of diameter vs. albedo from Table 3 is shown in Fig. 1. The most striking featureis the evidence of a trend where the smaller Trojans have higher albedos, or at least a higherlikelihood of having higher albedos. The Spearman rank-order correlation coefficient amongthe diameters and albedos of these 44 objects is − . . × − . In terms of the sum-squareddifference of the ranks the correlation is significant at the 3 . σ level.Since an absolute magnitude reported by the MPC (or by other agencies for that matter)could potentially be more uncertain than the 0.1 mag we have arbitrarily assumed – owingto uncertainty in color transformations, in phase darkening laws, and in weighting schemes– we have also analyzed the statistical significance of the apparent trend in Fig. 1 whileexcluding the 12 objects for which we did not have our own visible data. We believe this isa more robust analysis since it uses the results of more uniform datasets. In this case, theSpearman rank-order correlation coefficient is − . . × − . The significance of the sum-squared difference ofthe ranks is even higher, 3 . σ . So the trend is statistically significant regardless of whetherwe include 32 or 44 objects.In Fig. 2 we add the 32 albedos from our earlier work (Paper I) onto the same plot.Including these data with the best 32 gives us 64 data points, and the correlation appears evenmore pronounced. The Spearman rank-order correlation coefficient among the diameters andalbedos of these 64 objects is − . . × − . In terms of the sum-squared difference of the ranksthe correlation is significant at the 6 . σ level.It is clear from Fig. 2 that the average albedo of a small Trojan is larger than that of alarge Trojan. This is more readily demonstrated in Fig. 3, where the histograms of the twopopulations are compared. We can also see that the range of albedos is larger. Note that ourearlier work (Paper I) reported V-band albedos, so we have scaled those albedos to R-bandby using the average Trojan color V − R = 0 .
45 (Fornasier et al. 2007). To be quantitative,we compare the averages, medians, and standard deviations of the two populations in Table 10 –4. (We have listed separately the values for our whole sample of 44 and those values forthe 32 objects that have multiwavelength data.) Clearly the typical small-Trojan albedo ishigher than that of the large Trojans.Table 4 and Fig. 3 indicate that the range of albedos among small Trojans is wider thanthe range for the large Trojans, but we must be careful since the width of the distributionis at least partly artificial due to the lack of simultaneity in our datasets as described in § m from 0 to 0.3 mag. The distribution thentails off toward ∆ m = 0 . m = 0 .
24 mag and the median is∆ m = 0 .
22 mag. From this we take 0 .
24 mag to be the appropriate average ∆ m for theTrojans in our sample. That means that an optical magnitude would be at most ± . .
24 mag as a worst-case scenario, corresponding to a change in visibleflux density by a factor of 10 . × ( ± . = 0 .
80 or 1.25. To first order that would also be thefactor change in the albedo. So, if hypothetically all the small Trojans had a true albedo ofexactly 0.100, then we would expect to see a distribution that ranges from (0 . × .
80 =)0.080 to (0 . × .
25 =) 0.125. Clearly the histogram of small Trojan albedos is wider thanthis. In fact the observed albedos from 0.04 to 0.12 could only be explained with an average∆ m of about 0.6 mag. So unless the small Trojans have substantially higher typical axialratios than were measured by Mann et al. (2007), the spread of small Trojan albedos reallyis intrinsically wider than that of the large Trojans.We searched for correlations between albedo and other properties of the small Trojans.These comparisons are shown in Fig. 4, where we plotted albedos against three orbitalparameters and three observed parameters. The only panel suggesting a correlation is theinclination, in which higher albedo Trojans are more likely to have low inclination. Howeverthe Spearman rank-order correlation coefficient among the 32 multiwavelength objects isonly − . . . σ level. Adding in the 32 large-Trojan albedos from Paper I improves thecorrelation, but this is likely to be spurious since the Trojans in the two surveys do not haveoverlapping inclinations. 11 – It is important to consider whether the trend in Fig. 2 is a product of discovery bias.That is, perhaps we are measuring higher albedos because such small Trojans are morelikely to be discovered; after all, a Trojan of a given diameter with 0.12 albedo will be 1.09mag brighter than one with the same diameter but 0.044 albedo. If the high-albedo smallTrojans are near the limit of what can be discovered by asteroid surveys, then 1.09 magof difference would render a hypothetical low-albedo subpopulation invisible. On the otherhand, the situation is not quite this simple since a Trojan could have been discovered atanother lunation when it was brighter. Furthermore, the unknown rotational period andaxial ratio make predicting when a Trojan can and cannot be discovered difficult.A simple argument does suggest though that at least the fraction of high-albedo smallTrojans (however one wants to define “high”) is greater than that fraction among the largeTrojans. Only 3% (1/32) of the large Trojans from Paper I have albedos above the medianalbedo we have measured here, 0.117. If only 3% of all small Trojans in reality have albedosabove this value, then the asteroid surveys would have to have missed a vast population ofTrojans with 5 < D <
25 km, a population that is about 17 times larger than what hasactually been discovered. This seems unlikely.We can test this situation more rigorously however by assuming an overall albedo dis-tribution to the small Trojans and then determining what the measured albedo distributionwould be for the asteroids that are actually discovered by the asteroid surveys. To do thiswe created a virtual population of small Trojans and assigned them absolute magnitudes H such that the ensemble’s distribution of H matched that for the real Trojans as mea-sured by Jewitt et al. (2000). They found that for Trojans below a diameter of about 50km, the cumulative magnitude distribution N as a function of absolute magnitude H is N ( < H ) ∝ αH , with α = 0 . ± .
06. For our modeling, we assumed α = 0 .
40 precisely.The number of objects in the simulation was 1.26 million.We then created an albedo probability distribution P ( p ) to dictate what V-band albedoeach virtual object would be assigned. (We discuss P further below.) From this we calculatedthe diameters D for all virtual objects using D = 10 − . H × / √ p .Next, we assigned orbits to all virtual objects using a five-dimensional distributionof Trojan semimajor axis ( a ), orbital eccentricity ( e ), orbital inclination ( i ), argument ofperihelion ( ω ), and longitude of ascending node (Ω). This 5-D distribution was derivedempirically by extracting the orbital elements as compiled by the Minor Planet Center . URL
12 –For ease, we let each virtual object’s perihelion time t P be randomly chosen between 1992January 1 and 2004 January 1, i.e., sometime within a 12-year span (since 12 years is aboutone Trojan orbital period). From the orbital elements we could calculate each object’sheliocentric distance, geocentric distance, and phase angle over this 12-year interval. Incombination with H , and assuming a linear phase law of 0.04 mag/deg, we then calculatedeach object’s V-band magnitude m V over this span. This range of dates was chosen since itfalls within a period when the Spacewatch survey was surveying the sky down to m V ≈ m V = 20over the course of its orbit. This is a conservative choice in limiting magnitude; since ours isa simplistic model and does not explicitly take into account the actual sky coverage by thediscovery surveys or the robustness of their ability to detect low signal-to-noise asteroids, wedecided to pick a magnitude limit somewhat brighter than Spacewatch’s nominal limit.The result of the simulation is an ensemble of discovered, virtual objects that is a subsetof the whole group of objects. We then created a plot of diameter vs. albedo ( D vs. p R )for these discovered objects that can be compared to the plot of real observations in Fig. 1.(The R-band albedo p R was calculated from p by multiplying by 1.076 as in § P ( p ) as follows: P ( p ) = (cid:26) p ≤ C ′ e − ( p − p ) / σ p + C ′′ Π( p l , p l + p w ; p ) if p >
0, (1)where Π is the boxcar function. In other words, the albedo distribution had a gaussian,low-albedo component and a uniformly-distributed, high-albedo component. Specifically, p is the mean albedo of the low-albedo group; σ p is the standard deviation of the gaussian; p l is the lower bound of the high-albedo group; and p w is the albedo width of the high-albedogroup. (Note that for some parameter values, some objects that belong to the ostensiblylow-albedo gaussian could have albedos that overlap with those from the high-albedo uniformdistribution.) A fifth parameter, the fraction of objects with “high” albedo, f h , controls thevalue C ′′ : f h = Z p l + p w p l C ′′ Π( p l , p l + p w ; p ) dp = C ′′ p w . (2) 13 –The overall normalization R ∞−∞ P ( p ) dp = 1 controls the relative scale of C ′ and C ′′ , sothis setup has five parameters to investigate. Note that we have not assumed any trendbetween diameter and albedo.Our search through sample space is represented in Fig. 5. Each panel shows a contourplot of the nominal probability that our observed plot of D vs. p R (Fig. 1) and the simulatedplot are drawn from the same two-dimensional distribution. As Press et al. (1992) describe,probabilities greater than about 0.2 may not be precise due to the simplistic nature of thisformulation of the K-S test, but still do indicate similar distributions.The similarity of the contours in the panels of Fig. 5 indicates that the ‘best’ matchesare consistently near p ≈ .
07 to 0 .
12 and σ p ≈ .
01 to 0 .
06. The fraction of high-albedosmall Trojans f h in less constrained, but seems to be roughly under 30%. The extent inalbedo of that fraction is likewise not well constrained. This all depends somewhat on the a priori functional form of the distribution we have assigned, but the important and robustconclusion is that the small-Trojan albedo distribution is definitely not like that of the largeTrojans ( p ≈ . σ p ≈ . f h ≈ As stated in §
3, the phase slope parameter G influences the final results in Table 3.Re-running our thermal model for an assumed G of 0.15 instead of 0.05 results in almost nochange to the radii (at the ∼
10 meter level) and a reduction in the albedos by about 8 to9%. Such a small change would not alter our conclusions.More critical is the choice of η , since this certainly can have a significant effect on thecalculated values of both D and p R . To gauge the influence that our assumptions have onour results, we re-analyzed our photometry in Table 2 using η = 1 . η = 1 .
6, and η = 2 . .
94. Each value would assume that the small Trojans had successively higherthermal inertia, similar to what has been measured in several near-Earth asteroids (e.g.,Delb´o et al. 2003). These three trial values of η result in the diameters being (on average)15%, 30%, and 47% higher and the albedos being (on average) 24%, 41%, 54% lower thanwhat we present in Table 3. Thus, if η really were 2.0 then the albedos of the small Trojanswould be sufficiently small so that the median value would approach that of the large Trojans,0.050 vs. 0.044, and it would be less clear how significantly more reflective the small Trojanswould be. However, this would mean that the thermal behavior of the small Trojans would 14 –be radically different from the large Trojans. In effect, an incorrect assumption of η wouldnot nullify the conclusion that the small Trojans are different from the large ones, it canonly alter the way in which they are different.But is it likely that smaller Trojans have higher thermal inertia due to having lessregolith, or a large-grained regolith? The large Trojans ( D >
140 km) seem to have fine-grained silicates on their surfaces that produce mid-infrared emission bands (Emery et al.2006), indicating a fluffy regolith or a regolith where silicates are embedded in transparentgrains. These concepts are also consistent with the average η we found in Paper I. There isas yet no such detailed data on small Trojans (such as those in our current sample) to testwhether the regolith properties change as a function of size.Recent thermal studies of cometary nuclei are suggestive as a point of comparison,since the size matching between comets and small Trojans is reasonable, and since they areboth classes of primitive objects. As mentioned earlier ( § η for about 50 cometary nuclei observed at4 to 5 AU from the Sun is near unity. Such heliocentric distances are very near that of theTrojans. If the small Trojans are structurally similar to these comets, then an assumptionof η = 0 .
94 is quite reasonable.
As noted above, the published properties of the Trojans are broadly compatible withthose of the cometary nuclei. In particular, the Trojan optical color distribution resemblesthat of the cometary nuclei (Jewitt & Luu 1990) in that both are deficient in ultrared ma-terial known to coat the surfaces of many Kuiper belt objects (Jewitt 2002). The albedosof the larger Trojans (Paper I) are likewise similar to the albedos of cometary nuclei, andsuggest a carbonized, non-volatile surface composition. Comparison between the physicalproperties of the comets and the Trojans is especially interesting in the context of the Nicemodel, in which Trojans and Jupiter-family comets are both products of the Kuiper belt(Morbidelli et al. 2005). The depletion of the ultra-red matter on the comet nuclei and Tro-jans already argues either that this supposition is incorrect, or that the surface properties ofKuiper belt objects are modified after their removal from the Kuiper belt (Jewitt 2002). Asystematic difference in the albedos would demand a similar interpretation. 15 –The new results presented here tend to decrease the similarity with the comets, in thesense that when Trojans and nuclei of the same size are compared, the Trojan albedosare systematically higher. The strength of this statement is limited by the small sampleof cometary nuclei for which reliable albedo determinations exist, something soon to becorrected by an on-going survey of cometary nuclei (see Fern´andez et al. 2008).
Previous workers derived the size distribution of the Trojans based on the magnitudedistribution and an assumption of constant albedo. Jewitt et al. (2000) found a distributionconsistent with two power laws; for the largest objects, with diameters D ≥
70 km, theyfound that the differential size distribution’s power law index is a relatively steep q = 5 . ± D between 4 and 40 km, they measured q = 3 . ± .
3. The small-end ofthe distribution was also measured by Yoshida & Nakamura (2005) and Szab´o et al. (2007),who found similar values for q : 2 . ± . . ± .
25, respectively. However, now we arein a position to make a better conversion between absolute magnitude H and diameter D since we have a relationship between D and p R in Figs. 1 and 2. The higher albedos foundfor small Trojans imply smaller diameters than expected, which would result in a flatteningof the size distribution relative to the constant albedo case.To quantify this effect, we represent the D vs. p R trend in Fig. 2 by an ad-hoc function.We used the data in Fig. 2 to fit (by least-squares) the coefficients to the following 4th-orderpolynomial: p R ( D ) = X m =0 c m x m (3)where x = log( D/ D < p R at 0.3, and for D >
143 kmwe set a floor of p R = 0 . c = 1 . c = − . c = 2 . c = − . c = 0 . . (4)This function is plotted with the data in Fig. 2.We converted the differential size distribution provided by Jewitt et al. (2000) for theL4 swarm – their Eqs. 8 and 9 – back to a luminosity function, i.e. a function of H , usingtheir 0.04 assumed albedo. Using our Eq. 3, we could convert H to a more robust estimateof D and thus then derive a new size distribution.The result is shown in Fig. 6. The kink in our size distribution near D = 5 km is dueto the break in our p R ( D ) function at that diameter. The curvature to the middle segment 16 –between D = 5 km and D = 35 km is due to the curvature in p R ( D ) at those diameters, butin log-log space the segment approximates a power-law.The other implication of Fig. 6 is that the number of Trojans larger than a given sizeis lower than previously estimated (assuming that there is no trend of beaming parameterwith diameter). Figure 6 indicates that there are approximately 9 × Trojans in the L4swarm with diameter larger than 2 km, and about 3 × L4 Trojans with diameter largerthan 1 km. This is about a factor of two smaller than the estimate obtained by Jewitt et al.(2000) using a constant albedo and q = 3. Other estimates of the Trojan population (e.g.Yoshida & Nakamura 2005; Szab´o et al. 2007; Nakamura & Yoshida 2008) that assume aconstant albedo and use a similar magnitude distribution would have similar downwardcorrections to the population estimate. The observed albedo vs. diameter relation could have a number of causes, ranging fromthe profound to the insignificant. The degree of heating experienced by a solid body due tothe decay of embedded radioactive nuclei increases with the diameter, all else being equal.One hundred kilometer scale Trojans will experience a temperature increase from trappedradio-nuclei approximately 10 times larger than will Trojans only 10 kilometers in scale.Thus, it is tempting to think that the observed albedo vs. size relation might be an artifactof different degrees of metamorphism in the Trojans, assuming that these objects trappedsufficient quantities of short-lived radio-nuclei like Al and Fe to be appreciably heated.Arguing against this possibility is the size distribution of the Trojans, which resembles (atleast) two power laws intersecting at about 30 to 40 km diameter (i.e. neatly separating thesample in the present study from that in Paper I). Trojans larger than this are thought tobe survivors of a primordial population while those smaller than this are more likely to beproducts of past, shattering collisions. If so, the small objects in the present sample wereonce part of larger bodies that must have been radioactively heated, and no simple differencebased on the efficiency with which radiogenic heat can be trapped is expected.The albedo vs. diameter relation may instead suggest the action of some process in-volving collisions. The collisional lifetimes of small Trojans are short compared to the largerobjects. If the exposed surfaces of Trojan asteroids are progressively darkened, for exam-ple by the irradiation and dehydrogenization of hydrocarbons (e.g. Thompson et al. 1987;Moroz et al. 2004), then it is at least qualitatively reasonable to expect an albedo vs. diam-eter trend with the sense observed. 17 –Such a cause might also imply that there should be a color-diameter trend in the Tro-jans, since irradiation can change the reflectance slope as well as the albedo. Laboratoryresults indicate that the changes in the slope depend on dosages and on the original make-up of the surface (e.g. Moroz et al. 2003, 2004), so there may be no easy answer as towhat colors to expect on Trojans that have suffered various amounts of weathering. Ob-servationally, Jewitt & Luu (1990) found a trend where smaller Trojans (i.e. Trojans withsurfaces that are statistically younger) have redder surfaces, and this trend was corrob-orated for D-type asteroids by Fitzsimmons et al. (1994), by Lagerkvist et al. (2005) (forCybeles), and by Dahlgren et al. (1997) (for Hildas). Recent work on a wider sample ofTrojans has let some workers study colors of “background” Trojans as distinct from those ofTrojans in dynamical families. In particular, Fornasier et al. (2007) conclude that there isno statistically-significant trend between color and size, while Roig et al. (2008) argue thatamong the background population, it is the larger Trojans that are redder. In short, theobservational situation regarding a color vs. diameter relation currently remains unresolved.Future visible and near-IR datasets on a larger number of familial and non-familial Trojansand on Trojans down to small sizes may shed more light on this issue.In any case, in a scenario where collisions play a significant role in determining the sizesof the small Trojans that now exist in the swarms, one might expect the size distributionpower-law to more closely mimic the Dohnanyi power-law for collisional fragments. Fig. 6indicates however that we have now moved the small-size power-law to a shallower slope,away from the collisional equilibrium value. So while collisions likely are influencing thedistribution, there is as yet no simple explanation for Fig. 6.
5. Summary
We have measured the 24- µ m thermal emission from 44 small Jovian Trojans using theSpitzer Space Telescope and the R-band reflected sunlight from 32 of those to derive effectivediameters and albedos. Our sample covers diameters from 5 to 24 km, significantly smallerthan the large Trojans we sampled in an earlier survey ( D >
57 km; Paper I). We reach thefollowing conclusions: • The measured mean R-band geometric albedo of the small Trojans in our sample is0 . ± . . ± . . ± . . ± . . ± .
001 found for the large Trojans(Paper I). 18 – • The spread in R-band albedos among the small Trojans exceeds that of the largeTrojans, with a standard deviation of about 0 .
065 vs. 0 .
008 (Paper I). • The R-band geometric albedo decreases with increasing diameter in the 5 to 24 kmrange. This correlation is significant at the 3 . σ level, and becomes more significant(6 . σ ) when we include the large Trojans from our earlier work (Paper I). • The differences in albedo distribution between the large and small Trojans are unlikelyto be caused by either (a) the non-simultaneity in our optical/thermal data, or (b) adiscovery bias toward finding Trojans of high albedo in the first place. It is possiblethat the albedo differences are artifacts of using a size-independent infrared beamingparameter in interpreting the radiometry (and that the small Trojans have a differentensemble average thermal inertia than the large ones do), but we believe this possibilityto be unlikely. • The origin of the albedo-diameter relation is unknown but collisions, which shatter andcreate small bodies on much shorter timescales than large bodies, may be implicated. • The measured size dependence of the albedo tends to flatten the best-fit power law sizedistribution index relative to the value computed under the assumption of constantalbedo. We find that the differential power law index that best matches publishedsurvey data for objects in the 5 ≤ D ≤
30 km range is q ≈ .
8, whereas the valueunder the constant albedo assumption is q ≈ . • This flattened size distribution implies that there are about a factor of 2 fewer objectsof radius greater than 1 km than estimated when assuming a 0.04 constant albedo.For example, using the magnitude distribution reported by Jewitt et al. (2000), wefind that there are about 9 × L4 Trojans with radius greater than 1 km instead ofthe 1 . × inferred with the constant albedo assumption.We thank John Dvorak for operating the UH telescope, and for the helpful comments ofan anonymous referee and of Rachel Stevenson. This work is based in part on observationsmade with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory,California Institute of Technology under a contract with NASA. Support for this work wasprovided by NASA through an award issued by JPL/Caltech to YRF and DJ, and alsothrough Planetary Astronomy grant NNG06GG08G to DJ. We acknowledge the referencematerial provided by the Minor Planet Center. Facilities:
Spitzer (), UH:2.2m () 19 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
23 –Fig. 1.— Scatter plot of the 44 albedos and diameters derived in this survey. The 12 objectsfor which we used H are shown in grey; the 32 objects for which we have multiwavelengthdata are shown in black. The mean albedo of large Trojans as found by us (Paper I), andtranslated from V-band to R-band, is indicated with a horizontal dashed line. There is acorrelation of albedo with radius among the black points that is significant at the 3 . σ level. 24 –Fig. 2.— Combination of the radii and albedos from the current survey (diamonds) and fromour earlier work (squares; Paper I). Diamond greyscale is the same as Fig. 1. All 32 pointsfrom Paper I have been included here. Among the 64 black points there is a correlationof albedo with radius that is significant at the 6 . σ level. Horizontal dashed line indicatesthe mean large-Trojan albedo of 0.044. Solid piece-wise curve represents an ad hoc functionused to investigate the size distribution; see § σ level. 27 –Fig. 5.— Contour plots of the probability that the observed D -vs- p distribution seen inFig. 1 is drawn from the same distribution as that based on the simulations using the five-parameter model described in § f h , p l , and p w ; the values are written in the lower left. Contours correspond to probabilities of 10 − (outermost contour), 10 − , 0 .
01, 0 .
1, 0 .
3, and 0 .
5. 28 –Fig. 6.— Differential size distribution of Trojans. Dashed line is the distribution derived byJewitt et al. (2000) based on an assumed albedo of 0.04 that was size-independent. Solidline is our new derivation based on their survey data and the size-dependent albedo shownin Fig. 2. The equivalent power-law slopes, q , of each segment in both distributions areshown. 29 –Table 1. Target List and Observing CircumstancesNo. Name L n H Tel. UT Date UT r ∆ α (mag) yyyy-mm-dd at start (AU) (AU) ( ◦ )(58153) 1988 RH L5 13.2 S 2004-11-04 01:29:25 5.706 5.506 10.2” ” ” ” H 2005-04-07 07:06:09 5.749 5.627 10.0” ” ” ” H 2005-04-08 07:44:34 5.749 5.643 10.0(37572) 1989 UC L5 13.4 S 2004-11-10 08:55:22 5.431 5.368 10.8” ” ” ” H 2005-04-07 07:11:15 5.575 5.226 10.0” ” ” ” H 2005-04-08 07:49:39 5.576 5.242 10.0(58366) 1995 OD L4 13.7 S 2005-04-08 22:17:04 5.483 5.297 10.5” ” ” ” H 2005-06-30 07:31:15 5.473 4.499 3.4” ” ” ” H 2005-06-30 09:04:49 5.473 4.499 3.4(58475) 1996 RE L4 13.7 S 2005-04-06 11:08:11 5.150 4.835 11.0” ” ” ” H 2005-04-07 12:49:42 5.150 4.637 10.1” ” ” ” H 2005-04-08 12:11:49 5.150 4.625 10.0” ” ” ” H 2005-06-28 09:43:02 5.173 4.219 4.3” ” ” ” H 2005-06-28 10:14:52 5.173 4.219 4.3(192393) 1996 TT L4 13.8 S 2005-04-06 11:57:26 5.246 5.229 11.0(37789) 1997 UL L4 13.5 S 2005-04-08 23:20:30 5.300 5.099 10.9” ” ” ” H 2005-06-29 10:10:33 5.303 4.306 2.4” ” ” ” H 2005-06-29 12:01:19 5.303 4.306 2.4 · · · L4 14.0 S 2005-04-06 11:47:36 5.777 5.729 10.0 · · · L4 13.4 S 2005-04-06 15:51:08 5.718 5.253 9.3 · · · ” ” ” H 2005-04-07 12:32:22 5.718 5.073 8.2 · · · ” ” ” H 2005-04-08 11:18:39 5.718 5.060 8.1 · · · ” ” ” H 2005-06-28 09:18:45 5.709 4.830 5.6 · · · ” ” ” H 2005-06-28 09:52:58 5.709 4.830 5.6(40262) 1999 CF
L4 13.2 S 2005-04-06 12:07:00 5.967 5.981 9.6(59355) 1999 CL
L4 13.3 S 2005-05-19 13:21:44 5.277 4.716 9.7(60257) 1999 WB L4 13.4 S 2005-04-06 10:21:48 5.153 4.763 10.7” ” ” ” H 2005-04-07 12:44:35 5.153 4.569 9.6” ” ” ” H 2005-04-08 11:57:27 5.153 4.557 9.5” ” ” ” H 2005-06-28 09:27:00 5.171 4.250 5.2 30 –Table 1—ContinuedNo. Name L n H
Tel. UT Date UT r ∆ α (mag) yyyy-mm-dd at start (AU) (AU) ( ◦ )” ” ” ” H 2005-06-28 10:01:11 5.171 4.250 5.2(60322) 1999 XB L4 13.8 S 2005-03-10 01:30:56 5.391 5.224 10.7” ” ” ” H 2005-04-07 11:58:09 5.391 4.655 7.8” ” ” ” H 2005-04-08 10:49:01 5.391 4.644 7.7” ” ” ” H 2005-06-28 06:41:33 5.392 4.606 7.4” ” ” ” H 2005-06-28 08:03:45 5.392 4.606 7.4(192942) 2000 AB
L4 13.5 S 2005-04-06 11:28:35 5.425 5.313 10.7(60388) 2000 AY
L4 13.8 S 2005-09-23 23:13:27 5.365 4.957 10.4(162396) 2000 CV
L4 13.0 S 2005-05-13 07:11:06 5.376 4.851 9.7(60421) 2000 CZ L4 13.3 S 2005-05-19 16:00:01 5.349 5.126 10.8(62692) 2000 TE L5 13.3 S 2005-04-10 07:04:45 5.395 4.824 9.2” ” ” ” H 2005-04-07 07:25:08 5.395 4.955 10.0” ” ” ” H 2005-04-08 08:01:59 5.395 4.969 10.0(68112) 2000 YC
L4 13.4 S 2005-04-08 03:26:17 5.698 5.293 9.6” ” ” ” H 2005-04-07 12:39:16 5.698 5.126 8.7” ” ” ” H 2005-04-08 11:22:54 5.698 5.112 8.6” ” ” ” H 2005-06-28 09:34:54 5.699 4.776 4.7” ” ” ” H 2005-06-28 10:08:01 5.699 4.776 4.7(63193) 2000 YY
L4 13.2 S 2005-04-09 00:16:01 5.429 5.182 10.5” ” ” ” H 2005-04-07 13:31:51 5.428 5.014 10.0” ” ” ” H 2005-04-08 12:57:37 5.429 5.000 10.0” ” ” ” H 2005-06-29 09:37:46 5.471 4.490 3.1” ” ” ” H 2005-06-29 12:14:40 5.471 4.490 3.1(63259) 2001 BS L4 13.2 S 2005-04-06 11:37:43 5.094 4.979 11.4(88240) 2001 CG L4 13.4 S 2005-04-08 22:06:49 5.419 5.138 10.5” ” ” ” H 2005-04-07 13:20:37 5.418 4.972 9.9” ” ” ” H 2005-04-08 12:07:30 5.419 4.959 9.9” ” ” ” H 2005-06-29 07:57:07 5.433 4.482 4.2” ” ” ” H 2005-06-29 09:16:37 5.433 4.482 4.2(63284) 2001 DM L4 13.3 S 2005-04-08 22:57:38 6.075 5.888 9.5 31 –Table 1—ContinuedNo. Name L n H
Tel. UT Date UT r ∆ α (mag) yyyy-mm-dd at start (AU) (AU) ( ◦ )” ” ” ” H 2005-06-29 10:17:41 6.067 5.071 2.1” ” ” ” H 2005-06-29 12:08:45 6.067 5.071 2.1(63279) 2001 DW L4 13.4 S 2005-04-09 00:06:01 5.659 5.401 10.1” ” ” ” H 2005-06-29 08:14:53 5.672 4.691 3.0” ” ” ” H 2005-06-29 09:26:57 5.672 4.691 3.0(28960) 2001 DZ L4 13.3 S 2005-04-08 23:39:43 5.356 5.162 10.8” ” ” ” H 2005-06-29 10:34:59 5.336 4.338 2.3” ” ” ” H 2005-06-29 11:51:57 5.336 4.338 2.3(109266) 2001 QL
L5 13.4 S 2004-12-03 14:55:45 4.592 4.361 12.6” ” ” ” H 2005-04-07 07:35:26 4.726 4.176 10.8” ” ” ” H 2005-04-08 09:39:42 4.727 4.192 10.9(156222) 2001 UB L5 13.7 S 2004-11-05 18:28:57 5.334 5.349 10.9” ” ” ” H 2005-04-07 07:18:00 5.378 5.014 10.3” ” ” ” H 2005-04-08 07:54:47 5.378 5.030 10.4(156250) 2001 UM
L5 13.7 S 2004-12-03 15:05:49 5.122 4.933 11.4” ” ” ” H 2005-04-07 07:40:32 5.256 4.672 9.4” ” ” ” H 2005-04-08 09:43:56 5.257 4.688 9.5(64326) 2001 UX L5 13.4 S 2004-12-03 14:45:03 5.424 5.194 10.7” ” ” ” H 2005-04-07 07:47:17 5.445 4.918 9.4” ” ” ” H 2005-04-08 09:48:22 5.445 4.932 9.5” ” ” ” H 2005-06-30 06:20:27 5.452 6.111 7.7” ” ” ” H 2005-06-30 06:26:40 5.452 6.111 7.7” ” ” ” H 2005-06-30 06:34:27 5.452 6.111 7.7(158333) 2001 WW L5 13.7 S 2005-05-11 09:07:32 5.634 5.348 10.1” ” ” ” H 2005-06-30 06:05:03 5.660 6.281 7.7” ” ” ” H 2005-06-30 06:11:14 5.660 6.281 7.7 · · ·
L4 14.6 S 2005-03-10 16:54:36 5.607 5.530 10.3 · · · ” ” ” H 2005-04-07 12:03:20 5.615 4.942 8.1 · · · ” ” ” H 2005-04-08 11:00:30 5.615 4.930 8.0 · · · ” ” ” H 2005-06-28 07:00:34 5.633 4.773 6.0 32 –Table 1—ContinuedNo. Name L n H
Tel. UT Date UT r ∆ α (mag) yyyy-mm-dd at start (AU) (AU) ( ◦ ) · · · ” ” ” H 2005-06-28 08:13:04 5.633 4.773 6.0(43627) 2002 CL L4 13.2 S 2005-03-10 17:55:00 5.709 5.651 10.1” ” ” ” H 2005-04-07 12:10:05 5.710 5.052 8.1” ” ” ” H 2005-04-08 11:04:45 5.710 5.040 8.0” ” ” ” H 2005-06-28 07:08:31 5.708 4.838 5.7” ” ” ” H 2005-06-28 08:21:08 5.708 4.838 5.7(65179) 2002 CN
L4 13.5 S 2005-04-06 15:32:05 5.622 5.156 9.5” ” ” ” H 2005-04-07 12:15:30 5.622 4.976 8.3” ” ” ” H 2005-04-08 11:09:01 5.622 4.964 8.2” ” ” ” H 2005-06-28 07:39:49 5.615 4.737 5.7” ” ” ” H 2005-06-28 08:49:54 5.615 4.737 5.7(166115) 2002 CO
L4 13.9 S 2005-04-10 04:40:41 5.223 5.119 11.1 · · ·
L4 14.4 S 2005-04-06 15:41:46 5.209 4.734 10.2 · · · ” ” ” H 2005-04-07 12:20:35 5.209 4.555 8.9 · · · ” ” ” H 2005-04-08 11:13:16 5.209 4.542 8.8 · · · ” ” ” H 2005-06-28 07:16:28 5.219 4.342 6.2 · · · ” ” ” H 2005-06-28 08:31:21 5.219 4.342 6.2(65174) 2002 CW
L4 13.6 S 2005-04-06 10:41:52 5.227 4.954 10.9” ” ” ” H 2005-04-07 13:15:27 5.228 4.752 10.1” ” ” ” H 2005-04-08 12:48:53 5.228 4.738 10.0” ” ” ” H 2005-06-29 07:49:13 5.262 4.297 3.8” ” ” ” H 2005-06-29 09:08:36 5.262 4.297 3.8(65206) 2002 DB L4 13.4 S 2005-04-06 10:51:49 5.542 5.253 10.2” ” ” ” H 2005-04-07 13:01:46 5.542 5.053 9.5” ” ” ” H 2005-04-08 12:44:37 5.542 5.039 9.4” ” ” ” H 2005-06-29 07:41:36 5.549 4.595 4.0” ” ” ” H 2005-06-29 09:00:59 5.549 4.595 4.0(89913) 2002 EC L4 13.6 S 2005-04-06 10:31:52 5.671 5.365 9.9” ” ” ” H 2005-06-29 07:34:06 5.656 4.706 4.0” ” ” ” H 2005-06-29 08:53:29 5.656 4.706 4.0 33 –Table 1—ContinuedNo. Name L n H
Tel. UT Date UT r ∆ α (mag) yyyy-mm-dd at start (AU) (AU) ( ◦ )(65211) 2002 EK L4 13.5 S 2005-04-08 23:29:54 5.288 5.098 10.9” ” ” ” H 2005-06-30 08:02:27 5.282 4.285 2.4” ” ” ” H 2005-06-30 09:40:03 5.282 4.285 2.4(195258) 2002 EN L4 13.7 S 2005-05-13 07:01:27 5.179 4.678 10.2(65227) 2002 ES L4 13.3 S 2005-04-12 12:34:12 5.351 5.225 10.8” ” ” ” H 2005-06-30 08:18:12 5.363 4.355 1.6” ” ” ” H 2005-06-30 09:56:42 5.363 4.355 1.6(65217) 2002 EY L4 13.4 S 2005-04-10 18:21:17 5.349 5.232 10.8” ” ” ” H 2005-04-08 13:46:22 5.349 5.071 10.6” ” ” ” H 2005-06-30 08:26:52 5.356 4.346 1.4” ” ” ” H 2005-06-30 10:10:29 5.356 4.346 1.4(65250) 2002 FT L4 13.3 S 2005-04-13 02:29:44 5.946 5.820 9.7(183358) 2002 VM
L5 13.0 S 2004-12-03 17:38:59 5.360 4.947 10.3 · · · ” ” ” H 2005-04-07 07:30:20 5.438 5.076 10.2 · · · ” ” ” H 2005-04-08 09:34:36 5.439 5.092 10.2(58096) Oineus L4 13.7 S 2005-04-06 11:18:34 5.772 5.584 10.0” ” ” ” H 2005-04-07 13:40:30 5.772 5.377 9.4” ” ” ” H 2005-04-08 13:02:50 5.771 5.362 9.4” ” ” ” H 2005-06-29 10:03:33 5.752 4.764 2.7” ” ” ” H 2005-06-29 11:42:13 5.752 4.764 2.7Note. — Here L n indicates the swarm (Lagrange point) in which each object re-sides; H gives the absolute magnitude as listed by the Minor Planet Center at URL ; the “Tel.” column listswhich telescope was used: “S” = Spitzer Space Telescope, “H” = University of Hawaii 2.2-meter Telescope; r , ∆, and α list the heliocentric distance, geocentric/Spitzercentric distance,and geocentric/Spitzercentric phase angle as given by JPL’s Horizons system. 34 –Table 2. PhotometryNo. Name Tel. UT Date UT F or m R yyyy-mm-dd at start (mJy or mag)(58153) 1988 RH S 2004-11-04 01:29:25 13 . ± .
16” ” H 2005-04-07 07:06:09 20 . ± . . ± . S 2004-11-10 08:55:22 5 . ± .
06” ” H 2005-04-07 07:11:15 20 . ± . . ± . S 2005-04-08 22:17:04 6 . ± .
28” ” H 2005-06-30 07:31:15 20 . ± . . ± . S 2005-04-06 11:08:11 14 . ± .
24” ” H 2005-04-07 12:49:42 20 . ± . . ± . . ± . . ± . S 2005-04-06 11:57:26 4 . ± . S 2005-04-08 23:20:30 16 . ± .
48” ” H 2005-06-29 10:10:33 20 . ± . . ± . · · · S 2005-04-06 11:47:36 2 . ± . · · · S 2005-04-06 15:51:08 6 . ± . · · · ” H 2005-04-07 12:32:22 20 . ± . · · · ” H 2005-04-08 11:18:39 20 . ± . · · · ” H 2005-06-28 09:18:45 20 . ± . · · · ” H 2005-06-28 09:52:58 20 . ± . S 2005-04-06 12:07:00 5 . ± . S 2005-05-19 13:21:44 7 . ± . S 2005-04-06 10:21:48 10 . ± .
14” ” H 2005-04-07 12:44:35 20 . ± . . ± . . ± .
016 35 –Table 2—ContinuedNo. Name Tel. UT Date UT F or m R yyyy-mm-dd at start (mJy or mag)” ” H 2005-06-28 10:01:11 19 . ± . S 2005-03-10 01:30:56 43 . ± .
49” ” H 2005-04-07 11:58:09 19 . ± . . ± . . ± . . ± . S 2005-04-06 11:28:35 4 . ± . S 2005-09-23 23:13:27 10 . ± . S 2005-05-13 07:11:06 17 . ± . S 2005-05-19 16:00:01 13 . ± . S 2005-04-10 07:04:45 29 . ± .
38” ” H 2005-04-07 07:25:08 19 . ± . . ± . S 2005-04-08 03:26:17 6 . ± .
11” ” H 2005-04-07 12:39:16 20 . ± . . ± . . ± . . ± . S 2005-04-09 00:16:01 14 . ± .
44” ” H 2005-04-07 13:31:51 20 . ± . . ± . . ± . . ± . S 2005-04-06 11:37:43 9 . ± . S 2005-04-08 22:06:49 11 . ± .
28” ” H 2005-04-07 13:20:37 20 . ± . . ± . . ± . . ± . S 2005-04-08 22:57:38 4 . ± .
40 36 –Table 2—ContinuedNo. Name Tel. UT Date UT F or m R yyyy-mm-dd at start (mJy or mag)” ” H 2005-06-29 10:17:41 20 . ± . . ± . S 2005-04-09 00:06:01 4 . ± .
64” ” H 2005-06-29 08:14:53 20 . ± . . ± . S 2005-04-08 23:39:43 16 . ± .
38” ” H 2005-06-29 10:34:59 19 . ± . . ± . S 2004-12-03 14:55:45 12 . ± .
14” ” H 2005-04-07 07:35:26 20 . ± . . ± . S 2004-11-05 18:28:57 3 . ± .
05” ” H 2005-04-07 07:18:00 21 . ± . . ± . S 2004-12-03 15:05:49 7 . ± .
08” ” H 2005-04-07 07:40:32 21 . ± . . ± . S 2004-12-03 14:45:03 22 . ± .
26” ” H 2005-04-07 07:47:17 19 . ± . . ± . . ± . . ± . . ± . S 2005-05-11 09:07:32 4 . ± .
07” ” H 2005-06-30 06:05:03 20 . ± . . ± . · · · S 2005-03-10 16:54:36 3 . ± . · · · ” H 2005-04-07 12:03:20 21 . ± . · · · ” H 2005-04-08 11:00:30 20 . ± . · · · ” H 2005-06-28 07:00:34 20 . ± .
026 37 –Table 2—ContinuedNo. Name Tel. UT Date UT F or m R yyyy-mm-dd at start (mJy or mag) · · · ” H 2005-06-28 08:13:04 20 . ± . S 2005-03-10 17:55:00 4 . ± .
08” ” H 2005-04-07 12:10:05 20 . ± . . ± . . ± . . ± . S 2005-04-06 15:32:05 7 . ± .
22” ” H 2005-04-07 12:15:30 20 . ± . . ± . . ± . . ± . S 2005-04-10 04:40:41 3 . ± . · · · S 2005-04-06 15:41:46 2 . ± . · · · ” H 2005-04-07 12:20:35 20 . ± . · · · ” H 2005-04-08 11:13:16 20 . ± . · · · ” H 2005-06-28 07:16:28 20 . ± . · · · ” H 2005-06-28 08:31:21 20 . ± . S 2005-04-06 10:41:52 12 . ± .
15” ” H 2005-04-07 13:15:27 19 . ± . . ± . . ± . . ± . S 2005-04-06 10:51:49 9 . ± .
16” ” H 2005-04-07 13:01:46 20 . ± . . ± . . ± . . ± . S 2005-04-06 10:31:52 10 . ± .
16” ” H 2005-06-29 07:34:06 20 . ± . . ± .
036 38 –Table 2—ContinuedNo. Name Tel. UT Date UT F or m R yyyy-mm-dd at start (mJy or mag)(65211) 2002 EK S 2005-04-08 23:29:54 3 . ± .
35” ” H 2005-06-30 08:02:27 19 . ± . . ± . S 2005-05-13 07:01:27 14 . ± . S 2005-04-12 12:34:12 14 . ± .
20” ” H 2005-06-30 08:18:12 19 . ± . . ± . S 2005-04-10 18:21:17 11 . ± .
13” ” H 2005-04-08 13:46:22 19 . ± . . ± . . ± . S 2005-04-13 02:29:44 5 . ± . S 2004-12-03 17:38:59 7 . ± . · · · ” H 2005-04-07 07:30:20 20 . ± . · · · ” H 2005-04-08 09:34:36 20 . ± . . ± .
79” ” H 2005-04-07 13:40:30 20 . ± . . ± . . ± . . ± . F or m R ” column lists either the flux density F at a wavelengthof 23.68 µ m as observed by Spitzer or the Cousins R magnitude m R asobserved by the UH 2.2-meter Telescope. 39 –Table 3. Physical Parameters and Formal ErrorsNo. Name D (km) p R (58153) 1988 RH . ± .
10 0 . ± . . ± .
06 0 . ± . . ± .
20 0 . ± . . ± .
10 0 . ± . . ± .
06 0 . ± . . ± .
22 0 . ± . · · · . ± .
16 0 . ± . · · · . ± .
12 0 . ± . . ± .
08 0 . ± . . ± .
08 0 . ± . . ± .
08 0 . ± . . ± .
14 0 . ± . . ± .
04 0 . ± . . ± .
08 0 . ± . . ± .
08 0 . ± . . ± .
08 0 . ± . . ± .
12 0 . ± . . ± .
08 0 . ± . . ± .
22 0 . ± . . ± .
08 0 . ± . . ± .
16 0 . ± . . ± .
32 0 . ± . . ± .
58 0 . ± . . ± .
18 0 . ± . . ± .
06 0 . ± . . ± .
06 0 . ± . . ± .
06 0 . ± . . ± .
10 0 . ± . . ± .
08 0 . ± . · · · . ± .
14 0 . ± . . ± .
08 0 . ± .
016 40 –Table 3—ContinuedNo. Name D (km) p R (65179) 2002 CN . ± .
02 0 . ± . . ± .
06 0 . ± . · · · . ± .
14 0 . ± . . ± .
08 0 . ± . . ± .
10 0 . ± . . ± .
10 0 . ± . . ± .
28 0 . ± . . ± .
06 0 . ± . . ± .
10 0 . ± . . ± .
08 0 . ± . . ± .
10 0 . ± . . ± .
06 0 . ± . . ± .
50 0 . ± . D is the effective diameter and p R is theCousins R-band geometric albedo. For both quantities,the quoted error is a 1- σ formal error given the modelingassumptions and the photometric uncertainties. 41 –Table 4. Ensemble R-Band Geometric AlbedosGroup N Average Median Std. Dev. Source Excluding“Large” 31 0 . ± .
001 0 . ± .
001 0.008 Paper I 1 outlier“Small” 44 0 . ± .
003 0 . ± .
004 0.062 this work none“Small” 32 0 . ± .
004 0 . ± .