Comparison of the Physical Properties of the L4 and L5 Trojan Asteroids from ATLAS Data

Jupiter Trojans are minor planets that orbit 60 degrees ahead of (L4) and behind (L5) Jupiter in the 1:1 resonant Lagrange points. As of 2020 January, the number of known Trojans listed by the Minor Planet Center is 7673, with Nakamura & Yoshida (2008) predicting a total of 10⁵ Trojans with diameter (D) greater than 2 km across the two clouds. Of the Trojans listed in the MPC, 4952 orbit as part of the L4 cloud and 2721 orbit in the L5.

Our understanding of the Trojan population will be greatly enhanced by the forthcoming Lucy mission, which will explore targets in both clouds starting in 2025. The Nice model (Gomes et al. 2005) predicts that the current Trojan population may have formed much further from the Sun than their current location and may consist of material scattered from the outer solar system that is subsequently captured by Jupiter. This formation mechanism does not reproduce the difference in number between the two Trojan clouds: the L4 cloud contains a greater number of objects with D > 10 km than the L5 cloud by a factor of approximately 1.4 (Grav et al. 2011). Nesvorný et al. (2013) consider a capture mechanism involving excitation of Jupiter's orbit in the early evolution of the solar system due to the presence of a fifth giant planet. During the orbital instability resulting from encounters between Jupiter and this fifth planet, Jupiter's position and by extension its Lagrangian points move, resulting in the loss of primordial Trojans previously occupying these stable regions. The L4 and L5 clouds are then repopulated with material captured in the postmigration orbit of Jupiter. This material is proposed to be similar in origin to current outer solar system (e.g., Kuiper Belt Object) populations, although modeling by Nesvorný et al. (2013) predicts that most of the material was captured close to Jupiter's current orbit. The L4/L5 number asymmetry can be explained in this case if a giant planet passes through the L5 cloud in its motion leading to this cloud being preferentially depleted (Nesvorný et al. 2013).

In addition to a difference in the number of objects in each cloud the remaining properties of Trojan asteroids (for instance, the observed color/spectral dichotomy between less-red and more-red populations) also show some difference between the clouds. Szabó et al. (2007) observe a color dependence on inclination in both clouds independent of size, and match a power law to both distributions. The difference in color distribution between the clouds was explained as potentially due to the different number densities and could be solved by normalizing each cloud differently. Roig et al. (2008) observe a clearly different distribution of spectral slopes between the two clouds, suggesting that this is due to the presence of collisional families in the clouds, as considering only background objects yields identical distributions in both L4 and L5. Emery et al. (2011) discovered a bimodality in the spectral slopes of Jupiter Trojans between "red" and "less-red" objects, which correlates with the color bimodality observed at visible wavelengths. This effect is seen equally in both L4 and L5 clouds.

In this paper we present a brief description of the Asteroid Terrestrial-impact Last Alert System (ATLAS) and the data used in Section 2 and an overview of the shape distribution model used in Section 3. We present and discuss the color measurements, phase functions and rotation periods derived for Trojans in ATLAS in Sections 4 and 5 respectively.

The photometry data used in this study originate from survey observations performed between 2015 and 2018 by ATLAS.⁵ Currently consisting of two units both located in Hawai'i, ATLAS is designed to achieve a high survey speed per unit cost (Tonry et al. 2018). Its main purpose is to discover asteroids with imminent impacts with Earth that are either regionally or globally threatening in nature. To fulfill this, the two current ATLAS units scan the complete visible northern sky every night enabling it to make numerous discoveries in multiple astronomical disciplines, such as supernovae candidates discovery (Prentice et al. 2018), gamma-ray burst phenomena (Stalder et al. 2017), variable stars (Heinze et al. 2018), and asteroid discovery (Tonry et al. 2018). All detected asteroid astrometry and photometry are posted to the Minor Planet Center, while the supernova candidates are publicly reported to the International Astronomical Union Transient Name Server. The 5-sigma limiting magnitude (AB) per 30 sec exposure is 19.7 for both ATLAS filters.

The two ATLAS units are 0.5 m telescopes each covering 30 deg² field of view in a single exposure. The main survey mode mostly utilizes two custom filters, a "cyan" or c-filter with a bandpass between 420 and 650 nm and an "orange" or o-filter with a bandpass between 560 and 820 nm (as shown in Figure 1). For further details on ATLAS, ATLAS photometry, and the ATLAS All-Sky Stellar Reference Catalog, see Tonry et al. (2018), Heinze et al. (2018) and Tonry et al. (2018). Although this AB photometric system uses only two, relatively wide filters, the c − o color obtained from ATLAS detections can be a good initial diagnostic to distinguish among asteroid taxonomic types. Further detail on this methodology can be read in Erasmus et al. (2020).

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For this project, photometry data of 863 L4 Trojans and 380 L5 Trojans could be extracted from the ATLAS data set as of this writing. We typically have a median of 55 and 74 unique observations for each detected L4 and L5 Trojan respectively, with roughly 30% in c and 70% in o band for both clouds. For this study we limit the data set to objects that had at least 100 and 20 observations in the o- and c-filters, respectively. We also only considered objects that had at least one observation at a phase angle (Sun-Observer-Target angle), α, of 5° or lower and phase-angle coverage of at at least 6° to ensure reliable phase curve fits. These criteria distilled the ATLAS datset to 209 L4 Trojans and 133 L5 Trojans for which we had a median of ∼300 observations, a median for the minimum phase angle of 13, and a median phase-angle range of 96. For all analysis in this work, we first cast all observed magnitudes to a corresponding absolute H_c and H_o magnitude by removing the distance and phase-angle dependence for all objects and for both filters. Hereafter, all references to o- and c-filter data refers to the H_c and H_o magnitudes unless stated otherwise.

The statistical shape model employed here generates a synthetic population of triaxial ellipsoids with assumed shapes and spin-pole orientations based on input distributions. We generate synthetic observations of these objects at an observing cadence equivalent to that of the ATLAS survey. We then compare the resulting set of individual observations of synthetic objects from different shape and spin-state distributions to the observed data using the two-sampled Kolmogorov–Smirnov (K-S) test as well as Mann–Whitney and chi-squared fits for confirmation. Only relative changes in brightness due to geometry are considered; as such, we do not need to account for heliocentric distance or surface characteristics. We considered applying the method of McNeill et al. (2019) used for NEOs in Spitzer observations; however, to compare sets of rotational amplitudes rather than individual detections requires observations made contiguously rather than sparsely over a long range of time.

Previous distribution models applied to main belt asteroids and Near-Earth Objects have assumed a uniform spin frequency distribution from 1.0–10.9 day⁻¹ across all applicable size ranges, corresponding to rotational periods from the spin barrier at 2.2–24 hr (McNeill et al. 2016; McNeill et al. 2019). This assumption is reasonable given the flat distribution of measured rotational frequencies at small asteroid sizes (Pravec et al. 2002). For the Trojan populations, we also include a population of slow rotators.

For generated synthetic detections a uniformly distributed uncertainty value is selected between −0.05 and 0.05 magnitudes, consistent with the uncertainty values in the selected ATLAS data set and applied to the value. We do not include any effect of limb scattering and/or darkening, which would only be significant at phase angles larger than the maximum of 10° reached by Trojans.

As we are only concerned with the relative magnitude differences caused by differing shapes, the value of a can be fixed and the values of b and c can be varied using various distributions e.g., Gaussians, Lorentzians, and bimodal distributions. A range of synthetic populations are generated from different input shape and spin-pole distributions and compared to the observational data using the two-sample K-S and Mann–Whitney tests. Identical distributions would produce a value of p = 1 in these cases. The results of these statistical tests indicate how closely the synthetic population resembles the underlying population that ATLAS sampled.

To test the validity of the fits from our model, we applied it to a series of known distributions. We first generated a range of known distributions of the axis ratio b/a, truncated at b/a = 1, as b cannot be greater than a. We assume a = 1 in all cases. From each input shape distribution, pseudo-ATLAS data was generated accounting for the cadence of ATLAS observations and the typical phase-angle distribution for Jupiter Trojans. The model was applied to these distributions and the returned best-fit values compared with the known parameters. Using multiple two-sampled statistical tests, we find that the derived result is correct in mean aspect ratio, and Gaussian parameters where a Gaussian distribution was assumed. The width of this Gaussian is a free parameter ranging from 0.1–0.4, small variations in this and the center of the Gaussian distribution produce equally good fits. The mean elongation for these best fits remains consistent, however, and we favor it as a metric in this analysis.

4.1. Colors and Phase Curves

To determine the colors of each ATLAS object, we first cast all observed magnitudes of both the c- and o-filter data (see top panel of Figure 2) to reduced magnitude, H(α), by removing the influence of distance on the brightness of the body. This reduced magnitude depends only on the phase angle, α, of the observations Using the formulation by Bowell et al. (1989), we fit the H-G model to the reduced magnitudes to extract a fitted phase curve parameter (see middle panel of Figure 2 and values recorded in Table 1). The improved H-G1/G2 (cite Muinonen) better fits sparse data that spans both very small and large phase angles, and therefore also better fits the the most dramatic part of opposition surge in brightness at phase angle equal to zero. However, for this study, we have opted for the less-complex H-G model because we have many observations for each object but those observations do not span a very large phase-angle range (<10°, see Section 2) We also do not have any observations for most objects at a 0° phase angle. Our fitted H-G model is also used to cast all reduced magnitudes to absolute magnitudes H_c and H_o (see bottom panel of Figure 2) which are used for color determination and for the shape modeling analysis (see Section 4.3).

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Table 1.  ATLAS Colors and Phase Curve Parameters

No.	Object Name	Trojan Group	Ha	c–o color	G-parameter
			(mag)	(mag)
001	10247 Amphiaraos (6629 P-L)	L4	11.2	0.248 ± 0.065	0.392 ± 0.070
002	10664 Phemios (5187 T-2)	L4	11.3	−0.148 ± 0.107	0.285 ± 0.140
003	10989 Dolios (1973 SL1)	L4	11.3	0.320 ± 0.064	0.319 ± 0.105
004	11252 Laertes (1973 SA2)	L4	10.7	0.387 ± 0.050	0.482 ± 0.065
005	11351 Leucus (1997 TS25)	L4	10.9	0.355 ± 0.078	0.255 ± 0.184
006	11395 (1998 XN77)	L4	9.9	0.207 ± 0.029	0.054 ± 0.045
007	11396 (1998 XZ77)	L4	10.6	0.368 ± 0.043	0.330 ± 0.079
008	11397 (1998 XX93)	L4	10.2	0.303 ± 0.042	0.670 ± 0.122
009	11429 Demodokus (4655 P-L)	L4	10.4	0.471 ± 0.048	0.072 ± 0.087
010	1143 Odysseus (1930 BH)	L4	8.3	0.385 ± 0.013	0.298 ± 0.040
011	114694 (2003 FC99)	L4	11.8	0.372 ± 0.040	0.930 ± 0.295
012	12238 Actor (1987 YU1)	L4	10.9	0.276 ± 0.050	0.283 ± 0.104
013	12658 Peiraios (1973 SL)	L4	11.2	−0.130 ± 0.091	0.454 ± 0.166
014	12714 Alkimos (1991 GX1)	L4	10.2	0.425 ± 0.039	0.387 ± 0.103
015	12916 Eteoneus (1998 TL15)	L4	11.4	0.394 ± 0.050	0.456 ± 0.122
016	12917 (1998 TG16)	L4	11.4	0.138 ± 0.202	1.044 ± 0.291
017	12921 (1998 WZ5)	L4	11.0	0.252 ± 0.059	0.588 ± 0.150
018	12973 Melanthios (1973 SY1)	L4	11.4	0.289 ± 0.087	0.222 ± 0.146
019	12974 Halitherses (1973 SB2)	L4	11.3	0.336 ± 0.073	0.103 ± 0.134
020	13060 (1991 EJ)	L4	10.8	0.461 ± 0.069	0.659 ± 0.282
021	13062 Podarkes (1991 HN)	L4	11.2	0.422 ± 0.071	0.923 ± 0.143
022	13182 (1996 SO8)	L4	10.7	0.259 ± 0.051	0.238 ± 0.077
023	13183 (1996 TW)	L4	10.7	0.369 ± 0.035	0.472 ± 0.150
024	13184 Augeias (1996 TS49)	L4	11.1	0.416 ± 0.076	0.411 ± 0.147
025	13323 (1998 SQ)	L4	11.1	0.342 ± 0.069	0.393 ± 0.101
026	13331 (1998 SU52)	L4	11.4	0.594 ± 0.120	0.372 ± 0.229
027	13362(1998 UQ16)	L4	11.0	0.409 ± 0.035	0.310 ± 0.097
028	13366 (1998 US24)	L4	11.2	−0.023 ± 0.257	0.432 ± 0.240
029	13372 (1998 VU6)	L4	11.3	0.432 ± 0.065	0.363 ± 0.135
030	13383 (1998 XS31)	L4	11.3	0.295 ± 0.099	−0.082 ± 0.108
031	13385 (1998 XO79)	L4	10.9	0.411 ± 0.027	0.497 ± 0.093
032	13463 Antiphos (5159 T-2)	L4	11.2	0.349 ± 0.090	0.451 ± 0.233
033	13694 (1997 WW7)	L4	10.9	0.191 ± 0.077	0.366 ± 0.104
034	13782 (1998 UM18)	L4	11.6	0.279 ± 0.161	0.662 ± 0.462
035	1404 Ajax (1936 QW)	L4	9.3	0.386 ± 0.036	0.063 ± 0.068
036	14268 (2000 AK156)	L4	10.5	0.478 ± 0.094	0.814 ± 0.190
037	1437 Diomedes (1937 PB)	L4	8.2	0.294 ± 0.017	0.194 ± 0.024
038	14690 (2000 AR25)	L4	10.6	0.423 ± 0.053	0.533 ± 0.118
039	14707 (2000 CC20)	L4	11.3	0.252 ± 0.054	0.592 ± 0.126
040	15033 (1998 VY29)	L4	10.7	0.109 ± 0.179	0.346 ± 0.179
041	15398 (1997 UZ23)	L4	10.9	0.433 ± 0.032	0.218 ± 0.139
042	15436 (1998 VU30)	L4	9.1	0.358 ± 0.032	0.345 ± 0.049
043	15440 (1998 WX4)	L4	9.6	0.469 ± 0.018	0.344 ± 0.060
044	15521 (1999 XH133)	L4	11.2	0.361 ± 0.077	1.058 ± 0.288
045	15527 (1999 YY2)	L4	10.9	0.383 ± 0.038	0.110 ± 0.068
046	15529 (2000 AA80)	L4	11.4	0.272 ± 0.087	0.924 ± 0.294
047	15535 (2000 AT177)	L4	10.6	0.378 ± 0.063	0.090 ± 0.078
048	15536 (2000 AG191)	L4	11.3	0.148 ± 0.294	0.532 ± 0.406
049	15539 (2000 CN3)	L4	10.5	0.431 ± 0.026	0.650 ± 0.116
050	15651 Tlepolemos (9612 P-L)	L4	11.3	0.343 ± 0.105	0.457 ± 0.213
051	1583 Antilochus (1950 SA)	L4	8.6	0.357 ± 0.015	0.134 ± 0.050
052	16099 (1999 VQ24)	L4	10.7	0.340 ± 0.038	0.431 ± 0.068
053	1647 Menelaus (1957 MK)	L4	10.5	0.452 ± 0.037	0.261 ± 0.058
054	16974 Iphthime (1998 WR21)	L4	9.9	0.400 ± 0.023	0.419 ± 0.044
055	17351 Pheidippos (1973 SV)	L4	11.3	0.385 ± 0.155	0.966 ± 0.484
056	1749 Telamon (1949 SB)	L4	9.5	0.419 ± 0.030	0.242 ± 0.062
057	17874 (1998 YM3)	L4	11.7	0.437 ± 0.062	0.794 ± 0.144
058	18060 (1999XJ156)	L4	11.1	0.319 ± 0.092	0.178 ± 0.088
059	18062 (1999 XY187)	L4	10.9	0.465 ± 0.055	0.251 ± 0.119
060	18063 (1999 XW211)	L4	11.2	0.404 ± 0.043	0.682 ± 0.227
061	18071 (2000 BA27)	L4	11.6	0.314 ± 0.110	0.228 ± 0.173
062	18263 Anchialos (5167 T-2)	L4	11.4	0.334 ± 0.080	0.479 ± 0.123
063	1868 Thersites (2008 P-L)	L4	9.5	0.421 ± 0.021	0.398 ± 0.047
064	1869 Philoctetes (4596 P-L)	L4	11.1	0.229 ± 0.054	0.171 ± 0.098
065	19725 (1999 WT4)	L4	10.8	0.345 ± 0.038	0.364 ± 0.085
066	19913 Aigyptios (1973 SU1)	L4	11.2	0.345 ± 0.067	0.210 ± 0.092
067	20144 (1996 RA33)	L4	11.4	0.238 ± 0.143	0.399 ± 0.235
068	20424 (1998 VF30)	L4	10.6	0.423 ± 0.107	0.317 ± 0.134
069	20729 (1999 XS143)	L4	10.5	0.390 ± 0.060	0.496 ± 0.126
070	20738 (1999 XG191)	L4	11.6	0.343 ± 0.095	0.698 ± 0.260
071	21284 Pandion (1996 TC51)	L4	11.5	0.458 ± 0.069	0.635 ± 0.129
072	21372 (1997 TM28)	L4	11.8	0.059 ± 0.117	1.505 ± 0.544
073	2146 Stentor (1976 UQ)	L4	10.0	0.391 ± 0.024	0.507 ± 0.168
074	2148 Epeios (1976 UW)	L4	10.8	0.377 ± 0.034	0.343 ± 0.091
075	21595 (1998 WJ5)	L4	10.7	0.380 ± 0.028	0.199 ± 0.111
076	21599 (1998 WA15)	L4	11.4	0.053 ± 0.223	0.917 ± 0.387
077	21601 (1998 XO89)	L4	10.1	0.309 ± 0.031	0.425 ± 0.069
078	21900 Orus (1999 VQ10)	L4	10.0	0.364 ± 0.038	0.365 ± 0.059
079	22008 (1999 XM71)	L4	11.6	0.280 ± 0.093	0.476 ± 0.135
080	22014 (1999 XQ96)	L4	10.2	0.463 ± 0.087	0.495 ± 0.158
081	22052 (2000 AQ14)	L4	11.2	0.321 ± 0.081	0.305 ± 0.103
082	22054 (2000 AP21)	L4	11.3	−0.143 ± 0.184	0.354 ± 0.217
083	22055 (2000 AS25)	L4	11.1	0.430 ± 0.037	0.430 ± 0.120
084	22059 (2000 AD75)	L4	11.2	0.379 ± 0.036	0.445 ± 0.162
085	22149 (2000 WD49)	L4	10.3	0.388 ± 0.046	0.462 ± 0.124
086	22203 Prothoenor (6020 P-L)	L4	11.6	0.313 ± 0.034	0.231 ± 0.049
087	2260 Neoptolemus (1975 WM1)	L4	9.3	0.380 ± 0.020	0.353 ± 0.057
088	23075 (1999 XV83)	L4	11.0	0.066 ± 0.103	0.550 ± 0.185
089	23126 (2000AK95)	L4	11.7	0.448 ± 0.051	0.374 ± 0.081
090	23135 (2000 AN146)	L4	10.0	0.286 ± 0.039	0.028 ± 0.058
091	23269 (2000 YH62)	L4	11.5	0.458 ± 0.265	0.365 ± 0.208
092	23285 (2000 YH119)	L4	11.1	0.154 ± 0.084	0.554 ± 0.281
093	23480 (1991 EL)	L4	11.3	0.478 ± 0.110	1.493 ± 0.514
094	23622 (1996 RW29)	L4	11.4	0.333 ± 0.108	0.101 ± 0.146
095	23709 (1997 TA28)	L4	11.6	0.306 ± 0.137	0.285 ± 0.154
096	23958 (1998 VD30)	L4	10.2	0.378 ± 0.054	0.296 ± 0.100
097	23970 (1998 YP6)	L4	11.0	0.476 ± 0.108	−0.028 ± 0.160
098	24244 (1999 XY101)	L4	10.9	0.424 ± 0.084	0.932 ± 0.183
099	24275 (1999 XW167)	L4	11.2	0.446 ± 0.129	0.564 ± 0.170
100	24312 (1999 YO22)	L4	11.6	0.526 ± 0.118	0.314 ± 0.144
101	24313 (1999 YR27)	L4	10.8	0.425 ± 0.029	0.229 ± 0.107
102	24390 (2000 AD177)	L4	11.6	0.477 ± 0.130	1.209 ± 0.477
103	24403 (2000 AX193)	L4	11.3	0.330 ± 0.179	1.445 ± 0.596
104	24485 (2000 YL102)	L4	11.4	0.366 ± 0.042	0.367 ± 0.114
105	24486 (2000 YR102)	L4	11.1	0.338 ± 0.031	0.349 ± 0.068
106	24505 (2001 BZ)	L4	11.1	0.391 ± 0.086	0.217 ± 0.154
107	24506 (2001 BS15)	L4	10.7	0.431 ± 0.051	0.311 ± 0.145
108	24534 (2001 CX27)	L4	11.3	0.319 ± 0.046	0.341 ± 0.166
109	24537 (2001 CB35)	L4	11.1	0.526 ± 0.080	0.340 ± 0.292
110	2456 Palamedes (1966 BA1)	L4	9.2	0.389 ± 0.039	0.375 ± 0.089
111	24587 Kapaneus (4613 T-2)	L4	11.5	0.332 ± 0.095	1.921 ± 0.981
112	25895 (2000 XN9)	L4	11.1	0.410 ± 0.047	0.100 ± 0.165
113	25911 (2001 BC76)	L4	11.7	0.321 ± 0.099	1.185 ± 0.544
114	2759 Idomeneus (1980 GC)	L4	10.0	0.344 ± 0.025	0.400 ± 0.099
115	2797 Teucer (1981 LK)	L4	8.7	0.470 ± 0.016	0.280 ± 0.091
116	2920 Automedon (1981 JR)	L4	8.7	0.437 ± 0.022	0.479 ± 0.054
117	30102 (2000 FC1)	L4	10.8	0.358 ± 0.050	0.235 ± 0.137
118	3063 Makhaon (1983 PV)	L4	8.5	0.407 ± 0.016	0.416 ± 0.048
119	31835 (2000 BK16)	L4	11.5	0.342 ± 0.062	0.583 ± 0.227
120	3391 Sinon (1977 DD3)	L4	10.3	0.395 ± 0.080	0.560 ± 0.207
121	3548 Eurybates (1973 SO)	L4	9.8	0.333 ± 0.029	0.093 ± 0.063
122	3564 Talthybius (1985 TC1)	L4	9.4	0.452 ± 0.038	0.237 ± 0.055
123	35673 (1998 VQ15)	L4	11.6	0.045 ± 0.116	0.045 ± 0.117
124	3596 Meriones (1985 VO)	L4	9.3	0.405 ± 0.034	0.247 ± 0.118
125	36259 (1999 XM74)	L4	11.4	0.382 ± 0.074	0.419 ± 0.188
126	36267 (1999 XB211)	L4	10.8	0.389 ± 0.056	0.344 ± 0.124
127	3709 Polypoites (1985 TL3)	L4	9.1	0.343 ± 0.018	0.398 ± 0.054
128	37297 (2001 BQ77)	L4	11.6	0.370 ± 0.156	1.018 ± 0.438
129	37298 (2001 BU80)	L4	11.7	0.534 ± 0.255	0.098 ± 0.249
130	37714 (1996 RK29)	L4	12.2	0.340 ± 0.089	0.367 ± 0.169
131	3793 Leonteus (1985 TE3)	L4	8.8	0.119 ± 0.032	0.107 ± 0.059
132	3794 Sthenelos (1985 TF3)	L4	10.5	0.394 ± 0.032	0.056 ± 0.058
133	3801 Thrasymedes (1985 VS)	L4	11.1	0.456 ± 0.042	0.050 ± 0.111
134	38050 (1998 VR38)	L4	9.9	0.362 ± 0.021	0.314 ± 0.064
135	38607 (2000 AN6)	L4	11.7	0.168 ± 0.094	0.376 ± 0.150
136	38610 (2000 AU45)	L4	11.5	0.455 ± 0.131	0.964 ± 0.304
137	39264 (2000 YQ139)	L4	10.8	0.398 ± 0.028	0.525 ± 0.128
138	4007 Euryalos (1973 SR)	L4	10.3	0.447 ± 0.041	0.199 ± 0.098
139	4035 (1986 WD)	L4	9.6	0.437 ± 0.019	0.287 ± 0.059
140	4057 Demophon (1985 TQ)	L4	10.1	0.374 ± 0.067	0.324 ± 0.090
141	4060 Deipylos (1987 YT1)	L4	9.3	0.268 ± 0.019	0.181 ± 0.052
142	4063 Euforbo (1989 CG2)	L4	8.7	0.395 ± 0.020	0.405 ± 0.053
143	4068 Menestheus (1973 SW)	L4	9.5	0.385 ± 0.029	0.287 ± 0.058
144	4086 Podalirius (1985 VK2)	L4	9.2	0.319 ± 0.069	0.374 ± 0.146
145	41379 (2000 AS105)	L4	11.4	0.269 ± 0.116	0.450 ± 0.229
146	4138 Kalchas (1973 SM)	L4	10.1	0.301 ± 0.054	0.095 ± 0.069
147	42168 (2001 CT13)	L4	11.6	0.386 ± 0.117	0.020 ± 0.151
148	42367 (2002 CQ134)	L4	11.0	0.379 ± 0.070	1.469 ± 0.376
149	42554 (1996 RJ28)	L4	11.7	0.408 ± 0.072	0.196 ± 0.145
150	4489 (1988 AK)	L4	9.0	0.450 ± 0.021	0.232 ± 0.043
151	4501 Eurypylos (1989 CJ3)	L4	10.5	0.319 ± 0.051	0.217 ± 0.108
152	4543 Phoinix (1989 CQ1)	L4	9.7	0.368 ± 0.064	0.555 ± 0.114
153	46676 (1996 RF29)	L4	12.0	0.304 ± 0.054	0.449 ± 0.088
154	4833 Meges (1989 AL2)	L4	9.0	0.378 ± 0.038	0.195 ± 0.060
155	4834 Thoas (1989 AM2)	L4	9.1	0.468 ± 0.020	0.349 ± 0.052
156	4835 (1989 BQ)	L4	10.6	0.415 ± 0.076	0.671 ± 0.186
157	4836 Medon (1989 CK1)	L4	9.5	0.454 ± 0.023	0.298 ± 0.062
158	4902 Thessandrus (1989 AN2)	L4	9.9	0.427 ± 0.040	0.277 ± 0.070
159	4946 Askalaphus (1988 BW1)	L4	10.2	0.522 ± 0.029	0.450 ± 0.092
160	5012 Eurymedon (9507 P-L)	L4	10.6	0.338 ± 0.032	0.368 ± 0.069
161	5023 Agapenor (1985 TG3)	L4	10.4	0.320 ± 0.027	0.307 ± 0.080
162	5025 (1986 TS6)	L4	10.4	0.431 ± 0.029	0.511 ± 0.126
163	5027 Androgeos (1988 BX1)	L4	9.7	0.433 ± 0.034	0.284 ± 0.048
164	5028 Halaesus (1988 BY1)	L4	10.3	0.415 ± 0.048	0.305 ± 0.124
165	5041 Theotes (1973 SW1)	L4	10.7	0.331 ± 0.072	0.402 ± 0.108
166	5123 (1989 BL)	L4	10.0	0.363 ± 0.047	0.063 ± 0.063
167	5126 Achaemenides (1989 CH2)	L4	10.6	0.210 ± 0.058	0.349 ± 0.094
168	5209 (1989 CW1)	L4	10.3	0.377 ± 0.034	0.464 ± 0.074
169	5244 Amphilochos (1973 SQ1)	L4	10.5	0.297 ± 0.050	0.251 ± 0.090
170	5254 Ulysses (1986 VG1)	L4	9.2	0.368 ± 0.025	0.384 ± 0.052
171	5258 (1989 AU1)	L4	10.3	0.345 ± 0.046	0.553 ± 0.076
172	5259 Epeigeus (1989 BB1)	L4	10.4	0.503 ± 0.025	0.291 ± 0.084
173	5264 Telephus (1991 KC)	L4	9.5	0.386 ± 0.037	0.431 ± 0.085
174	5283 Pyrrhus (1989 BW)	L4	9.7	0.500 ± 0.043	0.545 ± 0.100
175	5284 Orsilocus (1989 CK2)	L4	10.1	0.447 ± 0.027	0.379 ± 0.085
176	5285 Krethon (1989 EO11)	L4	10.1	0.413 ± 0.057	0.314 ± 0.081
177	53436 (1999 VB154)	L4	11.4	0.361 ± 0.071	0.559 ± 0.141
178	5436 Eumelos (1990 DK)	L4	10.4	0.353 ± 0.050	0.438 ± 0.100
179	55571 (2002 CP82)	L4	12.0	0.321 ± 0.064	0.548 ± 0.135
180	5652 Amphimachus (1992 HS3)	L4	10.1	0.342 ± 0.036	0.171 ± 0.057
181	588 Achilles (A906 DN)	L4	8.3	0.450 ± 0.041	0.341 ± 0.083
182	60383 (2000 AR184)	L4	11.1	0.366 ± 0.109	0.287 ± 0.108
183	6090(1989 DJ)	L4	9.4	0.434 ± 0.033	0.054 ± 0.080
184	624 Hektor (A907 CF)	L4	7.3	0.420 ± 0.009	0.243 ± 0.044
185	63273 (2001 DH4)	L4	11.5	0.285 ± 0.072	0.348 ± 0.186
186	63286 (2001 DZ68)	L4	12.1	0.529 ± 0.183	0.081 ± 0.160
187	65097 (2002 CC4)	L4	12.0	0.419 ± 0.075	0.484 ± 0.237
188	65257 (2002 FU36)	L4	11.5	0.402 ± 0.098	0.577 ± 0.268
189	6545 (1986 TR6)	L4	10.2	0.334 ± 0.052	0.382 ± 0.084
190	659 Nestor (A908 FE)	L4	8.7	0.263 ± 0.030	−0.010 ± 0.022
191	67065 (1999 XW261)	L4	11.9	0.331 ± 0.061	1.936 ± 0.765
192	7119 Hiera (1989 AV2)	L4	9.7	0.440 ± 0.045	0.103 ± 0.083
193	7152 Euneus (1973 SH1)	L4	10.3	0.365 ± 0.034	0.285 ± 0.067
194	7543 Prylis (1973 SY)	L4	10.6	0.329 ± 0.033	0.297 ± 0.105
195	7641 (1986 TT6)	L4	9.5	0.427 ± 0.027	0.418 ± 0.072
196	8125 Tyndareus (5493 T-2)	L4	10.8	0.355 ± 0.035	0.418 ± 0.088
197	8241 Agrius (1973 SE1)	L4	11.2	0.366 ± 0.086	0.617 ± 0.130
198	8317 Eurysaces (4523 P-L)	L4	11.1	0.234 ± 0.063	0.088 ± 0.115
199	90337 (2003 FQ97)	L4	11.6	0.192 ± 0.058	0.459 ± 0.119
200	911 Agamemnon (A919 FB)	L4	7.9	0.453 ± 0.012	0.133 ± 0.076
201	9431 (1996 PS1)	L4	10.6	0.415 ± 0.028	0.545 ± 0.105
202	9694 Lycomedes (6581 P-L)	L4	10.7	0.353 ± 0.064	0.527 ± 0.100
203	9712 Nauplius (1973 SO1)	L4	10.9	0.263 ± 0.127	0.381 ± 0.191
204	9713 Oceax (1973 SP1)	L4	11.3	0.408 ± 0.191	0.465 ± 0.171
205	9790 (1995 OK8)	L4	10.9	0.324 ± 0.044	0.430 ± 0.081
206	9799 (1996 RJ)	L4	9.7	0.424 ± 0.041	0.326 ± 0.056
207	9817 Thersander (6540 P-L)	L4	11.5	0.256 ± 0.167	0.695 ± 0.252
208	9818 Eurymachos (6591 P-L)	L4	11.0	0.244 ± 0.061	0.292 ± 0.097
209	9857 (1991 EN)	L4	10.3	0.394 ± 0.057	0.240 ± 0.096
210	11089 (1994 CS8)	L5	10.7	0.316 ± 0.038	0.205 ± 0.075
211	11487 (1988 RG10)	L5	11.3	0.349 ± 0.076	0.426 ± 0.102
212	11509 Thersilochos (1990 VL6)	L5	10.1	0.426 ± 0.027	0.423 ± 0.081
213	11552 Boucolion (1993 BD4)	L5	10.1	0.263 ± 0.035	0.359 ± 0.058
214	11554 Asios(1993 BZ12)	L5	10.5	0.324 ± 0.016	0.336 ± 0.050
215	11663 (1997 GO24)	L5	10.9	0.383 ± 0.087	0.346 ± 0.146
216	1172 Aneas (1930 UA)	L5	8.2	0.491 ± 0.033	0.346 ± 0.063
217	1173 Anchises (1930 UB)	L5	8.9	0.313 ± 0.040	0.173 ± 0.047
218	11887 Echemmon (1990 TV12)	L5	10.8	0.448 ± 0.057	0.399 ± 0.162
219	12052 Aretaon (1997 JB16)	L5	10.6	0.406 ± 0.049	0.269 ± 0.081
220	12126 (1999 RM11)	L5	10.1	0.332 ± 0.050	0.640 ± 0.076
221	128299 (2003 YL61)	L5	11.5	0.391 ± 0.070	0.466 ± 0.114
222	12929 (1999 TZ1)	L5	10.0	0.392 ± 0.030	0.044 ± 0.069
223	15502 (1999 NV27)	L5	10.0	0.325 ± 0.027	0.273 ± 0.053
224	15977 (1998 MA11)	L5	10.4	0.483 ± 0.019	0.377 ± 0.076
225	16070 (1999 RB101)	L5	9.7	0.371 ± 0.020	0.285 ± 0.047
226	16560 Daitor (1991 VZ5)	L5	10.7	0.261 ± 0.042	0.160 ± 0.048
227	16667 (1993 XM1)	L5	10.8	0.332 ± 0.060	0.806 ± 0.192
228	16956 (1998 MQ11)	L5	10.7	0.316 ± 0.046	0.704 ± 0.121
229	17171 (1999 NB38)	L5	10.5	0.437 ± 0.034	0.623 ± 0.082
230	17172 (1999 NZ41)	L5	10.8	0.427 ± 0.042	0.292 ± 0.054
231	17314 Aisakos (1024 T-1)	L5	10.9	0.335 ± 0.024	0.370 ± 0.060
232	17365 (1978 VF11)	L5	10.5	0.279 ± 0.056	0.812 ± 0.208
233	17419 (1988 RH13)	L5	11.3	0.258 ± 0.041	0.418 ± 0.130
234	17492 Hippasos (1991 XG1)	L5	10.1	0.358 ± 0.022	0.509 ± 0.109
235	18046 (1999 RN116)	L5	10.5	0.380 ± 0.035	0.417 ± 0.075
236	18054 (1999 SW7)	L5	10.8	0.317 ± 0.044	0.373 ± 0.065
237	18137 (2000 OU30)	L5	11.0	0.380 ± 0.051	0.241 ± 0.054
238	18278 Drymas (4035 T-3)	L5	11.4	0.345 ± 0.059	0.895 ± 0.272
239	18493 Demoleon (1996 HV9)	L5	10.7	0.335 ± 0.053	0.251 ± 0.125
240	1867 Deiphobus (1971 EA)	L5	8.3	0.426 ± 0.022	0.257 ± 0.050
241	1870 Glaukos (1971 FE)	L5	10.6	0.305 ± 0.040	0.241 ± 0.051
242	1871 Astyanax (1971 FF)	L5	11.2	0.290 ± 0.057	0.429 ± 0.091
243	1872 Helenos (1971 FG)	L5	10.8	0.242 ± 0.038	0.123 ± 0.087
244	1873 Agenor (1971 FH)	L5	10.1	0.379 ± 0.029	0.484 ± 0.090
245	19020 (2000 SC6)	L5	10.5	0.317 ± 0.034	0.301 ± 0.085
246	2207 Antenor (1977 QH1)	L5	8.9	0.305 ± 0.021	0.137 ± 0.028
247	22180 (2000 YZ)	L5	10.2	0.558 ± 0.028	0.983 ± 0.237
248	2223 Sarpedon (1977 TL3)	L5	9.1	0.492 ± 0.023	0.608 ± 0.082
249	2241 Alcathous (1979 WM)	L5	8.5	0.387 ± 0.010	0.290 ± 0.041
250	23549 Epicles (1994 ES6)	L5	11.7	0.196 ± 0.072	0.655 ± 0.167
251	2357 Phereclos (1981 AC)	L5	8.9	0.404 ± 0.008	0.348 ± 0.031
252	2363 Cebriones (1977 TJ3)	L5	8.9	0.306 ± 0.035	−0.201 ± 0.044
253	24446 (2000 PR25)	L5	11.0	0.334 ± 0.047	0.423 ± 0.152
254	24448 (2000 QE42)	L5	11.4	0.303 ± 0.087	0.510 ± 0.125
255	24451 (2000 QS104)	L5	10.3	0.335 ± 0.029	0.926 ± 0.157
256	24453 (2000 QG173)	L5	11.2	0.250 ± 0.060	0.315 ± 0.082
257	24454 (2000 QF198)	L5	11.4	0.232 ± 0.102	0.297 ± 0.106
258	24470 (2000 SJ310)	L5	10.9	0.425 ± 0.049	0.348 ± 0.156
259	24471 (2000 SH313)	L5	11.1	0.277 ± 0.086	0.647 ± 0.388
260	25883 (2000 RD88)	L5	11.3	0.393 ± 0.063	0.144 ± 0.097
261	2674 Pandarus (1982 BC3)	L5	9.1	0.473 ± 0.033	0.419 ± 0.059
262	2893 Peiroos (1975 QD)	L5	9.0	0.333 ± 0.028	0.344 ± 0.047
263	2895 Memnon (1981 AE1)	L5	10.1	0.246 ± 0.031	0.121 ± 0.048
264	29603 (1998 MO44)	L5	11.1	0.272 ± 0.059	0.687 ± 0.133
265	29976 (1999 NE9)	L5	11.0	0.351 ± 0.043	0.310 ± 0.056
266	30504 (2000 RS80)	L5	11.4	0.187 ± 0.100	0.753 ± 0.141
267	30506 (2000 RO85)	L5	11.0	0.335 ± 0.049	0.288 ± 0.097
268	30704 Phegeus (3250 T-3)	L5	11.3	0.298 ± 0.068	0.421 ± 0.129
269	30705 Idaios (3365 T-3)	L5	10.4	0.410 ± 0.028	0.700 ± 0.119
270	30942 Helicaon (1994 CX13)	L5	11.4	0.322 ± 0.050	0.228 ± 0.086
271	31342 (1998 MU31)	L5	10.5	0.390 ± 0.038	0.359 ± 0.075
272	31344 Agathon (1998 OM12)	L5	10.9	0.271 ± 0.060	0.551 ± 0.110
273	31819 (1999 RS150)	L5	11.4	0.345 ± 0.078	0.521 ± 0.116
274	32339 (2000 QA88)	L5	11.7	0.296 ± 0.079	0.685 ± 0.187
275	32397 (2000 QL214)	L5	11.5	0.315 ± 0.105	0.148 ± 0.139
276	3240 Laocoon(1978 VG6)	L5	10.2	0.372 ± 0.041	0.567 ± 0.078
277	32435 (2000 RZ96)	L5	11.2	0.331 ± 0.078	0.714 ± 0.194
278	32440 (2000 RC100)	L5	11.4	0.360 ± 0.098	0.696 ± 0.146
279	32464 (2000 SB132)	L5	11.6	0.322 ± 0.094	0.316 ± 0.189
280	32475 (2000 SD234)	L5	10.8	0.306 ± 0.051	0.671 ± 0.216
281	32482 (2000 ST354)	L5	11.0	0.285 ± 0.061	0.070 ± 0.072
282	32499 (2000 YS11)	L5	10.5	0.303 ± 0.053	0.667 ± 0.121
283	32501 (2000 YV135)	L5	11.0	0.444 ± 0.049	0.091 ± 0.118
284	32615 (2001 QU277)	L5	11.0	0.330 ± 0.065	0.280 ± 0.093
285	32811 Apisaon (1990 TP12)	L5	11.3	0.329 ± 0.061	0.097 ± 0.083
286	3317 Paris (1984 KF)	L5	8.4	0.402 ± 0.010	0.241 ± 0.035
287	3451 Mentor (1984 HA1)	L5	8.5	0.157 ± 0.050	−0.006 ± 0.059
288	34642 (2000 WN2)	L5	10.8	0.345 ± 0.043	0.129 ± 0.082
289	34746 (2001 QE91)	L5	9.9	0.217 ± 0.039	0.125 ± 0.067
290	3708 (1974 FV1)	L5	9.3	0.478 ± 0.032	0.300 ± 0.050
291	37519 Amphios (3040 T-3)	L5	11.1	0.402 ± 0.050	0.229 ± 0.117
292	42277 (2001 SQ51)	L5	12.1	0.378 ± 0.057	0.413 ± 0.090
293	4348 Poulydamas (1988 RU)	L5	9.6	0.234 ± 0.022	0.134 ± 0.046
294	4707 Khryses (1988 PY)	L5	10.6	0.357 ± 0.046	0.647 ± 0.101
295	4708 Polydoros (1988 RT)	L5	9.9	0.383 ± 0.017	0.369 ± 0.039
296	4709 Ennomos (1988 TU2)	L5	8.6	0.270 ± 0.038	0.107 ± 0.062
297	4715 (1989 TS1)	L5	9.8	0.348 ± 0.050	0.465 ± 0.064
298	4722 Agelaos (4271 T-3)	L5	10.0	0.418 ± 0.035	0.529 ± 0.050
299	4754 Panthoos (5010 T-3)	L5	10.0	0.392 ± 0.018	0.436 ± 0.037
300	4791 Iphidamas (1988 PB1)	L5	10.0	0.526 ± 0.044	0.395 ± 0.100
301	4792 Lykaon (1988 RK1)	L5	10.1	0.353 ± 0.036	0.481 ± 0.062
302	4805 Asteropaios (1990 VH7)	L5	10.1	0.340 ± 0.022	0.235 ± 0.046
303	4827 Dares (1988 QE)	L5	10.5	0.389 ± 0.029	0.262 ± 0.053
304	4828 Misenus (1988 RV)	L5	10.4	0.325 ± 0.040	0.242 ± 0.048
305	4829 Sergestus (1988 RM1)	L5	11.2	0.335 ± 0.043	0.488 ± 0.102
306	4832 Palinurus (1988 TU1)	L5	10.0	0.408 ± 0.027	0.579 ± 0.056
307	48438 (1989 WJ2)	L5	10.9	0.404 ± 0.032	0.522 ± 0.152
308	4867 Polites (1989 SZ)	L5	9.8	0.420 ± 0.014	0.449 ± 0.045
309	48764 (1997 JJ10)	L5	11.5	0.431 ± 0.067	0.911 ± 0.232
310	5119 (1988 RA1)	L5	10.3	0.395 ± 0.044	0.464 ± 0.070
311	5120 Bitias (1988 TZ1)	L5	10.3	0.344 ± 0.030	0.500 ± 0.107
312	5130 Ilioneus (1989 SC7)	L5	9.8	0.398 ± 0.021	0.277 ± 0.049
313	51364 (2000 SU333)	L5	11.6	0.279 ± 0.049	0.421 ± 0.122
314	51365 (2000 TA42)	L5	10.7	0.309 ± 0.038	0.358 ± 0.160
315	5144 Achates (1991 XX)	L5	9.0	0.414 ± 0.014	0.437 ± 0.055
316	51958 (2001 QJ256)	L5	11.5	−0.022 ± 0.145	0.846 ± 0.327
317	51962 (2001 QH267)	L5	11.5	0.125 ± 0.103	0.977 ± 0.194
318	5233 (1988 RL10)	L5	11.5	0.344 ± 0.051	0.351 ± 0.066
319	54656 (2000 SX362)	L5	10.7	0.400 ± 0.045	0.284 ± 0.088
320	5476 (1989 TO11)	L5	10.6	0.366 ± 0.036	0.230 ± 0.075
321	55060 (2001 QM73)	L5	11.1	0.342 ± 0.047	0.745 ± 0.123
322	5511 Cloanthus (1988 TH1)	L5	10.3	0.446 ± 0.049	0.550 ± 0.093
323	55419 (2001 TF19)	L5	11.2	0.297 ± 0.041	0.478 ± 0.096
324	5638 Deikoon (1988 TA3)	L5	10.5	0.386 ± 0.034	0.265 ± 0.054
325	5648 (1990 VU1)	L5	9.7	0.350 ± 0.025	0.505 ± 0.091
326	56968 (2000 SA92)	L5	11.6	0.346 ± 0.105	0.939 ± 0.239
327	58008 (2002 TW240)	L5	11.6	0.403 ± 0.036	0.350 ± 0.100
328	5907 (1989 TU5)	L5	11.1	0.334 ± 0.044	0.522 ± 0.079
329	6002 (1988 RO)	L5	10.5	0.423 ± 0.038	0.387 ± 0.088
330	617 Patroclus (A906 UL)	L5	8.2	0.287 ± 0.017	0.299 ± 0.045
331	68444 (2001 RH142)	L5	11.5	0.244 ± 0.085	0.734 ± 0.158
332	6997 Laomedon (3104 T-3)	L5	10.6	0.336 ± 0.055	0.143 ± 0.087
333	7352 (1994 CO)	L5	10.0	0.308 ± 0.034	0.152 ± 0.043
334	76857 (2000 WE132)	L5	11.0	0.338 ± 0.061	0.673 ± 0.227
335	7815 Dolon (1987 QN)	L5	10.2	0.378 ± 0.024	0.315 ± 0.059
336	82055 (2000 TY40)	L5	11.7	0.304 ± 0.105	0.363 ± 0.123
337	884 Priamus (A917 SU)	L5	8.7	0.438 ± 0.017	0.376 ± 0.044
338	9023 Mnesthus (1988 RG1)	L5	10.2	0.375 ± 0.032	0.233 ± 0.048
339	9030(1989 UX5)	L5	11.1	0.384 ± 0.046	0.608 ± 0.088
340	9142 Rhesus (5191 T-3)	L5	10.6	0.388 ± 0.049	0.491 ± 0.098
341	9430 Erichthonios (1996 HU10)	L5	11.3	0.352 ± 0.046	0.886 ± 0.148
342	99943 (2005 AS2)	L5	11.6	0.276 ± 0.091	0.134 ± 0.083

Note.

^aH magnitude was obtained from https://ssd.jpl.nasa.gov/horizons.cgi.

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A large contribution to the remaining scatter in the H_c and H_o magnitudes is the potential brightness variation due to the asteroid's rotation so the c–o color of each object is extracted by determining the median H_c and H_o magnitudes of a randomly selected 50% subset of the c- and o-filter data, respectively, and repeating that process 10 times and calculating the the average. The final c–o color is defined as the difference between the average of the median magnitudes. The uncertainty in the color value incorporates the standard deviation of the median magnitudes in the previous step. The bottom plots of Figure 2 show examples of the results of this procedure with the c–o color values and uncertainty displayed in top left corner of each plot and recorded in Table 1.

4.2. Rotation Periods

4.3. Shape Distribution

The L4 and L5 Trojan clouds were considered separately and shape distributions for each population were determined. Due to the unknown spin-pole distribution of the Jupiter Trojans we assume both a case where all objects have spin poles perpendicular to the ecliptic and a case where all objects have spin-pole latitudes θ = 50°. Although the real shape distribution of the population is highly dependent on the spin-pole latitudes observed, any difference between the two clouds will remain regardless of the assumption used. Therefore, it is emphasized that although the mean values for elongation in the two clouds themselves may not be meaningful, any disparity between the L4 and L5 values is a real effect.

For the L4 cloud we obtain a best-fit mean elongation of $\tfrac{b}{a}=0.77\pm 0.02$ and for L5 we obtain a value $\tfrac{b}{a}=0.86\,\pm 0.02$. Figure 3 shows the p-value obtained from the K-S test for a wide range of shape distributions for both L4 and L5 clouds plotted against the average elongation of the distribution. In this case the assumption of perpendicular spin poles is used. Figure 4 shows the same information in the case of spin-pole latitudes ≈50°. Potential mechanisms for this difference will be discussed in Section 5.2.

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We considered the possibility of carrying out this modeling for individual families within the Trojans; however, the number of family members in this data set is insufficient for this analysis. This represents a potential avenue for study when future surveys, e.g., LSST, come online.

5.1. Comparison of Colors and Phase Curve Parameters between L4 and L5

In general, both the L4 and L5 objects have c–o colors consistent with the expected D-type taxonomy's c–o color (see Figure 5). However, the median color of the L4 cloud appears at a slightly higher c–o color (∼0.41 mag) than that of the L5 cloud (∼0.39 mag). While the median color of the L4 is slightly higher, the color distribution of the L4 objects is much broader containing both redder objects and also objects with low C- or X-like c–o colors (i.e., objects with less-red or even neutral slopes). Using a two-sample K-S test on the two distributions for suggests that the colors are not drawn from identical distributions but we cannot rule this out for slope parameter, G. (p-values: 0.01 for c–o color, 0.71 for slope parameter G).

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We attemped to verify this color discrepancy using data from the Sloan Digital Sky Survey (SDSS; Ivezic 1998). The color–color plots for the L4 and L5 Trojans from SDSS are given in Figure 6. From these measurements, the L4 Trojans do show a broader distribution in a* than the L5 Trojans. Here, a* is a principal component used in asteroid color analysis from SDSS as defined by Ivezić et al. (2002). Again in the case of the SDSS data, the K-S test suggests that these two populations cannot have been drawn from the same distribution. To assess whether this difference could be due to the presence of an abundance of Eurybates family objects in our sample, we cross-checked our list of targets with a list of family members and found only several objects in common. Szabó et al. (2007); Roig et al. (2008) and Emery et al. (2011) all offer evidence of color bimodality in the Trojan population, yet none draw clear conclusions about differences in color abundances between the two clouds. Szabó et al. (2007) find a difference in color distribution between L4 and L5, however, this is removed through normalization due to the different number densities of two clouds. Roig et al. (2008) observe a color bimodality, and see a difference in the distributions of each cloud but removing family members from their analysis they find the two clouds to be identical. As only a few Eurybates family members are present in our data, they alone cannot be producing this effect. Figure 6 shows the i–z and a* colors for all objects shared in our ATLAS sample and SDSS archival data. The bimodality is not readily visible in these plots, however, limiting this to only the brightest targets (H < 12) does show this effect. This is due to large uncertainties on objects on Trojans where H > 13. We do not observe color bimodality in ATLAS c–o measurements as the wavelength ranges here span the "kink" in the spectra of "less-red" Trojans (Emery et al. 2011). This has the effect of making the expected c–o colors for each group closer together, preventing the bimodality from being observed.

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5.2. Potential Explanations for the Difference in Apparent Elongation between L4 and L5

We consider a range of mechanisms which may produce the difference in shape distribution between the two Trojan clouds and assess their validity, from least likely to most likely.

5.2.1. A Difference in Spin-pole Distribution

As previously stated, the assumption of the spin-pole distribution of the population is important in obtaining its overall shape distribution. In the investigation we have assumed that both clouds have identical spin-pole distributions. However, if this is not the case, a difference in spin-pole distribution could be invoked to explain the difference in shape distribution between the two clouds. Assuming a spin-pole distribution where the poles are aligned toward the observers produces a shape distribution that appears more spherical, while a more randomly oriented distribution more accurately reveals an underlying elongated shape distribution.

Keeping the spin-pole distribution of the more elongated L4 cloud constant we vary the spin-pole distributions of the L5 cloud in order to try to produce identical results. This is presented in Figure 7. We find that in order to produce the same best-fit shape distribution the required spin-pole distribution of the more elongated L4 cloud is for all objects to have spin-pole latitudes parallel to the ecliptic while the L5 cloud objects are all aligned to spin poles perpendicular to the ecliptic. We reject this solution as too unlikely. Although a difference in spin-pole distribution cannot by itself explain the difference in shape distribution, it is possible that it may be a contributing factor along with a stronger mechanism and as such we cannot rule out a difference in the spin-pole distributions of the two clouds.

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5.2.2. An Abundance of Slow Rotators

The spin-rate distribution is also a key component of the shape distribution model. If the spin-rate distribution assumed under-represents slow rotating asteroids then partial lightcurves for these objects may be mistaken for much lower amplitude objects with a shorter rotation period. For example, if the model spin-rate distribution allows a maximum of 10% of the population to be slow rotators and the real abundance is 20%, then the overflow, i.e., half of the slow rotators in the population, will be treated as low-amplitude average rotators. A significant proportion of these objects could result in a best-fit shape distribution skewed toward more spherical shapes than are really present in the population.

Assuming the more elongated L4 cloud to have a fixed spin-rate distribution we vary the distribution for the L5 cloud by artificially injecting a proportion of slow rotating objects (P > 100 h) into the model and obtaining the best-fit shape distribution in each case. Figure 8 shows how the derived shape distribution of the L5 cloud varies depending on the proportion of slow rotators assumed. Here, the spin-pole distribution of both clouds is kept constant. A excess in the proportion of slow rotators alone of 30–40% is required in the L5 cloud to bring both clouds to the same mean axis ratio. A proportion of slow rotators has been identified among Jupiter Trojans, with estimates from Szabó et al. (2017) and Ryan et al. (2017) suggesting a proportion of slow rotators (P > 100 h) of ∼15%. However, there has been no evidence of a disparity in this proportion between the two Trojan clouds (French et al. 2015). As this is a relatively large difference in the proportion of slow rotators we consider this to be unlikely. Previous work using the Transiting Exoplanet Satellite Survey (TESS) has shown a proportion of slow rotators in the main asteroid belt, though one much smaller than that required here (McNeill et al. 2019).

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5.2.3. Collisional Effects

If the difference in shape shown in the two clouds is a real difference in shape and not a function of differing rotational properties (spin-pole orientations, rotation periods) between the two clouds, this may imply a different collisional evolution within each cloud. The Trojan clouds are often considered to be a collisionless environment, but this is generally in reference to external collisions, i.e., collisions between a Trojan and an object from outside the population. Trojans can undergo collisions with Hilda objects, however, Trojan–Trojan collisions will dominate interactions. This is backed up by the presence of a collisional family in the L4 Trojan cloud dynamically linked to (3548) Eurybates (Brož & Rozehnal 2011). These Eurybates objects are likely to be C-type objects; however, there is only a very small proportion of these objects in our data set and these can be easily excluded.

The L4 cloud contains more objects than the L5, so it therefore follows that collisional interactions in this cloud will be more frequent. This difference may produce a systematic difference in shape distribution between the two populations. From Dell'Oro et al. (1998) we have Equation (1), which gives the expected timescale over which subcatastrophic collisions occur in each of the two Trojan clouds where _Pi is the intrinsic collision probability of each cloud, R is the radius of the target objects, and N(r) is the number of objects with a radius greater than r:

$$\begin{eqnarray}&&\tau =\displaystyle \frac{1}{{P}_{i}{R}^{2}N(r)}.\end{eqnarray} \tag{ 1 }$$

We calculate τ for projectile radii r larger than 1 km, corresponding to the lower completeness limit of the Trojan size distribution for each Trojan cloud as determined by Yoshida & Nakamura (2008).

We assume average impact velocities of 5.06 and 4.96 km s⁻¹ for the L4 and L5 clouds, respectively (Dell'Oro et al. 1998). We consider the case of a D > 2 km projectile colliding with a target of variable size. In each case we calculate the Trojan–Trojan collisional timescale for this object, and hence the number of collisions it would experience in 4 Gyr, if it were situated in both the L4 and L5 clouds. This is shown in Figure 9.

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The collisional timescale of objects in the L4 cloud is shorter than the L5 regardless of the size of the target simply due to the increased number density of the population. As the L4 cloud is more collisionally evolved, we conclude that if collisions are responsible for the shape difference between the two clouds then subcatastrophic collisions must make a population more elongated over time. Both Domokos et al. (2009) and Henych & Pravec (2015) simulated the effect of subcatastrophic collisions on the elongation of small asteroids (D < 20 km). They demonstrated that the cumulative effect of collisions should lead to an increase in the target objects elongation, occurring over shorter timescales at smaller sizes. However, it is worth noting that Henych & Pravec (2015) found that the estimated timescales for this process to occur are significantly longer than the collisional disruption timescales for the asteroids in question, a discrepancy not fully explained by the longer collisional timescales of objects in the Trojan clouds. Henych & Pravec (2015) simulate the time taken for an object to go from a 2:1 axis ratio to a 3:1 axis ratio due to erosion from subcatastrophic collisions. For a 10 km object, this is found to be of order 1 Gyr, a longer timescale than the collisional lifetime of such an object in the main belt. Due to the relatively lower number density of the Trojan clouds compared with the main belt, the collisional lifetime in this region will be longer. However, until a similar modeling work of subcatastrophic collisional erosion is carried out for the Trojan population it is impossible to reliably compare these timescales.

Work by Wong et al. (2014) and Wong & Brown (2015) discovered a dependence of the abundances of "less-red" Trojans and "red" Trojans with diameter. This effect showed that there were a greater proportion of "less-red" objects at small sizes. This trend is hypothesized to be collisional in nature. Wong et al. (2014) assume both Trojan sub-groups to have similar interior composition, catastrophic collisions of each type of object will produce spectroscopically identical resultant bodies which will resemble "less-red" objects. On a sufficient timescale this leads to the increasing abundance of less-red objects and the depletion of red objects. If the color difference observed is due to only the surface of "red" and "less-red" objects, due to e.g., space weathering, it follows that subcatastrophic collisions will also potentially produce a similar effect by exposing underlying material. The collisions considered in this paper are limited to impactors of D > 2 km due to the completeness of the known size distribution for Trojans. Smaller objects will contribute to this effect and when surveys carry out further observations of small Trojans to improve this size distribution this will represent an interesting avenue for future work in collisional modeling.

Using data from the ATLAS survey, we derive phase functions and c − o colors for 266 Trojans. The colors obtained here from ATLAS show different distributions of colors for the L4 cloud than the L5. We also present shape distributions derived for each of the Trojan clouds. The L4 population appears to show a more elongated shape distribution than the L5 cloud. We rule out that this difference could be solely a result of different spin-pole distributions between the two clouds. We also rule out a difference in the abundance of slow rotators (P > 100 h) between the two clouds as unrealistic. This leaves as the most plausible explanation the different collisional environments in the two clouds. The collisional timescale in the L4 cloud is shorter, and therefore this cloud is more collisionally evolved. We conclude that collisions may have made this population more elongated on average. This shape discrepancy should be confirmed using observations from The Rubin Observatory Legacy Survey of Space and Time (LSST) which will obtain observations for a larger population of Jupiter Trojans (approximately 280,000 over the lifetime of the survey; LSST Science Collaboration et al. 2009). Furthermore, comparisons of the apparent cratering records for L4 and L5 targets as measured by the Lucy mission will provide a good test of this proposed hypothesis.

We thank the anonymous referees for their input, which has led to significant improvement of this manuscript. Regent Innovation Fund. This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. ATLAS is primarily funded to search for near-Earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO search include images and catalogs from the survey area. The ATLAS science products have been made possible through the contributions of the University of Hawaii Institute for Astronomy, the Queen's University Belfast, the Space Telescope Science Institute, and the South African Astronomical Observatory. The authors thank Bill Bottke for discussions which improved this manuscript.

Facilities: ATLAS (Tonry et al. 2018).

Software: astropy (Astropy Collaboration et al. 2013).