Comparison of the physical properties of the L4 and L5 Trojan asteroids from ATLAS data

Jupiter has nearly 8000~known co-orbital asteroids orbiting in the L4 and L5 Lagrange points called Jupiter Trojan asteroids. Aside from the greater number density of the L4 cloud the two clouds are in many ways considered to be identical. Using sparse photometric data taken by the Asteroid Terrestrial-impact Last Alert System (ATLAS) for 863 L4 Trojans and 380 L5 Trojans we derive the shape distribution for each of the clouds and find that, on average, the L4 asteroids are more elongated than the L5 asteroids. This shape difference is most likely due to the greater collision rate in the L4 cloud that results from its larger population. We additionally present the phase functions and $c-o$ colours of 266~objects.


INTRODUCTION
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 January, 2020, the number of known Trojans listed by the Minor Planet Center is 7673, with Nakamura & Yoshida (2008) predicting a total of 10 5 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 post-migration orbit of Jupiter. This material is proposed to be similar in origin to current outer Solar System (e.g., Kuiper Belt Object) populations, although modelling 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 colour/spectral dichotomy between less red and more red populations) also show some difference between the clouds. Szabó et al. (2007) observe a colour-dependence on inclination in both clouds independent of size, and match a power law to both distributions. The difference in colour distribution between the clouds was explained as potentially due to the different number densities and could be solved by normalising 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 colour 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 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 colour measurements, phase functions and rotation periods derived for Trojans in ATLAS in Sections 4 and 5 respectively.

ASTEROID TERRESTRIAL-IMPACT LAST ALERT SYSTEM (ATLAS)
The photometry data used in this study originate from survey observations performed between 2015 and 2018 by the Asteroid Terrestrial-impact Last Alert System (ATLAS) 1 . 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 deg 2 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 420-650 nm and an "orange" or o-filter with a bandpass between 560-820 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) andTonry et al. (2018) . Although this AB photometric system uses only two, relatively wide filters the c − o colour 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).
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-andc-filter respectively. We also only considered objects that had at least one observation at a phase angle (Sun-Observer-Target angle), α, of 5 degrees or lower and phase angle coverage of at at least 6 degrees 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 1.3 • , and a median phase angle range of 9.6 • . 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.

SHAPE DISTRIBUTION MODEL
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 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 h to 24 h (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, 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 Kolmogorov-Smirnov 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.
In order 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 since 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 centre of the Gaussian distribution produce equally good fits. The mean elongation for these best fits remains consistent, however, and we favour it as a metric in this analysis.

Colours and Phase Curves
To determine the colours 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 since we have many observations for each object but those observations do not span a very large phase angle range (< 10 degrees, see Section 2) We also do not have any observations for most objects at a zero degree 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).
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 colour 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 colour is defined as the difference between the average of the median magnitudes. The uncertainty in the colour 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 colour values and uncertainty displayed in top-left corner of each plot and recorded in Table 1.

Rotation Periods
Rotation periods were extracted from the ATLAS data by generating Lomb-Scargle periodograms (Lomb 1976;Scargle 1982) of each target's o-filter photometric data (as there are more o measurements than c measurements). Targets that generated periodograms containing a peak with a false-alarm probability 10 −10 were flagged to have a potentially extractable rotation period. Both o-and c-filter data of those targets were folded with the period corresponding to strongest periodogram peak and visually inspected to ascertain the quality of the fold (for instance, we retained an extracted period if the periodogram-independent c-filter data also folded commensurately with the ofilter data). Using this methodology we report rotation periods for 16 L4 and 25 L5 Trojans of which 27 of those have previously reported periods in the Asteroid Light Curve Database 1 (LCDB; Warner et al. 2009, Updated 2020 June 26). Our extracted periods match 20 out of the 27 objects that had previously reported periods. Aliasing ambiguity in the extracted rotation period is a common problem with ATLAS data due to the diurnal cadence of the ATLAS observations which means that in most cases the ATLAS sampling frequency is lower than the rotation frequencies we are trying to resolve. In some cases in the ATLAS data, aliases of the actual rotation period can also have similar peak strengths in the periodograms and it becomes difficult to unambiguously extract a period. We extract the ±2-, ±1-, or ±0.5-day alias periods of the strongest periodogram derived period for the 7 Trojans where our period did not match the literature period and found that in all cases the LCDB period was one of these aliases. This effect acts in both directions, and our derived period is also then an alias of the LCDB period. As these 7 objects have multiple listings of the same period in the LCDB and have been assigned a quality code U = 3 (defined as an unambiguous period solution) we defer to the literature in these cases. In Table 2 we report the 41 periods we could extract and also show the LCDB periods. Where our period does not match the LCDB period we indicate our best matching alias period and the corresponding alias window.

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 emphasised 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 b a = 0.77±0.02 and for L5 we obtain a value b a = 0.86±0.02. Figure 3 shows the p-value obtained from the KS 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.
We considered the possibility of carrying out this modelling for individual families within the Trojans, however, the number of family members in this dataset is insufficient for this analysis. This represents a potential avenue for study when future surveys, e.g. LSST, come online.

Comparison of colours and phase curve parameters between L4 and L5
In general, both the L4 and L5 objects have c-o colours consistent with the expected D-type taxonomy's c-o colour (see Figure 5). However, the median colour of the L4 cloud appears at a slightly higher c-o colour (∼ 0.41 mag)   Figure 3 but now using the assumption of spin pole latitude θ = 50 • than that of the L5 cloud (∼ 0.39 mag) (compare the moving-average curves Figure 5). While the median colour of the L4 is slightly higher, the colour distribution of the L4 objects is much broader containing both redder objects and also objects with low C-or X-like c-o colours (i.e. objects with less-red or even neutral slopes). Using a twosample Kolmogorov-Smirnov test on the two distributions for suggests that the colours are not drawn from identical distributions but we cannot rule this out for slope parameter, G. (p-values: 0.01 for c-o colour, 0.71 for slope parameter G) We attemped to verify this colour discrepancy using data from the Sloan Digital Sky Survey (SDSS; Ivezic 1998). The colour-colour 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 colour analysis from SDSS as defined by Ivezić et al. (2002). Again in the case of the SDSS data, the KS test suggests that these two populations can not 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.  2008) observe a colour 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* colours 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 colour 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 colours for each group closer together, preventing the bimodality from being observed.

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.

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 can not rule out a difference in the spin pole distributions of the two clouds. 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.

An abundance of slow rotators
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).   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.

Collisional effects
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 P i 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: 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 4Gyr, if it were situated in both the L4 and L5 clouds. This is shown in Figure 9.
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. Since the L4 cloud is more collisionally evolved we conclude that if collisions are responsible for the shape difference between the two clouds then sub-catastrophic 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 object's 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 which is 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 sub-catastrophic 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 to the main belt, the collisional lifetime in this region will be longer. However, until a similar modelling work of sub-catastrophic 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 hypothesised to be collisional in nature. Wong et al. (2014) assume both Trojan subgroups 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 colour difference observed is due to only the surface of 'red' and 'less-red' objects, due to e.g. space weathering, it follows that sub-catastrophic 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 modelling.

CONCLUSIONS
Using data from the ATLAS survey, we derive phase functions and c − o colours for 266 Trojans. The colours obtained here from ATLAS show different distributions of colours 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.

Acknowledgments
We thank the anonymous referees for their input which has led to significant improvement of this manuscript. A.M. was supported by the Arizona Board of Regents' 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.