THE COLOR–MAGNITUDE DISTRIBUTION OF SMALL JUPITER TROJANS^*

After bias-subtracting the images, we flat-fielded them using the twilight flats taken at the start of the first night of observation. For every chip image (10 per exposure, 40 per observed field, 5880 for the entire survey), we calculated the astrometric solution by first creating a pixel position catalog of all the bright sources in the image. This was done using Version 2.11 of SExtractor (Bertin & Arnouts 1996), with the threshold for source detection set at a high value (typically 30 or higher, depending on the seeing, in units of the estimated background standard deviation). The source catalog was then passed to Version 1.7.0 of SCAMP (Bertin 2006), which matches objects in the source catalog with those in a reference catalog of stars and computes the astrometric projection parameters for each chip image; we used the 9th Data Release of the Sloan Digital Sky Survey (SDSS DR9) as our reference catalog. In order to assess the quality of the astrometric solution from SCAMP, we compared the corrected on-sky position of stars in each image with the position of matched reference stars and found typical residual rms values much less than 01. Likewise, we calculated the position scatter between matched stars from pairs of images taken in the same observing field and found typical rms values less than 005.

Next, we passed the distortion-corrected images through SExtractor again, this time setting the detection threshold at 1.2 times the background standard deviation; we also required detected sources to consist of at least two adjacent pixels with pixel values above the detection threshold. These conditions were chosen to minimize the detection of faint non-astrophysical sources in the background noise as well as the loss of possible moving objects of interest. The resulting list contains the right ascension (R.A.) and declination (decl.) of all detected objects within each image. To search for moving object candidates, we fit orbits through sets of four source positions, one from each of the four images in an observing field, using the methods described in Bernstein & Khushalani (2000). A source was flagged as a moving object candidate if the resulting χ² value from the fit was less than 10. We reduced the number of non-Trojan moving object candidates flagged in this procedure by only considering sets of source positions consistent with apparent sky motion $| v|$ less than $25^{\prime\prime} /\mathrm{hr}.$ For comparison, typical sky motions of known Trojans during the time of our observations lie in the range $14\leqslant | v| \lt 22^{\prime\prime} /\mathrm{hr}.$

Each moving object candidate was verified by aligning and blinking 100 × 100 pixel stamps clipped from each of the four images in the vicinity of the object. We rejected all non-asteroidal moving object candidates (e.g., sets of four sources that were flagged by the previous orbit-fitting procedure, but that included cosmic ray hits and/or anomalous chip artifacts). We also rejected asteroids that passed in front of background stars, coincided with a cosmic ray hit in one or more image, or otherwise traversed regions on the chip that would result in unreliable magnitude measurements. These objects numbered around 10, of which only 2 had on-sky positions and apparent sky motions consistent with Trojans (see Section 2.2). Therefore, the removal of these objects from our Trojan data set is not expected to have any significant effect on the results of our analysis. Through the procedures described above, we arrived at an initial set of 1149 moving objects.

Observations were taken when the L4 point was near opposition, where the apparent sky motion of an object, $| v| ,$ is roughly inversely related to the heliocentric distance. Since the short observational arc of ∼70 minutes prevented an accurate determination of the heliocentric distance from orbit-fitting, we resorted to using primarily sky motion to distinguish Trojans from Main Belt Asteroids and Hildas. Figure 2 shows the distribution of apparent velocities in R.A. and decl. for all numbered minor bodies as calculated from ephemerides generated by the JPL HORIZONS system for UT 2014 February 27 12:00 (roughly the middle of our first night of observation). We have shown only objects with on-sky positions in the range of our observing fields ($130^\circ \leqslant \lambda \lt 190^\circ$ and $-8^\circ \leqslant \beta \lt 8^\circ$). A large majority of the objects with $| v| \lt 20^{\prime\prime} /\mathrm{hr}$ are Trojans, while the relative portion of non-Trojan contaminants increases rapidly in the range $20^{\prime\prime} /\mathrm{hr}]\leqslant | v| \lt 22^{\prime\prime} /\mathrm{hr}.$ We made an initial cut in angular velocity to consider only objects in the range $14\leqslant | v| \lt 22^{\prime\prime} /\mathrm{hr}.$

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The contamination rate varies across the surveyed area as a function of apparent sky motion, as well as position on the sky. We carried out a more detailed contamination analysis by applying a rough grid in the space of $| v| ,\;$λ, and β spanning the range $14\leqslant | v| \lt 22^{\prime\prime} /\mathrm{hr},\;$ $130^\circ \leqslant \lambda \lt 190^\circ ,$ and $-8^\circ \leqslant \beta \lt 8^\circ .$ We then binned all numbered objects contained in the JPL HORIZONS system into this three-parameter space and computed the Trojan fraction $\gamma \;\equiv$ [numbered Trojans]/[all numbered objects] in each bin. For bins containing no numbered objects, we assigned a value γ = 0. Figure 3 shows the results of this analysis. We note that while the non-Trojan contamination rate over the whole surveyed area among objects with apparent sky motions between 20 and $22^{\prime\prime} /\mathrm{hr}$ is high, there are regions in the sky where the Trojan fraction is 1. Similarly, for objects with $| v| \lt 20^{\prime\prime} /\mathrm{hr},$ there are regions near the edges of the ecliptic longitude space covered by our survey where the non-Trojan contamination rate is very high.

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In this paper, our operational Trojan data set includes only those objects detected in regions of the $(\lambda ,\beta ,| v| )$ space with Trojan fraction $\gamma =1$ (i.e., zero expected contamination). The resulting data set contains 557 Trojans. Choosing a slightly more lenient Trojan selection criterion (e.g., including objects detected in bins that do not contain numbered objects, where the Trojan fraction is technically unknown) was not found to significantly affect the main results of our magnitude and color distribution analysis.

The apparent magnitude m of an object detected in our images is given by $m={m}_{0}-2.5{\mathrm{log}}_{10}(f)={m}_{0}+{m}_{s},$ where f is the measured flux, m₀ is the zero-point magnitude of the corresponding chip image, and the survey magnitude has been defined as ${m}_{s}\equiv -2.5{\mathrm{log}}_{10}(f).$ The flux of each object was calculated by SExtractor through a fixed circular aperture with a diameter of 5 pixels. We computed the zero-point magnitude for each chip image by matching all bright, non-saturated sources detected by SExtractor to reference stars in the SDSS DR9 catalog and fitting a line with slope one through the points $({m}_{s},m).$ The maximum allowed position difference for a match between image and reference stars was set at 05, which resulted in an average of ∼150 matched stars per chip image. We used Sloan g-band and i-band reference star magnitudes to calibrate images taken in the g' and i' filters of Suprime-Cam, respectively. Consequently, the calculated apparent magnitudes of our detected Trojans are effectively Sloan g and i magnitudes, which greatly facilitates the comparison of our computed colors with those derived for Trojans in the SDSS Moving Object Catalog (see Section 3.2).

Both the error in the zero-point magnitude σ₀ and the error in the measured flux σ_f (reported by SExtractor) contribute to the error in the apparent magnitude σ, which we calculate using standard error propagation methods: $\sigma =\sqrt{{\sigma }_{0}^{2}+{(2.5({\sigma }_{f}/f)/\mathrm{ln}(10))}^{2}}.$ We set the g magnitude of each Trojan detected in our survey to be the error-weighted mean of the apparent magnitudes calculated from the two g-' Suprime-Cam images, i.e., $g=\left({\displaystyle \sum }_{k=1}^{2}{m}_{g,k}/{\sigma }_{g,k}^{2}\right)/\left({\displaystyle \sum }_{k=1}^{2}1/{\sigma }_{g,k}^{2}\right),$ with the corresponding uncertainty in the g magnitude defined by ${\sigma }_{g}^{2}=1/\left({\displaystyle \sum }_{k=1}^{2}1/{\sigma }_{g,k}^{2}\right);$ the i magnitude and uncertainty of each Trojan were defined analogously.

As mentioned previously in Section 2.2, orbit-fitting over the short observational arcs prevented us from precisely determining the orbital parameters of the detected objects. Therefore, we did not directly convert the apparent magnitudes to absolute magnitudes using the best-fit heliocentric distances. Instead, we considered the difference between apparent V-band magnitude and absolute magnitude H of known Trojans, as computed by the JPL HORIZONS system for the time of our observations. By fitting a linear trend through the computed apparent sky motions $| v|$ as a function of the magnitude difference values, we obtained an empirical conversion between the sky motion and V − H. Since V − H depends also on the viewing geometry, we divided the ecliptic longitude range $130^\circ \leqslant \lambda \lt 190^\circ$ into 5^◦ bins and derived linear fits separately for known Trojans within each bin. We calculated the apparent V-band magnitudes of the detected Trojans from our survey using the empirical mean colors g − r = 0.55 and V − r = 0.25 reported in Szabó et al. (2007). Then, we translated these V values to H using the measured sky motions and our empirical V − H conversions.

From the absolute magnitude, the diameter of a Trojan can be estimated using the relation $D=1329\times {10}^{-H/5}/\sqrt{{p}_{v}},$ where D is in units of kilometers, and we assumed a uniform geometric albedo of p_v = 0.04 (Fernández et al. 2009). The brightest Trojan we detected in our survey has H = 10.0, corresponding to a diameter of 66.5 km, while the faintest Trojan has H = 20.3, corresponding to a diameter of 0.6 km.

An analysis of the magnitude distribution of a population can be severely affected by detection incompleteness. Varying weather conditions during ground-based surveys lead to significantly different detection completeness limits for observations taken at different times, resulting in a non-uniform data set and biases in the overall magnitude distribution. Since we required a moving object candidate to be detected in all four images taken in an observed field, the epoch with the highest seeing in each set of four exposures determined the threshold magnitude to which the survey was sensitive in that particular field. Here we set the seeing value of each chip image to be the median of the full-width half-max of the point-spread function for all non-saturated stars, as computed by SExtractor. If the highest seeing value across a four-image set was high, then the object detection pipeline would have missed many of the fainter objects positioned within the field. This is because the signal-to-noise of objects of a given magnitude located on the worst image of the set would be significantly lower than that of the same objects located on an image taken at good seeing. As a result, the magnitude distribution of objects detected in an observing field with high-seeing images would be strongly biased toward brighter objects, and when combined with the full body of data, would affect the overall magnitude distribution.

Figure 4 shows the highest measured seeing for each group of four chip images containing a detected Trojan, plotted with respect to the time of first epoch. A large portion of our first night of observation was plagued by very high seeing—exceeding 30 at times. We chose a cutoff seeing value of 12 and defined a filtered Trojan set containing only Trojans detected in images with seeing below this value.

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We used signal-to-noise to establish a limiting magnitude for our magnitude distribution analysis. Instead of attempting to generate an empirical model of the detection efficiency for the faintest objects, we elected to stipulate a conservative S/N threshold of 8. We defined the upper magnitude limit of our filtered Trojan data set to be the magnitude of the brightest object (detected at seeing less than 12) with ${\rm{S}}/{\rm{N}}\lt 8.$ The limiting magnitude was determined to be H = 16.4. The final filtered Trojan set contains 150 objects, and when fitting for the total magnitude distribution, we assume that this set is complete (i.e., is not characterized by any size-dependent bias).

The color c of each Trojan is defined as the difference between the g and i magnitudes: $c\equiv g-i.$ When calculating the uncertainty of each color measurement, we must consider the effect of asteroid rotation in addition to the contribution from photometric error. The oscillations in apparent amplitude seen in a typical asteroidal rotational light curve arise from the non-spherical shape of the object and peak twice during one full rotation of the asteroid (see, for example, the review by Pravec et al. 2002). The average observational arc of ∼70 minutes for each object may correspond to a significant fraction of a characteristic light curve oscillation period, which can consequently lead to a large variation in the apparent brightness of an object across the four epochs.

In the absence of published light curves for faint Trojans, we must develop a model of the rotational contribution to the color uncertainty in order to accurately determine the total uncertainty of our color measurements. We took advantage of the fact that we obtained two detections of an object in each filter to estimate the effect of asteroid rotation empirically. We considered the difference between the two consecutive g'-band magnitude measurements ${\rm{\Delta }}g\equiv {g}_{1}-{g}_{2}.$ The standard deviation in Δg values, σ_Δg, contains a contribution from the photometric error given by the quadrature sum of the individual magnitude uncertainties as defined in Section 2.3: ${\sigma }_{{\rm{\Delta }}g,\mathrm{phot}}=\sqrt{{\sigma }_{g,1}^{2}+{\sigma }_{g,2}^{2}}.$ Figure 5 compares the standard deviation of Δg values measured in 0.5 mag bins (blue squares) to the corresponding error contribution from the photometric uncertainty only, ${\sigma }_{{\rm{\Delta }}g,\mathrm{phot}}$ (black crosses).

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From the plot, it is evident that an additional contribution attributable to rotation is needed to account for the uncertainty in Δg seen in our data. We observe that the discrepancy between σ_Δg and the photometric error contribution is large at low magnitudes, where the photometric error is small, and decreases with increasing magnitude and increasing photometric error. This overall trend suggests that the total error in Δg can be empirically modeled as a quadrature sum of the photometric contribution and the rotational contribution, i.e., ${\sigma }_{{\rm{\Delta }}g}=\sqrt{{\sigma }_{{\rm{\Delta }}g,\mathrm{phot}}^{2}+{\sigma }_{{\rm{\Delta }}g,\mathrm{rot}}^{2}},$ where the rotational contribution is magnitude-invariant. We found that a rotational contribution of ${\sigma }_{{\rm{\Delta }}g,\mathrm{rot}}\sim 0.08$ gives a good match with the measured standard deviation in Δg, and likewise for ${\rm{\Delta }}i.$ In Figure 5, the green squares show the binned Δg standard deviation values calculated from our empirical model combining both photometric and rotational contributions.

The rotational contribution to the color uncertainty was estimated as a quadrature sum of the rotational contributions to Δg and Δi: ${\sigma }_{c,\mathrm{rot}}\sim 0.11\;\mathrm{mag}$. The total color uncertainty is therefore given by:

$$\begin{eqnarray}&&{\sigma }_{c}=\sqrt{{\sigma }_{g}^{2}+{\sigma }_{i}^{2}+{\sigma }_{c,\mathrm{rot}}^{2}},\end{eqnarray} \tag{ 1 }$$

where σ_g and σ_i are the errors in the g and i magnitudes derived from the flux and calibrated zero-point magnitude errors only (see Section 2.3).

We note that the effects of bad seeing and detection incompleteness discussed in Section 2.4 are not expected to affect the bulk properties of the Trojan color distribution (e.g., mean color), except at the faintest magnitudes, where there is a slight bias toward redder objects. This is because we used the measured g magnitudes to convert to absolute magnitudes H. For objects with a given g magnitude, redder objects are slightly brighter in the i'-band (lower i magnitude) than less-red objects. The predicted effect is small, since the characteristic difference in g − i color between objects in the red and less-red populations is only ∼0.15 mag (see Section 3.2). In our color analysis, we strove to avoid this small color bias by limiting the analysis to objects with H magnitudes less than 18, thereby removing the faintest several dozen objects from consideration.

The cumulative magnitude distribution N(H) of all objects in the filtered Trojan set (Section 2.4) as a function of H magnitude is shown in Figure 6. The main feature of the magnitude distribution is a slight rollover in slope to a shallower value at around H ∼ 15. Following previous analyses of the magnitude distribution of Trojans (e.g., Jewitt et al. 2000), we fit the differential magnitude distribution, ${\rm{\Sigma }}(H)={dN}(M)/{dH},$ with a broken power law

$$\begin{eqnarray}{\rm{\Sigma }}({\alpha }_{1},{\alpha }_{2},{H}_{0},{H}_{b}| H)=\left\{\begin{array}{ll}{10}^{{\alpha }_{1}(H-{H}_{0})}, & {\rm{for}}\;{\rm{}}H\lt {H}_{b}\\ {10}^{{\alpha }_{2}H+({\alpha }_{1}-{\alpha }_{2}){H}_{b}-{\alpha }_{1}{H}_{0}}, & {\rm{for}}\;{\rm{}}H\geqslant {H}_{b},\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where the power law slope for brighter objects α₁ changes to a shallower faint-end slope α₂ at a break magnitude H_b. The parameter H₀ defines the threshold magnitude for which ${\rm{\Sigma }}({H}_{0})=1.$

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We fit the model curve in Equation (2) to our filtered Trojan magnitude distribution using a maximum likelihood method similar to the one used in Fraser et al. (2008) for their study of Kuiper Belt objects. We defined a likelihood function L that quantifies the probability that a random sampling of the model distribution will yield the data:

$$\begin{eqnarray}&&L({\alpha }_{1},{\alpha }_{2},{H}_{0},{H}_{b}| {H}_{i})\propto {e}^{-N}\displaystyle \prod _{i}{P}_{i}.\end{eqnarray} \tag{ 3 }$$

Here, H_i is the H magnitude of each detected Trojan and ${P}_{i}={\rm{\Sigma }}({\alpha }_{1},{\alpha }_{2},{H}_{0},{H}_{b}| {H}_{i})$ is the probability of detecting an object i with magnitude H_i given the underlying distribution function Σ. N is the total number of objects detected in the magnitude range under consideration and is given by

$$\begin{eqnarray}&&N={\displaystyle \int }_{-\infty }^{{H}_{\mathrm{max}}}\eta (H){\rm{\Sigma }}({\alpha }_{1},{\alpha }_{2},{H}_{0},{H}_{b}| H)\;{dH},\end{eqnarray} \tag{ 4 }$$

where in the case of our filtered Trojan set we have ${H}_{\mathrm{max}}=16.4.$ We have included the so-called "efficiency" function $\eta (H)$ that represents an empirical model of the incompleteness in a given data set and ensures that the best-fit distribution curve corrects for any incompleteness in the data. Our filtered Trojan data set was defined to remove incompleteness contributions from both bad seeing and low signal-to-noise, so when fitting the broken power law to the corresponding magnitude distribution, we set $\eta =1.$

We used an affine-invariant Markov Chain Monte Carlo (MCMC) Ensemble sampler (Foreman-Mackey et al. 2013) with 50,000 steps to estimate the best-fit parameters and corresponding 1σ uncertainties. The magnitude distribution of the filtered Trojan set is best-fit by a broken power law with parameter values ${\alpha }_{1}=0.45\pm 0.05,\;$ ${\alpha }_{2}={0.36}_{-0.09}^{+0.05},\;$ ${H}_{0}={11.39}_{-0.37}^{+0.31},\;$ and ${H}_{b}={14.93}_{-0.88}^{+0.73}.$ The cumulative magnitude distribution for this best-fit model is plotted in Figure 6 as a solid black line. We note that while the difference between the two power-law slopes is small, it is statistically significant: Marginalizing over the slope difference ${\rm{\Delta }}\alpha ={\alpha }_{1}-{\alpha }_{2},$ we obtained ${\rm{\Delta }}\alpha ={0.10}_{-0.08}^{+0.09},$ which demonstrates that the difference between the two power-law slopes is distinct from zero at the 1.25σ level.

We also fit the magnitude distribution of known bright Trojans contained in the MPC catalog by repeating the analysis described in Wong et al. (2014), this time including only L4 Trojans. Fitting the distribution of bright L4 Trojans likewise to a broken power law of the form shown in Equation (2), we obtained a best-fit distribution function with ${\alpha }_{0}^{\prime }={0.91}_{-0.16}^{+0.19},\;$ ${\alpha }_{1}^{\prime }=0.43\pm 0.02,\;$ ${H}_{0}^{\prime }={7.22}_{-0.25}^{+0.24},$ and ${H}_{b}^{\prime }={8.46}_{-0.54}^{+0.49}.$ We have denoted the best-fit parameters for the MPC Trojan magnitude distribution with primes to distinguish them from the best-fit parameters for the Subaru Trojans; the subscripts on the slope parameters α are adjusted to reflect the magnitude ranges they correspond to in the overall distribution. The cumulative magnitude distribution of MPC L4 Trojans through H = 12.3 (corrected for catalog incompleteness following the methodology described in Wong et al. 2014) is shown in Figure 7 along with the best-fit curve.

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Earlier studies of L4 Trojans in this size range have reported power law slopes that are consistent with our values: Yoshida & Nakamura (2008) fit MPC L4 Trojans with magnitudes in the range 9.2 < H < 12.3 and derived a slope of 0.40 ± 0.02. We note that their fit did not take into account the incompleteness in the MPC catalog, which we estimated in Wong et al. (2014) to begin at H = 11.3. Therefore, the slope value in Yoshida & Nakamura (2008) is somewhat underestimated. Jewitt et al. (2000) carried out a survey of L4 Trojans and computed slopes of 0.9 ± 0.2 and 0.40 ± 0.06 over size ranges corresponding to our reported ${\alpha }_{0}^{\prime }$ and ${\alpha }_{1}^{\prime }$ slopes, respectively. Our best-fit power law slopes calculated for bright MPC L4 Trojans are consistent with these previously published values at better than the 1σ level.

We combined the magnitude distributions of faint Subaru Trojans and brighter cataloged Trojans to arrive at the overall magnitude distribution of L4 Trojans, shown in Figure 7, where the cumulative absolute magnitude distribution of faint Trojans was approximately scaled to match the overall number of MPC Trojans at H ∼ 12. The error bars indicate 95% confidence bounds derived from Poisson errors on the Subaru survey data and reflect the uncertainty associated with scaling up the survey magnitude distribution to approximate the magnitude distribution of the total L4 population. The combined Subaru and MPC data sets cover the entire magnitude range from H = 7.2 to a limiting magnitude of ${H}_{\mathrm{max}}=16.4.$ The best-fit power law distribution slopes in the region containing the overlap between the MPC and Subaru data sets (${\alpha }_{1}^{\prime }=0.43\pm 0.02$ and ${\alpha }_{1}=0.45\pm 0.05$) are statistically equivalent, indicating that the magnitude distribution between H ∼ 10 and H ∼ 15 is well-described by a single power law slope. The overall magnitude distribution is characterized by three distinct regions: the brightest L4 Trojans have a power law magnitude distribution with slope α₀ = 0.91. At intermediate sizes, the magnitude distribution rolls over to a slope of α₁ ∼ 0.44. Finally, the faintest objects detected in our Subaru survey are characterized by an even shallower magnitude distribution slope of α₂ = 0.36. These three regions are separated at the break magnitudes ${H}_{b}^{\prime }={8.46}_{-0.54}^{+0.49}$ and ${H}_{b}={14.93}_{-0.88}^{+0.73},$ which correspond to Trojans of size ${135}_{-27}^{+38}\;\mathrm{km}$ and ${7}_{-2}^{+3}\;\mathrm{km}$, respectively.

In a previous study, Yoshida & Nakamura (2005) detected 51 faint L4 Trojans near opposition using the Suprime-Cam instrument and found the magnitude distribution to be well-described by a broken power law with a break at around H ∼ 16 separating a brighter-end slope of 0.48 ± 0.02 from a shallower faint-end slope of 0.26 ± 0.02. The methods used in the Yoshida & Nakamura (2005) fits differed from ours in several ways: the data was binned prior to fitting, and the two slopes were determined from independent fits of the bright and faint halves of their data. To better compare the distribution of Trojans studied by Yoshida & Nakamura (2005) with the best-fit distribution we derived from our Subaru survey, we reanalyzed their data using the techniques described in this paper. Fitting all of the Yoshida & Nakamura (2005) magnitudes through H = 17.9 (90% completeness limit) to a broken power-law, we obtained the two slopes ${\alpha }_{1,{\rm{Y}}\&amp;{\rm{N}}}={0.44}_{-0.06}^{+0.07}$ and ${\alpha }_{2,{\rm{Y}}\&amp;{\rm{N}}}={0.26}_{-0.04}^{+0.06}$ and a roll-over at ${H}_{b,{\rm{Y}}\&amp;{\rm{N}}}={15.11}_{-1.02}^{+0.89}.$ These values are consistent with the corresponding best-fit values for α₁, α₂, and H_b from the analysis of our Subaru survey data at better than the 1σ level.

Previous spectroscopic and photometric studies of Trojans have noted bimodality in the distribution of various properties, including the visible (Szabó et al. 2007; Melita et al. 2008; Roig et al. 2008) and near-infrared (Emery et al. 2011) spectral slope, as well as the infrared albedo (Grav et al. 2012). Wong et al. (2014) demonstrated that the bimodal trends were indicative of two separate populations within the Trojans, which are referred to as the less-red and red populations, in accordance with their relative colors. It was further shown that the magnitude distributions of these two color populations are distinct, with notably different power-law slopes in the magnitude range H ∼ 9.5–12.3.

We repeated the color population analysis presented in Wong et al. (2014), using only L4 Trojans. Objects were categorized as less-red or red primarily based on their visible spectral slopes, which were derived from the g, r, i, and z magnitudes listed in 4th release of the SDSS Moving Object Catalog (SDSS-MOC4). We estimated the incompleteness in color categorization via the fraction of objects in each 0.1 mag bin that we were able to categorize as either less-red or red; when fitting the magnitude distributions of the color populations up to a limiting magnitude of H = 12.3, the categorization incompleteness was factored into the efficiency function η along with the estimated incompleteness of the MPC catalog. See Wong et al. (2014) for a complete description of the methodology used.

The magnitude distributions of the color populations were fit using the same techniques that we applied in the total magnitude distribution fits in Section 3.1. The red population magnitude distribution is best-fit by a broken power law with ${\alpha }_{1}^{{\rm{R}}}={0.84}_{-0.16}^{+0.22},{\alpha }_{2}^{{\rm{R}}}={0.33}_{-0.03}^{+0.04},{H}_{0}^{{\rm{R}}}={7.30}_{-0.26}^{+0.30},$ and ${H}_{b}^{{\rm{R}}}={8.82}_{-0.44}^{+0.35}.$ The less-red population has a magnitude distribution more consistent with a single power law: ${\rm{\Sigma }}({\alpha }_{1},{H}_{0}| H)={10}^{{\alpha }_{1}(H-{H}_{0})};$ here we computed the following best-fit parameters: ${\alpha }_{1}^{\mathrm{LR}}={0.61}_{-0.06}^{+0.07}$ and ${H}_{0}^{\mathrm{LR}}={8.18}_{-0.29}^{+0.31}.$ Figure 8 shows the cumulative magnitude distributions for the less-red and red populations, scaled to correct for incompleteness, along with the best-fit curves. The error bars indicate the 95% confidence bounds derived from the incompleteness correction. The best-fit slopes calculated above for the L4 color populations are consistent within the errors to the corresponding values in Wong et al. (2014) derived from the color analysis of both L4 and L5 Trojans.

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To determine whether the previously studied bimodality in color among the brighter Trojans carries through to the fainter Trojans from our Subaru observations, we constructed a histogram of the g − i color distribution for all Trojans contained in the SDSS-MOC4 catalog and compared it to the histogram of g − i colors for Trojans detected in our Subaru survey. As discussed in Section 2.4, the bulk distribution of colors is not expected to be affected by detection incompleteness due to bad seeing, and we include all Trojans brighter than H = 18. The two histograms are plotted in Figure 9. The bimodality in the color histogram of brighter SDSS-MOC4 objects is evident. We fit a two-peaked Gaussian to the color distribution of brighter SDSS-MOC4 Trojans and found that the less-red and red populations have mean g − i colors of ${\mu }_{1}=0.73$ and ${\mu }_{2}=0.86,$ respectively.

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On the other hand, while there is some asymmetry in the color distribution of faint Trojans detected by our Subaru survey, there is no robust bimodality. This can be mostly attributed to the large contribution of asteroid rotation to the variance in the color measurements, the magnitude of which is comparable to the difference between the mean red and less-red colors (see Section 2.5). As such, it is not possible to categorize individual Trojans from our Subaru observations into the less-red and red populations based on their colors and construct magnitude distributions of the color populations, as was done in Wong et al. (2014) for the brighter SDSS-MOC4 Trojans.

Instead, we considered bulk properties of the distribution. In particular, we calculated the mean g − i color as a function of H magnitude for the combined set of SDSS-MOC4 Trojans and faint Trojans from our Subaru observations in order to assess whether the resulting trend is consistent with extrapolation of the best-fit color–magnitude distributions obtained previously for the brighter objects (i.e., curves in Figure 8). Here, we assumed that the mean colors of the two color populations are invariant across all magnitudes. The mean g − i color and uncertainty in the mean for the data were computed in 1 mag bins and are plotted in Figure 10 in blue. We see that the mean color is consistent with a monotonically decreasing trend with increasing magnitude, or equivalently, decreasing size. To derive the extrapolated mean color values from the best-fit color–magnitude distributions, we calculated the expected mean color at each bin magnitude as a weighted mean ${c}^{\prime }(H)=({\mu }_{1}\times {{\rm{\Sigma }}}^{\mathrm{LR}}(H)+{\mu }_{2}$ × ${{\rm{\Sigma }}}^{{\rm{R}}}(H))/({{\rm{\Sigma }}}^{\mathrm{LR}}(H)$ + ${{\rm{\Sigma }}}^{{\rm{R}}}(H)),$ where ${\mu }_{\mathrm{1,2}}$ are the mean g − i colors of the less-red and red populations, respectively, as derived previously from fitting the color distribution of SDSS-MOC4 Trojans. The functions ${{\rm{\Sigma }}}^{\mathrm{LR}}(H)$ and ${{\rm{\Sigma }}}^{{\rm{R}}}(H)$ are the best-fit differential magnitude distributions for the less-red and red populations presented earlier. The resulting model mean color values are denoted in Figure 10 by red dots.

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The extrapolated mean colors show very good agreement with the mean colors derived from the data. This suggests that the best-fit magnitude distributions for the less-red and red color populations shown in Figure 8 continue past the limiting magnitude of the MPC data analysis (H = 12.3) and likely extend throughout most of the magnitude range covered by our Subaru observations. We conclude that the fraction of less-red objects in the overall L4 Trojan population increases steadily with increasing magnitude (decreasing size), and that L4 Trojans fainter than H ∼ 16, or equivalently, smaller than ∼4 km in diameter, are almost entirely comprised of less-red objects.