Small Jupiter Trojans Survey with the Subaru/Hyper Suprime-Cam^*

The data are processed with the HSC data reduction/analysis pipeline hscPipe (version 3.8.5), developed by the HSC collaboration team based on the Large Synoptic Survey Telescope (LSST) pipeline software (Ivezić et al. 2008; Axelrod et al. 2010). We run only the single-frame processing, i.e., the data processes for each exposure, including image correction, source detection, measurement, and calibration. First, the images are reduced by standard procedures such as bias subtraction, trimming of the overscan/prescan regions, flat fielding, defect removal, and sky background subtraction. Next, the pipeline detects sources from the corrected images; performs source measurements of centroids, shapes, and photometry with several methods; and compares the coordinates and fluxes with the matched objects in a reference catalog for astrometric and photometric calibration. We use the SDSS DR9 catalog with the source fluxes corrected by the data of the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) 1 (PS1) survey (Schlafly et al. 2012; Tonry et al. 2012; Magnier et al. 2013). The PS1 r magnitude (${r}_{\mathrm{PS}1}$) is converted into the HSC r magnitude (${r}_{\mathrm{HSC}}$) using a color term equation determined by the HSC software team as

$$\begin{eqnarray}{r}_{\mathrm{HSC}} & = & {r}_{\mathrm{PS}1}+0.00280+0.02094\,({r}_{\mathrm{PS}1}-{i}_{\mathrm{PS}1})\\ & & -0.01878\,{({r}_{\mathrm{PS}1}-{i}_{\mathrm{PS}1})}^{2}.\end{eqnarray} \tag{ 1 }$$

Finally, hscPipe estimates the astrometric solution and photometric zero point (in the AB magnitude system) and creates a source catalog with the measured values (e.g., centroids, shapes, fluxes) for each CCD image.

We used the source catalogs produced by hscPipe for the mechanical detection of moving objects. We developed an algorithm allowing us to efficiently extract moving objects from the source catalogs with the following procedures: (i) removing suspected cosmic rays and saturated sources based on the flags added by the pipeline processes, (ii) excluding the sources that have identical coordinates among three visits as stationary objects, and (iii) searching for combinations of sources from each visit whose positions have a pattern corresponding to uniform linear motion on the sky.

For selective capture of JTs and Hildas, we limit the motion range to be searched to cover the motion distribution derived from orbits with an eccentricity of 0.0–0.3 and inclination of $0^\circ$–40° at the survey field, as shown by the area enclosed by the dotted black lines in Figure 2. We regard the detected moving objects with motion vectors located in this area as JT/Hilda candidates. Finally, all of the images of candidate sources are visually inspected.

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The apparent velocities along the ecliptic longitude/latitude of the detected JT/Hilda candidates are plotted in Figure 2. We overplotted motion distributions of synthetic orbits for JTs and Hildas randomly generated from probability distributions based on the orbital distributions of known objects obtained from the Minor Planet Center (MPC) database⁴ in Figure 2. One can see that the detected objects are divided into two swarms and the motion distribution well matches that of the synthetic JT/Hilda objects. We defined the boundary between these two groups as motions of circular orbits with a semimajor axis of 4.5 au (dashed red line). The slower group corresponding to JTs consists of 631 objects, while the faster group corresponding to Hildas consists of 130 objects.

In this paper, we focus on the investigation of the size distribution of JTs. The analysis for the Hilda candidates will be reported in another paper (T. Terai & F. Yoshida 2017, in preparation).

As mentioned in the previous section, hscPipe measures various parameters of the detected sources, including centroid positions. However, the centroid for a moving object with an elongated shape may be inaccurate because the barycenter of the intensity profile is sensitive to an instability of seeing and/or transparency during the exposure. Therefore, instead of using the hscPipe measurement, we independently determine the center positions of the detected JTs by making a ${\chi }^{2}$ fitting of object models to the image data. The model is generated from integration of Gaussian profiles with the center shifting with a constant motion equal to the measured velocity, as shown in Figure 3. The FWHM of the individual Gaussian profile is given as the typical seeing size of the image.

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Using the fixed center position, we measure the total flux of each object by aperture photometry with the sinc-interpolation technique developed by Bickerton & Lupton (2013). The aperture flux is computed from a weighted sum of pixel values over an aperture A as

$$\begin{eqnarray}&&{f}_{A}=\displaystyle \sum _{i,j}\ {w}_{{ij}}\ {p}_{{ij}},\end{eqnarray} \tag{ 2 }$$

where p_ij is the discretely sampled flux given from a continuous flux function $p(x,y)$ and w_ij is the weighting term represented by

$$\begin{eqnarray}&&{w}_{{ij}}=\int {\int }_{A}\ \displaystyle \frac{\sin (\pi (x-{x}_{i}))}{\pi (x-{x}_{i})}\ \displaystyle \frac{\sin (\pi (y-{y}_{j}))}{\pi (y-{y}_{j})}\ {dxdy}.\end{eqnarray} \tag{ 3 }$$

The aperture is formed into the shape of a trailed image with a radius of 20 with the velocity of each moving object (see Figure 3). If there is a star/galaxy within the aperture area, we perform the same photometry again after background subtraction with a reference image of another visit via point-spread function matching. The measured flux is converted into apparent magnitude using the photometric zero point estimated by hscPipe. The representative magnitude is determined as the weighted mean of the values at all visits with the photometric errors.

As shown in Table 1, the time intervals of the exposures are ∼18 minutes and 36–56 minutes between the first and second and the second and third visits, respectively. The brightness variation due to asteroid rotation could cause an additional uncertainty in the apparent magnitude measurement. To assess the effect of asteroid rotation, we estimated the magnitude differences among the visits, ${\rm{\Delta }}{m}_{r}={m}_{r,i+1}-{m}_{r,i}\ (i=1,2)$, where ${m}_{r,i}$ is the r-band magnitude at the ith visit, for each object as an index representing both the photometric and rotational contributions (Wong & Brown 2015). As performed in WB2015, the standard deviations of ${\rm{\Delta }}{m}_{r}$ contained in 0.5 mag bins of absolute magnitude (see Section 4.1) were compared with the medians of the photometric errors of the same objects. We found that there is no significant difference between the two values in most of the bins, indicating a rotational contribution of ∼0.05 mag or less. Since this is much smaller than the uncertainty of the estimated heliocentric distance corresponding to 0.08 mag for absolute magnitude (see Section 3.4), the size distribution is likely not affected by the rotational contribution. Thus, we exclude the effect of rotational magnitude variation from consideration in this survey.

The observation arcs of the detected JTs are less than 80 minutes, too short to determine their orbits. However, the survey field was located close to the opposition, allowing us to approximate the orbital elements from the measured sky motions assuming circular orbits with relatively small uncertainties. We estimated the semimajor axis (or heliocentric distance) and inclination of each object using the expressions presented in Terai et al. (2013).

It is necessary to evaluate the accuracy of heliocentric distance in our orbit approximation, since the value is used for calculation of the absolute magnitude. We conducted a Monte Carlo simulation of the orbit fitting using the synthetic object generator described in Section 3.2. Figure 4(a) shows a comparison of heliocentric distance between the generated and estimated orbits. It displays that the estimated heliocentric distances have a systematic deviation from the 1:1 line. This is because the sky motion of an elliptically orbiting object at the perihelion side is slower than that of a circularly orbiting object with an equal heliocentric distance, while the sky motion of an elliptically orbiting object at the aphelion side is faster than that of a circularly orbiting object with an equal heliocentric distance. The correlation between the actual and estimated heliocentric distances (${r}_{\mathrm{act}}$ and ${r}_{\mathrm{est}}$, respectively, in au) is fitted with a linear function as ${r}_{\mathrm{est}}=0.521\,{r}_{\mathrm{act}}\,+\,2.48$ (solid line in Figure 4(a)). Figure 4(b) is the same as Figure 4(a) but with the estimated heliocentric distances corrected by this function. The systematic error can be removed from the estimated heliocentric distances of the detected JTs by applying this correction. The statistical error of the corrected heliocentric distance is 0.09 au. This causes uncertainties in the absolute magnitude and body diameter of 0.08 mag and 4%, respectively.

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For an accurate investigation of the size distribution of the JT population, it is critical to evaluate the completeness of the detection as a function of apparent magnitude. The sensitivity depends on the vignetting effect increasing with distance from the FOV center, as well as the time variation of airmass and sky condition. We examined the detection efficiency for JTs in all frames on a CCD-by-CCD basis with the following processes: (i) creating a synthetic blank image reproducing the sky background of the original image, (ii) implanting synthetic moving objects corresponding to JTs with a given flux produced in the manner described in Section 3.3 into the generated image at (i), (iii) processing this image with the reverse order of the data reduction procedure as a pseudo-raw image, and (iv) running hscPipe for the pseudo-raw image and counting the number of detected JT sources.

The above procedure is repeated for synthetic JTs with apparent magnitude m_r from 24.0 to 25.4 mag in increments of 0.2 mag. We represent the detection efficiency curve for a single CCD image with a sum of the two functions for proper fitting as

$$\begin{eqnarray}&&\eta (m)=\displaystyle \sum _{k=1,2}\displaystyle \frac{{\epsilon }_{k}(m)}{2}\left[1-\tanh \left(\displaystyle \frac{m-{m}_{50}}{{w}_{k}}\right)\right],\end{eqnarray} \tag{ 4 }$$

where m₅₀ is the magnitude at half maximum and w_k is the transition width. In addition, ${\epsilon }_{k}(m)$ shows the fraction of each term, given by

$$\begin{eqnarray}&&{\epsilon }_{1}(m)=1-{\epsilon }_{2}(m)=\displaystyle \frac{1}{2}\left[1-\tanh \left(\displaystyle \frac{m-{m}_{50}}{w}\right)\right].\end{eqnarray} \tag{ 5 }$$

Actually, the necessary condition for identifying a moving object is to detect it from all three of the visits. The expected detection efficiency for JTs in each field is given by

$$\begin{eqnarray}&&\eta (m)=\displaystyle \prod _{i=1}^{N}{\eta }_{i}(m),\end{eqnarray} \tag{ 6 }$$

where ${\eta }_{i}(m)$ is the individual detection efficiency of the ith visit (N = 3).

Figure 5 shows the detection efficiency of the CCDs located near the center, middle, and edge of the HSC's FOVs (red, green, and blue lines, respectively) taken at all 17 of the survey fields. You can see that the lowest detection efficiency occurred on one of the CCDs near the edge. We defined the lowest detection efficiency of 0.5 (i.e., 50% detection) as the detection limit of our survey, which is 24.4 mag in apparent magnitude.

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The absolute magnitude of the detected JTs in the r band is estimated from the equation presented by Bowell et al. (1989),

$$\begin{eqnarray}&&{H}_{r}={m}_{r}-5\mathrm{log}(r{\rm{\Delta }})-P(\theta ),\end{eqnarray} \tag{ 7 }$$

where r and Δ are the heliocentric and geocentric distances, respectively; and P(θ) is the phase function at a phase angle θ given by

$$\begin{eqnarray}&&P(\theta )=-2.5\mathrm{log}[(1-G){{\rm{\Phi }}}_{1}+G{{\rm{\Phi }}}_{2}].\end{eqnarray} \tag{ 8 }$$

Here, G is the slope parameter, and ${{\rm{\Phi }}}_{1}$ and ${{\rm{\Phi }}}_{2}$ are the phase functions at a phase angle θ given by

$$\begin{eqnarray}{{\rm{\Phi }}}_{i} & = & \exp (-{A}_{i}{[\tan (\theta /2)]}^{{B}_{i}}),i=1,2,\\ {A}_{1} & = & 3.33,{A}_{2}=1.87,{B}_{1}=0.63,{B}_{2}=1.22.\end{eqnarray} \tag{ 9 }$$

We could not measure the P(θ) of each object because our survey was done at only a single phase angle. Therefore, we assumed a constant slope parameter of G = 0.15.

The absolute magnitude can be converted into body diameter D by

$$\begin{eqnarray}&&\mathrm{log}D=0.2{m}_{\odot ,r}+\mathrm{log}(2{r}_{\oplus })-0.5\mathrm{log}p-0.2{H}_{r},\end{eqnarray} \tag{ 10 }$$

where ${m}_{\odot ,r}$ is the apparent r-band magnitude of the Sun (−26.91 mag; Fukugita et al. 2011), ${r}_{\oplus }$ is the heliocentric distance of the Earth (i.e., 1 au) in the same unit as D, and p is the geometric albedo. We assumed a constant albedo of p = 0.07 for JTs based on an analysis of the NEOWISE data (Grav et al. 2012), which shows a mean albedo of 0.07 ± 0.03 across all sizes, consistent with the C, P, and D taxonomic classes in the JT population.

Figure 6 shows the plot of the heliocentric distance and absolute magnitude. The dashed line represents the apparent magnitude of the detection limit, m_r = 24.4 mag. To avoid a detection bias caused by the decrease in brightness with increasing distance from the Sun and Earth, we defined the outer edge of JTs as r = 5.5 au, where m_r = 24.4 mag corresponds to H_r = 17.4 mag, and selected objects located in the region of r $\leqslant$ 5.5 au and H_r $\leqslant$ 17.4 mag as an unbiased sample. The extracted sample contains 481 objects. Compared with WB2015, who detected over 550 JTs but only analyzed an unbiased sample of 150 objects with H_V = 7.2–16.4 mag (H_V is the absolute magnitude in the V band), our survey obtained more than three times as many unbiased sample JTs.

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Figure 7 shows the cumulative size distribution (CSD) of JTs in the unbiased sample as a function of H magnitude with a bin width of 0.5 mag, which is sufficiently larger than the typical H_r uncertainty (≲0.2 mag). The cumulative number is corrected by the detection efficiency as

$$\begin{eqnarray}&&N(\lt H)=\displaystyle \sum _{j:{H}_{j}\lt H}\displaystyle \frac{1}{\eta ({m}_{j})},\end{eqnarray} \tag{ 11 }$$

where m_j and H_j are the apparent and absolute magnitudes of object j, respectively. The error bars are given by the Poisson statistics.

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We found that the CSD in H_r > 13.0 mag can be represented by a single-slope power law, not a broken power law as claimed by WB2015. This allows the differential H distribution, ${\rm{\Sigma }}(H)={dN}(H)/{dH}$, to be fitted by

$$\begin{eqnarray}&&{\rm{\Sigma }}(H)={10}^{\alpha (H-{H}_{0})},\end{eqnarray} \tag{ 12 }$$

where α is the power-law slope and H₀ is given as ${\rm{\Sigma }}({H}_{0})=1$. When the CSD is expressed as

$$\begin{eqnarray}&&N(\gt D)\propto {D}^{-b},\end{eqnarray} \tag{ 13 }$$

where $N(\gt D)$ is the number of objects larger than D in diameter, the power-law index b is converted into α by $b=5\alpha$.

We used a maximum-likelihood method (e.g., Bernstein et al. 2004) for fitting Equation (12) to the H distribution of the unbiased JT sample. The likelihood function is given by

$$\begin{eqnarray}&&L(\alpha ,{H}_{0})\,\propto \,{e}^{-\tilde{N}}\displaystyle \prod _{j}\,\eta ({m}_{j})\,{\rm{\Sigma }}({H}_{j}| \alpha ,{H}_{0}).\end{eqnarray} \tag{ 14 }$$

Here $\tilde{N}$ is the expected number of detected objects in the survey, which is estimated by

$$\begin{eqnarray}&&\tilde{N}=\int \,{dH}\,\eta (m(H))\,{\rm{\Sigma }}(H| \alpha ,{H}_{0}),\end{eqnarray} \tag{ 15 }$$

where m(H) is the apparent magnitude approximated as H + 5 $\mathrm{log}(r{\rm{\Delta }})$ with r = 5.2 au and Δ = 4.2 au. Uncertainties in the fitted parameters are estimated from repeated fitting to synthetic object samples generated from the actual objects based on the measurement errors.

We obtained the best-fit power-law slope of α = 0.37 ± 0.01, corresponding to b = 1.84 ± 0.05. This value agrees with the result of Yoshida & Nakamura (2005), b = 1.89 ± 0.10 in D = 2–10 km, as well as the faint-end slope with $\alpha ={0.36}_{-0.09}^{+0.05}$ in H_V = 14.9–16.4 mag shown by WB2015, though the precision is highly improved compared with those studies.

The best-fit power law is plotted in Figure 7 as a dashed line. The data are very close to the fitted line over H_r ≳ 13.0 mag, indicating no evidence of the power-law break at ${H}_{V}={14.93}_{-0.88}^{+0.73}$ reported by WB2015. We concluded that L4 JTs have a single-slope power-law size distribution in the range of 13.0 ≲ H_r ≲ 17.0.

Finally, we compared the CSD obtained from our sample with that of known L4 JTs from the MPC catalog consisting of 4,080 objects. Figure 8 shows that MPC JTs exhibit an almost constant increase with absolute magnitude in log scale over H_V ∼ 10.0–14.0 mag, indicating that the completeness limit is likely to be located at H_V ∼ 14.0 mag. We assumed the V − r color of 0.25 mag (Szabó et al. 2007) and scaled the CSD of our sample to the cumulative number of the MPC sample at H_V = 14.0 mag. We also added the CSD from the L4 JT sample containing 93 objects presented by Jewitt et al. (2000, hereafter JTL00), which is normalized at H_V = 15.0 mag.

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As seen in Figure 8, the three CSDs match each other well, except for the faint ends of the MPC and JTL00 samples that seem to reach beyond the completeness limit. The combination of those CSDs shows that there is likely to be a power-law break around H_V = 13 mag. We fitted a broken power law to the combined CSD with H_V > 12.0 mag. The function is given as

$$\begin{eqnarray}{\rm{\Sigma }}(H)=\left\{\begin{array}{ll}{10}^{{\alpha }_{1}(H-{H}_{0})}, & \mathrm{for}\,\,H\lt {H}_{\mathrm{break}}\\ {10}^{{\alpha }_{2}H+({\alpha }_{1}-{\alpha }_{2}){H}_{\mathrm{break}}-{\alpha }_{1}{H}_{0}}, & \mathrm{for}\,\,H\geqslant {H}_{\mathrm{break}},\end{array}\right.\end{eqnarray} \tag{ 16 }$$

where ${\alpha }_{1}$ and ${\alpha }_{2}$ are the power-law slopes for the brighter and fainter objects, respectively, with the break magnitude ${H}_{\mathrm{break}}$ as a border. We found the best-fit parameters of ${\alpha }_{1}$= 0.50 ± 0.01, ${\alpha }_{2}$= 0.37 ± 0.01, and ${H}_{\mathrm{break}}={13.56}_{-0.06}^{+0.04}$ (see Figure 9). The goodness of fit is evaluated by the Anderson–Darling test (Anderson & Darling 1952; Press et al. 1992),

$$\begin{eqnarray}&&D=\mathop{\max }\limits_{-\infty \lt x\lt \infty }\displaystyle \frac{| S(x)-P(x)| }{\sqrt{P(x)[1-P(x)]}},\end{eqnarray} \tag{ 17 }$$

where S(x) is the observed CSD and P(x) is the fitted function. The statistic D was calculated to be 0.078, which cannot reject the null hypothesis for equality between S(x) and P(x) even at a 40% significance level.

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The size distribution of SSSBs has been evaluated by the index of power-law distribution (α or b in Equations (12) and (13)).The first survey of the size distribution of L4 JTs was conducted by JTL00 using the University of Hawaii 2.2 m telescope. This survey determined the index, b = 2.0, or α = 0.4 in the range of H = 11–16 mag. At their survey period, the number of cataloged JTs was still small. The ASTORB catalog was only completed up to H_V = 9 or so. Therefore, there was a gap in the range of size distributions between the cataloged JTs and small JTs detected from the survey by JTL00. Later, Szabó et al. (2007) investigated the SDSS data (the third release of the Moving Object Catalog; MOC3) and found the index, b = 2.2, or α = 0.44 in the range of H_V = 10–13.5 mag. They determined that all JTs with H_V < 12.3 mag had already been discovered and listed in the ASTORB file. By combining the ASTORB file, SDSS MOC3 data, and survey result by JTL00, the size distribution of L4 JTs had been revealed up to ${H}_{V}\sim 16$ mag. Up to now, the surveys for JTs were done by 2 m class telescopes.

In order to examine the size distribution of JTs smaller than those of JTL00, Yoshida & Nakamura (2005) searched for small JTs in the data set taken by the Subaru Telescope in 2001 February. This is equivalent to the first L4 JT survey done by an 8 m class telescope (Yoshida et al. 2003). They detected JTs in the range 14 < H_V (mag) < 17.7, corresponding to 2 < D (km) < 10 (assuming an albedo of 0.04) and estimated the CSD. The slope index b was 1.89 for the entire size range. They also suggested that the size distribution may have a break around H_V = 16 mag. In this case, the slope index changes from b = 2.39 for H_V < 16 mag to b = 1.28 for H_V > 16 mag.

Yoshida & Nakamura (2008) used a data set of the MBA survey taken by the Subaru Telescope in 2001 October to search for L5 JTs (Yoshida & Nakamura 2007). This is equivalent to the second survey of JTs by the Subaru Telescope but is the first survey for L5 JTs. They noticed that the L4 and L5 swarms may have different size distributions. Although both surveys were important to determine the faint end of the size distribution of L4 and L5 JTs, the number of detections of JTs in both surveys was small: 51 L4 JTs and 62 L5 JTs. This is because the surveys were not dedicated surveys for JTs; their main purpose was detection of MBAs. Therefore, we guess that the determination accuracy of the size distribution cannot be very high because of small-number statistics. Robust results using a large sample of JTs have been long awaited.

Recently, WB2015 conducted a dedicated survey of L4 JTs by the Subaru Suprime-Cam. Their survey detected JTs in the range 7.2 < H_V (mag) < 16.4. By using the cataloged L4 JTs and their detected ones, WB2015 found that there are two break points in the CSD of JTs up to H_V = 16.4 mag: one at H_b1 = 8.46 and one at H_b2 = 14.9. They also found that the power-law slopes of b are ${4.55}_{-0.80}^{+0.95}$ for the largest population of L4 JTs, ∼2.2 for the middle population, and ${1.80}_{-0.45}^{+0.25}$ for the faint end of the population (see Figure 7 in WB2015). In this work, meanwhile, the break point in the CSD corresponding to H_b2 in WB2015 was slightly shifted to a brighter magnitude (H_break = 13.56), and the slope index at the faint end was b = 1.84 ± 0.05. The slope indices between WB2015 and this work are consistent within the range of error bars. The break point at H = 16 found by Yoshida & Nakamura (2005) was not seen in WB2015 or this work. It seems to be a fluctuation induced from small-number statistics.

Wong et al. (2014), who investigated the SDSS data set, found that the ${R}_{{\rm{g}}}$ and ${\mathrm{LR}}_{{\rm{g}}}$ in the JT swarm have different size distributions, and WB2015 found that most of the smaller JTs in their sample belong to the ${\mathrm{LR}}_{{\rm{g}}}$. Based on these findings, WB2015 proposed a possibility that the fragmentation of the red objects by collisional evolution creates the less-red objects, then causes a difference of the size distribution between R_g and LR_g. However, there is another possibility: that the difference of the size distributions in the different color groups is caused by the difference of the chemical materials related to different formation regions. For example, Yoshida & Nakamura (2007) reported that the S- and C-complex with the faint MBAs (H_R ≳ 15 mag) detected by the Subaru Telescope with the Suprime-Cam show the different size distributions. This finding probably reflects a difference in compositions or origins of the two groups in the population.

We could not confirm WB2015's findings, because our HSC survey used only the rband for detecting JTs. This is because we wanted to find as many JTs as possible during one night of observation (the filter exchange of the HSC takes about 40 minutes). Further multicolor observations are needed to understand collisional evolution in a population that consists of different taxonomic types or composition groups.

Since we estimated the size distribution of the smaller JTs by using the 481 JTs with H_R ≳ 17.4 mag, corresponding to $D\,\lesssim$ 2 km (assuming an albedo of 0.07; Grav et al. 2012) detected by Subaru + HSC, our knowledge of the size distribution of JTs now reaches to D ∼ 2 km. Therefore, we can compare the size distributions of JTs and MBAs down to 2 km in diameter.

For a more detailed comparison of the size distributions of JTs and MBAs, we made R plots (relative plots). This method was devised by the Crater Analysis Techniques Working Group (Crater Analysis Techniques Working Group et al. 1979) to better show the size distribution of craters. When a sufficient data set is available, the R plot provides a more sensitive comparison of size distributions than the cumulative plot. Strom et al. (2005, 2015) compared the size distributions of near-Earth asteroids (NEAs), MBAs, and the Moon's young/old craters on the R plots. They found that the impactor population that made lunar highland craters in the old era is different from the population that made lunar mare craters in the relatively young era, and they concluded that the source of impactors in the inner solar system region had changed from MBAs to NEAs around 3.8 Gyr ago. Here we used the same method as Strom et al. (2005, 2015) and made the R plots for inner (2.0 < a (au) < 2.6), middle (2.6 < a (au) < 3.0), and outer (3.0 < a (au) < 3.5) MBAs and JTs separately in Figure 10. We used the same data sets of MBAs as Strom et al. (2005, 2015), and we added the data sets newly obtained by the Subaru Telescope (Yoshida et al. 2011), AKARI survey (Usui et al. 2011), and WISE survey (Masiero et al. 2011) to the R plots. See the caption of Figure 10 for details of the data sets used. In Figure 10, the scale of the vertical axis is arbitrary. We connected all data sets smoothly. At first, we noticed that a wavy structure seen in the R plot of the MBAs does not show up on the R plot of JTs. According to the numerical simulations of collisional evolution for L4 JTs by de Elía & Brunini (2007), a wavy structure on the size distribution of JTs cannot be reproduced by a certain set of collisional parameters. This implies that the collisional parameters of the JT population are different from those of the MBA population. Another numerical simulation suggests that the impact frequency and relative velocity of objects among MBAs and JTs are not so different (Davis et al. 2002). Therefore, the difference in the shape of the size distribution is probably caused by a difference of composition and/or internal structure between the MBA and JT populations, suggesting that they are from different origins.

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We also noticed that there is a remarkable dip around a few tens of km in the R plots of the inner and middle MBAs. The dip becomes shallower in the R plots of the outer MBAs, and it disappears in the R plots of the JTs. It is well known that a majority of inner-belt objects belong to the S-complex and a majority of outer-belt objects belong to the C-complex. The difference of the size distributions in the main-belt regions may be related to the difference of composition and/or inner structure between the S- and C-complex.

Bottke et al. (2005) argued that the knee of the size distribution in MBAs around 120 km in diameter is a fossil from the early violent collisional evolution era before planet migration. For the MBAs, the knee around 120 km is shown in all three regions, while the shape of the size distribution of JTs is rather flat around 120 km in diameter. This probably suggests that MBAs and JTs were completely different populations before planet migration happened. From these facts mentioned above, we can say that MBAs and JTs originated from different regions.

The shallow dip around a few tens of km in the R plot of the outer main belt may be related to a materials/inner-structure difference between the S- and C-complex, as mentioned above. However, there is a possibility that it may be caused by the influence of implanting objects whose size distribution has no dip around a few tens of km, such as JTs in the era of the LHB; namely, the outer objects, such as Kuiper Belt objects (KBOs), were captured into the JT region as the Nice model suggested, but the objects have reached even into the outer main belt. In order to investigate how the outer objects were captured into the JT region and the main-belt region during the LHB, further numerical simulations with a high spatial resolution would be needed.

In addition to the simulations, the investigation of the size distribution of small TNOs would be necessary. To confirm the prediction from the latest planet migration models, we need a direct comparison of the size distributions of JTs and TNOs. However, the surveys of faint TNOs have been insufficient to compare the size distributions of JTs and TNOs in the same size range. Morbidelli et al. (2009) predicted that the absolute magnitude distribution of hot KBOs should become Trojan-like steep at $6\lt H$(mag) < 9. Fraser et al. (2014) reported that the size distribution of JTs is similar to that of hot KBOs, but their analysis was limited to only large KBOs. The detection limit of ground-based observations such as the Subaru Telescope is usually $r\sim 25$ mag. This means that we can detect KBOs larger than $D\sim 110$ km for the classical TNOs assuming 40 au and an albedo of 0.04 or $D\sim 60$ km for the scattered TNOs assuming 30 au and an albedo of 0.04. Thus, it is impossible to find TNOs smaller than 50 km with the capability of the current ground-based telescopes. In this situation, the common size range of JTs and TNOs is too narrow to compare the properties of the size distributions. Therefore, we need other input sources, such as the size distribution of craters on icy satellites or Pluto. New Horizons provided the crater size distribution on Pluto and Charon (Schlafly et al. 2016). This size distribution looks flat on the R plot. The craters provide us with useful data to derive the size distribution of much smaller TNOs, and then we can compare the size distributions of JTs and TNOs.

We calculated the surface number density (SND) using the 481 JTs, which is the complete sample set from our survey (H_r ≲ 17.4 mag and r ≲ 5.5 au). Figure 11 shows the SND of each field with the distance (d) of each field from the L4 point. One can see a trend that the SND is larger at the field closer to the L4 point. The average SNDs for 11.0 $\lt \,d$ (°) < 13.5 (the averaged distance ($\bar{d}$) is 123) and for 16.0 < d (°) < 20.0 ($\bar{d}$ is 180) are 22.7 $\pm \,$ 3.3 and 17.9 ± 4.2 deg⁻², respectively. Since our survey of the two fields in d = 14°–16° had been done with bad seeing, we excluded those fields for the calculation of the SNDs. Note that our survey area was located at 10°–20° longitude behind the L4 point. By contrast, the previous survey with Subaru + Suprime-Cam done by Yoshida & Nakamura (2005, 2008) with almost the same limiting magnitude was done at ∼30° longitude ahead of the L4 point and obtained an SND of 14.8 deg⁻². In order to estimate the population of the entire L4 swarm based on the SNDs, we need further observations at different longitudes against the L4 point.

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