The Population of Kilometer-scale Retrograde Jovian Irregular Moons

After all rates and PAs were searched over all CCDs, the objects we detected were compared with the implanted moons. Any implanted object that was matched (within a tight tolerance) to a detection was labeled "found." Any object found that was unable to be matched to an implanted object became a candidate moon.

The fraction of implanted objects found, as a function of magnitude, over the whole field is very close to 1 on the bright end. Despite a typical moving source's path crossing several background galaxies, stellar halos, and/or bad-pixel columns, essentially all sources brighter than 25th magnitude were easily recovered. The detection efficiency then starts to drop around m_r  = 25 and falls to 0.5 by m_r  = 25.7 (see Figure 1, black points and the blue curve). We fit this fraction with a hyperbolic tangent function $\eta ({m}_{r})=\tfrac{A}{2}\left(1-\tanh \left(\tfrac{{m}_{r}-\mu }{\delta }\right)\right)$, where A ≃ 1 is the fraction of bright objects that are detected, μ is the magnitude where the fraction drops to A/2 ≃ 0.5, and δ is a "width" of the drop. The curve of best fit for the implanted moons (blue line in Figure 1) has the parameters A = 0.998, μ = 25.69, and δ = 0.31, which we use as our detection efficiency. For the luminosity function study we term the "characterization limit" to be where the detection efficiency drops to 0.5. Therefore, our characterization limit is m_r  = μ = 25.7; if brighter than this limit we have confidence that we can accurately debias the detected sample. Performing the same fit on various subregions of our field enabled the detection of a tiny drop of about 0.1 mag in the 50% limit going from the side of the field that is furthest from Jupiter to the closest. Thus there is an uncertainty of 0.05 mag on our characterization limit.

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The fraction of implanted moons that were found as a function of the rate and PA is mostly constant except for small drop offs (about 10%) at the extreme minimum and maximum implanted values. This drop off is due to the implanted ranges being slightly larger than the search ranges. Since only a small fraction of our moon candidates have rates and/or PAs outside our search ranges, any effects caused by these drop offs will be negligible.

During our search activities, 55 moon candidates were discovered. Of these candidates, three were traveling faster than the fastest rate search, which raised doubt as to whether they are Jovian moons. Examining the distribution of on-sky rates of our detections, there is a clear separation between these three fast moving objects (green histogram in Figure 2) and the rest of the detections (red histogram). ³ We compared our detections with all minor bodies in the International Astronomical Union (IAU) Minor Planet database that are within 3 deg of our field center at the time of observation and have PAs between 238° and 252° (a range that encompasses the known moons). The three fast moving detections are right on the tail of the asteroid distribution (25''–41'' hr⁻¹), which leads us to believe these objects are in fact slow-moving asteroids and will thus not be included in our analysis of the Jovian luminosity function. The region in which our moon candidates lie, 15''–23'' hr⁻¹, contain no known minor planet, which is one line of evidence that strongly indicates our candidates are indeed Jovian moons.

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Of the candidates, four were beyond the characterization limit of m_r  = 25.7 and another one of the candidate moons was found outside our search area by chance while performing astrometry and photometry on a different object. ⁴ The three fast, four faint, and one extra candidates are not included in the characterized sample, which thus consists of 47 moons. A full list of the characterized and uncharacterized detections are found in Tables 1 and 2, respectively. The cumulative number of our detections as a function of magnitude is shown in Figure 1 (red line). To debias the detections, we weighted them by the detection efficiency. Each detection became 1/η detections, where η is the detection efficiency at the object's measured magnitude. Only near the characterization limit does the cumulative number of debiased detections (green line) becomes noticeably different from the unbiased detections, due to η ≃ 1, except at the sharp drop off near the limit; because we have chosen to only debias down to the 50% detection efficiency, our population estimate is little affected by the efficiency correction.

We provide the astrometric entries (in standard Minor Planet Center (MPC) format) in Table A1.

It was obvious that most of the irregular moon detections found in this pencil-beam search would be beyond the magnitude limit of other available data (because it is only another shift-and-stack pencil-beam study that could to descend beyond 24th magnitude for these rapidly moving targets). Nevertheless, we were able to identify some additional observations beyond the night of detection. We believe that essentially every object in the 15''–23'' hr⁻¹ rate range is a Jovian moon.

Although there are hundreds of bright (visible on single images) asteroids in this field, the only two objects with m_r  < 22.5 that were in the field and in this rate range turn out to be the two brightest known Jovians in the field: Hermippe and Erinome (see Table 1). We believe that if main-belt asteroids were generating any significant confusion into the rate cut, it is highly likely some of them would have been bright, given how shallow the main-belt luminosity function is at these magnitudes.

Jup LIX (magnitude ∼23.2) had a 2 yr arc from observations in 2016 and 2017 and was sufficiently close to its predicted position that identification was trivial.

In the Jovian rate range, two more objects turned out to be Jup LI and LII, which is fortunate because Alexandersen et al. (2012) discovered them in this very same data set. We used no knowledge of the prior existence of any irregular in our search; we have only later identified detections in our rate range with previously known moons. With a better photometric calibration now available, we believe the original magnitudes of JLI and JLII were first reported as ≃0.5 mag brighter than they should be.

Jup LXIX was a less trivial identification. The 2017 and 2018 discovery astrometry was confined to fewer dark runs than LIX and in fact its predicted sky position was just off this 2010 data set's coverage. However, we identified j21r113a26 as close in magnitude and sky rates while being only about 4.1' away and thus believed this was a recovery of the moon. We determined this moon should then be on archival recovery observations reported in Alexandersen et al. (2012) that were targeted to track Jup LI and LII; we located and measured it. The MPC confirmed that these observations all link together, and we have thus increased the observed arc from 1 yr to 8 yr.

The object j22r94a24 was located on Palomar 5 m observations taken on the same calendar date that overlaps with this CFHT's field's coverage; these observations start roughly two hours before the CFHT data and overlap in time.

For completeness, we mention that the high-precision orbits of Harpalyke and Eurydome placed them inside this square degree, but they occupied CCD gaps at the time of our observations and were thus not detected. Note that this gap coverage is corrected for below in our estimates of the total population. Our field contained nine known moons with roughly 20% of the area lost in the shift-and-stack process; having two non-detected moons is thus unsurprising. Lastly, there are two Jovians with temporary designations (S/2003 J 9 and S/2003 J 12) whose ephemeris, based on a few-month arc from 2003, is on the field but with an enormous error (of order thousands of arcseconds according to analysis of Jacobson et al. 2012). It is possible that these moons are actually among our detections, but we have not yet been able to establish a linkage.

To summarize, every object we detected in our Jovian irregular rate cut that could get additional observations has turned out to be a Jovian irregular, leaving little credence for the argument that the fainter detections in this rate range are not Jovians. While there remains a small possibility that one of these detections happens to be a small Centaur passing close to Jupiter, our main goal is a luminosity function analysis to estimate the size distribution and the total Jovian irregular population down to magnitude 25.7; an interloper or two will have a negligible effect on these analyses.

We performed a single exponential least-squares fit on the binned differential luminosity function of both our debiased detections, from m_r  = 23.75 to 25.75, and on the known retrograde moons in the MPC database, starting where the size distribution starts to ramp up, m_r  = 21.85, to just before it rolls over, 23.05 (the reason for using just the retrograde population is explained in Section 3.5). The resulting logarithmic slopes we get from the fit for our detections and the known retrograde moons are α = 0.29 ± 0.15 (the orange line in Figure 3) and 0.6 ± 0.3, respectively. The uncertainties were obtained by generating differential luminosity functions with random numbers for each bin drawn from a Poisson distribution with the original number in each bin being the expected number of occurrences. Single exponential least-squares fits were performed on 10,000 such luminosity functions, producing a distribution of slopes; we used the full width at half maximum of this distribution as our uncertainty bounds. The two slopes are different at a level between 1 and 2σ, providing weak evidence that the luminosity function for retrograde Jovians changes to a shallower slope at m_r  ∼ 23.5. This is not a surprise, as simulations (Bottke et al. 2010) produce "waves" in the luminosity function as the exponential index fluctuates around the collisional equilibrium value of α = 0.5 (Dohnanyi 1969) as one moves to fainter magnitudes/smaller sizes. These waves propagate down the luminosity function as the largest objects in the finite distribution are disrupted; the fact that α is close to 0.5 usually indicates that this portion of the size distribution has "relaxed" to near equilibrium.

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We assume at this point that the vast majority of our detections have retrograde orbits. This assumption is based on (1) the edge of the field that is closest to Jupiter is about a degree away, which is approximately the projected apocenter distance of most (8 of 10) of the known direct irregular moons, and (2) only a small fraction of all known moons are direct (10/71). As such, we will make the approximation that all moons in our field are retrograde. Our analysis will thus focus on providing an estimate Jupiter's retrograde population.

We can get an estimate on the total number of Jovian irregulars if we know the fraction of the population in our field's Jovian offset at any time. We counted the average number of known retrograde moons in a field with the same size and with the same on-sky offset from Jupiter for 10 different oppositions (2009–2019). Note that over this 10 yr time interval the moons complete many orbits and thus "lose memory" of where they were discovered (most were discovered before 2004). On average there were 6.9 known moons in this field. Accounting for the 20% of sky area lost due trimming and chip gaps, on average we will detect 5.5 known retrograde moons (of 61). Thus the multiplier going from our sample to the full populations is 11 ± 5; the uncertainty value comes from the standard deviation of number of known moons over the 10 oppositions. If there were biases induced in the fraction of known retrograde moons in our field due to where the detections surveys found many orbits ago (which we think unlikely), there could be a systematic present but we believe this is likely small compared to our factor of 2 estimate.

Our estimated total population of retrograde moons overlaps nicely with the known population from m_r  ≈ 22–23 (see Figure 3). Beyond m_r  ≈ 23.2 the known population flattens out and diverges from our estimated population, indicating that the current completion limit is m_r  ≈ 23.2. This is in agreement with the fact that the brightest object we were unable to match to a known object has m_r  = 23.3.

Our results produce an estimate of 160 ± 60 retrograde Jovian moons with m_r  < 24, which agrees with the Sheppard & Jewitt (2003) prediction of 100 down to the same limit (although their prediction includes retrograde and direct moons). We estimate there are 600 retrograde Jovian irregulars (within a factor of 2) down to 25.7 magnitude. Using an albedo of 0.04 (the average retrograde Jovian albedo from Grav et al. 2015) and the distance of Jupiter at the time of observation produces a radius of 0.4 km for a magnitude of 25.7. We get a 10% error in the radius, which is dominated by the fact we do not know if the moon is half a Hill sphere in front or behind Jupiter. If retrograde Jovians are almost complete down to m_r  = 23, then the known retrograde population being well within the error bars of our population estimate suggests that we have overestimated the size our error bar; our R > 0.4 km estimate's uncertainty may thus be less than a factor of 2.

We posit that the overall completion limit for known direct Jovians would likely be at a brighter magnitude that for the retrogrades. Direct Jovian moons have smaller semimajor axes compared to retrograde moons, resulting in direct moons spending more time close to Jupiter. We speculate that this makes direct moons harder to detect and could contribute to the relative lack of direct Jovians compared to retrograde.