A DEEP SEARCH FOR ADDITIONAL SATELLITES AROUND THE DWARF PLANET HAUMEA

HST Program 12243 obtained 10 orbits of observations over the course of ∼15 hr in 2010 July. This program used the Wide Field Camera 3 (WFC3) UVIS imager with the F350LP (long-pass) filter. The primary goal of these observations was the detection of a mutual event between the inner moon Namaka and Haumea (RB09). In order to produce high cadence time series photometry of the proposed mutual event (and to avoid saturation), exposures were limited to ∼45 s. To prevent a costly memory buffer download, only a 512 × 512 subarray of the full WFC3 camera was used with a field of view of ∼20.5 arcsec. HST tracked at Haumea's rate of motion (except for controlled dithering) to maintain its position near the center of the field of view throughout the observations. The geocentric distance to Haumea at the time of observations was 50.85 au. At this distance, 1'' corresponds to 36,900 km, 1 WFC3 pixel (0.04 arcsec) corresponds to 1475 km, and the entire subarray field of view corresponds to ∼750,000 km.

Parts of the last two orbits were affected by the South Atlantic Anomaly. This caused a portion of the these orbits to lose data entirely, and another portion was severely affected by cosmic rays and the loss of fine pointing precision. The most offensive frames were discarded for the purpose of the satellite search, leaving 260 individual exposures.

The center of Haumea was identified by eye in combination with a 2D Gaussian fitting routine. With these well-determined preliminary Haumea locations, all images were co-registered to Haumea's position. In this Haumea-centric frame, cosmic rays and hot pixels were identified by significant changes in brightness at a particular position using robust median absolute deviation filters. A detailed and extensive image-by-image investigation of cosmic rays by eye confirmed that this method was very accurate at identifying cosmic rays and other anomalies. Furthermore, the automatic routine did not flag the known objects (Haumea and its two moons: Hi'iaka and Namaka) as cosmic rays, nor were any other specific localized regions identified for consistent masking (i.e., putative additional satellites were not removed).

TinyTim software⁶ was used to generate local PSF models. As in RB09, these PSF models were then fit to the three known objects using standard ${\chi }^{2}$ minimization techniques (Markwardt 2009). This identified the best-fit locations and heights of the scaled PSFs, with bad pixels masked and thus not included in the ${\chi }^{2}$ calculation. The astrometric positions of the known satellites relative to Haumea seen by this method were in clear accord with their projected orbital motion from RB09. Despite Namaka's nearness to Haumea (which was purposely chosen, as the goal of the observations was to observe a mutual event), it is easily distinguishable in the first few orbits.

The best-fit PSFs are visually inspected and found to be good fits for all images. These PSFs are then subtracted, removing a large portion of the three signals, but leaving non-negligible residuals; these residuals are caused by imperfect PSFs (note that Haumea may be marginally resolved in some images) and standard Poisson noise. Using the updated best-fit centers of Haumea, these PSF-subtracted images are re-coregistered to Haumea's best-fit location (though the additional shifts from the preliminary 2D Gaussian centers are very small). As described below, these same PSF-subtracted images are used to perform the nonlinear shift-and-stack.

Moons with negligble motion relative to Haumea can be identified in this image by stacking all the observations in the Haumea-centered reference frame (including fractional pixel shifts implemented by IDL's fshift). Throughout this investigation, we create stacks by performing a pixel-by-pixel median of the images, which is less sensitive to cosmic rays, bad pixels, and errors in the PSF subtraction of the known bodies. This results in a small decrease in sensitivity, as, assuming white noise, the noise level grows a factor of $\sqrt{\tfrac{\pi }{2}}$ faster when using the median over the mean. This will correspond to only an $\sim 20\%$ difference in brightness sensitivity, or an $\sim 10\%$ difference in radius, and we find this acceptable so that the other effects mentioned above can be mitigated. A portion of this stacked image around Haumea is shown in Figure 1. Detailed investigation of this deep stack by eye by each co-author yielded no clear satellite candidates. The median stacks were also automatically searched using the IDL routine find, which uses a convolution filter with an FWHM of 1.6 pixels to identify positive brightness anomalies. Detections with signal-to-noise ratios (S/Ns) of five or greater were examined; none were found that were consistent with an additional satellite (e.g., having a PSF-like shape). Scaling from the S/N of Haumea and using a more conservative detection limit S/N of 10, this places a limit on non-moving satellites as fainter than about $V\simeq 27.1$, corresponding to Haumea satellites with radii less than about 8 km (see Section 5).

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HST has observed Haumea during many programs for multiple reasons. Program 11518 was proposed to obtain astrometry of both moons and is five independent orbits of observations spread over two weeks (RB09). Although it would be interesting to investigate the possibility of combining these data in a long-baseline nonlinear shift-and-stack, given the existence of other more sensitive data sets, we investigated only the single-orbit median-stacked images. Motion during a single 45-minute HST orbit is small compared to the PSF width, even for the shortest satellite orbital periods.

HST Program 11971 was five consecutive orbits and HST Program 12004 was seven consecutive orbits, both attempts to observe the last satellite–satellite mutual events. The latter program was within a few weeks of the HST 4th Servicing Mission but was still executed. Unfortunately, for 6.5 of the 7 orbits, the STIS shutter was closed and no on-sky data were taken.

For the 5-orbit Wide Field Planetary Camera 2 observation of Program 11971, we median-stacked images centered on Haumea and searched for additional sources by eye and using IDL's find as described above. We investigated stacks of individual orbits and of the entire five-orbit sequence and found no sources consistent with additional satellites. Though the nonlinear shift-and-stack method below could fruitfully be applied to these observations, the WFC3 observations are considerably deeper and we opted to focus on our best data set.

Finally, we obtained some long-duration (∼5 hr) observations of Haumea using the Laser Guide Star Adaptive Optics system at the Keck Observatory. Co-registered stacks of this data also showed no clear additional satellites, though the known satellites were very easily detected.

In a typical shift-and-stack search for KBOs, the putative sky tracks are selected from a grid of the six degrees of freedom needed to describe an object in a Keplerian orbit (PK10, Bernstein et al. 2004). For the purposes of a KBO satellite, particularly that of a primary with other known satellites, it is convenient to instead sample the space of Keplerian orbital elements relative to the primary: semimajor axis (a), eccentricity (e), inclination (i), longitude of the ascending node (Ω), argument of periapse (ω), and mean anamoly at epoch (M). Sampling in this space allows for direct control over the types of orbits that are searched, making it straightforward to exclude unphysical motions. In our case, we also benefit from a well-known mass of the primary; if this is not known, a variety of plausible values could be sampled for the generation of sky tracks. For this search, a and e were randomly sampled from orbits with semimajor axes between 5310 and 368,000 km and eccentricities less than 0.5, while the orbital angles i, Ω, ω, and M were allowed to assume any value. All parameters were chosen from the sample space uniformly, with the exception of a, which was sampled on a log scale to increase the likelihood of sampling an orbit in the regime of fast-moving satellites.

The lower bound on a is a constraint imposed by our sensitivity of detection. This limit corresponds to 3.75 pixels (15 mas) on the WFC3, at which distance from the center of Haumea the subtraction noise is considerable enough to make reliable detections difficult (see Figure 1). The upper bound on a is set much larger than the semimajor axes needed to shift images (as opposed to investigation of the unshifted stack). At a distance where the satellite's maximum velocity would cause it to travel less than one PSF FWHM over the course of the 15 hr observational period (here ∼27 m s⁻¹), shift-and-stack is unnecessary, giving the upper limit of $a\simeq 150,000$ in our search. This semimajor axis is ∼3 times the semimajor axis of Hi'iaka, whose motion in these frames is detectable, but $\lesssim 0.5$ pixels. For the upper limit on a, we doubled this number to be conservative.

Much of this orbital parameter space can be excluded on physical grounds, reducing the number of shift rates necessary to well-sample the space. Any putative orbit that crossed paths with the known satellites was rejected, as was any orbit with periapsis less than 3000 km. These weak restrictions on orbital elements did not appreciably affect the selection of shift-and-stack parameters and additional tests (described below) show that we are sensitive to objects on practically any orbit with a semimajor axis $\gtrsim 10,000$ km.

Having created a bank of physically plausible orbits, we then generate a set of shift rates with which to create composite images to search for satellites. Orbital parameters were converted into sky coordinates relative to Haumea, right ascension (ΔR.A.) and declination (Δdecl.), for each image, as described in RB09. We assumed an instantaneous Keplerian orbit for the position of the satellite as this is an excellent approximation over the course of our observations. In our case, orbital acceleration was quite important, as we desired to search orbital periods down to ∼40 hr, of which the 15 hr observational arc is a sizable fraction. Therefore, the sky positions ΔR.A. and Δdecl. were fit with quadratic polynomials in time, which we found were sufficient to accurately describe the nonlinear motion in every case.

In order to minimize the number of sky tracks, we eliminated tracks which were similar to one another, as suggested by PK10. To determine if two tracks were similar, we focused on the final requirement that the shift rates localize the flux of satellite so that it can be identified in the stacked image. If the flux of a satellite traveling along the second orbit would be well-localized by the shift rates of the first, then there would be considerable overlap of the flux between images when shifted according to the rates of the first. This criterion can be quantified by calculating the overlap fraction between two shift-and-stack rates using the reasonable assumption that the WFC3 PSF is nearly Gaussian, with FWHM of 0.067 arcsec ($\approx 1.7$ pixels). For a pair of shift rates, the overlap was defined for each image in the data set as the integral of the product of two such Gaussians separated by the difference in the two rates (ΔR.A. and Δdecl.) at the time of that image. We call this the overlap between two orbits as it is calculated from the product of two overlapping PSFs, but it is distinct from the concept of the overlap in co-added images. If the median overlap (normalized to 1 for perfect coincidence) was greater than a pre-specified threshold, it was considered that a sufficient fraction of the flux of the proposed satellite would have been collected by the stack of an existing sky track, and the new track was rejected as unnecessary.

The goal is to build up a bank of sky tracks known to be mutually distinct. After accepting the first track into the bank, each subsequent track was compared to the previously selected tracks in the bank using the above overlap criterion. We experimented with different overlap threshold criteria and found the overall results mostly insensitive to the specific value chosen. In general, we required a median overlap of less than 0.7 with each previously accepted shift-and-stack tracks to accept the proposed track as distinct enough to add to the bank.

By drawing from a large set of orbits covering the desired search space, this method efficiently builds a bank of mutually distinct shift rates that are also the most relevant (PK10). However, unlike a grid search, random orbital draws can continue indefinitely. Thus, we also require a "stopping criterion" to decide when the bank is large enough for practical use. To determine the upper limit on the number of necessary shift rates, we noted that the sample space saturates quickly; that is, the rate of acceptance drops off drastically after 10–15 shift rates are chosen. Consequently, the number of orbits rejected between successive accepted rates grows very quickly. Our criterion for a dense sampling was that the number of rejected rates between successive accepted rates was at least equal to the total number of rates rejected so far. Put another way, the selection was stopped when the acceptance of each new shift rate required doubling the number of sampled orbits, which typically occurred after testing hundreds of thousands of random orbits. This is an exponentially slow process, which suggests that once past this threshold, we have reached the limits of our rate selection method. Practically speaking, we found that this stopping criterion still generated enough shift rates to recover injected sources with a complete variety of orbits.

Together with the above ranges for satellite orbital parameters, this method yielded only ∼35 sufficiently distinct orbits over which to search. Considering the size of the parameter space (linear and quadratic terms for shifts in both x and y directions), it might seem surprising that so small a set of potential orbits spans the space. However, a large portion of the space consists of short, almost entirely linear tracks, where the quadratic corrections are of limited importance. The only strongly quadratic orbits are very near to Haumea, which also have large linear rates. In other words, there are strong correlations between the allowed linear and quadratic coefficients of physically plausible tracks. The result is a relatively small number of nonlinear shift rates that efficiently cover the desired search space (PK10); these tracks are illustrated in Figure 2. Contrasted with the orbital element motivated sampling presented here, the number of shift rates in the case of a quadratic sky motion grid search would have been much larger.

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With our bank of non-degenerate sky tracks, we can now perform the nonlinear shift-and-stack procedure. Each track corresponds to a specific set of ΔR.A. and Δdecl. values of a putative satellite relative to Haumea. We used adxy, a routine from the IDL Astro Library which uses astrometric data from the image headers, to convert these sky coordinates into on-image pixel positions, thus yielding the desired pixel shifts.

The prepared images were shifted (including fractional pixel shifts implemented by IDL's fshift) and stacked using the pixel-by-pixel median of the images as described above. In preparation for the automated search, many images were investigated in detail by eye.

To test the sensitivity of this method, artificial sources were implanted into the images with a range of random brightnesses. Their positions and rates of motion were determined by orbits randomly drawn from the same space mentioned above (but without restriction of non-crossing orbits with the known moons). The implants were generated by scaling from the actual PSF of Haumea (when brightest). This source was implanted into the images at the pixel positions corresponding to the randomly chosen orbit. A subimage of 200 × 200 pixels was used for the search: the outer reaches of this subimage have objects that are practically not moving (see Section 3.1) and any object beyond this region would have the same detectability threshold as stationary objects.

This was done for a large number of orbits, with a new set of images created for each. Stacks were generated in the exact same manner as the real images, with the same ∼35 shift rates making new median stacks for each new set of images. These stacks were inspected using the same automated search routine (IDL's find). To distingush detections of implanted sources from the detection of the three known bodies, we examined the output of the search for the sets of stacks with no sources implanted. All detections here were due to known bodies, and the positions were used to establish a mask with three regions, one for each known body, to reject detections that were not due to implanted sources. In this way, detections could be automatically classified as a recovery of an implant or as a false positive due to the known bodies. These automated classifications were extensively verified with an investigation that included searching by eye and were found to be very robust.

Due to the application of a threshhold S/N by the find routine, objects in the vicinity of Haumea, while still far enough and bright enought to be seen by eye, may be rejected by the routine itself (not our masks). The presence of the nearby primary leads to an artificially high computed background noise level, which reduces the computed S/N significantly, causing the object to appear below threshhold. Any stacks with sources at risk of being left undetected due to this effect were searched by eye by multiple coauthors, and any that were detected in this processes were considered to be recovered for the purposes of our results, shown below.

The success rates of finding implanted objects places constraints on additional satellites of Haumea: any recovered implanted source represents a satellite that we can say with reasonable certainty is not present in the Haumea system.

At a semimajor axis of ∼10,000 km, the maximum separation of a satellite from Haumea would be 6.9 WFC3 pixels. Within 7 pixels ($\lesssim 4$ PSF FWHM) of Haumea, it is very difficult to recover objects due to imperfect subtraction of Haumea's PSF. It is possible that an empirical PSF subtraction would perform better for recovering very close-in satellites, but we do not consider such an approach here. As can be seen in Figure 4, there is the expected anti-correlation between the brightness of an object that can be recovered and the separation from Haumea: close in, only brighter objects can be found.

However, there are dynamical reasons to expect that this region is nearly devoid of satellites. Due to Haumea's highly triaxial shape, the orbital region near Haumea is strongly perturbed and long-term stable orbits are difficult to maintain. According to Scheeres (1994), periods less than about 10 times the spin period are unlikely to be stable due to primary-spin-satellite-orbit resonances. In Haumea's case, this is exacerbated by the additional effects of tidal evolution and other dynamically excited satellites (Canup et al. 1999; Ćuk et al. 2013; Cheng et al. 2014b). An orbital period that is 10 times the spin period corresponds to a semimajor axis of about 5000 km (about five times the long-axis radius of Haumea). While about twice as distant as the Roche radius, for long-term dynamical stability, we consider this the inner limit.

Even if satellites were originally found in such short orbits, it is possible that long-term tidal evolution would have moved them to a more detectable distance. A detailed analysis by Ćuk et al. (2013) calls into question the idea first proposed that the satellites tidally evolved outward from orbits near the Roche lobe. While extensive tidal evolution might not have taken place, it is worth noting that scaling the tidal evolution from the properties of the other satellites (Brown et al. 2005), indicates that even for the smallest satellites we could have detected (which evolve the shortest distance due to tides), tidal evolution would have placed them near or beyond the ∼5000 km detection threshold.

There remains a range of semimajor axes from 5000–10,000 km that could potentially harbor very small undetected satellites, which would be somewhat protected from dynamical and tidal instability. By lying well within Haumea's PSF, these satellites also generally evade detection. Furthermore, some satellites would not have been detected if they had an orbital phase placing them at undetectably small distances (although this is mitigated somewhat by observations at a variety of times). Overall, it is difficult to hide stable inner ($a\lesssim 10,000$ km) Haumean satellites with radii $\gtrsim 30$ km.

At semimajor axes between about 10,000 and 350,000 km lies the region of satellites near where the other two moons are detected (at semimajor axes of 25,600 km for Namaka and 49,900 km for Hi'iaka, RB09). At this distance, contamination from Haumea is negligible and the main limitation to detecting satellites is insufficient S/N or falling beyond the edge of the image. By using the nonlinear shift-and-stack, we maximize the search depth, particularly closer to Haumea.

The search depth can be reported as relative brightness (in magnitudes and flux) and as the radius of a spherical satellite assuming the same albedo as Haumea. As is usual for such deep searches, the recovery rate is a function of magnitude (Figures 3 and 4). We reliably detect satellites at −9.2 mag (0.0002 relative brightness, radius of 10 km), our recovery rate is roughly 50% at −10 mag (0.0001 relative brightness, radius of 8 km), and our best case recovery is at −10.4 mag (0.00007 relative brightness, radius of 6 km). Following typical practice, we summarize the recovery depth using the 50% recovery rate. Note that it is possible that the albedo of the satellites is even higher; using Haumea family member 2002 TX300's measured albedo of 0.9 (Elliot et al. 2010) instead of Haumea's presumed 0.7 albedo (Lockwood et al. 2014) would imply a radius detection threshold of only ∼7 km (or ∼5 km in the best case).

While close approaches to Hi'iaka and Namaka as projected on the sky would result in a missed detection for faint objects, this is generally unlikely (even for orbits coplanar with the known satellites which are near edge-on, RB09). Close approaches to Hi'iaka and Namaka are negligibly unlikely to happen at more than one epoch,⁷ thus any missed detection would be mitigated for moderately bright objects by the non-detection of satellites in other data sets. Thus, we expect that this region of the Haumean system does not contain undiscovered satellites larger than ∼8 km in radius.

Our results compare favorably with the current state of knowledge regarding the small satellites of Pluto. From the New Horizons flyby, we now have detailed knowledge of the albedoes (about 0.5) and sizes of the small satellites: ∼10 km for Styx and Kerberos and ∼40 km for Nix and Hydra (Stern et al. 2015). As Figure 3 shows, we predict that a satellite of apparent magnitude relative to Haumea, similar to that of Hydra or Nix around Pluto (−8.7 and −9.2 mag, respectively), would fall above our detection limit. With the higher expected albedo (0.7) of Haumean satellites, we would have detected objects as large as Styx and Kerberos. We conclude that Haumea very likely does not contain small satellites similar to Pluto's.

Satellites with semimajor axes beyond 350,000 km may not have been detected in the Program 12243 WFC3 data due to the small field of view employed for the subarray observations. Other HST and Keck observations that were not as deep covered a larger area and were also searched for satellites. We estimate that satellites larger than about 40 km in radius (again assuming an albedo similar to Haumea's) would have been detected even several tens of arcseconds away by, e.g., the WFPC2 observations (with a field of view of 162''). Because the motion of satellites in this region is negligible over the relevant timescales, the shift-and-stack method is not necessary.

Using half the size of Haumea's Hill sphere at perihelion as an estimate of the full region of stable satellites (Shen & Tremaine 2008), the semimajor axis of the most distant stable satellites would be about 4.6 × 10⁶ km or 124''. About half of this volume has been covered down to 40 km in radius.

For comparison, the Program 12243 deep observations covered separations up to about 10'' around Haumea or 350,000 km. Thus, this limit on very small intermediate-range satellites corresponds to about 0.5% of the stable region radius.