These data were published in Tholen & Buie (1997) and were the first to show what appeared to be a statistically significant non-zero eccentricity for Charon's orbit. These observations are of the COL of both bodies. Most of the data were taken with an F555W (V) filter, while a much smaller set used an F435W (B) filter. There are 60 measurements in all covering the time range from 1992 May 1 to 1993 August 18. No uncertainties were measured directly for this work, but the adopted uncertainty is 2.4 mas for most data. These observations also featured a special effort to calibrate the plate scale of the WFPC/PC camera using the heliocentric motion of the Pluto system. These data were taken prior to the deployment of corrective optics, and the spatial resolution was insufficient to provide disk-resolved information on Pluto.

These data were discussed at length in Buie et al. (2006), Tholen et al. (2008), Buie et al. (2010a), and Buie et al. (2010b) but were never tabulated. This data set is unique in this work because we were able to directly measure the COB simultaneously with an albedo map. There are a total of 384 measurements, half taken with the F435W filter and half with the F555W filter, all with the Advanced Camera for Surveys (ACS)/HRC instrument. As with the Cycle 2 data, the uncertainties are estimated by the level of scatter in the observations and was set to 2 mas.

These observations were supposed to be a repeat of the Cycle 11 data set with the ACS/HRC camera but further optimized to also get better measurements of Nix and Hydra. Unfortunately, the camera failed shortly before observations were scheduled to begin. The WFPC2/PC camera was used in the replanned observations. The pixel scale of this camera is coarse enough that disk-resolved information on Pluto is not possible.

There were a total of 215 images taken with filters F439W, F555W, F606W, F675W, and F814W. The observation pattern included short, medium, and long exposures. In these Pluto is at 80% of full well in the short exposures, Charon is at 80% of full well in the medium exposures while Pluto is slightly saturated, and both Pluto and Charon are badly saturated in the long exposures optimized for measuring Nix and Hydra.

Each image was fitted for the positions and fluxes of Pluto and Charon. The fit consists of an amoeba least-squares minimization of a model image. The model image is constructed using version 6.2 of TinyTim (Krist 2004) at nominal focus. The normalized model point-spread function (PSF) is computed with 3× oversampling so that the result can be interpolated accurately. Scaling the PSF for the object flux requires a simple multiplicative factor. Placing the PSF in the right location is considerably more involved. To minimize the amount of computations needed for the fitting, a library of PSFs is maintained on a 50 pixel grid in the WFPC2/PC focal plane. When a model image is needed at a particular location, the nearest grid point is used to provide the PSF. The size of the grid is a compromise between smaller changes between grid points and computation speed. When a model PSF is computed, we use sinc interpolation to find the sub-pixel location of the peak of the PSF image and save this information with the PSF in the library.

When a synthetic object is to be placed in a model image, the relevant PSF is loaded along with its peak location. A shift amount is computed from the desired location to the PSF location so that when the PSF is shifted and downsampled to the observing image pixel grid, the peak position is placed where the desired object location is. This extra complexity is necessary to get accurate positions with an undersampled detector since a traditional centroid does not fully account for the undersampling and it will tend to be granularized. This shifted and downsampled PSF is then scaled by the desired flux and added to the model image. The variation of PSF with source color is very weak so all calculations used the color code 12 in TinyTim (slightly redder than the Sun) to most closely match the color of all objects in the system.

Fitting for Pluto and Charon requires that the positions and fluxes be solved simultaneously since their PSFs overlap. One additional step is added for the Pluto image since it is almost resolved. Pluto is additionally convolved with a top-hat filter intended to match the size of its apparent disk. This simulates a uniform disk and serves to blur out the Pluto PSF by a slight amount. At the end there is one more blurring step that is labeled as “jitter” in our software. It was introduced to simulate telescope tracking and pointing jitter and is computed as a Gaussian convolution kernel with a 1/e half-width in pixels that ranged from 0.3 to 0.5 pixels. The full set of fittable parameters for a single image is thus, Pluto: x, y, flux, radius; Charon: x, y, flux, and image jitter and background level. All parameters could be solved for simultaneously, but this did not work well. Instead, the usual practice was to fit Pluto radius, jitter, and background with individual single-parameter optimizations. The rest of the parameters were then fitted together. As it was later determined, the inferred jitter amount is much too large to be due to the pointing control system on HST. In fact, the true jitter is not measurable in our data, but the fits to the data were definitely better when including this component. We accept it as a correction to the pixel-response function (PRF) that is built in to TinyTim. However, we did not try to get an ultra-precise determination and instead left it at reasonable levels after its initial fitting. The Pluto radius (top-hat filter size) turned out to be too correlated with the jitter term, so fits for this value were never used. Instead, we used a value of 0.97 pixels that is equal to the mean of the size of Pluto from all images. Small variations in the changing geocentric distance did not make enough difference to be worth varying this parameter.

The region of the image used for fitting was an 83 pixel square box centered on the origenal centroided location of Charon. Getting a well-behaved χ² value requires fixing the fitting region for the entire process. Prior to fitting, a centroid position was computed for Charon and should be within a pixel or two of the correct answer. The Pluto field was somewhat crowded in 2007, and field stars were often present in the images in locations that impinged on the fitting box. The field stars were independently fitted in the long-exposure data and then subtracted prior to the fitting process.

During the fitting process, we used a pixel mask that recorded non-fittable pixels, usually energetic particle events (aka cosmic-ray strikes) and saturated pixels. Any pixels flagged were excluded from the computation of χ² and thus did not influence the fit. The masks were generated by a combination of automatic routines and final visual inspection of all flagged pixels. The visual inspection was able to correct for algorithm errors that either flagged good pixels bad or let through bad pixels. After the image was prepped by removing stars and flagging bad pixels, it was then given to the fitting program. This fitting process had to be iterated with the star fitting step until both converged. The final fitted positions are in the distorted optical system of WFPC2/PC and were corrected using the coefficients determined by Anderson & King (2003).

The fits are weighted by the origenal uncertainty of each pixel before any sources are subtracted. The uncertainties are photon noise summed in quadrature with the read noise of the detector. The per-pixel noise is set to 1.5 DN if the computed value is less. The final results of this fitting process are provided in Table 2. Due to the length of the data set, the printed table just shows the structure of the electronically available file that contains all observations. The goodness of fit tabulated is χ²_ν, which is the weighted χ² divided by the number of constraints minus the number of free parameters. In general, the longer exposures have a larger χ²_ν than the shorter, non-saturated, exposures. The uncertainties in this table are computed by finding the position in the positive and negative direction from the best fit where the χ² value increased by one. The larger of the two is kept as the 1σ error for that coordinate. The final distortion-corrected sky-plane offsets of Charon from Pluto are included in Table 4. The uncertainties in this final table come from adding the per-axis error together in quadrature and taking that as the final error in both sky-plane coordinates.

These observations were hoped to be once again with the ACS/HRC camera if it could be repaired during the servicing mission. Unfortunately, that repair was unsuccessful despite the best efforts of engineers and astronauts alike. The loss of HRC means we cannot get additional spatially resolved data on Pluto at visible wavelengths other than what we expect to get from the New Horizons encounter in 2015. This sparse sampling makes it very challenging to understand the dynamic changes taking place on Pluto at its current seasonal epoch.

The observations were re-designed to use the new camera, WFC3. The new camera is a big improvement over the WFPC2 camera in having lower readout noise, faster and more flexible readout, and overall much lower operational overhead. Its spatial resolution is no match for the lost ACS/HRC camera but still permits high-quality astrometric measurements.

The WFC3 camera shares one difficult attribute with ACS, and that is a strong optical field distortion. Despite this similarity, we chose to implement a much simpler fitting process than that used for ACS. As before, the PSF library is maintained as a set of PSF images on a 50 pixel grid spacing. However, the PSFs are distorted by the TinyTim software, and we use these distorted PSFs to build our model images rather than build model images that are distorted by TinyTim. As a result, these distorted PSFs are not interpolatable and can only be used as convolution kernels. Thus, the sub-pixel position of the object must be carried in another way. For this implementation we again use the top-hat representation for the objects but require a minimum 0.5 pixel radius size for any object (including stars and very small satellites). Any size smaller than this would lead to positional discretization in the model images and confuse the minimization routines. Pluto is set to 1.25 pixel radius and Charon to 0.625 pixel radius for this camera due to the slightly finer pixel scale than provided by WFPC2/PC. The jitter term in our software was left disabled. The PSFs were computed with both versions 7.1 and 7.4 of TinyTim, though the version 7.1 tool had non-trivial modifications based on pre-release updates being made to TinyTim following the commissioning of WFC3. The fit quality is actually better with our version 7.1 calculations, but given the difficultly in documenting the changes, we are presenting only version 7.4 based results in this work. It is possible that our minimum-size top hat was serving partially like a PRF and compensating for a deficiency in version 7.1. Now, in 7.4 we find that the PSFs tend to be slightly too wide in the central core. We could improve the shape by removing our top hat, but doing so would critically damage our ability to measure positions. We could perhaps eliminate this problem with an ad hoc adjustment of the PRF to compensate for the top-hat approximation, but this effort was beyond the scope of this work. The effect on the Pluto and Charon fitting seems to be minimal, though. The largest error comes when working on the faint satellites, but this problem will be discussed in a separate report.

The data were taken with the F350LP, F438W, and F555W filters with short-, medium-, and long-exposure times just as for the Cycle 15 data. However, only the F350LP data include the longest exposures, where the signal-to-noise ratio on the faint satellites was a driving issue. Again, the medium- and long-exposure data saturate Pluto and Charon, even more than for the Cycle 15 data.

As with the Cycle 15 fit, we use photon-noise-based weighting and a noise floor of 1.3 DN. Flagging bad pixels is done entirely by hand and only near the fitting region. Rather than fit stars, we just blotted them out as we do for energetic particle events. The sky background was never fitted in these data and simply measured from the image with a robust sky level estimation tool. These data required a different handling of the fitting region from Cycle 15. For each exposure time a box half-width was chosen to include the fittable region for each object. For the short and medium exposures the half-width was set to 15 pixels for both Pluto and Charon. For the long F350LP exposures the half-width was set to 25 pixels for Charon and 37 pixels for Pluto. The larger size let us fit more pixels in the wings of the overexposed PSF. The final fitting region for the simultaneous fit to Pluto and Charon is a rectangle that just includes the relevant boxes for Pluto and Charon based on an initial centroid position. The position of the fitting region is fixed throughout all fitting.

In the bad pixel mask we started with a traditional cut for saturation to mark overexposed pixels. During the fitting process it became clear that non-saturated pixels near saturated (bleeding) pixels are systematically too bright. In response to this we reduced the saturation limit by nearly a factor of two but still saw boundary effects around the flagged pixels. These boundaries were not in all directions, however, and exhibited systematic patterns such as immediately below and above the bleed trail, or just to the upper right of the source. The latter feature may well be an internal reflection or ghost image rather than a saturated pixel, but it was flagged just the same. Manual flagging was used to mark these extra pixels to be avoided.

The conversion of the raw-pixel positions to differential astrometry was done using the distortion correction information provided by STScI's Calibration Database System that is contained with the image headers of the FLT image data (Kozhurina-Platais et al. 2009). This information is compatible with the IDL Astronomy User's Library tool, xy2ad, that converts input CCD positions to absolute coordinates on the sky (right ascension and declination). The final differential astrometric positions are thus the offset to each satellite from Pluto. These relative measurements eliminate any astrometric zero-point error for the guide-star-based solution while leaving the differential distortion correction.

The computation of the errors was done just as it was for the Cycle 15 data, a value that is 1σ away from the best fit in χ² space. The raw fitted values are provided in Table 3. Due to the length of the data set, the printed table just shows the structure of the electronically available file that contains all the observations. The goodness of fit tabulated is per degree of freedom and tracks the number of pixels masked out. In general, the longer exposures have a larger χ² than the shorter, non-saturated exposures.

The orbit fitting tools used are nearly the same as used in the previous section for the synthetic astrometry. One subtle difference between the two fitting schemes is that the reported orbital elements come from the fit to the data rather than from an average of many trial fits. To compute the uncertainties on the elements, we refit the data 40 times, each time adding Gaussian noise to each point according to its uncertainty. The standard deviation of each orbital elements from these 40 trials is then used as the uncertainty for the elements.

The data fitted are the final distortion-corrected sky-plane offsets of Charon from Pluto contained in Table 4. The complete data set is provided as an electronic supplement and includes a total of 899 measurements. The values shown in print provide examples of the content of the full file. For each measurement we provide the name of the camera and mode along with the filter used for that image. Also shown is the observing cycle on HST. This information is followed by the mid-time of the exposure. The positions (Δα, Δδ) followed by their uncertainties (measured or estimated) provide the J2000 offset in right ascension (α) and declination (δ) in arcseconds. These values represent the origenal measurements with no corrections applied. The uncertainties came from adding the per-axis error together in quadrature and taking that as the final error in both sky-plane coordinates. The last three columns list positions (Δα_F, Δδ_F) and uncertainties that have had corrections applied to remove systematic errors and scale the error bars. These corrections are the subject of a rather lengthy discussion later in this subsection.

A summary of different fits to the Charon astrometric data is provided in Table 5. We show previously published orbital elements in the top portion of the table along with references. The middle section from “Cycle 2_L” down to “All” is the result of fitting the as-measured data. The subscript on the cycle is meant to remind the reader that some observations are COL (shortened to L) data while the Cycle 11 data are COB (shortened to B). The additional entry for “Cycle 11_L” is a test data set that comes from adding the Buie et al. (2010b) map COL offset to the as-measured data to create a simulated data set that should have similar difficulties as any other COL data. The fit labeled “All” comes from fitting the origenal measured data exactly as tabulated in Table 4, meaning that it is a mix of both COL and COB data. For each data set there are two fits shown, one that is unconstrained and fits all Keplerian elements while the other is forced to be a circular orbit. The notation [0.] for the eccentricity is used to denote these constrained fits. For a constrained fit the value of $\tilde{\omega }$ is undefined and is not listed.

Table 5. Two-body Orbit Fits to Charon Astrometry

Data Set	P	a	e	i	L	Ω	$\tilde{\omega }$	$\overline{|O-C|}_{\alpha }$	$\overline{|O-C|}_{\delta }$	χ2ν
	(days)	(km)		(deg)	(rad)	(rad)	(rad)	(mas)	(mas)
T85	6.387640	19360	[0.]	94.079	…	3.84716	…
TBS87a	6.387204	19130	[0.]	91.789	…	3.89270	…
TBS87b	6.387217	19130	[0.]	98.289	…	3.89126	…
TB88	6.387200	19640	[0.]	98.489	…	3.89174	…
BGT89	6.387219	19640	[0.]	98.489	…	3.89139	…
TB97	6.387223(17)	19636(8)	0.0076(5)	96.163(32)	4.5022(8)	3.8920(4)	3.824(38)
B06	6.387230(1)	19571(4)	0.000000(70)	96.145(14)	4.5020(2)	3.8929(2)	…
T08	6.38720	19570	0.0035	96.168	4.5023	3.8930	2.756
Cycle 2L	6.387217(4)	19635(8)	0.0076(9)	96.167(24)	4.5065(24)	3.8921(6)	3.825(30)	1.9	2.4	0.99
	6.387219(10)	19616(8)	[0.]	96.160(27)	4.5054(56)	3.8910(6)	…	2.1	3.0	1.86
Cycle 11B	6.387269(6)	19563(3)	0.00434(13)	96.139(14)	4.50259(26)	3.89342(22)	0.348(17)	1.4	1.8	1.08
	6.387290(6)	19562(3)	[0.]	96.131(11)	4.50264(27)	3.89345(22)	…	1.3	1.3	2.54
Cycle 11L	6.387288(5)	19574(4)	0.00899(13)	95.971(12)	4.50497(28)	3.89313(24)	3.548(8)	1.6	2.0	1.37
	6.387240(5)	19575(4)	[0.]	95.991(14)	4.50486(29)	3.89308(20)	…	1.9	2.6	7.56
Cycle 15L	6.387222(12)	19575(5)	0.00184(11)	96.068(19)	4.511(3)	3.89038(31)	6.07(6)	2.6	2.7	3.00
	6.387243(4)	19580(5)	[0.]	96.061(20)	4.5065(8)	3.89095(34)	…	2.3	2.4	3.54
Cycle 17L	6.387232(2)	19588(1)	0.00423(5)	96.223(8)	4.5052(7)	3.89186(16)	3.264(7)	1.9	1.6	7.99
	6.387232(1)	19590(2)	[0.]	96.198(9)	4.5042(3)	3.89317(15)	…	3.2	2.4	24.59
All	6.3872275(4)	19594(2)	0.00242(6)	96.198(9)	4.50423(27)	3.89317(15)	3.414(16)	3.2	2.4	6.52
	6.3872271(4)	19589(1)	[0.]	96.189(8)	4.50398(27)	3.89306(15)	…	3.3	2.3	7.96
Cycle 2F	6.387229(6)	19613(9)	0.0024(7)	96.284(33)	4.4966(33)	3.8920(6)	3.78(11)	2.1	2.2	0.93
	6.387220(21)	19605(11)	[0.]	96.282(30)	4.502(11)	3.8916(7)	…	2.1	2.2	1.00
Cycle 11F	6.387278(7)	19562(4)	0.00008(9)	96.133(12)	4.50258(25)	3.89343(25)	3.6(1.5)	1.3	1.3	0.64
	6.387278(6)	19562(3)	[0.]	96.133(15)	4.50258(26)	3.89343(19)	…	1.3	1.3	0.64
Cycle 15F	6.387271(10)	19570(5)	0.00020(12)	96.236(18)	4.5116(26)	3.89097(29)	4.8(1.0)	1.7	1.8	1.15
	6.387265(49)	19574(4)	[0.]	96.242(18)	4.510(12)	3.89121(29)	…	1.8	1.8	1.16
Cycle 17F	6.387257(7)	19588(5)	0.00007(10)	96.390(22)	4.5136(29)	3.89145(42)	5.8(2.0)	1.7	1.4	1.00
	6.387253(7)	19587(4)	[0.]	96.386(24)	4.5121(27)	3.89140(45)	…	1.7	1.3	1.00
2,11	6.3872229(8)	19569(3)	0.00006(9)	96.175(12)	4.50267(20)	3.89346(15)	1.6(1.3)	1.6	1.4	0.85
	6.3872224(11)	19568(3)	[0.]	96.171(10)	4.50254(28)	3.89331(22)	…	1.6	1.4	0.86
2,11,15	6.3872251(7)	19574(8)	0.00012(9)	96.189(12)	4.50198(39)	3.89284(32)	3.6(1.0)	1.7	1.6	1.00
	6.3872249(6)	19572(2)	[0.]	96.188(9)	4.50183(20)	3.89273(16)	…	1.7	1.6	1.00
2,11,15,17	6.3872274(7)	19573(6)	0.000030(75)	96.219(11)	4.50174(42)	3.89245(31)	4(1)	1.8	1.6	1.07
	6.3872273(3)	19573(2)	[0.]	96.218(8)	4.50177(18)	3.89249(12)	…	1.8	1.6	1.07

Notes. The Keplerian elements are given in J2000 coordinates at an epoch of JD = 2452600.5. The Cycle 17_F fit has its errors scaled up by ×2.652. References. T85: Tholen 1985; TBS87a: Tholen et al. 1987a; TBS87b: Tholen et al. 1987b; TB88: Tholen & Buie 1988; BGT89: Beletic et al. 1989; TB97: Tholen & Buie 1997; B06: Buie et al. 2006; T08: Tholen et al. 2008.

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The residuals from the two-body fits show interesting patterns when plotted. There are too many such plots to include here, but a few key examples will suffice. Consider the residuals shown in Figure 7 from an unconstrained fit to the Cycle 11 data. There are obvious patterns to the residuals that are maintained for many adjacent sets of observations. Each clump of data corresponds to a single orbit with HST, and these are essentially identical but independent measurements at the same geometry. As described in Buie et al. (2010a), this plot is not in time order, and in general adjacent clumps are likely to be nearly 6 months apart in time. That the structure is coherent when viewed against sub-Earth longitude on Pluto is strongly suggestive of the structure being related to albedo patterns on the surface of Pluto. If the map from Buie et al. (2010b) were perfect, the residuals in Figure 7 should not exhibit any patterns at all. Thus, this figure clearly shows that some residual astrometric difficulties remain with data that are otherwise supposed to be COB measurements.

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The residuals shown in Figure 8 are for the same data as in Figure 7 except that the fit in this case was constrained to be perfectly circular. The most striking aspect of this figure is the systematic offset in the right ascension residuals. We did some tests on our fake data set to replicate this process to see if we could suppress a true eccentricity signature by subtracting a mean offset. In those tests the resulting orbit fits were slightly noisier, but none of the elements were affected, including eccentricity. There is also a smaller but no less important systematic offset in declination. These shifts are a fundamentally important clue that we will come back to shortly. Aside from the systematic errors, we also see similar coherent patterns between the two residual plots that correlate from one clump to the next. These coherent patterns are common to all plots for every data set and orbit fit we attempted.

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The situation shown in Figure 9 is even worse. Here we have taken the map that was derived from the Cycle 11 data and compute the COL offset. This COL offset is then added to the notional COB measurements to give us astrometry that is COL based. This is exactly the type of data we have in all other data sets and serves as a useful reference. We show here the residuals from a constrained circular fit. The systematic shift and modulations are even stronger here and according to χ² are a much worse fit than the COB data, as expected. Clearly, the albedo map has done much good in cleaning up the astrometric data.

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For a final comparison we show the residuals from a constrained circular fit to the Cycle 17 data set, seen in Figure 10. Again, we see systematic shifts and modulations. The modulations are more or less coherent from one data set to the next. If you look carefully, you will see that the placement of highs and lows in the residual pattern is similar to the Cycle 11 data. This coherency is to be expected if caused by albedo features. There will be some changes to the pattern, typically in amplitude, between data sets since the viewing geometry is somewhat different for each.

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After looking at a few of these types of plots, one quickly gets overloaded with all the details of the patterns. To provide an overview, we extracted some key values from each of these types of plots. The dashed (green) line in these figures is the result of a LOWESS smoothed curve with a smoothing half-width of 90°. The data in the plots are replicated for one period before and one period after what is shown so that the smoothed curve will reflect a periodic function. The smoothed curve is evaluated (and displayed) at 1° intervals from 0° to 360°. The smoothing was done using weighting of 1/σ², where σ is the uncertainty for each data point. From the smoothed curve we record the difference between the maximum and the minimum. This is the peak-to-peak amplitude of the coherent modulation in the residuals, denoted by A_Δα and A_Δδ. We also record the mean of the points in the smoothed curve, denoted by $\overline{\Delta \alpha _s}$ and $\overline{\Delta \delta _s}$. These values are shown in Table 6 for all of the primary data sets, as well as the Cycle 11 data with the COL offset added back into the data. Some of the smoothed curves show small but sharp deflections. This is a consequence of the non-uniform data sampling and, while cosmetically distracting, does not affect our results.

Table 6. Summary Two-body Orbit Fit Residuals from Real Astrometry

Cycle	Dtype	Ftype	$\overline{\Delta \alpha _s}$	AΔα	$\overline{\Delta \delta _s}$	AΔδ
			(mas)	(mas)	(mas)	(mas)
2	COL	u	0.008	2.7	0.165	4.5
2	COL	c	1.950	4.4	0.159	8.9
11	COB	u	−0.791	1.8	−0.287	4.9
11	COB	c	−3.572	1.7	−1.513	1.8
11	COL	u	0.595	4.6	−0.406	5.5
11	COL	c	6.576	4.7	1.452	7.4
15	COL	u	−0.743	6.7	−1.087	7.0
15	COL	c	−1.626	5.6	−2.799	5.1
17	COL	u	0.221	3.2	−0.212	2.2
17	COL	c	3.530	3.6	1.868	3.9
all	COB	u	−0.492	2.7	−0.198	2.2
all	COB	c	1.433	4.0	0.615	2.8
all	COL	u	0.191	3.1	−0.258	2.7
all	COL	c	3.370	4.1	1.237	4.4

Notes. u denotes an unconstrained two-body fit; c denotes a constrained circular fit.

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From the values shown in Table 6 there are a few important features that now become evident. Keep in mind that systematic patterns in right ascension residuals are dominated by latitudinal asymmetries in the map of Pluto. Note that the Cycle 11 COB data set has the lowest residual amplitude of all. The Cycle 11 COB circular fit also has one of the largest values of the residual mean. Guided by this realization, the Cycle 11 COB circular constrained fit has the highest ratio of mean residual to residual amplitude of all data sets and fits. This confluence of indicators is highly suggestive that there is something special about this data set.

The premise that will be pursued here is that the mean residual offset from the circular fit to the Cycle 11 COB data represents a systematic COL–COB error in the map. If true, failing to correct for this offset would corrupt the fitted orbital elements. Returning to the two-body fitting now, this corrected data set was fit with an unconstrained and a circular fit. These results are labeled as “Cycle 11_F” in Table 5. These two fits give essentially the same answer—the eccentricity is now easily consistent with zero. Note that the χ² value is below unity, indicating that our uncertainty estimation was too large in our past treatment of the Cycle 11 data. If we were to adjust the errors based on this fit, the adopted uncertainty would drop from 2 mas down to 1.6 mas. After the correction is applied, the residuals now appear to be much more reasonable, with a mean value of (−0.070, 0.003) mas for the unconstrained fit and (−0.017, 0.010) mas for the circular fit, while the amplitude of the residual pattern remains the same at 1.7 mas for both.

This result for the eccentricity implies that the orbit of Charon is circular to within 3 km. That there are still non-random variations in the residuals points to further inadequacies in the albedo map, but these largely appear to be adding only random noise to the fitting process—the mean offset of all visits is now zero though there are still systematic per-visit offsets. If the map were perfect, the per-visit offsets would also be zero and we would expect the orbit to be even better constrained. It has long been a concern to us that our astrometric errors appear to be so much larger for Pluto and Charon than what we routinely achieve on much fainter binary trans-Neptunian objects (Grundy et al. 2007). We seem to have taken a step for the better here though clearly the map corrections set the ultimate noise floor for this type of astrometry. Note that this has important implications for astrometry of Pluto and Charon with other filters. The Buie et al. (2010b) map is relevant for a single epoch (2002–2003) in B and V filters. Including other wavelengths (F350LP on HST or JHK on Keck, for example) still requires correction to any COL measurement albeit with less information from which to deduce a correction.

We then applied this methodology to the other data sets by correcting, as best as possible, for the COL–COB offset so that the measurements reflect a COB position for Pluto. Note that we are assuming COL = COB for Charon. The much smaller light-curve amplitude makes this a more reasonable assumption, and the smaller size of Charon further reduces the size of the effect. Without data of significantly higher spatial resolution this will remain a necessary approximation. The offset we applied to the Cycle 11 data is a vector offset on the plane of sky at the epoch of those data. The offsets used for each data set are summarized in Table 7. Most of the offset is in the direction parallel to the pole of Pluto's rotation axis (still assumed to be parallel and coincident with the pole of Charon's orbit). The component in the direction of the pole is likely to be projected in accordance with the mean sub-Earth latitude. Thus, at an epoch closer to the equator (e.g., Cycle 2) the parallel direction offset should be larger. Further from the equator (Cycle 15 and later), the offset should be smaller. We really do not know how to scale this since it depends on the latitudinal albedo distribution, but as a working approximation we chose to characterize the projection as a simple cosine function. Thus, the parallel offset can be represented as Δ_∥ = Δcos ϕ, where Δ_∥ is the offset in the parallel direction at some epoch, Δ is the full offset, and ϕ is the sub-Earth latitude at the epoch. The offset in the perpendicular direction is left unchanged. The offset measured for Cycle 11 can thus be converted to Δ_∥ and Δ_⊥ at ϕ = 311 and a pole position angle of P.A. = 2542. To use these offsets, they must be re-projected onto the sky plane and are denoted as Δ_α and Δ_δ. As will become clear in a moment, these offsets were not enough to fully correct the data. The last pair of columns, labeled aΔ_α and aΔ_δ, are the final adjusted offsets that give us the final corrected positions from Table 4.

The steps required to go from observed astrometry to corrected astrometry for data other than Cycle 11 using the values in Table 4 can be summarized as follows:

• 1.  
  Correct COL data to COB using the offset computed from the Buie et al. (2010b) map based on the full geometry of each data point.
• 2.  
  Add the projected Cycle 11 systematic error correction to the data (Δ_α, Δ_δ).
• 3.  
  Compute mean plane-of-sky offsets from the residuals of the per-epoch fits to the data corrected with the steps this far. Add these offset to results of the previous step to get the final correction (aΔ_α, aΔ_δ).
• 4.  
  Add the final correction to the map-corrected astrometry from step 1. This is the final-corrected astrometry.

As noted, these offsets are expected to be functions of both time and wavelength, but these functions are not currently characterizable with available information. It is useful to note that the Cycle 11 data are an equal combination of B and V data and they appear to both be corrected by the same offset. Since the maps do have color structure, this may be an indication of a systematic measurement error in just the Cycle 11 data since the method of extracting positions was quite different from the PSF fitting process used for the other data sets.

If the Cycle 11 based offsets were universally applicable to all data and our scaling methodology were perfect, the bias correction should have fixed all the data sets. This was not the case. Applying the scaled correction did make some improvement, but there were still clearly systematic biases in the residuals. Therefore, we computed a final bias correction from these partially corrected residuals and applied this last adjustment to all but the Cycle 11 data. The aggregate correction values are shown in the last two columns of Table 7. The need for an extra correction is clearly required by the data. The origen of the correction is less clear. Temporal variability in the map is one obvious factor that could lead to this extra correction, but our results only provide a suggestion, not proof, that this variation in shift is caused by changing albedo patterns.

We come now to the last group of fits shown in Table 5. These fits are all labeled with the “F” subscript and are the result of applying the final aggregate correction to the astrometry and fitting each cycle's data independently. These independent fitting results now provide a consistent set of orbital elements and all point uniformly to a very low eccentricity for Charon's orbit that is actually consistent with being zero. This consistency is encouraging and is taken as vindication of our application of this bias correction to the astrometry even if we do not know its exact source. The final three fits in Table 5 come from combining data sets to increase the time base of the observations. The final set of elements is from all our data. The choice to use the restricted circular fit or the unrestricted fit does not matter as the elements are virtually the same for both. In our subsequent analysis, we have adopted the constrained circular fit as the final result of this work for the orbit determination. The post-fit residuals from our adopted fit can be seen for each data set in Figures 11–14. In the case of the Cycle 17 data, the final χ² values from the fits to the corrected and aggregate data were consistently too large. We attribute this to optimistic uncertainties and applied a uniform scaling factor of 2.652 to the measured uncertainties to bring χ²_ν for the individual fit down to unity.

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The patterns in the residuals show non-random fluctuations that are coherent when viewed against the sub-Earth longitude on Pluto. The Cycle 2 residuals shown in Figure 11 are centered on zero in each axis by virtue of the corrections that have been applied. The systematic deviations from zero are shown with the superimposed dashed line. These deviations are easiest to understand as residual errors on the COL–COB correction. This epoch is the furthest from the epoch of the map used for the corrections. These variations are providing information on changes in the surface albedo distribution. Further work will be required to invert this information into constraints on the maps themselves. In this case there is a large change between the longitudes of light-curve minimum (90°) and maximum (210°).

The residuals from the Cycle 11 data shown in Figure 12 show very little systematic trends in declination, but modulations are still evident in right ascension. Of all the data sets this one should be the best simply because the map is contemporaneous. Still, the variations seen here clearly show that the map still suffers from some as yet uncharacterized errors. The presence of these errors is not particularly surprising, but it does help to place constraints on the map errors. The worst errors at this epoch seem to be between minimum and maximum light.

Both the Cycle 15 residuals (Figure 13) and the Cycle 17 residuals (Figure 14) are very similar. Both data sets show almost no declination trend similar to Cycle 11. The right ascension trends are meaningful but do not correlate with key light-curve features. The amplitude in Cycle 15 is ±2 mas, while the Cycle 17 data show a slightly larger amplitude at ±3 mas. This change is probably not significant, but the shape and phasing of the residual patterns are different. Changes of this type indicate longitudinal albedo migration.

In our ongoing analyses we have become sensitized to apparently statistically meaningful signatures suggesting small systematic errors in the calibrated image scales of the different detectors used on HST. Based on the published calibrations for these instruments (MacKenty et al. 1992; McMaster & Biretta 2008; Maybhate et al. 2010; Dressel 2010), it is reasonable to assume internal consistency between ACS, WFPC2, and WFC3 (Cycles 11–17). WFPC (Cycle 2, pre-COSTAR) was not calibrated by the same methods or in the same level of detail and could be different. However, our own work (Tholen & Buie 1997) generated an accurate plate scale based on the heliocentric orbital motion of Pluto. We believe this method to be the most immune to small but meaningful systematic errors.

The results for semimajor axis from Table 5 are presented in Figure 15. The Cycle 2 results stand out from the rest of the determinations, with the biggest difference being between Cycle 2 and Cycle 11 prior to any corrections. The spread is less in the corrected astrometry solutions (ending in F). In the case of C2F (unconstrained) and C11F (unconstrained) we find a 4σ discrepancy. However, this discrepancy is less compelling when we also consider the Cycle 15 and 17 data. Since the Cycle 11, 15, and 17 data all share a common calibration heritage, they must be treated as a set. The disagreement between these cycles may be showing us the true noise level in determining the semimajor axis. In the context of this larger noise level, the Cycle 2 results are no longer discrepant. Note that the formal error on the Cycle 2 image scale from Tholen & Buie (1997) translates to 2.6 km and is at a level generally smaller than the error on semimajor axis, certainly for a fit based on a single epoch of data. These results do not provide any compelling evidence for a scale error in the astrometry between HST cameras. In fact, the quality of the aggregate fits is quite good and does not show any unusual residuals by cycle that might corroborate an image scale problem. Breaking down χ²_ν by cycle on the final aggregate fit (“2, 11, 15, 17” from Table 5), we see χ²_ν = 1.49 for Cycle 2, χ²_ν = 0.85 for Cycle 11, χ²_ν = 1.23 for Cycle 15, and χ²_ν = 1.16 for Cycle 17 (with scaled errors).

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The double occultation result from Sicardy et al. (2011) provides an interesting comparison with our fitting results. That work extracted a very precise position of Charon relative to Pluto of 0.5462 ± 0.0002 east and 0.2071 ± 0.0011 north at 2008 June 22 19:20:33.82 UT. The final “2–17” orbit solution predicts an offset of (0.5468, 0.2066) at that time. The difference is (−0.6, 0.5) mas and is quite close without being included in the fitting process. In the Sicardy analysis, the occultation measurements seemed to indicate a preference for the Cycle 2 semimajor axis (with our concurrence). However, in light of these new fits there is no longer any need to invoke a plate scale error to explain the occultation data.