OBSERVATIONS AND ANALYSIS OF MUTUAL EVENTS BETWEEN THE URANUS MAIN SATELLITES^*

Besides the mutual events among the Jupiter Galilean moons, up to now the only other observed intersatellite occultations and eclipses were those of the Saturn system in 1995 (see, for example, Aksnes et al. 1984; Franklin et al. 1991; Arlot et al. 1992, 1997; Thuillot et al. 2001). The Jovian observations were used to improve the parameters of the dynamical theories of the Jupiter satellites (see, for example, Aksnes & Franklin 2001; Lainey et al. 2004). From the Saturn events, very precise satellite positions were obtained (Vasundhara et al. 2003; Noyelles et al. 2003).

Every 42 years, the Earth and the Sun pass through the orbital plane of the Uranus main satellites. The last Uranus equinox was in 1966. From 2007 to 2009, the Earth and the Sun are once more crossing the equatorial plane of Uranus. Since it will not occur again until 2050, this is a unique opportunity to observe mutual events among the Uranus five main satellites: Miranda, Ariel, Umbriel, Titania, and Oberon. Such observations can improve the ephemeris of the satellites and help us to map their surfaces (Christou 2005; Arlot et al. 2006).

Concerning the ephemeris, its match with contemporary intersatellite astrometric measurements reaches 50 mas for the four larger Uranus satellites and 100 mas for Miranda (Jones et al. 1998; Veiga & Vieira Martins 1999; Shen et al. 2002; Veiga et al. 2003). On the other hand, the errors on the determination of the centroids by modern techniques are as small as 20 mas. This evidences the presence of systematic errors and points out the need for improvements on the parameters of the dynamical models. Since the accuracy of the so-called astrometric photometry can be better than 10 mas, the observation of mutual events is a powerful tool to meet these requirements. Furthermore, in the determination of the parameters of the current theories of the Uranus system, most of the used astrometric positions of the satellites were obtained with their orbital planes faced-on. Now, in contrast, the orbital planes are edge-on, a scenario that favors the determination of inclinations.

Concerning the satellite surfaces, a systematic mapping was made by Voyager 2 in 1986 (Smith et al. 1986). However, at that time, their subsolar points were seen from near the South Pole. Now these points are near the equator and so the regions of the northern hemisphere are currently visible.

In order to provide precise predictions for the Uranus events, a number of tables were published by Christou (2005) and Arlot et al. (2006). These predictions are based on the analytic model GUST 86 (Laskar & Jacobson 1987) and on the numerical model LA06 (Lainey 2008). Using those tables, we have successfully observed seven foreseen events visible from the Laboratório Nacional de Astrofísica, Itajubá, Brazil (IAU code 874). Note that for the current Uranus equinox, only three events have been analyzed and published so far at the time of this writing. One was an occultation of Umbriel by Oberon (Hidas et al. 2008), another was an eclipse of Titania by Umbriel (Arlot et al. 2008), and a third was a predicted occultation of Oberon by Miranda which in fact was not detected (Birlan et al. 2008). A report on two other occultations—Titania and Ariel by Umbriel—is also in press (Miller & Chanover 2009). The one involving Ariel was also observed by us (see Section 4).

In the following, we describe our observations in Section 2 and the reduction procedures in Section 3. The results are presented in Section 4 and discussed in Section 5. In the Appendix, we describe in detail our general model for occultations and eclipses, including the expressions and derivatives used for fitting the observed light curves.

The observations were carried out with the 1.6 m, f/10, Perkin-Elmer reflector with a Ritchey–Chretien optics located at the Laboratório Nacional de Astrofísica, Itajubá, Brazil (λ = +45°35', ϕ = −22°32', h = 1860 m, IAU code = 874). The used detector was a thin, back-illuminated EEV CCD (model 02-06-1-206) with 385 ×  578 pixels² of 03 (22 μm). A near-infrared filter with a passband of 7200–1000 Å was used to minimize the Uranus scattered light. This filter associated with the CCD color sensitivity gives an effective passband of 7200–9000 Å which cuts the blue light from the planet (Karkoschka 1998). The CCD electronics allows for simultaneous integration and readout tasks during sequential exposures. Yet, to completely remove the time loss in between the exposures due to readout overhead, it was necessary for each event to optimally define a small window on the CCD array, so as to include just the two involved targets plus another satellite as flux calibrator. The planet was also integrally included in to allow for the digital coronagraphy of the images (see the following section).

From the foreseen events (Arlot et al. 2006), we successfully observed the light curves for five occultations and two eclipses. The predicted visibility of the recorded events is given in Table 1. They are the event date, the midtime (in UT), the event type (occ for occultation and ecl for eclipse, T for total and P for partial), the target satellites of the event (A, Ariel; U, Umbriel; T, Titania; O, Oberon; M, Miranda), and the flux drop (R band) of the occulted or eclipsed satellite. Observational information is also presented in this table: the solar phase angle, the distance to the planet center, the duration of the event, the mean zenithal distance, the exposure time, the number of fitted points (CCD frames), and the seeing.

Zenithal distances were always smaller than 60°, being smaller than 30° for four of the events. Seeing was smaller than 15 for three events. Exposure time varied from 1 to 3 s in a compromise between the signal-to-noise ratio (S/N) of the satellites' fluxes as opposed to the time resolution and to the contamination of the scattered light, which depends on the distance of the satellites from the planet and on the sky transparency. Acquisition started several minutes before the predicted start time of the event and continued for several minutes after its end. In all cases, a good sampling in time resolution was achieved with a high number of frames obtained during the event itself.

All five larger Uranus satellites, including Miranda, were eventually recorded on an occultation or on an eclipse. Among the occultations, only Titania was not observed. Among the eclipses, neither Umbriel nor Oberon was observed. The satellite distances from the Uranus center varied from 34 to 14'' (Uranus radius was 2''). Event durations varied from 2.5 to 14.5 minutes. Solar phase angles were about 15 for all the events but the last one in the year. The predicted types of all the events were confirmed by the analysis of the light curves, as will be seen in the following sections.

In the following, we describe the digital coronagraphy applied to the images, the differential aperture photometry, and the fit of the occultation/eclipse model to the observed light curves. The software routines used for these tasks are part of the photometric module of the astrometric/photometric reduction package Platform for Reduction of Astronomical Images Automatically (PRAIA; Assafin 2006; Assafin et al. 2007).

3.1. Digital Coronagraphy

One of the main difficulties in the reduction of the images is the contamination of the scattered-light halo surrounding Uranus over the nearby satellites. Many processes were tested before the campaign of the mutual events (Assafin et al. 2008) and finally a digital coronagraphy method was adopted. This method consists of computing the bright source light profile within a given area of the digitized field. After this is done, the original image is subtracted from that profile and a new, digitally coronagraphed image is obtained.

The procedure is as follows. The count estimates for the light of the bright source are calculated for each pixel, at their exact geometric center. For each pixel (i, j), the distance d from its geometric center to the (xc, yc) central coordinate of the bright source is recorded. Then a ring of 1 pixel width, radius = d, and center (xc, yc) is established around the bright source center. Next, a histogram of the counts is produced based on the counts of each ring pixel (m, n). The (m, n) counts are weighed according to the small displacements between the geometric center of the (m, n) ring pixels and the exact ring circumference of radius = d. The mode of the histogram is then recorded as the best count estimate for the source light at the pixel (i, j). The procedure is repeated until all pixels (i, j) within the working area around the source are processed. The obtained source light profile is then subtracted from the original image to obtain the coronagraphed image of the field. The more accurate the (xc, yc) center is, the smaller the residual scattered light originating from the bright source becomes, and the better the coronagraphy result is.

This accurate centering is accomplished by applying the above procedure for an array of (xc, yc). First, an initial (xc, yc) estimate is obtained, based on a two-dimensional circular Gaussian fit. Then, starting from an array of 3 × 3 pixels around this bright center, we determine the pixel, then the subpixel (xc, yc) coordinates that give the least scattered residuals around the bright source figure. The procedure is repeated until the scattered residual does not change by more than a given convergence value (adopted as 1%). Of course, a better initial (xc, yc) estimate and a better computation of the scattered residuals are obtained if the entire planet image is present in the frame, as in our case. After we compute the most accurate center for the bright source by this method, we use the final derived coordinates (xc, yc), and apply the coronagraphy procedure once more; this time over an area large enough to encompass the objects of interest, in our case the Uranus satellites.

One should note that one advantage of the process is that it is relatively insensitive to the presence of diffraction spikes, bright objects nearby the source (like satellites), saturation artifacts in the source image or even to the existence of anomalous darker regions. This is because higher or lower count pixels will have low frequencies along the rings and thus will not affect the mode of the count histograms.

In Figure 1, we present a sample image of the August 14th occultation of Oberon by Umbriel. From left to right, we display the original image, the image of the computed planet brightness profile, and finally the output image digitally coronagraphed.

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3.2. Photometry

3.3. Modeling of Occultations and Eclipses and Fitting of the Observed Light Curves

The light curves of all the observed events are plotted in Figure 2. The differences between the observed points (normalized fluxes of target satellites) and the fitted curve are also shown. For display purposes, only the fraction of the curve encompassing the duration of the mutual event is presented.

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One distinct set of light curves comprises the three occultations from August. These events had long durations (more than 5 minutes) and involved the satellites Ariel, Umbriel, and Oberon, at distances from Uranus of about six times its radius. They were observed in good conditions of seeing (2'' or smaller) and with satisfactory exposure time (3 s). The fits were very good with the normalized χ² smaller than 0.05. The errors in the central time, impact parameter, and relative speed were, respectively, about 1 s, 6 km, and 0.06 km s⁻¹ or less. The occultation of Ariel by Umbriel on August 19th was also independently observed by Miller & Chanover (2009). They have obtained parameters and errors quite similar to our fits for this event.

Another set comprises the three light curves including Miranda. Events including this body will always be more difficult because of its relative faintness and proximity to Uranus as compared with the other satellites. The durations of the events with Miranda were always smaller than 3 minutes and were always within three planetary radii of Uranus. Seeing conditions varied from very good (08) to regular (26). The observation made on October 8th with seeing 26 was a total occultation of Miranda by Ariel. Its fitted impact parameter had an error of 82 km, which is not bad, but even so was 10 times larger than the typical errors obtained for the other events. The event on October 12th presented a very small but clear flux drop, successfully observed and measured due to very good seeing conditions (08).

The observation on November 28th was an eclipse of Titania by Ariel. The sky condition of this event was poor with seeing 38. The points of the light curve are very scattered and the errors in the determination of all parameters are relatively large with respect to the other fits.

The model parameters from the fittings of all seven observed events are presented in Table 3, together with their predicted values. In this table, "A" indicates our results. Predicted GUST86 (G) parameters were taken from Arlot et al. (2006) and from Christou (2005). LA06 (L) parameters were extracted from Arlot et al. (2006). The fixed albedo ratios used were extrapolated from Karkoschka (2001) to match the solar phase angle and filter passband of the observations. Also listed are the central time of the event, the impact parameter and the relative speed. For the three fitted parameters, their errors are presented as well as the mean error of the fit (i.e., the standard deviation of the observed minus fitted curve residuals) and the normalized χ² of the fitted curve. For comparison purposes, the absolute and apparent radii of the satellites as taken from the Voyager 2 spacecraft (Thomas 1988) are also given at the bottom of the table.

As can be seen from Table 3, all curves were satisfactorily fitted as the normalized χ² was always smaller than 1. The statistical significance was not smaller than 0.99 for all fittings. The relative speed and the impact parameter were fitted with errors ten times smaller than their values. In all events, almost all the values found are closer to those predicted from LA06. Even so, the differences are still significant, being frequently much larger than three times the parameter errors.

We observed and reduced seven mutual events of the Uranus satellites and compared the results with two different predictions. Like in Hidas et al. (2008), we find that our observations are in closer agreement with LA06, with offsets not larger than 200 km, thus within the LA06 estimated errors of a few hundred kilometers (Lainey 2008). However, considering the small errors obtained for the dynamical parameters, we conclude that our results can be used to further improve the ephemerids of the main satellites. In fact, the errors were 2.9 s for the central time, 7.6 km (0.54 mas) for the impact parameter (excluding the event on October 8th), and 0.18 km s⁻¹ (0.013 mas s⁻¹) for the relative speed.

These results were achieved due to two important reduction steps, carried out by the use of the PRAIA reduction package. The first one was to perform a digital coronagraphy of the images to flatten the sky flux regardless of the proximity to the planet. The second one was to use a simple model to describe partial and total occultations and eclipses. In fact, with the fitting of the model to the light curves we could evaluate the correlations between the parameters and determine that the correlation between impact parameter and albedo ratio was about 95%. So, like in Hidas et al (2008), we used albedo ratio values extrapolated from Karkoschka (2001). Nevertheless, these values may not yet represent perfectly well the actual albedo ratios since the conditions of our observations, such as wavelength passband, solar phase angle, and the part of the surface observed of every satellite, were not the same as those of the Karkoschka observations.

Our results can be improved if we determine directly the albedo ratios from the images obtained before and after the events. In fact, for almost all events, we observed the Uranus system for many hours. Since the rotation velocity of the satellites is very small, the photometry of the satellites—once photometrically resolved—should yield albedo ratios very similar to those of the corresponding mutual event.

Our light curve analytic model also allows for the evaluation of local albedo variations. In fact, a decentered spot with distinct albedo in the surface of the occulted body will result in an asymmetric light curve (see Wasserman et al. 1976). Taking the derivative to the first order of the normalized flux ratio with respect to the albedo ratio from our model, one can easily estimate this asymmetry as a function of the spot's diameter and albedo. The actual detection of such spots is nevertheless limited to light curves of extreme quality derived from very high S/N observations. For instance, using the model for the August 13th occultation, it would take a spot with 17% of the occulted body radius and with twice its albedo to create a flux discrepancy (asymmetry) of twice the observed flux ratio error for measurements before and after the event midtime.

Penumbra widths were estimated in less than 2 km and thus were not considered here. On the other hand, improvements can be made including the solar phase angle in the model. In fact, in Table 1, we can see that the phases of the satellites were typically around 15, but for the highest value of 277, involving the largest moon Titania, we get a positional shift of about 21 km (1.5 mas). This is a small contribution considering the impact parameter values alone, but is more than three times the typical errors of 0.2 mas from our best observations.

M.A. acknowledges grant E-26/170.686/2004/FAPERJ and grants 306028/2005-0 and 478318/2007-3 (CNPq). F.B.R. thanks the financial support by the CAPES. J.I.B.C. acknowledges grant 151392/2005-6/CNPq and grant E-26/100.229/2008/FAPERJ. A.H.A. thanks grant 307126/2006-4 ("Produtividade em Pesquisa" CNPq). R.V.M. acknowledges grant 304124/2007-9/CNPq.

A simple geometrical model was developed to fit the light curves of occultations and eclipses, partial, total, annular, or central. Here, we describe in detail the analytical expressions and their derivatives with respect to the parameters of the model. The expressions shown in this appendix can be readily used in nonlinear least-squares procedures to fit the observed light curves of any mutual event in general. First, we obtain the expressions for occultations. Then, it is explained why the same formalism developed for occultations also applies for the eclipses.

In the following, we always refer to apparent sizes regardless of the actual sizes of the bodies. The basic hypothesis is that the satellites are illuminated spheres which scatter the light equally in all directions (Lambert law). So, the satellites illuminated by the sun light are seen from the Earth-like two gray disks. Let us then consider the geometry of a partial occultation as shown in Figure 3, where two disks S₁ and S₂ of radius R₂ < R₁ partially intercept each other. The intercepted area A of occultation is then

$$\begin{equation*} A = S({\rm O_{1}BC}) - \Delta ({\rm O_{1}BC}) + S({\rm O_{2}BC}) - \Delta ({\rm O_{2}BC}), \end{equation*}$$

where S(O₁BC) and S(O₂BC) are the areas of the circular sectors of the disks S₁ and S₂, respectively, and Δ(O₁BC) and Δ(O₂BC) are the areas of the triangles. But

$$\begin{eqnarray*} &&S({\rm O_{1}BC}) = \alpha _{1}{R_{1}}^{2},\quad S({\rm O_{2}BC}) = \alpha _{2}{R_{2}}^{2}, \\ &&\Delta ({\rm O_{1}BC}) = \mbox{$\frac{1}{2}$}{R_{1}}^{2}\sin {2\alpha _{1}}, \quad \Delta ({\rm O_{2}BC}) = \mbox{$\frac{1}{2}$}{R_{2}}^{2}\sin {2\alpha _{2}},\nonumber \end{eqnarray*}$$

so that

$$\begin{equation*} A = {R_{1}}^{2}\big(\alpha _{1}-\mbox{$\frac{1}{2}$}\sin {2\alpha _{1}}\big) + {R_{2}}^{2}\big(\alpha _{2}-\mbox{$\frac{1}{2}$}\sin {2\alpha _{2}}\big).\nonumber \end{equation*}$$

Let k₁ and k₂ be the albedos of the satellites 1 and 2. Then, when the larger body S₁ occults the smaller S₂, their total flux F_1o2 is

$$\begin{eqnarray*} F_{1o2} &=& k_{1}\pi {R_{1}}^2+k_{2}\pi {R_{2}}^2 - k_{2}{R_{1}}^{2}\big(\alpha _{1}-\mbox{$\frac{1}{2}$}\sin {2\alpha _{1}}\big)\\ && - k_{2}{R_{2}}^{2}\big(\alpha _{2}-\mbox{$\frac{1}{2}$}\sin {2\alpha _{2}}\big).\nonumber \end{eqnarray*}$$

Now, let F₁₊₂ be the sum of the fluxes of the satellites 1 and 2 without occultation. Thus, the normalized flux F_occ of the two satellites, i.e., the total flux F_1o2 divided by F₁₊₂, when the larger body S₁ occults the smaller S₂, is given by

$$\begin{equation*} F_{\rm occ} = \frac{F_{1o2}}{F_{1+2}} = 1 - \frac{{R_{1}}^2\big(\alpha _{1}-\mbox{$\frac{1}{2}$}\sin {2\alpha _{1}}\big)+ {R_{2}}^{2}\big(\alpha _{2}-\mbox{$\frac{1}{2}$}\sin {2\alpha _{2}}\big)}{\frac{k_{1}}{k_{2}}\pi {R_{1}}^2+\pi {R_{2}}^2}.\nonumber \end{equation*}$$

The angles in F_occ are easily calculated from the distance d = O₁O₂ and from the satellites' apparent radii. Considering the triangle O₁O₂B, from the cosine law we have for α₁ and α₂

$$\begin{eqnarray} &&\cos \alpha _{i} = \frac{ {R_{i}}^{2} - {R_{j}}^2 + d^{2}}{2 R_{i} d}, \end{eqnarray} \tag{ A1 }$$

where i = 1 or 2 and j = 2 or 1, respectively.

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Note that in Figure 3 the distance d is larger than R₂. However, for d < R₂, the intercepted area A between the disks is now given by

$$\begin{equation*} A = S({\rm O_{1}BC}) - \Delta ({\rm O_{1}BC}) + S({\rm O_{2}BC}) + \Delta ({\rm O_{2}BC}).\nonumber \end{equation*}$$

Even so, the expressions for F_1o2 and thus for F_occ are still valid, since $\alpha _{2} > \frac{\pi }{2}$.

Now, if the smaller satellite S₂ passes in front of the larger one S₁, the formal expressions deduced above still hold for F_occ, once we replace the albedos k₁ by k₂ and vice versa.

In the case of a total occultation, when d < R₁ − R₂, there are two situations depending on what body is in front of the other. When it is the smaller satellite S₂ being occulted by the larger one S₁, then we have

$$\begin{equation*} F_{\rm occ} = \frac{F_{1o2}}{F_{1+2}} = \frac{\mbox{$\frac{k_{1}}{k_{2}}$}\pi {R_{1}}^2}{\mbox{$\frac{k_{1}}{k_{2}}$}\pi {R_{1}}^2+\pi {R_{2}}^2}\nonumber \end{equation*}$$

and when it is the smaller satellite S₂ which is in front of the larger one S₁

$$\begin{equation*} F_{\rm occ} = \frac{F_{1o2}}{F_{1+2}} = 1 - \frac{\mbox{$\frac{k_{1}}{k_{2}}$}\pi {R_{2}}^2}{\mbox{$\frac{k_{1}}{k_{2}}$}\pi {R_{1}}^2+\pi {R_{2}}^2}.\nonumber \end{equation*}$$

The dynamics of occultations and of eclipses is defined by the relative motion between the satellites. This is given by

$$\begin{eqnarray} &&d^{2} = x^{2} + v^{2}{(t - t_{0})}^{2}, \end{eqnarray} \tag{ A2 }$$

where x is the impact parameter, v is the relative speed between the satellites, and t₀ is the central time. Therefore, these three dynamical parameters can be associated with the geometry of the occultation through the distance d between the satellites, which in turn determines the flux ratio F_occ of the observed light curve. Thus, the model of the light curve depends on four parameters: the albedo ratio $\mbox{$\frac{k_{1}}{k_{2}}$}$, x, v, and t_o. In compact form, the model can be written as

$$\begin{eqnarray} F &=& \frac{F_{1o2}}{F_{1+2}} = 1 - \frac{A}{B} \nonumber \\ k &=& {\frac{k_{1}}{k_{2}}} \nonumber \\ A &=& {R_{1}}^{2}\left(\alpha _{1}-{\frac{1}{2}}\sin {2\alpha _{1}}\right) + {R_{2}}^{2}\left(\alpha _{2}-{\frac{1}{2}}\sin {2\alpha _{2}}\right) \nonumber\\ B &=& k\pi {R_{1}}^2+\pi {R_{2}}^2 \nonumber \\ C &=& \frac{1}{B}\left(\frac{{R_{1}}^2}{d}\sin {2\alpha _{1}}+\frac{{R_{2}}^2}{d}\sin {2\alpha _{2}}-2R_{1}\sin {\alpha _{1}}\right.\nonumber\\ &&\left.-2R_{2}\sin {\alpha _{2}}\vphantom{\frac12}\right) \nonumber \\ c &=& \pi {R_{1}}^2, \end{eqnarray} \tag{ A3 }$$

where the angles α are computed from Equation (A1), using Equation (A2). As before, in the case that the larger satellite S₁ is behind the smaller one S₂, we must exchange the albedos indices in k in Equations (A3).

We usually need the derivatives to the first order to solve for the model parameters in a nonlinear least-squares procedure to fit the flux ratio of the observed light curve (Bevington 1969). We start with initial parameter values taken from the JPL's Horizons ephemeris service (Giorgini et al. 1996) and, using the derivatives, we solve for the small parameter increments in an iterative least-squares procedure until the parameter values converge. These derivatives are not difficult to calculate and so we present here only the results. Thus, in the case of partial occultations, when the larger body S₁ occults the smaller one S₂, we have

$$\begin{eqnarray} \frac{dF}{dk} & = & c \frac{A}{{B}^{2}} \nonumber \\ \frac{dF}{dx} & = & \frac{C}{d} x \nonumber \\ \frac{dF}{dv} & = & \frac{C}{d} v {(t - t_{0})}^{2} \nonumber \\ \frac{dF}{d{t}_{0}} & = & -\frac{C}{d} {v}^{2} (t - t_{0}). \end{eqnarray} \tag{ A4 }$$

In total occultations, we have two cases for the derivative with respect to the albedo ratio. If the smaller satellite S₂ is being occulted, then

$$\begin{eqnarray} &&\frac{dF}{dk} = \frac{{\pi }^{2}{R_{1}}^{2}{R_{2}}^{2}}{B} \end{eqnarray} \tag{ A5 }$$

and, in contrast,

$$\begin{eqnarray} &&\frac{dF}{dk} = \frac{{\pi }^{2}{R_{2}}^{4}}{B}. \end{eqnarray} \tag{ A6 }$$

All expressions developed here for the occultations are rigorously valid for eclipses, once we replace the occulting/occulted body radius and albedos by those of the eclipsing/eclipsed body. The reason is that both eclipsing and eclipsed objects cannot be resolved in the images, due to their relative proximity as compared to their distance from the observer. Therefore, the flux of the eclipsing body is inevitably added to that of the eclipsed one in the photometric measurements. In this case, the resulting measured total flux F_1e2 and the normalized flux F_ecl of eclipses can then be described exactly with the same formalism of the total flux F_1o2 and normalized flux F_occ of occultations. Finally, it is easily verified that either annular or central events are also taken into account in the above formalism. In practice, Equations (A1)–(A6) can thus be used in any case for fitting light curves of mutual events.

An issue is the strong correlation between the impact parameter and the albedo ratio. In the present work, we did not adjust albedo ratios in the light curve fittings, but rather took fixed values and only adjusted the other three purely dynamical parameters.

Note that an eclipse is an occultation as seen from the Sun so that, for ephemeris-improving purposes, a correct interpretation of the fitted parameters must be done according to the type of mutual event. For example, when considering the listed central times in occultations we must discount the light times of each satellite with respect to the observer, but in the case of eclipses the light time between the two satellites must also be considered.

It must be emphasized that the solar phase angle and the penumbra in the model above were not considered. In fact, they are small for the Uranus mutual events.