THE ORBITS OF THE URANIAN SATELLITES AND RINGS, THE GRAVITY FIELD OF THE URANIAN SYSTEM, AND THE ORIENTATION OF THE POLE OF URANUS^*

Sir William Herschel discovered the first two Uranian satellites, Titania and Oberon, in 1787 (Herschel 1787). More than 60 yr later, in 1851, the next two, Ariel and Umbriel, were found by William Lassell (Lassell 1851). In 1948, after nearly a hundred years, photographs taken by Gerard Kuiper revealed the fifth satellite, Miranda (Kuiper 1949). The U. S. Naval Observatory (USNO) based their ephemerides (Wilkins & Springett 1961) for Ariel, Umbriel, Titania, and Oberon on elements developed by Newcomb (1875) with improvements from Struve (1913); no ephemeris was provided for Miranda.

Shortly after Miranda's discovery, Harris (1949) determined its orbital elements from a series of 17 photographic observations. He also updated the elements of the other four satellites from a combination of the early visual and the later photographic observations through 1948. Following Harris's work, photographic observations of the Uranian system continued to be collected. Dunham (1971) used all of the photographic measurements over the period 1914–1966 to determine new elements for all five satellites. C. Veillet carried out a 5 yr observing campaign of the satellites from 1977 to 1982. He combined his new data with an extensive set of the previously published visual and photographic data to revise the elements (Veillet 1983b).

Jacobson et al. (1986) reported on the ephemerides that were employed for the navigation of the Voyager 2 spacecraft through its encounter with Uranus in 1986 January. They based those ephemerides on a numerical integration initially fit to Earth-based photographic astrometry spanning 1973–1985. The data set included Veillet's measurements and those acquired at the Lick, McDonald, McCormick, USNO Washington, and USNO Flagstaff observatories. During the encounter the ephemerides were periodically updated using spacecraft imaging data. In the dynamical model for the satellite orbits as well as the spacecraft trajectory, the orientation of the Uranian pole and the zonal harmonics in the Uranus gravity field were taken from the analysis of the occultations of the inclined elliptical Uranian rings (Elliot & Nicholson 1984). For the post-encounter trajectory reconstruction and final satellite ephemerides, the pole and gravity harmonics were replaced by those found in the ring occultation analysis of French et al. (1986). Jacobson et al. (1992) gives the GMs (GM is the product of the Newtonian constant of gravitation G and the body's mass M) of Uranus and the satellites determined as part of the post-encounter Voyager 2 data analysis.

Shortly after the Voyager 2 flyby, Laskar (1986) published a general theory describing the motion of the satellites. This theory was subsequently fit to the Voyager 2 imaging data and an extensive Earth-based observation set covering 1911–1986 (Laskar & Jacobson 1987). The theory fit adopted the Uranus pole and gravity harmonics from the ring occultation analysis, but redetermined the planet and satellite GMs, finding little difference from those previously published. Ephemerides derived from the theory were adopted by the USNO (Seidelmann 1992).

Lazzaro (1991) also developed an analytical theory for the satellite orbits which he compared to the existing published results but did not fit to actual observations. He concluded that his theory would not lead to an improvement over Laskar's ephemeris.

Taylor (1998) complied an extensive catalog of Uranian satellite observations and fit a numerical integration to photographic, CCD, meridian circle, and Voyager 2 astrographic data (derived from the spacecraft imaging; Jacobson 1992). His data set covered 1977–1995. He used the pole orientation and Uranus J₄ from French et al. (1988) and the GMs of Ariel and Umbriel from Jacobson et al. (1992), but determined the Uranus J₂, the Uranus GM, and the other satellite GMs as part of the data fit. His estimated values for the Uranus J₂ and the Uranus and Miranda GMs closely agreed with the previously published results, but for Titania and Oberon he found significantly smaller GMs.

Most of the previous analyses of the satellite and ring orbits as well as the Voyager 2 mission operations were carried out in the FK4/B1950 reference system. Jacobson & Rush (2008) produced a new trajectory reconstruction for the Voyager 2 Uranus encounter analysis in the modern International Celestial Reference Frame (ICRF). This work required numerically integrated satellite ephemerides in that same frame. To develop those ephemerides, they extended the satellite astrometry to 2006 and re-determined the planet and the satellite GMs. Moreover, they included the ring occultations in their data fit and re-estimated the Uranus pole and gravity harmonics. The new pole and harmonics differed little from French et al. (1988), and the GMs matched Jacobson et al. (1992), with the exception of the Titania and Oberon GMs which were a closer match to Taylor (1998).

Lainey (2008) fit astrometric observations for the period 1948–2006 to a numerical integration. He took the planet and satellite GMs from Jacobson et al. (1992) and the Uranus gravity harmonics from French et al. (1988), but he used the pole that had been updated with additional ring occultations (Mason et al. 1992). Emelyanov & Nikonchuk (2013) carried an analysis similar to that of Lainey but extended the observation period back to the satellite discovery observations in 1787 and forward to 2008.

In this article we report on a revision of the work of Jacobson & Rush (2008), but we focus on the orbits of the satellites and rings rather than the spacecraft trajectory. We extend the span of the astronomical observations through 2013 and add the ring stellar occultations from 1992. We also expand our satellite system to include Puck, the largest of the 10 small satellites discovered by Voyager 2 (Smith et al. 1986); Puck's orbit is interior to that of Miranda. We include Puck because it is the only one of the 10 that has been observed from the Earth and because we have a number of Puck positions measured relative to Miranda with the Hubble Space Telescope (HST). These relative observations help to constrain Miranda's orbit.

Our dynamical model contains both gravitational and non-gravitational forces; the former affect the motion of the satellites, rings, and spacecraft, whereas the latter affect only the spacecraft. Sources of the gravitational forces are the Sun, the solar system planets, and the Uranian satellites; Puck is assumed to be massless due to its small size and unknown mass. The gravitational effects of the rings are ignored because the ring mass is unknown but presumed small. The gravity field of the planet is represented by the standard spherical harmonic expansion of its gravitational potential; we retain the second, fourth, and sixth zonal harmonics. JPL planetary ephemeris DE430 (Folkner et al. 2014) provides the positions and GMs of the Sun and planets. The non-gravitational forces acting on the spacecraft are caused by solar radiation pressure, non-isotropic thermal radiation from the RTGs (Pu²³⁸ radioactive thermal generators which provide electrical power), trajectory correction maneuvers, and accelerations induced by the operation of the attitude control thrusters. Details on these forces may be found in Jacobson et al. (1992) and Jacobson & Rush (2008).

Peters (1981) provides the equations of motion for the orbits of the satellites, and Moyer (2000) gives them for the spacecraft. Both sets of equations are formulated in Cartesian coordinates centered at the Uranian system barycenter and referenced to the ICRF. The satellite orbits were integrated with a variable order, variable step size, Gauss-Jackson method (Jackson 1924), and the spacecraft trajectory was integrated with JPL's Orbit Determination Program (Moyer 1971; Ekelund 1980) which is a spacecraft navigation software set in use at JPL.

We represented the rings by precessing ellipses inclined to the Uranus equator; Appendix A describes the ring model in detail. The orientation of Uranus's pole is defined by its ICRF Right Ascension and declination which are specified in terms of trignometric series. The series were developed by fitting the angles to the numerically integrated polar motion of a rigid body Uranus with torques from the Sun and the satellites (see Appendix B).

The primary data used to determine the satellite orbits are the satellite astrometry derived from telescopic observations made at astronomical observatories and HST over the time period 1911–2013 and the imaging from the Voyager 2 spacecraft. Taylor (1998) gives a comprehensive overview of the Earth-based astrometry from 1815 to 1997. He used part of the extensive set of data spanning 1982–1998 acquired at the Laboratório Nacional de Astrofísica (LNA/MCT) in Brazil, but he overlooked some of the 1983–1988 observations and could not include the post-1994 observations which had not yet been published. Lainey (2008) and Emelyanov & Nikonchuk (2013) fit the later re-reduction of the entire set of LNA/MCT astrometry (Veiga et al. 2003), as did we. Beginning in 1995 and continuing through the present, the USNO has been observing the outer planets and their satellites with the Flagstaff astrometric scanning transit telescope (FASTT; Stone et al. 1996). The FASTT Uranian satellite astrometry began in 1998 and constitutes a large percentage of the Earth-based astrometry acquired since that time. Both Lainey and Emelyanov & Nikonchuk fit the FASTT data for 1998–2005; we fit the data through 2013. Another ongoing observing program is the meridian transit circle observations at the Bordeaux observatory. Arlot et al. (2008) provides the Uranian satellite positions for 1997–2005; Lainey fit the pre-publication data for 1997–2003, but Emelyanov & Nikonchuk ignored these data. For the period of 1998–2007, an observational program was carried out at observatories near Beijing and Shanghai (Qiao et al. 2013) yielding astrometric positions of the five main satellites; these data were published after Lainey's work; Emelyanov & Nikonchuk included them.

The equator of Uranus and the orbital planes of its satellites and rings are strongly tilted relative to Uranus's orbital plane. Every 42 yr the Uranian pole appears normal to the Earth line of sight and the satellite and ring system can be viewed edge on. This unique geometry occurred in 2007 and provided astronomers the opportunity to observe mutual occultations and eclipses of the satellites. Christou et al. (2009), Mallama et al. (2009), and Arlot et al. (2013) derived high precision astrometric measurements from these mutual events. Emelyanov & Nikonchuk were the first to fit these data.

On 2001 September 8 Titania occulted the Hipparcos star 106829 and on 2003 August 1 occulted star TYC 5806-696-1 (Widemann et al. 2009). Nearly 70 observing stations attempted to record the first event, and the second event was recorded at two sites. We used the ingress and egress times from the best 27 records of the first event and from the two records of the second.

In 1994 Pascu et al. (1998) used HST to observe Puck together with seven of the other small satellites discovered by Voyager 2. They produced positions of Puck relative to Miranda. Descamps et al. (2002) made the first Earth-based observations of Puck in early 1999 with the adaptive optics system of the European Southern Observatory (ESO) at La Silla, Chile. They followed with additional observations in late 1999 and again in late 2000. In conjunction with the Puck measurements, they also obtained simultaneous positions of Miranda and Ariel. We used the data in the form of Puck and Ariel positions relative to Miranda. Showalter & Lissauer (2006) investigated the Uranian ring-moon system with HST during 2003–2005, observing Puck, Ariel, Miranda, and several of the other small satellites. They followed up with more observations in 2006 (M. R. Showalter 2007, private communication). Again we constructed Puck and Ariel positions relative to Miranda. In 2004 Veiga & Bourget (2006) employed a specially designed Coronagraph at LNA/MCT to image Puck together with the five main Uranian satellites. These constitute the second set of Puck observations that have been obtained from the Earth.

The original spacecraft imaging data are pictures of the satellites against a stellar background. The stars in the images provide the inertial reference for the observed direction from the spacecraft to the satellite. Jacobson (1992) converted the imaging data to an equivalent astrometric form that depended upon a particular Voyager 2 trajectory; Taylor, Lainey, and Emelyanov & Nikonchuk all used that data. However, since we redetermined the spacecraft trajectory as well as the satellite orbits, we returned to the data in their original form. They cover the 1985 August 22–1986 January 31 timeframe.

The ring orbits were found using occultation measurements in the form of the occultation ingress and egress times and the radii of the occulting rings. These are stellar occultations seen from Earth-based observatories during the period 1980–1992, stellar occultations seen by the Voyager 2 spacecraft during the Uranus encounter, and radio occultations of the Voyager 2 spacecraft also observed during the Uranus encounter. We took the observations from French et al. (1986, 1987, 1988, 1996), Elliot et al. (1987), and Millis et al. (1987). We also included the measurements made in 1982 at Palomar observatory reported by P. D. Nicholson (1986, private communication). When processing the stellar occultation data we took the star positions from the UCAC2 star catalog (Zacharias et al. 2004) and the Hipparcos catalog (Perryman et al. 1997). The occultations observed by the spacecraft or of the spacecraft also help to determine the spacecraft trajectory.

The Voyager 2 trajectory determination relied on coherent two-way and three-way spacecraft Doppler and ranging data. These were the same as was used by Jacobson & Rush (2008), covering the period from 1985 August 22 to 1986 February 13. The carrier for the radio data was transmitted to the spacecraft at S band (2200 MHz) and transponded back to the tracking stations at both S band and X band (8400 MHz). However, X band was the primary downlink frequency. We calibrated the Doppler and range for the effects of the seasonal changes in Earth's troposphere. We also applied calibrations for Earth's ionosphere beginning in 1985 November; prior to that time no calibrations were available. When Doppler tracking was available at both S and X band after 1985 December 25, differences between measurements at those bands provided calibrations for the interplanetary plasma (note: these calibrations were recently discovered and had not been used previously). For range data prior to 1985 December 25 we relied on a solar corona model to account for plasma delays, but for the Doppler we applied no plasma calibrations. We also included spacecraft delta differential one-way range (ΔDOR) data (Thornton & Border 2000). These provide a measure of the angular separations between Voyager 2 and nearby quasars; we used the latest IERS quasar locations in our analysis.

4.1. Method of Solution

We determined the orbits of the satellites, rings, and spacecraft by adjusting parameters in the dynamical model to obtain a weighted least-squares fit to the observational data. The parameters for the satellites and rings were

• 1.  
  the epoch position and velocity of each satellite and
• 2.  
  the elements of each ring.

The spacecraft parameters were

• 1.  
  the epoch position and velocity,
• 2.  
  the thrust magnitude and direction for the large trajectory correction maneuver (TCM13),
• 3.  
  impulsive velocity changes for the small maneuvers, and
• 4.  
  general non-gravitational accelerations.

The common gravitational parameters were

• 1.  
  the GMs of the Uranian system and the satellites;
• 2.  
  the J₂, J₄, and J₆ gravitational harmonics of Uranus; and
• 3.  
  the Right Ascension and declination of Uranus's pole at epoch J2000.

To obtain an adequate fit to the observations we also had to adjust a number of parameters affecting the observation models:

• 1.  
  Observer dependent position angle and separation biases in the early visual observations which account for the systematic errors known as the observer's "personal equation"; separate biases were determined for each observation year to allow for variations in observing conditions.
• 2.  
  A position angle bias in the photographic data of Sytinskaja (1930) which we presume to be related to the origin of his angle measurement.
• 3.  
  Yearly observer dependent biases in the absolute astrometric positions to account for small errors remaining in the DE430 Uranus planetary ephemeris; the FASTT data were actually used in the preparation of DE430, but were processed with an earlier version of the Uranian satellite ephemeris.
• 4.  
  Corrections to the orientation and scale factor in the HST data from M. R. Showalter (2007, private communication); separate corrections were determined for each observation frame.
• 5.  
  Corrections to the orientation and scale factor for each frame in the data from Descamps et al. (2002); the original data calibration was done using computed positions of the bright satellites from Laskar & Jacobson (1987).
• 6.  
  The positions of the stars being occulted by Titania and the rings; these cover both the actual errors in the star catalog and the DE430 ephemeris errors.
• 7.  
  Observer dependent ring occultation timing offsets.
• 8.  
  Tracking station dependent range biases to account for calibration errors and small errors in DE430; DE430 was actually fit to the ranges but a different Voyager 2 trajectory was used in the data processing.
• 9.  
  The position of the quasar in the ΔDOR data; as with the ranges these data were used to determine DE430 but with a different spacecraft trajectory; small corrections to the quasar position cover actual errors in that position as well as the DE430 error.
• 10.  
  Monthly corrections to the day and night ionosphere Doppler and range delays; these corrections were applied because the calibrations from the Voyager 2 era were not as accurate as modern calibrations, moreover, for some of the early data no calibrations were available.
• 11.  
  Spacecraft camera pointing angles for each picture.
• 12.  
  Satellite dependent phase corrections for the spacecraft imaging data.

We grouped the astrometric data, mutual events, transits, and occultations according to type, observatory, and observing period and then assigned data weights through an iterative process to be consistent with the root-mean-square (rms) of the residuals of each group.

Similarly, we set separate Doppler data weights for each tracking pass to correspond to an accuracy consistent with the rms of the residuals for that pass; we next applied a scale factor to the rms to account for the fact that the Doppler noise is not a white-noise process (Folkner 1994). The scale factor, which is 0.468(86400/τ)^1/3 with τ being the Doppler sample interval in seconds, preferentially weights the Doppler at the diurnal frequency (86400 s). However, it yields a weight that is too conservative for the close encounter data. For the Uranus and Miranda close flyby passes, we adopted the Doppler whitening algorithm (Mackenzie & Folkner 2006) that was developed for Cassini gravity analysis. We selected the range data weights to be consistent with an accuracy of 10 m; the estimated range biases, with an a priori uncertainty of 1 km, accounted for systematic errors. The DDOR data weights were chosen to represent a timing delay accuracy of 2 ns which translates into a position error of roughly 180 km at Uranus.

We computed the spacecraft imaging data weights from the algorithm:

$$\begin{equation*} W^{2} = W^{2}_{{\rm min}} + \left(C \cdot d\right)^{2}, \end{equation*}$$

where W is the weight applied, W_min is minimum weight, C is an apparent diameter scale factor, and d is the apparent diameter as seen from the spacecraft (it depends upon the range to the satellite). We used 0.25 pixels for the minimum weight and 0.05 for scale factor for all satellites.

To carry out the fit we first formed separate square root information (SRI) arrays (Bierman 1977) for each observation set. All of the spacecraft data were packed into a single array which was obtained by processing them with a batch-sequential, square-root information filter that treated the non-gravitational accelerations and camera pointing angles as white noise. The accelerations were batched at 1 day intervals with additional batches at the times of the spacecraft attitude changes. The pointing angles were batched by picture. We combined all of the separate SRI arrays and applied a singular value decomposition algorithm to the composite SRI array to obtain the parameter values.

During our estimation process we constrained the corrections to the occultation star positions by the quoted uncertainties in their catalog positions; an analogous constraint was placed on the quasar position based on the uncertainty of its ICRF location. A priori uncertainties were also included for the spacecraft maneuvers and non-gravitational accelerations: 5% on the thrust magnitude and 0095 on the direction of TCM13, 1–5 mm s⁻¹ on the impulsive velocity changes, and 5× 10⁻⁷–5× 10⁻⁶ mm s⁻² on the accelerations.

4.2. Satellite Orbits

Table 1 contains the satellite state vectors determined from the orbit fit; 16 digits are quoted to facilitate future orbit integrations. The mean orbital elements in Table 2, obtained by fitting precessing ellipses to the integrated orbits over 1900–2100, give a geometric description of orbits. The semimajor axis is a, the eccentricity is e, the inclination is i, the mean longitude is λ, the longitude of periapsis is ϖ, and the longitude of the ascending node is Ω. All of the orbits have non-zero eccentricities and inclinations with Miranda having a significant inclination. Whitaker & Greenberg (1973) first uncovered the unexpectedly "large" inclination for Miranda together with a somewhat large eccentricity. Subsequent analysis by Veillet (1983a) confirmed the inclination but concluded that the eccentricity was too large, most likely due to systematic errors in some of the early observations. Our semimajor axes, eccentricities, and inclinations are comparable those in the GUST86 theory (Laskar & Jacobson 1987).

Table 1. State Vectors at 1985 August 1 (TDT)

Satellite	Position	Velocity
	(km)	(km s−1)
Ariel	−185785.2177189803	−0.384730129923274
	42477.81018746200	−1.393752472818678
	−2109.273462727150	5.325004225204424
Umbriel	−176566.9475784755	−3.350588413391897
	89016.12833946147	−0.153568184837806
	−176154.9418970623	3.273855499527411
Titania	−221240.1941919138	−3.049048602775958
	145452.9878060127	0.138409610142017
	−346697.1461249496	1.991437563896877
Oberon	−155108.4287158760	−2.962407899779837
	181606.6634411168	0.385864135361727
	−532879.3651011410	0.993694238058708
Miranda	−127430.9607930668	−0.422514450329333
	23792.64617013941	−1.271890082631948
	−3464.554580724168	6.552338419694388
Puck	−24369.49145882789	−7.667367460395494
	27011.79870872380	1.014093590954378
	−77882.20359704649	2.753303665833902

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Table 2. Mean Equatorial Orbital Elements at 2000 January 1.5 (TDT)

Element	Ariel	Umbriel	Titania	Oberon	Miranda	Puck
a (km)	190930.	265982.	436282.	583449.	129858.	86005.
e	0.00122	0.00394	0.00123	0.00140	0.00135	0.00019
i (deg)	0.0167	0.0796	0.1129	0.1478	4.4072	0.3562
λ (deg)	203.0922	251.2562	281.5845	352.5701	328.6961	266.0933
ϖ (deg)	83.3294	352.9610	228.3816	212.8435	256.3256	98.9184
Ω (deg)	289.7415	195.4845	26.4013	30.4840	100.7027	216.0360
$\dot{\lambda }$ (deg day−1)	142.8356506	86.8688753	41.3514187	26.7394835	254.6906573	472.5445452
$\dot{\varpi }$ (deg yr−1)	6.2308	2.8428	0.9993	0.2680	20.0409	80.8938
$\dot{\Omega }$ (deg yr−1)	−6.0839	−2.6316	0.0208	−0.4210	−20.2470	−80.8624

Notes. The longitudes are measured from the intersection of the Uranus equator and the ICRF reference plane.

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The mean element rates confirm that our integrated orbits for Miranda (V), Ariel (I), and Umbriel (II) are nearly commensurate

$$\begin{equation*} \dot{\lambda }_{V}-3\,\dot{\lambda }_{I}+2\,\dot{\lambda }_{II} = -0.0785 \, {\rm deg/day.} \end{equation*}$$

Greenberg (1976) investigated the consequences of this near commensurability and showed that its dominant effect was a periodic perturbation in Miranda's longitude proportional to the product of the masses of Ariel and Umbriel. We find that in our integrated Miranda orbit the pertubation has an amplitude of roughly 3500 km and a period of about 12.55 yr.

Table 3 contains the rms of the residuals of the astrometric observations of the five main satellites grouped by source. The table entries give a summary of the number and type of the observations and the time span of the observation set. The observation type indicators are:

• Δρ, ρΔθ  
  planet relative separation and position angle (position angle residuals are scaled by the associated separation);
• Δαcos δ, Δδ  
  the differential Right Ascension scaled by the cosine of the declination of the reference body (another satellite or the planet) and the differential declination;
• α, δ  
  the Right Ascension and declination of the satellite.

Table 3. Statistics of the Residuals for the Astrometric Observations

Source	Dates	Type	No.	rms	Type	No.	rms
Filar micrometer
Aitken (1912)	1911	ρΔθ	23	0279	Δρ	23	0314
Barnard (1912)	1911	ρΔθ	23	0440	Δρ	46	0328
Barnard (1915)	1913	ρΔθ	12	0309	Δρ	12	0239
Harris (1949)	1913–1948	Δαcos δ	62	0173	Δδ	61	0182
Aitken (1914)	1914	ρΔθ	24	0252	Δρ	24	0327
Barnard (1916)	1915	ρΔθ	18	0298	Δρ	18	0292
Barnard (1919)	1916–1918	ρΔθ	45	0342	Δρ	42	0324
Barnard (1927)	1919–1922	ρΔθ	53	0353	Δρ	54	0287
Hall (1921)	1920	ρΔθ	21	0321	Δρ	21	0347
Hall & Bower (1923)	1922	ρΔθ	11	0126	Δρ	11	0231
Struve (1928)	1927–1928	ρΔθ	41	0333	Δρ	41	0237
Steavenson (1948)	1947–1948	ρΔθ	23	0441
Steavenson (1964)	1949	ρΔθ	7	0202
Photographic
Nicholson (1915)	1914	Δαcos δ	19	0301	Δδ	19	0433
Sytinskaja (1930)	1926	ρΔθ	62	0530	Δρ	62	0438
van Biesbroeck (1970)	1948–1964	Δαcos δ	757	0158	Δδ	757	0149
Whitaker & Greenberg (1973)	1960–1973	ρΔθ	16	0179
Tomita & Soma (1979)	1964–1977	ρΔθ	95	0363	Δρ	123	0443
van Biesbroeck (1970)	1966	Δαcos δ	43	0202	Δδ	43	0172
van Biesbroeck et al. (1976)	1966	Δαcos δ	55	0148	Δδ	55	0123
Soulie (1968)	1966–1967	Δαcos δ	4	0398	Δδ	4	0377
Soulie (1972)	1968–1969	Δαcos δ	57	0412	Δδ	57	0388
Soulie (1975)	1970–1971	Δαcos δ	30	0497	Δδ	30	0334
Soulie (1978)	1973–1974	Δαcos δ	67	0423	Δδ	67	0444
Mulholland (1985)	1974–1982	Δαcos δ	215	0085	Δδ	215	0083
Walker et al. (1978)	1975–1977	Δαcos δ	51	0050	Δδ	51	0042
Veillet (1983b) (Pic du Midi)	1977–1982	Δαcos δ	427	0090	Δδ	427	0103
Veillet (1983b) (OHP)	1977–1982	Δαcos δ	58	0157	Δδ	58	0171
Veillet (1983b) (ESO)	1977–1982	Δαcos δ	448	0057	Δδ	448	0049
Veillet (1983b) (CFH)	1977–1983	Δαcos δ	146	0053	Δδ	146	0097
Harrington & Walker (1984)	1979–1983	Δαcos δ	283	0056	Δδ	283	0055
C. Veillet (1985, private communication)	1982–1984	Δαcos δ	568	0068	Δδ	568	0063
Debehogne et al. (1981)	1980	Δαcos δ	22	0583	Δδ	23	0452
Veiga et al. (2003)	1982–1988	Δαcos δ	1395	0095	Δδ	1397	0077
Standish (1996)	1983–1986	Δαcos δ	481	0210	Δδ	481	0190
R. S. Harrington (1985, private communication)	1985	Δαcos δ	41	0025	Δδ	41	0036
Walker & Harrington (1988)	1985–1986	Δαcos δ	104	0038	Δδ	104	0041
Chanturiya et al. (2002)	1987–1994	Δαcos δ	91	0557	Δδ	91	0576
Kulyk et al. (1990)	1990	Δαcos δ	40	0291	Δδ	40	0239
CCD
Pascu et al. (1987)	1981–1985	Δαcos δ	76	0137	Δδ	76	0108
Veiga et al. (2003)	1989–1998	Δαcos δ	5221	0080	Δδ	5221	0089
Jones et al. (1998)	1990–1991	Δαcos δ	494	0038	Δδ	489	0031
Shen et al. (2002)	1995–1997	Δαcos δ	442	0054	Δδ	442	0056
Stone & Harris (2000)	1998	α	100	0155	δ	102	0137
Qiao et al. (2013)	1998–2007	α	2358	0081	δ	2358	0097
Stone (2000)	1999	α	117	0134	δ	118	0131
Descamps et al. (2002)	1999–2000	α	60	0003	δ	60	0013
W. M. Owen (2009, private communication)	1999–2009	α	186	0062	δ	184	0110
Stone (2001)	2000–2001	α	245	0143	δ	248	0133
B. Gladman (2001, private communication)	2001	Δαcos δ	6	0341	Δδ	6	0271
Stone (2005)	2002–2005	α	230	0118	δ	232	0132
Garradd & McNaught (2003)	2003	α	6	0079	δ	6	0116
M. R. Showalter (2007, private communication)	2003–2006	Δαcos δ	180	0006	Δδ	179	0017
Veiga & Bourget (2006)	2004	α	287	0072	δ	287	0078
Izmailov et al. (2007)	2005	α	20	0068	δ	20	0145
Monet (2007)	2005–2006	α	71	0146	δ	71	0152
Khovritchev (2009)	2007	Δαcos δ	22	0073	Δδ	22	0091
Harris (2013)	2007–2013	α	337	0139	δ	337	0167
Transit
Arlot et al. (2008)	1997–2005	α	227	0153	δ	227	0185
Mutual Events
Christou et al. (2009)	2007	Δαcos δ	1	0001	Δδ	2	0008
Mallama et al. (2009)	2007	Δαcos δ	2	0038	Δδ	2	0009
Arlot et al. (2013)	2007–2008	Δαcos δ	34	0010	Δδ	34	0017
Titania stellar occultation timing
Widemann et al. (2009)	2001, 2003		58	0150

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Since the cross-line-of-sight Titania velocity in 2001 and 2003 varied between 12 km s⁻¹ and 15 km s⁻¹, the stellar occultation timing residual rms corresponds to about a 2 km position error. When fitting the occultations we also corrected the star positions rather than the position of Uranus. Table 4 lists the star positions from Widemann et al. (2009) together with our estimated corrections; the corrections are consistent with the a priori uncertainties in the star locations and that of Uranus.

Table 4. Titania Occultation Star Catalog Positions and Corrections

No.	Cat. R.A.	Cat. Decl.	Δαcos δ	Δδ
	(deg)	(deg)	(mas)	(mas)
Hipp. 106829	324.55809702	−14.91006013	14.3 ± 3.0	8.0 ± 3.7
TYC 5806-696-1	333.97730960	−11.61565510	38.5 ± 3.2	18.4 ± 4.1

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Table 5 gives the astrometric residual statistics for Puck, and Table 6 gives the residual rms for the Voyager 2 imaging.

Table 5. Statistics of the Residuals for the Puck Astrometric Observations

Source	Dates	Type	No.	rms	Type	No.	rms
Pascu et al. (1998)	1994	ρΔθ	31	0012	Δρ	32	0012
Descamps et al. (2002)	1999–2000	α	30	0032	δ	30	0143
M. R. Showalter (2007, private communication)	2003–2006	Δαcos δ	152	0004	Δδ	152	0013
Veiga & Bourget (2006)	2004	α	135	0102	δ	135	0091

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Table 6. Voyager Imaging Residuals rms

Object	No.	Sample	Line	Object	No.	Sample	Line
Ariel	109	0.263	0.178	Oberon	64	0.167	0.336
Umbriel	103	0.215	0.278	Miranda	116	0.241	0.221
Titania	65	0.192	0.267	Puck	49	0.177	0.145
Stars	879	0.253	0.250

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The dominant source of error in determining the satellite orbits is the random and systematic errors in the observations. We can mitigate observational error to some degree by including a variety of observations from many observers and by weighting observation sets differently. The presumption is that observers' errors are independent and that different data types are subject to different errors. Consequently, the systematic errors cancel to some extent or can be identified, e.g., the "personal equation" in the visual observations, and removed. Differential weighting gives more strength to data having lower random errors thereby reducing the random error contribution to the orbit error. Given that the data are properly weighted and systematic errors minimized, the formal errors from the data fit provide a reasonable measure of the uncertainties in the orbits. In practice, we have found that actual orbit errors are rarely twice our formal errors; we gauge actual errors by the size of the orbit change required to fit newly acquired data. Table 7 contains the maximum formal uncertainties in the 2000–2050 timeframe displayed in the orbit radial (R), tangential (T), and normal to the orbit plane (N) directions. Orbital period errors cause the tangential error to grow at the rate ($\dot{{{{T}}}}$). The errors in the other two directions remain nearly constant, except for Miranda. Because of Miranda's inclination, the error in the orbit precession rate causes about a 2 km yr⁻¹ increase in the N direction error.

Table 7. Satellite Orbit Uncertainties for the Period 2000–2050

Satellite	R	T	N	$\dot{{{{T}}}}$
	(km)	(km)	(km)	(km yr−1)
Ariel	15	150	30	3
Umbriel	20	150	50	3
Titania	35	160	50	3
Oberon	30	160	60	3
Miranda	6	250	25–125	5
Puck	6	150	30	3

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4.3. Ring Orbits

Table 8 gives the elements, precession rates, and formal 1σ errors that we found for the Uranian rings. Also included are the number of observations of each ring and the rms of the residuals. The elements and rms are in good agreement with those in French et al. (1988). The addition of the 1992 occultation appears to have little effect on the elements. To properly fit the occultations, it was necessary to account for systematic offsets between the time bases used at several observatories, the Deep Space Station at Tidbinbilla (DSS43), and for the Voyager photopolarimeter (French et al. 1988). Table 9 contains our estimated offsets.

Table 8. Ring Elements Referred to the Uranus Equator, Number of Observations (N), and rms of the Residuals

Element	6	5	4	α
a (km)	41837.27 ± 0.30	42234.85 ± 0.20	42570.99 ± 0.21	44718.43 ± 0.22
e (× 103)	1.012 ± 0.008	1.898 ± 0.005	1.060 ± 0.007	0.761 ± 0.006
i (deg)	0.062 ± 0.003	0.055 ± 0.002	0.032 ± 0.001	0.015 ± 0.001
ϖ (deg)	242.50 ± 0.72	170.04 ± 0.38	126.98 ± 0.39	333.33 ± 0.44
Ω (deg)	11.49 ± 1.41	286.40 ± 0.87	89.81 ± 2.45	61.06 ± 5.14
$\dot{\varpi }{}$ (deg yr−1)	1008.767 ± 0.052	975.767 ± 0.046	948.935 ± 0.044	798.166 ± 0.029
$\dot{\Omega }{}$ (deg yr−1)	−1006.775 ± 0.052	−973.875 ± 0.046	−947.124 ± 0.044	−796.786 ± 0.029
N	30	39	39	44
rms (km)	0.33	0.24	0.33	0.36
Element	β	η		λ
a (km)	45661.05 ± 0.18	47176.02 ± 0.27	51149.21 ± 0.30	50024.16 ± 0.96
e (× 103)	0.441 ± 0.004	0.003 ± 0.009	7.930 ± 0.007	0.0
i (deg)	0.005 ± 0.001	0.001 ± 0.001	0.001 ± 0.001	0.0
ϖ (deg)	224.83 ± 0.65	321.24 ± 103.53	214.591 ± 0.154
Ω (deg)	311.60 ± 12.54	186.11 ± 54.12	171.69 ± 104.374
$\dot{\varpi }{}$ (deg yr−1)	741.742 ± 0.024	661.372 ± 0.023	497.941 ± 0.018
$\dot{\Omega }{}$ (deg yr−1)	−740.513 ± 0.024	−660.345 ± 0.023	−497.284 ± 0.018
N	41	35	44	4
rms (km)	0.21	0.50	0.60	0.80

Notes. The epoch for the longitudes is 1977 March 10 20:00:00 (UTC).

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Table 9. Observatory Time Offsets

Station	Offset (s)	Station	Offset (s)
European Southern Obs.a	−0.072 ± 0.063	European Southern Obs.b	−0.127 ± 0.024
European Southern Obs.c	0.062 ± 0.029	Las Campanas Obs.	−0.030 ± 0.022
Pic du Midi	3.685 ± 0.033	Tenerife Ingress	−0.070 ± 0.035
Tenerife Egress	0.368 ± 0.035	DSS43	−0.011 ± 0.023
PPSBPer	−0.054 ± 0.053	PPSSSgr	0.450 ± 0.526

Notes. ^a1980 August 15, ^b1982 April 22, ^c1992 July 11.

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The work of French et al. (1986) and French et al. (1988) was done in the FK4/1950.0 system using star locations in that system. We replaced those locations with ICRF compatible locations from the Hipparcos and UCAC2 star catalogs. In our processing we applied parallax and proper motion corrections and estimated corrections to the star positions. Since we did not correct the Uranus ephemeris, the changes in the star positions also accounted for Uranus ephemeris errors. Table 10 gives star positions from the appropriate catalog and our corrections.

Table 10. Star Catalog Positions and Corrections

Star	Cat.	Cat. R.A.	Cat. Decl.	Δαcos δ	Δδ
		(deg)	(deg)	(mas)	(mas)
S77	Hipp.	219.54921290	−14.95473933	7.308 ± 0.054	−5.913 ± 0.131
KM11	UCAC2	233.40997800	−18.90128950	−294.172 ± 0.151	27.147 ± 0.091
KM12	UCAC2	229.54172530	−17.99479950	−196.954 ± 0.065	125.594 ± 0.211
KME13	Hipp.	237.10643560	−19.77402446	−3.149 ± 0.042	22.473 ± 0.119
KME14	Hipp.	242.14934740	−20.80743248	8.837 ± 0.053	−21.867 ± 0.086
KME15	UCAC2	241.79331860	−20.74504340	120.387 ± 0.039	23.800 ± 0.095
KME16	UCAC2	240.36678420	−20.48854530	1.543 ± 0.052	97.397 ± 0.078
KME17B	Hipp.	247.63035990	−21.74199001	−35.879 ± 0.047	4.279 ± 0.054
U23	UCAC2	256.37848620	−22.87389030	97.926 ± 0.038	33.601 ± 0.058
U25	UCAC2	255.59000530	−22.80714590	64.089 ± 0.041	− 291.815 ± 0.047
U28	UCAC2	261.49121500	−23.29306250	105.295 ± 0.031	3.573 ± 0.057
U103	UCAC2	287.39833590	−22.91141370	−6.625 ± 0.075	13.173 ± 0.044
σ Sgr	Hipp.	283.81633188	−26.29651960	0.0	0.0
β Per	Hipp.	47.04231934	+40.95561187	0.0	0.0

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4.4. Spacecraft Trajectory

The Voyager 2 Uranus flyby was designed to place the spacecraft on a post-Uranus trajectory to Neptune; the time of closest approach was selected to satisfy Uranus science requirements. Navigation targeting for the flyby was specified in terms of coordinates in the B-plane, a plane perpendicular to the incoming trajectory asymptote and passing through the center of the planet. The coordinate axes, denoted the T and R axes, are perpendicular and parallel to the ecliptic, respectively. Table 11 contrasts the Voyager 2 navigation results reported shortly after the Uranus encounter by the navigation team (Taylor et al. 1986) with those found in the ICRF reconstruction (Jacobson & Rush 2008) and with those from our current analysis. The table contains B-plane coordinates of the encounter position, B·T and B·R, the position error ellipse semi-major (SMAA) and semi-minor (SMIA) axes and orientation angle (θ) measured from the T axis, and the time of closest approach (TCA) and its error; the errors are all formal 1σ errors. The agreement among the results is good. Our formal errors are reduced somewhat due to revisions in the data weighting and processing procedures and improved ephemerides.

Table 11. Voyager 2 Navigation Performance—Uranus B-plane

Source	B·R	B·T	SMAA	SMIA	θ	TCA	σTCA
	(km)	(km)	(km)	(km)	(deg)	(H:M:S)	(s)
Taylor et al. (1986)	25384	128680	1	1	70	17:59:46.5	0.07
Jacobson & Rush (2008)	25384.9	128678.8	0.9	0.1	101	17:59:46.544	0.004
This work	25385.5	128678.5	0.6	0.05	101	17:59:46.531	0.002

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Figure 1 displays the post-fit Doppler residuals giving an idea as to the data quantity, noise level, and fit quality. The gap in the early December data corresponds to the solar conjunction period. The Doppler noise during the encounter period on January 24 is higher because the data there were compressed to a 5 s sample interval rather than the longer 5 minute interval used outside the encounter period. We found that our fit to the Doppler as well as to the range and ΔDOR data (not shown) was slightly better than that in Jacobson & Rush (2008) (see Figure 2 of the reference). As a consequence, we believe that this analysis has yielded a small improvement in Voyager 2 trajectory.

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4.5. Gravity and Pole

In this article we report the results of a comprehensive analysis of the orbits of the main Uranian satellites and rings and of the gravity field of the Uranian system. In determining the satellite orbits, we extended the observational data arc used in previously published orbits to include astrometry through the end of 2013. We also included the mutual events observed in 2007. For the ring orbits we extended the data arc to the Earth-based observations made in 1992. Our gravity field parameter determination required the combination of the satellite astrometry, ring occultations, and radiometric tracking of the Voyager 2 spacecraft during its passage through the Uranian system. We found that an accurate estimate of the Uranian system GM, i.e., the planet GM, could only be made from the Voyager 2 data. Adding the astrometry yielded the GMs of the satellites, and adding the occultations determined the zonal harmonics and pole direction. However, we also found consistent but lower accuracy values of the harmonics and pole from just the combination of the astrometry and Voyager 2 data.

Our satellite and ring orbits are well determined. Additional astrometry and occultations in the coming decades should provide moderate improvement. With respect to the gravity field, problems remain. The GMs of Ariel and Umbriel are correlated as are the Uranus zonal harmonics. We believe that more Earth-based astrometry will not break these correlations. However, additional ring occultation measurements may moderately improve the determination of J₂, J₄, and the pole orientation. Any significant improvement will require another spacecraft mission to the system. We recommend a mission similar to Galileo or Cassini, i.e., an orbiter with close satellite flybys. The orbiter must have low altitude Uranus periapsis passages for the tracking data to be sensitive to the Uranus gravitational harmonics and pole direction.

The Uranus pole precession rate is about 1.3 mas yr⁻¹ which is too small to be detected with the current data set (we were unable to determine it within the context of this analysis), but it may measurable in the future with a decade or more of high precision, ≈1 mas, absolute satellite astrometry. With the precession known we can infer the Uranus moment of inertia, γ.

Ephemerides for all of the satellites are available electronically from the JPL Horizons online solar system data and ephemeris computation service¹ and from NASA's Navigation and Ancillary Information Facility². The ephemeris designation is URA111.

The research described in this publication was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We wish to thank Richard French for his helpful discussions concerning the ring orbit model and occultation data processing.

We define a ring plane coordinate system in terms of equinoctial coordinates (Broucke & Cefola 1972); the unit vectors along the coordinate axes are

$$\begin{eqnarray*} [\hat{\mathbf {f}},\hat{\mathbf {g}},\hat{\mathbf {w}}] &=& \frac{1}{1+p^{2}+q^{2}}\nonumber\\ &&\times \left[\begin{array}{@{}ccc@{}} 1-p^{2}+q^{2}& 2\,pq &2p \\ 2pq & 1+p^{2}-q^{2}&-2q \\ -2p & 2q &1-p^{2}-q^{2}\end{array} \right], \end{eqnarray*}$$

where p = tan (I/2)sin Ω, q = tan (I/2)cos Ω, I is the inclination of the ring plane to the planet equator, and Ω is the node of the ring plane on the planet equator; Ω is measured from the intersection of the planet equator and the ICRF reference equator. The unit vector $\hat{\mathbf {w}}$ is the pole of the ring plane. The rotation of an arbitrary vector x between planet equatorial and ring plane coordinates is

$$\begin{equation*} \mathbf {x}_{{\rm equatorial}} = \mathsf {R}_{3}\left({-\Omega }\right)\mathsf {R}_{1}\left({-I}\right)\mathsf {R}_{3}\left({+\Omega }\right)\mathbf {x}_{{\rm ring\, plane}}. \end{equation*}$$

$\mathsf {R}_{j}\left({\phi }\right)$ is the standard rotation of a vector by angle ϕ about coordinate axis j. The ring plane pole and the other ring plane coordinate axes in the ICRF system are

$$\begin{eqnarray*} \hat{\mathbf {W}}&=& \mathsf {R}_{3}\left({270{^\circ }-\alpha }\right)\mathsf {R}_{1}\left({\delta -90{^\circ }}\right)\hat{\mathbf {w}}\\ \hat{\mathbf {F}}&=& \mathsf {R}_{3}\left({270{^\circ }-\alpha }\right)\mathsf {R}_{1}\left({\delta -90{^\circ }}\right)\hat{\mathbf {f}}\\ \hat{\mathbf {G}}&=& \mathsf {R}_{3}\left({270{^\circ }-\alpha }\right)\mathsf {R}_{1}\left({\delta -90{^\circ }}\right)\hat{\mathbf {g}}, \end{eqnarray*}$$

where α and δ are the ICRF right ascension and declination of the planet pole. If the ICRF position of a ring particle is ${\boldsymbol {\rho }}$, and we define the direction $\hat{\boldsymbol {\rho }}= {\boldsymbol {\rho }}/|{\boldsymbol {\rho }}|$ then

$$\begin{equation*} \hat{\boldsymbol {\rho }}\cdot \hat{\mathbf {F}}= \cos \lambda \qquad \hat{\boldsymbol {\rho }}\cdot \hat{\mathbf {G}}= \sin \lambda \qquad \hat{\boldsymbol {\rho }}\cdot \hat{\mathbf {W}}= 0, \end{equation*}$$

where λ is the true longitude measured from the $\hat{\mathbf {F}}$ axis. By construction λ is equal to the "broken" angle v + ϖ; v is the true anomaly measured from periapsis, and ϖ is the longitude of periapsis.

For an elliptical ring, the radial position in the ring plane is

$$\begin{equation*} r = \frac{a(1-h^{2}-k^{2})}{(1+k\cos \lambda +h\sin \lambda )}, \end{equation*}$$

where a is semi-major axis of the ring, e is eccentricity of the ring, h is e sin ϖ, and k is e cos ϖ. If we allow the ring node and periapsis to precess, i.e.,

$$\begin{equation*} \varpi = \varpi _{0}+ \dot{\varpi }\,t \qquad \Omega = \Omega _{0}+ \dot{\Omega }\,t, \end{equation*}$$

where t is time from epoch, we find

$$\begin{eqnarray*} h &=& h_{0}\cos (\dot{\varpi }\,t)+k_{0}\sin (\dot{\varpi }\,t) \\ k &=& k_{0}\cos (\dot{\varpi }\,t)-h_{0}\sin (\dot{\varpi }\,t)\\ p &=& p_{0}\cos (\dot{\Omega }\,t )+q_{0}\sin (\dot{\Omega }\,t ) \\ q &=& q_{0}\cos (\dot{\Omega }\,t )-p_{0}\sin (\dot{\Omega }\,t ), \end{eqnarray*}$$

with h₀, k₀, p₀, snd q₀ being the values of h, k, p, and q at epoch t = 0.

The longitude rates are computed from the secular rate expressions from Borderies-Rappaport & Longaretti (1994) augmented with the perturbations due to external satellites based on the secular perturbations in Brouwer & Clemence (1961):

$$\begin{eqnarray*} \dot{\varpi }&=&\left(\frac{\mu _{0}}{a^{3}}\right)^{1/2}\left[ \frac{3}{2}J_{2}\left(\frac{R}{a}\right)^{2}\left(1+2e^{2}-2\sin ^{2}I\right)\right.\\ && - \frac{15}{4}J_{4}\left(\frac{R}{a}\right)^{4}+\frac{105}{16}J_{6}\left(\frac{R}{a}\right)^{6} -\frac{45}{32}J_{2}J_{4}\left(\frac{R}{a}\right)^{6} \nonumber \\ &&\left.+\frac{27}{64}J_{2}^{3}\left(\frac{R}{a}\right)^{6} +\frac{1}{4}\sum _{i=1}^{M}\frac{\mu _{i}}{\mu _{0}}\gamma _{i}^{2}b_{3/2}^{\left(1\right)}\left(\gamma _{i}\right)\right ] \\ \dot{\Omega }&=& -\left(\frac{\mu _{0}}{a^{3}}\right)^{1/2}\left [ \frac{3}{2}J_{2}\left(\frac{R}{a}\right)^{2}\left(1+2e^{2}-\frac{1}{2}\sin ^{2}I\right)\right.\\ && -\frac{15}{4}J_{4}\left(\frac{R}{a}\right)^{4} -\frac{9}{4}J_{2}^{2}\left(\frac{R}{a}\right)^{4} +\frac{105}{16}J_{6}\left(\frac{R}{a}\right)^{6}\nonumber \\ && +\frac{315}{32}J_{2}J_{4}\left(\frac{R}{a}\right)^{6} +\frac{351}{64}J_{2}^{3}\left(\frac{R}{a}\right)^{6}\nonumber\\ &&\left. +\frac{1}{4}\sum _{i=1}^{M}\frac{\mu _{i}}{\mu _{0}}\gamma _{i}^{2}b_{3/2}^{(1)}(\gamma _{i})\right ], \end{eqnarray*}$$

where μ₀ is the planet GM, μ_i is the GM of satellite i, J_n is n'th zonal gravity harmonic of the planet, R is the planet's equatorial radius, γ_i = a(1 + 1/2e²)/a_i, a_i is the orbital radius of satellite i, and $b_{j}^{(s)}(\gamma _{i})$ is the j'th Laplace coefficient of degree s; it is function of γ_i.

The equations of the rotational motion of Uranus torqued by the Sun and five satellites are

$$\begin{eqnarray} {\dot \mathbf H_{0}} &=& 3 m_{0}J_{2} R^{2}\left[\frac{\mu _{\odot }}{{{\rm r}_{0}^{3}}} (\hat{\mathbf {r}}_{0}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {r}}_{0}{\boldsymbol {\times }}\hat{\mathbf {k}})\right.\nonumber\\ &&\left. +\sum _{j=1}^{5}\frac{\mu _{j}}{{{\rm r}_{j}^{3}}} (\hat{\mathbf {r}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {r}}_{j}{\boldsymbol {\times }}\hat{\mathbf {k}})\right], \end{eqnarray} \tag{ B1 }$$

where H₀ is the angular momentum of Uranus, $\hat{\mathbf {k}}$ is the Uranus pole vector, R is the Uranus's equatorial radius, J₂ is Uranus's second zonal gravity harmonic, μ_☉ is the GM of the Sun, m₀ is the mass of Uranus, ${{{{\rm r}_{0}^{}}}}$ is the distance of Uranus from the Sun, $\hat{\mathbf {r}}_{0}$ is the direction vector from Uranus to the Sun, μ_j is the GM of satellite j, ${{{{\rm r}_{j}^{}}}}$ is the distance of satellite j from Uranus, $\hat{\mathbf {r}}_{j}$ is the direction vector from Uranus to satellite j.

If we average over the orbital motion of the Sun and satellites about Uranus, we find that

$$\begin{equation} (\hat{\mathbf {r}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {r}}_{j}{\boldsymbol {\times }}\hat{\mathbf {k}}) = -\frac{1}{2}(\hat{\mathbf {h}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {h}}_{j}{\boldsymbol {\times }}\hat{\mathbf {k}}), \end{equation} \tag{ B2 }$$

where $\hat{\mathbf {h}}_{j}$ is the unit vector normal to the Uranus orbit or to the Uranus centered orbit of satellite j. Consequently, the long term Uranus rotational motion is described by

$$\begin{eqnarray} {\dot \mathbf H_{0}} &=& -\frac{3}{2}m_{0}J_{2} R^{2}\left[\vphantom{\sum _{j=1}^{5}} \frac{\mu _{\odot }}{{{\rm r}_{0}^{3}}} (\hat{\mathbf {h}}_{0}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {h}}_{0}{\boldsymbol {\times }}\hat{\mathbf {k}})\right.\nonumber\\ &&\left. +\sum _{j=1}^{5}\frac{\mu _{j}}{{{\rm r}_{j}^{3}}} (\hat{\mathbf {h}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {h}}_{j}{\boldsymbol {\times }}\hat{\mathbf {k}})\right]. \end{eqnarray} \tag{ B3 }$$

Assuming that the angular momentum of Uranus is dominated by its spin angular momentum we have

$$\begin{equation} \mathbf {H}_{0} = m_{0}R^{2}\gamma s\hat{\mathbf {k}}= m_{0}R^{2}\gamma s\left[\begin{array}{@{}c@{}} \cos \alpha \cos \delta \\ \sin \alpha \cos \delta \\ \sin \delta \end{array} \right], \end{equation} \tag{ B4 }$$

with s being the planet rotation rate and γ its axial moment of inertia factor. For a constant rotation rate,

$$\begin{eqnarray} {\dot \mathbf H}_{0} &=& m_{0}R^{2}\gamma s\left(\dot{\alpha }\cos \delta \left[\begin{array}{@{}c@{}} -\sin \alpha \\ \cos \alpha \\ 0 \end{array} \right] +\dot{\delta }\left[\begin{array}{@{}c@{}} -\cos \alpha \sin \delta \\ -\sin \alpha\sin \delta\\ \cos \delta \end{array} \right] \right)\nonumber\\ &=& m_{0}R^{2}\gamma s\left(\dot{\alpha }\cos \delta \hat{\mathbf {f}}{}+ \dot{\delta }\hat{\mathbf {g}}{}\right). \end{eqnarray} \tag{ B5 }$$

Because of the orthogonality of the vectors $\hat{\mathbf {f}}{},\hat{\mathbf {g}}{},\hat{\mathbf {k}}$, we find

$$\begin{eqnarray} \dot{\alpha }\cos \delta &=&-\frac{3}{2}\left(\frac{J_{2}}{\gamma s}\right)\left[ \frac{\mu _{\odot }}{{{\rm r}_{0}^{3}}} (\hat{\mathbf {h}}_{0}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {h}}_{0}{\boldsymbol {\cdot }}\hat{\mathbf {g}})\right. \nonumber\\ &&\left.+\sum _{j=1}^{5}\frac{\mu _{j}}{{{\rm r}_{j}^{3}}} (\hat{\mathbf {h}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {h}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {g}})\right] \end{eqnarray} \tag{ B6 }$$

$$\begin{eqnarray} \dot{\delta } &=& \frac{3}{2}\left(\frac{J_{2}}{\gamma s}\right)\left[ \frac{\mu _{\odot }}{{{\rm r}_{0}^{3}}} (\hat{\mathbf {h}}_{0}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {h}}_{0}{\boldsymbol {\cdot }}\hat{\mathbf {f}})\right.\nonumber\\ &&\left. +\sum _{j=1}^{5}\frac{\mu _{j}}{{{\rm r}_{j}^{3}}} (\hat{\mathbf {h}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {k}})(\hat{\mathbf {h}}_{j}{\boldsymbol {\cdot }}\hat{\mathbf {f}})\right]. \end{eqnarray} \tag{ B7 }$$

We numerically integrated Equations (B6–B7) over the 1000 yr period from 1600 to 2600. The solar and satellite masses, distances, and orbit normals were taken from the ephemerides. We set J₂ = 3510.7 × 10⁻⁶, γ = 0.2269 (Hubbard & Marley 1989), and s = 501.1600928 deg day⁻¹ (Desch et al. 1986). The initial pole orientation angles (at J2000) were α = 77309980 and δ = 15172395. The Figures 2 and 3 display the change in the right ascension and declination over the 1000 yr period. We obtained the following analytical representation of the orientation angles by fitting a trigonometric series to the integration.

$$\begin{eqnarray*} \alpha &=& 77{.\!\!^\circ }309980 + 0{.\!\!^\circ }000173\,T + 0{.\!\!^\circ }000895\sin S1\\ && + 0{.\!\!^\circ }000180\sin S2 + 0{.\!\!^\circ }000098\sin S3 \\ && + 0{.\!\!^\circ }000075\sin S4 - 0{.\!\!^\circ }000818\sin S5 \\ \delta &=& 15{.\!\!^\circ }172395 + 0{.\!\!^\circ }000019\,T + 0{.\!\!^\circ }000851\cos S1\\ && + 0{.\!\!^\circ }000173\cos S2 + 0{.\!\!^\circ }000094\cos S3\\ && + 0{.\!\!^\circ }000072\cos S4 + 0{.\!\!^\circ }000818\cos S5 \end{eqnarray*}$$

with

$$\begin{eqnarray*} S1 &=& 328{.\!\!^\circ }616724 + 26{.\!\!^\circ }9601\,T,\\ S2 &=& 259{.\!\!^\circ }275089 + 2024{.\!\!^\circ }7285\, T, \\ S3 &=& 102{.\!\!^\circ }827444 + 182{.\!\!^\circ }8030\,T,\\ S4 &=& 185{.\!\!^\circ }361668 + 276{.\!\!^\circ }4108\,T\quad S5 = 137{.\!\!^\circ }359959 \end {eqnarray*}$$

where T is Julian centuries and d is days from J2000; the added the S5 term forces the series to sum to 0 at epoch. The series can be simplified by adding the S5 term to the constant term.

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