Ephemerides of the main Uranian satellites

MNRAS 436, 3668 3679 (2013) Advance Access publication 2013 October 24 doi:10.1093/mnras/stt1851 Ephemerides of the main Uranian satellites N. V. Emelyanov 1,2 and D. V. Nikonchuk 3 1 Sternberg Astronomical Institute, M. V. Lomonosov Moscow State University, 13 Universitetskij prospect, 119992 Moscow, Russia 2 Institut de Mécanique Céleste et de Calcul des Éphémérides Observatoire de Paris, UMR 8028 du CNRS, 77 avenue Denfert-Rochereau, F-75014 Paris, France 3 Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russia Accepted 2013 September 26. Received 2013 September 26; in original form 2013 August 20 ABSTRACT A new model of motions of the five main Uranian satellites is developed. The model is based on all published observations made since the dates of the satellites discoveries until 2008. The corresponding periods are 220 yr for Titania and Oberon, 160 yr for Ariel and Umbriel and 60 yr for Miranda. To fit the parameters of the satellites motion, 30 139 observations were used, including those made by the Voyager 2 spacecraft as well as astrometric results of the photometric observations of mutual occultations and eclipses of the main Uranian satellites in 2007 2008. The model is elaborated by the numerical integration of equations of motions of the satellites where all necessary perturbating factors were taken into consideration. Basing on the model, new ephemerides of the main Uranian satellites were generated for the period from 1787 to 2031. The ephemerides are put to the MULTI- SAT ephemeris server. The root-mean-square residual of observed topocentric positions of the satellites and their ephemeris positions is 0.43 arcsec. Taking into account the weighting factors of the observations, this value is 0.12 arcsec. An attempt is also made to define from observations the parameters of the mechanical energy dissipation of the satellites motion. The dissipation can be caused by both tides in the planet and by the tides in the satellites bodies. Approximate values of the quadratic terms in the orbital longitudes are obtained. These values are equal to (0.64 ± 0.11) 10 11 d 2 for Ariel, (0.08 ± 0.24) 10 11 d 2 for Umbriel, (0.29 ± 0.09) 10 11 d 2 for Titania, (0.32 ± 0.07) 10 11 d 2 for Oberon and (7.56 ± 1.15) 10 11 d 2 for Miranda. The obtained parameters reveal deceleration in the orbital motion of the satellites, which results from the influence of the tides raised on the interior of Uranus that is rotating faster than the satellites. Key words: astrometry ephemerides planets and satellites: individual: main Uranian satellites. 1 INTRODUCTION Theories of motion of planets and satellites are necessary to produce ephemerides of the celestial bodies and to carry out space missions. The main difficulty in the problem of the motions of the main Uranian satellites is their mutual interactions. Solar perturbations are by several orders smaller than those caused by mutual interactions. The influence of the dynamical oblateness of the planet is also small and comparable to the satellites mutual interactions only in the case of the satellite that is the closest to the planet. The five main Uranian satellites are positioned in the following order of increasing distance to the planet: Miranda, Ariel, Umbriel, E-mail: emelia@sai.msu.ru Titania and Oberon, the two latter being the first discovered by William Herschel in 1787. At the end of the 1980s, an analytical theory of the main Uranian satellites called GUST86 (Laskar 1986; Laskar & Jacobson 1987) was built. The theory was based on observations made between 1911 and 1986 including those by the Voyager 2 spacecraft. In this theory, the mutual interactions of the satellites were taken into consideration by the classical Laplace Lagrange secular perturbation method. The terms caused by the planet s oblateness were added to the equations for the secular perturbations. Short-period terms of the first order and resonance terms of the second order caused by the satellites mutual interactions were also taken into account. Other perturbations were not considered. The disturbing function was expanded up to second degrees in inclinations and eccentricities which ensured computation of linear perturbating terms. It was the C 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

Ephemerides of the main Uranian satellites 3669 GUST86 ephemeris of the main Uranian satellites that have been long used in the ephemeris service of the Jet Propulsion Laboratory (JPL) in the USA (Giorgini et al. 1996). After 1986, a large amount of new observations of the main Uranian satellites was accumulated including those of high accuracy. It turned out that the discrepancies between the GUST86 ephemeris and the new observations were improperly great. The first model of motion of the main Uranian satellites built by numerical integration of the equations of motion was elaborated by Taylor (1998). However, the model was based on observations made at a small interval of time, between 1977 and 1995. Lainey (2008) elaborated a new model of motion of the main Uranian satellites. His model was constructed by numerical integration of the differential equations for the satellites motion. The right-hand sides of the equations involved the following perturbing effects: the second- and fourth-degree zonal harmonics in the expansion of the potential function of Uranus, mutual attractions of the satellites and solar perturbations. To refine his model, Lainey (2008) used Earth-based observations of the main Uranian satellites made between 1948 and 2006 as well as observations carried out by the Voyager 2 spacecraft in 1985 1986. The ephemerides of the main Uranian satellites constructed by Lainey (2008) for the period 1995 2015 are available online via MULTI-SAT natural planetary satellites ephemeris server (Emel yanov & Arlot 2008). The ephemerides for the main Uranian satellites built by the numerical integration of the equations of motion were also elaborated in JPL (Rush & Jacobson 2007) using Earth-based observations made from 1911 to 2006 and Voyager 2 observations. These ephemerides were included into JPL s HORIZONS ephemeris server (Giorgini et al. 1996). Because of some properties of orbital motions of satellites, the accuracy of ephemerides becomes worse as the date of the ephemeris moves away from that of the last observation used to fit the model of satellites motion. Moreover, it is known that the accuracy of the ephemerides can be improved by expanding the interval of observations used for constructing the ephemerides (Emelyanov 2010). In the case of the main Uranian satellites, improvement of the ephemerides can be obtained by using all available observations, including those made in the past centuries provided that, when fitting the orbital parameters by the least-squares method (LSM), appropriate weights are assigned to them. In this paper, a new model of motion of the main Uranian satellites is proposed constructed by numerical integration of equations of motion of the satellites and fitted to their observations. The main difference between our model and previous ones is that, to refine the satellite orbits, all observations of the main Uranian satellites available for the time being, from the date of their discovery in 1787 to 2008, i.e. at a 220-yr time interval, as well as astrometric results of the photometric observations of the main Uranian satellites mutual occultations and eclipses in 2007 2008 were used. 2 MODEL OF MOTION AND METHOD TO REFINE THE ORBITS To construct the ephemerides of the five main Uranian satellites, we solved differential equations of motion of these satellites where coordinates in planetocentric non-rotating Cartesian reference frame with directions of the axes corresponding to the International Celestial Reference Frame (ICRF) were used. The reference frame s identity with the ICRF is ensured by reducing the results of observations to the ICRF. Coordinates of the Sun were computed in the ICRF as well by using the INPOP10 planetary ephemeris (Fienga et al. 2011). The right-hand sides of the equations involve perturbations from (i) mutual attractions of the satellites; (ii) oblateness of Uranus (only second- and fourth-degree zonal harmonics); (iii) the Sun. Constant direction of the north pole of Uranian rotation with respect to the ICRF was taken according to the report of the IAU Working Group (Archinal et al. 2011). The differential equations of motion of the satellites were taken in the following form: r i = G(m 0 + m i )r i r 3 i + N+1 j=1,j i ( ) r j r i Gm j 3 r j ij rj 3 + G(m 0 + m i )RF 2,4 (R T r i ) N + Gm j RF 2,4 (R T r j ), (1) j=1,j i where G is the universal gravitational constant, m 0 the Uranian mass, m i the mass of the ith satellite (i = 1, 2,..., N), r i the position vector of the ith satellite in the given planetocentric reference frame (ICRF), N the number of the satellites involved (here N = 5). If j = N + 1, we have r N+1 and m N + 1 (m N+1 = m Sun ) the position vector and mass of the Sun, respectively. In addition, ij is the distance of the ith satellite to the body j, R is a matrix of transformation from planetocentric equatorial coordinates to the ICRF coordinate system and F 2,4 (r p ) is a part of the planetary disturbing acceleration vector caused by the second- and fourth-degree zonal harmonics, referred to the planetocentric equatorial reference frame and expressed as a function of a satellite s planetocentric equatorial coordinates composing the vector r p. The matrix R depends on geocentric equatorial coordinates of the north pole of Uranus α 0, δ 0, and the vector F 2,4 (r p ) depends on the Uranian equatorial radius r 0 as well as on J 2 and J 4 zonal harmonic coefficients of the second and fourth degree, respectively. Table 1 gives the adopted values of the parameters used in the equations of the satellites motion, as well as adopted values of the astronomical unit (au) and the speed of light c. Table 1. Constants used in this paper for constructing the theory of motion of the main Uranian satellites on the basis of their observations. Constant Value Source r 0 26 200 km J 2 3.34343 10 3 French et al. (1988) J 4 2.885 10 5 Gm 0 5793 939.3 km 3 s 2 Gm 1 90.3 km 3 s 2 Gm 2 78.2 km 3 s 2 Jacobson (1992) Gm 3 235.3 km 3 s 2 Gm 4 201.1 km 3 s 2 Gm 5 4.4 km 3 s 2 Gm Sun 132 712 440 055.0 km 3 s 2 Fienga et al. (2011) c 299 792.458 km s 1 au 149 597 870.70 km α 0 257. 311 Archinal et al. (2011) δ 0 15. 175

3670 N. V. Emelyanov and D. V. Nikonchuk Numerical integration of the equations was carried out by the Everhart method with adaptive stepsize (Everhart 1974). Numerical integration of the equations of motion inevitably leads to accumulation of errors. Therefore, to ensure the reliability of results as well as to provide additional control of their accuracy, we also used the numerical integration method proposed by Belikov (1993) where, at each step, solution is expanded as a series of Chebyshev polynomials. Since the latter method significantly differs from that of Everhart, it should be expected that the accuracy of the solution can thus be controlled. Comparison of the solutions obtained by both methods made it clear that they coincide with a great reserve of accuracy (compared to that of observations) over a 250-yr time interval. Orbits were refined by the LSM with Lagrange orbital elements taken as refined parameters of the satellites motion: n, λ = M + ω +, h = e sin π, k = e cos π, p = sin I sin, 2 q = sin I cos, 2 where n is the mean motion, λ the mean longitude, M the mean anomaly, e the eccentricity, I the inclination, π the longitude of pericentre and the longitude of the ascending node. These denotations are used for each satellite, satellite s index is omitted. The elements I and are referred to the planetocentric equatorial coordinate system. The z-axis of this system points to the south pole of Uranus, and the x-axis is directed to the ascending node of this system s reference plane over the Earth equator. Such a choice of a reference frame provides prograde motion of the main Uranian satellites. It is significant to note the reasons why Lagrange elements were taken as refined parameters. The matter is that if Cartesian coordinates and velocities are taken as refined parameters, the refinement process based on observations spanning more than 20 yr won t converge. However, if Lagrange elements are taken, the refinement process converges in four to five iterations. To fit orbital parameters of satellites, we needed partial derivatives of observed functions with respect to the fitted parameters. These partial derivatives were defined as relations of a function s finite increment to that of its argument. Thus, we integrated the equations of motion several times, each time giving a finite increment to each parameter. Before fitting Lagrange elements of satellite orbits to real observations, the parameters were fitted by using satellite coordinates (referred to the Earth equator) computed from Lainey s ephemerides (Lainey 2008) over a 20-yr time interval (the maximum time interval of these ephemerides). In result, after parameters were refined, for all the satellites, the root-mean-square residual ( O C ) did not exceed 0.001 arcsec, which corresponds to 15 km in rectangular coordinates. Such a concordance in results shows that the model of the perturbing forces acting on satellites adopted in our paper is identical with that adopted in Lainey (2008). However, the main problem was in fitting orbital parameters of the satellites to all published observations with the aim to produce the most reliable ephemerides. It should be noted that, after the orbital parameters of satellites were fitted, the final generation of ephemerides of the five main Uranian satellites was carried out for the time interval from 1787 to 2031. The ephemerides are represented as data files containing the coefficients of the Chebyshev series expansion for rectangular planetocentric coordinates, each covering time interval of 1 d. The terms in the expansion series were taken up to the 12th order of polynomials. The data format corresponds to that used in the MULTI-SAT ephemeris server (Emel yanov & Arlot 2008). 3 THE OBSERVATIONS USED TO FIT THE ORBITS Our aim was to produce the ephemerides of the main Uranian satellites based on observations covering the maximum time interval. We tried to find all published observations of the major Uranian satellites beginning from the date when first two satellites were discovered by William Herschel in 1787. In many cases the published results of observations were entered into the data files manually. Some small portions of observations were taken from the Natural Satellites Data Base (NSDB; Arlot & Emelyanov 2009). These observations were delivered by the observers directly to the NSDB without publication. Depending on the way the observations were carried out, they can be attributed to the following types. 3.1 Micrometric observations This type of observations supposes that a satellite s topocentric angular distance and position angle relative to the planet or other satellite are measured. These two coordinates are often measured at different moments of time. In such cases, we considered them as two separate observations. The results of the observations were referred to the reference frame with the equator and equinox of date. We converted them to the reference frame of the J2000 epoch. The residuals between observed and computed coordinates ( O C ) for micrometric observations were calculated using the formula σ = (S o S c ) 2 + [S c (P o P c )] 2, where S o and S c are the observed and computed values of angular distance, P o and P c are the observed and computed values of position angle, respectively. The instances when either only angular distance or only position angle was measured were considered as separate observations, so that expression for σ included only one summand. 3.2 Absolute observations Here, a satellite s right ascension α and declination δ are obtained from the measurements of their positions relative to stars and star coordinates in catalogues. Therefore, errors in the star coordinates given in the catalogues directly enter into the errors of the satellite absolute observations. In addition, computed right ascensions and declinations of satellites directly include the errors of the adopted model of planetary motion. Hence, differences of observed and calculated satellite coordinates have systematic errors. Preliminary analysis of these differences demonstrated that the residuals averaged over different groups of observations are significantly less than their root-mean-square values. That is why we did not exclude systematic errors from absolute observations that could be obtained from preliminary analysis. The residuals between observed and calculated coordinates ( O C ) for absolute observations are given by σ = [(α o α c )cosδ] 2 + (δ o δ c ) 2, where α o, δ o are observed values of right ascensions and declinations, and α c, δ c are their calculated values. Almost all absolute observations that were found have already been referenced to the ICRF. Only 24 observations published by

Ephemerides of the main Uranian satellites 3671 Debehogne, Freitas-Mourao & Vieira (1981) were referenced to the coordinate system of the FK4 star catalogue. In the latter case, the coordinates were first converted into the J2000 reference frame in accordance with the IAU resolution, using procedure described by Aoki et al. (1983). After that absolute coordinates of the satellites were referenced to the ICRF using rotation angles taken from Feissel & Mignard (1998). 3.3 Relative observations These observations of satellites suppose that the differences in right ascensions and declinations of either satellite and planet or those of two satellites are measured. When necessary, the observations were referred to the reference frame with the equator and equinox of the J2000 epoch. Besides photographic observations and observations made using CCD detectors, differences between angular coordinates of two satellites can be also obtained from photometric observations of their mutual occultations. These observations give differences in topocentric coordinates that can be interpreted as relative coordinates. In some cases we have only observed position angle of one satellite relative to another one. For relative observations the residuals between observed and calculated coordinates ( O C ) are defined by σ = [( (α) o (α) c )cosδ] 2 + ( (δ) o (δ) c ) 2, where (α) o, (δ) o are observed differences in right ascensions and declinations, (α) c, (δ) c are their calculated values. 3.4 Pseudo-heliocentric observations In this case, astrometric data are obtained by processing photometric observations of mutual eclipses of the satellites. For a certain mean moment of the phenomenon, the differences between heliocentric angular coordinates of two satellites referred to the Earth equator are given. When using these data, it is necessary to take into consideration the fact that, because of differing time of light propagation, satellite positions are defined for different instants of time: position of the eclipsing satellite is observed at the instant t 2 while that of the eclipsed one is observed at the instant t 1 so that t 2 < t 1 < t 0,wheret 0 is an instant of observation. We used astrometric data obtained from photometric observations of the main Uranian satellites made during the international campaign of observations in 2007 2008 (Arlot et al. 2013). These data are 34 astrometric measurements of differences in topocentric and heliocentric right ascensions and declinations of satellite pairs and four measurements of relative position angle. Five measurements of the same type were taken from the results in Mallama et al. (2009). 3.5 Observations by the Voyager 2 spacecraft The data were taken from the paper by Jacobson (1992) that gives right ascensions and declinations of the satellites in the coordinate system with origin at the spacecraft as well as the spacecraft s rectangular planetocentric coordinates referred to the reference frame with the Earth equator and equinox of the FK4/B1950 epoch. The observed coordinates of the satellites and the spacecraft were converted to the FK5/J2000 system and then to the ICRF in accordance with the procedures described earlier for the case of absolute observations. When using observations published by Jacobson (1992), it is necessary to take into account the fact that the moments of observations given there are instants when images were registered by photodetector, while measured coordinates of the satellites and the spacecraft are given for an instant of time when photons were emitted from the observed satellite. The differences between these moments, i.e. the periods of light propagation, were published along with the observations. The Voyager 2 observations are specific in that they were made from relatively short distances to the satellites, the distances being significantly different for different instants of time. Hence, the same errors in the measurements of the angular coordinates of the satellites obtained by the spacecraft and by Earth-based observatories result in different errors in the satellites orbital positions. These differences are rather significant because Voyager 2 observations were taken at ranges between 1 and 2 10 8 km, while Earth-based observations of the Uranian satellites are made at a mean distance of about 2875 Mkm. To have the possibility to compare errors of different types of observations, we normalized (multiplied) residuals between observed and computed angular coordinates ( O C ) for Voyager 2 observations by ρ/2875, where ρ is the distance (in million km) of the spacecraft to a satellite. 3.6 Following description concerns all used observations Let σ i be the residual between observed and computed coordinates, where i is the index of observation. In fact, this variable is the angular distance between observed and computed topocentric satellite positions. Such definition of residuals allows us to compare the accuracy of observations of different types. The refinement of the orbital parameters of the main Uranian satellites is performed by determining corrections to the parameters. The corrections are computed by using the LSM based on the conditional equations. Each observation provides us with one or two conditional equations. It is to note that, when using the LSM, we assume that the errors of observations are random and independent of each other. This allows us to consider the covariance matrix of observation errors to be a diagonal matrix. It is necessary to know the diagonal elements of the matrix (with an unknown multiplying factor). However, since we do not know the covariance matrix of observation errors in advance, when composing the normal equations of the LSM, we tried to find weighting factors in such a way that it could give us the identity covariance matrix of observation errors. This is obtained by assuming the weighting factors to be inversely proportional to the squared observation errors. Therefore, our task was to define observation errors that were not known a priori. This aim can be reached by arranging observations in groups so that each group includes observations that are presumably of equal accuracy. First it is necessary to make preliminary fitting of orbital parameters to all the observations assuming that the observations have equal accuracy. This gives us the residuals σ i of each observation. Then it is necessary to define the root-meansquare residual values σ k for each group (k is the group s index). These values give us the observation errors that can be used for assigning weighting factors. Thus, each observation of one group has the same weighting factor, different groups having differing weighting factors. For a group k, the weighting factor can be obtained by w k = 1 σ 2. k

3672 N. V. Emelyanov and D. V. Nikonchuk Table 2. The number of used observations, intervals of observations and the number of revolutions made by each satellite during the given interval of observations. Satellite Number of The initial and final Interval Number of observations dates of observations (yr) revolutions U1 Ariel 5469 1847/09/15, 2008/01/04 160.303 23 230 U2 Umbriel 5510 1847/10/02, 2008/01/03 160.252 14 124 U3 Titania 10 077 1787/02/16, 2008/01/04 220.884 9267 U4 Oberon 15 325 1787/02/16, 2008/01/04 220.884 5992 U5 Miranda 3997 1948/02/15, 2008/01/04 59.886 15 475 Here we assume that observations of one type performed using the same procedures at a certain observatory have the same accuracy. The quality of each observation was tested by comparing it with the GUST86 ephemerides (Laskar & Jacobson 1987). If observed satellite position deviated from that computed from the ephemerides by more than 10 arcsec, the observation was assumed to be faulty and was thrown off. For some publications, all the observations published therein were rejected. Such discrepancies with the ephemerides can be explained by inaccuracies in the descriptions of observations. For this reason, we did not use the data published in Carlsberg Meridian Catalogues la Palma (1999), Mulholland et al. (1979), Soulie (1968, 1972, 1975, 1978), van Biesbroeck (1970) and van Biesbroeck et al. (1976). A total of 161 observations from other publications were rejected as well. In result, after all selected data were collected, there were 30 169 observations in total. Using preliminary fitting of orbital parameters to the 30 169 observations, deviations between observations and ephemerides were calculated. For 30 observations deviations were greater than 5 arcsec, and these observations were thrown off. As a result, we used a total of 30 139 observations. The number of observations and intervals of observations of each satellite are given in Table 2. It is easy to see that the total of observations given in the table significantly exceeds the aforementioned number of used observations. This is due to the fact that for some relative observations the differences in positions of two satellites were measured, and this observation was numbered among observations of each of two satellites. The data base of observations used in our work has rather heterogeneous character. Its full composition is given in Table 3. Basing upon the information on observations, they were split into groups. In total, there are 52 groups of observations. All observations in a group have the same weighting factor obtained from σ k (the root-meansquare residual in the group), which presented in Table 3. Abridged references to the sources of data are also presented, detailed information being given in the References section. The observations taken from the NSDB (Arlot & Emelyanov 2009) are designated as Communication to NSDB without any bibliographic reference. The column Type in Table 3 gives one of the following types of observations: rel (relative), abs (absolute), microm (micrometric), phot (photographic), CCD (CCD observations), Voyager-2 (Voyager 2 observations), phen (astrometric results of photometric observations of mutual occultations and eclipses), occ (mutual occultations), ecl (mutual eclipses) and occ-p (only position angle was defined from mutual occultations). The following abbreviation are used for some observatories: ESO European Southern Observatory La Silla, OHP Observatoire de Haute-Provence, CTO Cerro Tololo Observatory, La Serena and USNO United States Naval Observatory. 4 FITTING OF PARAMETERS AND ESTIMATIONS OF EPHEMERIS QUALITY The fitting of the orbital parameters of the five major Uranian satellites was carried out by using methodology and data described above. First, fitting was made in assumption that observations have the same accuracy. Thus, residuals were obtained used to assign weighting factors to the groups of observations. Then, fitting of orbital parameters was repeated using obtained weighting factors. After that, residuals were calculated again and weighting factors were reassigned. The resulting root-mean-square residuals for each group of observations are given in Table 3. These are the data that were used to make a final fitting of the parameters and produce satellite ephemerides for the time interval from 1787 to 2031. The quality of the obtained ephemerides is defined by several conditions. First of all, we must have a solution of differential equations of motion with accuracy no worse than that of observations. This condition should be fulfilled for the whole time interval for which we need to have ephemerides including the time interval of observations. To evaluate accuracy of the numerical integration, we integrated the equations of motion of the five Uranian satellites over a 250-yr time interval beginning from the date of the first observation and then, using coordinates and velocities obtained for the final instant of time, integration was performed backwards to the initial instant of time. Results of such integration were compared with the initial conditions. It turned out that for Miranda, the closest satellite to Uranus, the differences in coordinates did not exceed 0.7 km. For the remaining satellites, the differences proved to be significantly less. For the average distance from Earth to Uranus the angular distance of 1 milliarcseconds (mas) corresponds to the difference in Cartesian coordinates of about 15 km. Hence, the accuracy of solution of the equations of motion expressed in topocentric angular coordinates is about 0.05 mas, which is much better than the accuracy of the observations that were used. Thus, it is clear that the accuracy of the ephemerides is defined first of all by the accuracy of observations and time interval of observations. In addition, the ephemeris accuracy is influenced by the uniformity of distribution of observations over time and orbit as well as by correct choice of weighting factors. The time intervals of observations for each satellite are given in Table 2, and the rootmean-square residuals for each group of observations can be found in Table 3. The distribution of residuals of all observations of each satellite over time is plotted in Fig. 1. Fig. 2 presents distribution of residuals over time for all the observations of the five major Uranian satellites made by the Voyager 2 spacecraft. The quality of the used observations can be estimated from the variable σ, the root-mean-square residual of all observations. Since, when fitting orbital parameters of satellites, we assigned weighting factors to observations, the contributions of errors of different observations into the ephemerides error are different. Hence, to evaluate the accuracy of observations taking into account the weighting factors, the variable σ w was introduced which is defined as the square root of a weighted average of the squared residuals of all observations, that is N w i σi 2 σ w = i=1. N w i i=1 When fitting the satellite orbits to observations, it is important to work out criteria for rejecting faulty observations. For each

Ephemerides of the main Uranian satellites 3673 Table 3. Composition of observations and description of groups of observations. Here k is the group s index, N k the number of observations in the group, σ k the root-mean-square residual in the group, T k the interval of observations. The column Type gives one of the types of observations as described in the text. The names of the observatories are preceded by their IAU observatory codes (if they are known). k Type Publication Place of observations T k N k σ k 1 microm, rel Herschel (1835) Windsor 1787 1832 102 2.149 2 microm, rel Lamont (1840) 532 Munich 1837 1837 26 1.639 3 microm, rel Lassel (1847, 1848a,b, 1849, 1850, 1851, Valetta, Liverpool, Malta 1847 1865 484 1.263 1852a,b, 1853, 1857, 1864, 1867) 4 microm, rel von Asten (1872) 084 Pulkovo 1847 1871 275 1.273 5 microm, rel Winlock & Pickering (1888) 802 Harvard 1866 1868 58 1.922 6 microm, rel Rosse (1875) Birr Castle 1872 1874 50 1.772 7 microm, rel USNO (1878, 1880, 1882, 1883, 1884, 1885a,b, 787 USNO before 1893, 786 1874 1922 1937 0.675 1887, 1889, 1899), Hall (1876, 1877, 1878, 1911, USNO since 1893 1921, 1922), Hall & Bower (1923), Eppes (1913, also see Eppes 1901, 1902, 1907), Frederickson (1908), Frederick & Hammond (1905), Dinwiddie (1904) 8 microm, rel Holden (1881) 787 USNO 1877 1877 19 1.200 9 microm, rel Hough (1881) 768 Dearborn 1880 1880 39 0.652 10 microm, rel Henry & Henry (1884a,b,c,d) 007 Paris 1883 1884 139 0.814 11 microm, rel Perrotin (1887) 020-Nice 1887 1887 25 0.327 12 microm, rel Schaeberle (1895, 1897), Barnard (1896), Hussey 662 Mount Hamilton 1894 1914 1297 0.368 (1902), Aitken (1898, 1899, 1901, 1904, 1905, 1909, 1912, 1915) 13 microm, rel Hammond (1911) 786 USNO 1900 1907 930 0.611 14 microm, rel Barnard (1909, 1912, 1915, 1916, 1919) 754 Williams Bay 1907 1918 526 0.466 15 microm, rel Hall (1929) 786 USNO 1908 1925 581 0.529 16 phot, rel Rosanof (1925) 192 Tashkent 1913 1914 16 1.216 17 phot, rel Nicholson (1915) 662 Mount Hamilton 1914 1914 20 0.586 18 microm, rel Barnard (1927) 754 Yerkes 1919 1922 112 0.456 19 phot, rel Sytinskaja (1930) 192 Tashkent 1926 1926 62 0.757 20 microm, rel Struve (1928) 536 Berlin 1927 1928 82 0.408 21 phot, rel Steavenson (1948, 1964) 503 Cambridge 1947 1949 38 0.675 22 phot, rel van Biesbroeck (1970) 711 McDonald 1948 1964 804 0.353 23 phot, rel Whitaker & Greenberg (1973) 693 Tucson 1948 1973 133 0.246 24 phot, rel Tomita & Soma (1979) 371 Tokyo-Okayama 1964 1977 155 1.090 25 phot, rel van Biesbroeck et al. (1976) 695 Kitt Peak, 693 Tucson 1966 1966 55 0.225 26 phot, rel Walker, Christy & Harrington (1978), Harrington & Walker (1984), 689 USNO-Flagst., 807 CTO 1975 1986 653 0.216 Walker & Harrington (1988) 27 phot, rel Veillet & Ratier (1980) 586 Pic du Midi 1977 1979 58 0.154 28 phot, rel Veillet (1983b) 586 Pic du Midi, 511 OHP, 1977 1982 1260 0.133 809 ESO, 568 Mauna Kea 29 phot, abs Debehogne et al. (1981) 809 ESO 1980 1980 24 0.898 30 phot, rel Veillet (1983a) 809 ESO, 586 Pic du Midi 1980 1981 112 0.157 31 CCD, rel Pascu et al. (1987) 687 Flagstaff 1981 1985 76 0.150 32 phot, rel Veiga et al. (1987), Veiga & Vieira Martins (1994) 874 Itajuba 1982 1988 2285 0.141 33 CCD, abs Veiga, Vieira Martins & Andrei (2003) 874 Itajuba 1982 1998 8518 0.293 34 Voyager-2 Jacobson (1992) Voyager 2 1985 1986 445 0.026 35 phot, rel Chanturiya, Kisseleva & Emelianov (2002) 119 Abastuman 1987 1994 188 0.628 36 CCD, rel Veiga & Vieira Martins (1995, 1999) 874 Itajuba 1989 1998 4258 0.076 37 phot, rel Abrahamian et al. (1993) 123 Byurakan 1990 1990 32 0.887 38 phot, abs Yizhakevich et al. (1990) 188 Majdanak 1990 1990 58 0.594 39 phot, rel Jones, Taylor & Williams (1998) 950 La Palma 1990 1991 514 0.065 40 CCD, abs Stone (2001), USNO Planet. Ephemeris Data (2007) 689 Flagstaff 1998 2006 503 0.242 41 CCD, abs Qiao et al. (2013) 337 Sheshan, formerly Zo-Se 1998 2006 2358 0.138 42 CCD, rel Communication to NSDB by Owen (1999, 2001) 673 Wrightwood 1999 2001 15 0.276 43 CCD, abs USNO Planetary Ephemeris Data (2006) 689 USNO Flagstaff 1999 2005 396 0.250 44 CCD, rel Communication to NSDB (2003) 413 Siding Spring Observatory 2003 2003 6 0.130 45 CCD, abs Veiga & Bourget (2006) 874 Itajuba 2004 2004 287 0.587 46 CCD, abs Izmailov et al. (2007) B05 Barybino 2005 2005 20 0.175 47 CDD, rel Izmailov et al. (2007) B05 Barybino 2005 2005 14 0.258 48 phen, occ-p Arlot et al. (2013) See publication 2007 2007 4 0.020 49 CCD, abs Khovritchev (2009) F65 Haleakala-Faulkes 2007 2007 29 0.152 50 phen, occ, ecl Mallama et al. (2009), Arlot et al. (2013) See publication 2007 2008 39 0.027 51 CCD, rel Communication to NSDB by Kisseleva (2009) 084 Pulkovo 2007 2008 10 0.215 52 CCD, abs Communication to NSDB by Kisseleva (2009) 084 Pulkovo 2007 2008 12 0.153

3674 N. V. Emelyanov and D. V. Nikonchuk Figure 1. Residuals of observations for each satellite. Figure 2. The residuals of observations made by the Voyager 2 spacecraft normalized by geocentric distance. observation the residual σ was calculated, which is the angular distance between the observed and computed satellite positions. We defined a certain limit value σ lim so that observations with the σ greater than this value were rejected. As mentioned above, initially the value of σ lim was taken to be 5 arcsec. If we reject observations using the more strict criteria, for example if we take σ lim = 0.5 arcsec, too many observations providing us with the data on the satellites motion will be thrown away and the time interval of the used observations can became shorter. In result, with such a rejection of observations, by using the LSM, we can obtain very small standard errors on the parameters of motion, but both refined parameters of motion and ephemerides will deviate from their real values more significantly than in the case of more complete set of observations. The probability theory does not give us strict criteria for rejecting faulty observations to get optimum evaluations of parameters. That is why we had to act at our own discretion. Trying to find some auxiliary information that could provide us with the necessary criteria, we refined the parameters of the satellites motion with different values of σ lim. After a certain deal of observations was rejected, the root-mean-square residuals of each group of observations were recalculated for each σ lim and weighting factors were reassigned. The results are given in Table 4. It is to note that in this table observations of one satellite relative to another are included among observations of either of the satellites. However, in the column Number of used observations they are counted only once. It should be noted also that the root-mean-square residuals of groups σ k given in Table 3 are calculated with σ lim = 5.0 arcsec. Table 4 shows that the root-mean-square residuals of observations taking into account the weighting factors (σ w ) do not significantly differ if the criteria for rejection of observations are changed. Note that the accuracy with which the parameters of the satellites motion are defined from the covariance matrix of parameters do not significantly differ with various values of σ lim either. This makes us conclude that, for the three criteria for rejection of observations, we have roughly the same accuracy with which parameters of motion are defined. Since with σ lim = 5.0 arcsec we have greater number of observations that span the greater time intervals, it was decided to take this limit value for generating the final variant of the ephemerides of the main Uranian satellites. Refinement of parameters of motion using different criteria for rejecting observations provides us with several variants of the ephemerides having the same accuracy but based on different set of observations. Comparison of these variants of ephemerides allows us to obtain some additional estimations of their accuracy. We compared the ephemerides produced with σ lim = 5.0 and 1.0 arcsec at the time interval from 1787 to 2031. For a series of time instants with constant stepsize angular distances between geocentric satellite positions obtained from the two variants of ephemerides were calculated. The results of the comparison for each of the five satellites are presented in Fig. 3. It is seen that at the time interval between 2000 and 2031 the ephemerides do not differ by more than 7mas. It is interesting to compare our ephemerides of the main Uranian satellites with those produced earlier by other authors. We took the ephemerides obtained by Lainey (2008) and, using the MULTI- SAT ephemeris server (Emel yanov & Arlot 2008), computed the geocentric equatorial coordinates of each satellite for a series of instants within the time interval of Lainey s ephemerides. For each obtained position we calculated residuals with our ephemerides (in the same manner as the residuals of observations were calculated), using final variant of our ephemerides where σ lim = 5.0 arcsec. The obtained residuals of both ephemerides for each of the five satellites are shown in Fig. 4. The most reliable estimations of the ephemeris accuracy can be obtained by the method of the repeated variation of errors or set of observations with subsequent refinement of the model of the

Ephemerides of the main Uranian satellites 3675 Table 4. The root-mean-square residuals of observations calculated with different values of σ lim. Unweighted values of σ and corresponding values of σ w taking into account the weighting factors are calculated on the basis of observations of all five satellites. Time interval of observations (yr) Number of observations of satellites Number of the used observations 5.0 U1 Ariel U2 Umbriel U3 Titania U4 Oberon U5 Miranda 160.3 160.3 220.9 220.9 59.9 5469 5510 10 077 15 325 3997 1.0 U1 Ariel U2 Umbriel U3 Titania U4 Oberon U5 Miranda 156.2 156.2 218.0 215.8 59.9 0.5 U1 Ariel U2 Umbriel U3 Titania U4 Oberon U5 Miranda 156.1 156.2 216.8 215.8 59.9 σ lim (arcsec) Satellite σ (arcsec) σw (arcsec) 30 139 0.430 0.121 5378 5423 9705 14 910 3971 29 229 0.270 0.117 5183 5185 8712 13 969 3839 27 063 0.203 0.109 Figure 3. Comparison of the ephemerides obtained with σ lim = 5.0 and 1.0 arcsec. The vertical axis gives the angular distance between the satellite s geocentric positions obtained from both ephemerides. Figure 4. Residuals between Lainey s ephemerides and our ephemerides for all the five satellites. satellites motion, recalculation of the ephemerides and defining the ephemerides dispersions for each given instant of time. This can be done in the same way as in Desmars et al. (2009) and Emelyanov (2010). These procedures are planned to be done in the future. For some problems of practical importance it is necessary to know approximate satellite coordinates at large time intervals. To solve such problems, we took the most simple model of motion the circular orbit in the equatorial plane of the planet, and, using our ephemerides, defined the most appropriate parameters of the model for 245-yr time interval including the time interval of the satellites observations. To describe the satellite motion, we use the planetocentric coordinate system where the main plane is that of the planet s equator. According to the decision of the IAU Working Group on Cartographic Coordinates and Rotational Elements (Archinal et al. 2011), the direction of the north pole is that forming an acute angle with the direction to the north pole of the invariable plane of the Solar system. For Uranus, this direction has negative declination so that the planet s rotation and orbital motion of its main satellites in this reference frame turn out to be retrograde. This presents some inconvenience in describing the motion of the satellites using customary formulae and parameters. That is why we consider the motion of the satellites in the coordinate system referenced to the Uranian equator where the positive direction of the z-axis coincides with that of the Uranian south pole. The x-axis points towards the ascending node of the given coordinate system over the Earth s equator. Thus, the

3676 N. V. Emelyanov and D. V. Nikonchuk Table 5. The parameters of the simplified model of the main Uranian satellites obtained by using the generated ephemerides. Satellite a (km) λ 0 N 0 (d 1 ) Period (d) U1 Ariel 190 929.789 657 4.191 290 417 89 2.492 952 336 2207 2.520 379 237 05 U2 Umbriel 265 984.008 927 2.550 274 819 30 1.516 147 852 7029 4.144 177 163 18 U3 Titania 436 281.937 093 2.848 933 837 86 0.721 718 319 6095 8.705 869 224 13 U4 Oberon 583 449.534 238 5.474 511 499 64 0.466 692 006 2040 13.463 237 474 94 U5 Miranda 129 848.114 137 0.508 309 927 24 4.445 190 522 8990 1.413 479 416 64 longitudes of the main satellites are calculated relative to this axis and increase in time. According to this simplified model, as mentioned above, a satellite has constant distance to the planet s centre a and moves along the planet s equator, its longitude being the function of time defined by λ = λ 0 + N 0 (t t 0 ). Using the generated ephemerides we obtained the parameters a, λ 0, N 0 and orbital periods of the satellites, which are given in Table 5. The time is given in the TT time-scale where the initial epoch t 0 is JD 244 5700.5. Although this simplified model is approximate and eccentricities and inclinations of orbits are neglected, such an ephemeris can be used at rather large time intervals beyond the limits of the period of the satellites observations. 5 EVALUATION OF THE ORBITAL DECELERATION OF THE SATELLITES The major planetary satellites raise tides on the planet s interior. If the planets were perfectly elastic bodies, the tides would not change the energy of the satellites orbital motion. However, viscosity of planetary material causes the tidal waves to shift in relation to the direction towards satellite. This shift, in its turn, creates additional force acting upon satellite along its trajectory and leading to the acceleration or deceleration of its orbital motion. Hence, positive or negative dissipation of mechanical energy appears, orbital longitude acquires a term quadratic in time and semimajor axis obtains a perturbing term linear in time. The sign of the coefficient of the quadratic term in longitude depends on the ratio of the planetary rotation velocity to the velocity of the satellite s orbital motion. If a planet rotates faster than its satellite, which is the case of the Moon and major Uranian satellites, the force raised by the tides on the planet s interior acts in the direction of the satellite s motion so that the coefficient of the quadratic term in longitude is negative and semimajor axis increases. If a satellite revolves in orbit faster than the planet rotates, the coefficient of the quadratic term in longitude is positive. This is the case of the Martian moon Phobos that is falling to Mars. Lainey et al. (2009) investigated the influence of tides upon the motion of the Jupiter s Galilean satellites. The coefficients of the quadratic terms in longitude were defined from astrometric observations of the satellites in conjunction with refinement of their orbital parameters. Significant progress in this area was achieved owing to the using of astrometric results of photometric observations of mutual occultations and eclipses of the satellites. These results have better accuracy in contrast to direct astrometric observations. Observations of Io s motion made it clear that it has orbital acceleration, although tides raised on Jupiter should cause Io to decelerate its motion since Jupiter rotates faster than Io. Not only does the abovementioned paper by Lainey et al. (2009) give explanation of this effect but it also provides evaluations of viscosity coefficients of both planet and satellite. It was found that, because of the tides raised on the body of the satellite itself, Io loses more energy than acquires from the tides raised on the body of Jupiter. In the case of the main Saturnian satellites we see a more complex picture of the way in which mechanical energy of motion changes. Lainey et al. (2012) found strong dissipation of the mechanical energy of orbital motion of these satellites caused by the tides in the Saturn s interior. The motion of Enceladus is a particular case in this problem. The study of mechanical energy dissipation of orbital motion of satellites is of great interest because it can be used to define physical parameters of satellites as well as to investigate heating balances of their interiors. Acceleration or deceleration of a satellite s orbital motion can be discovered from its astrometric observations provided that accuracy is sufficiently high and time interval of observations is long enough. It is necessary to introduce an additional parameter of motion describing the effect in consideration and to fit it to observations along with other parameters. In an analytical theory of satellite motion one of the following constants could be taken as such a parameter: ˆλ, the coefficient of the quadratic term in the expression for perturbations in satellite s orbital longitude: δλ = ˆλ(t t 0 ) 2 ; ˆn, the coefficient of the linear term in perturbations of mean motion: δn = ˆn(t t 0 ); â, the coefficient of the linear term in perturbations of semimajor axis: δa = â(t t 0 ), where t is time and t 0 is an initial epoch. When performing numerical integration of the equations of motion, influence of tides can be approximately taken into consideration by adding to the right-hand sides of the equations the perturbing acceleration directed opposite to satellite s velocity vector and having the following components: F x = ˆk V x V, F y = ˆk V y V, F z = ˆk V z V, where V x, V y, V z are satellite s velocity projections, V the absolute value of velocity and ˆk is a constant. The constant ˆk, which we call the coefficient of satellite decelerarion, can be both positive and negative. Newtonian equations for the Keplerian elements (Duboshin 1975) give the following relationships between these coefficients: ˆλ = 3 2 ˆk a, ˆk ˆn = 3 a, ˆk â = 2 n.

Ephemerides of the main Uranian satellites 3677 Table 6. The coefficients of deceleration ˆk of the main Uranian satellites obtained with different criteria for rejection of faulty observations, and averages over the three variants. The errors (±) are defined as maximum deviations from the averages over the three variants. The coefficients ˆλ (for changes in longitude) and â (for changes in semimajor axes of orbits) are calculated from the averaged values of ˆk. Average ˆλ â Satellite ˆk 10 6 km d 2 ˆk 10 6 10 11 10 6 σ lim = 5.0 arcsec σ lim = 1.0 arcsec σ lim = 0.5 arcsec km d 2 d 2 km d 1 U1 Ariel 0.70 1.04 0.73 0.82 ± 0.22 0.64 ± 0.11 0.66 U2 Umbriel 0.10 0.25 0.60 0.15 ± 0.45 0.08 ± 0.24 0.20 U3 Titania 0.89 0.58 1.06 0.84 ± 0.26 0.29 ± 0.09 2.33 U4 Oberon 1.00 1.25 1.53 1.26 ± 0.27 0.32 ± 0.07 5.40 U5 Miranda 5.60 6.49 7.55 6.55 ± 1.00 7.56 ± 1.15 2.95 To the six refined parameters of motion of each satellite we added the coefficient of satellite decelerarion ˆk i,wherei is the satellite s index. Initial values of the coefficients were assumed to be zeroes. After all the parameters were fit to all available observations, the coefficients of satellite deceleration were obtained for each satellite. These calculations were carried out for three values of σ lim (the criterion for rejection of faulty observations) with recalculation of weighting factors for each set of observations (as described above). The computed values of the coefficients for different sets of observations are given in Table 6. Standard errors of ˆk i defined from the LSM turned out to be rather small: 0.2 10 6 km d 2 for Miranda, while for the remaining satellites standard errors are not greater than 0.04 10 6 km d 2. However, these values of errors produced with the LSM do not represent real accuracy with which the coefficients of satellite deceleration were defined. It is to note that, after additional refined parameters were introduced, there was only slight decrease in the root-meansquare residual σ and in the corresponding value of σ w. Thus, with σ lim = 5.0 arcsec we have σ = 0.430 arcsec and σ w = 0.120 arcsec; with σ lim = 1.0 arcsec we have σ = 0.270 arcsec and σ w = 0.116 arcsec. For σ lim = 0.5 arcsec the residuals are σ = 0.203 arcsec and σ w = 0.108 arcsec. If we compare these values with those given in Table 4, we see that the differences are about 1 per cent. In such cases refined parameters significantly depend on the set of used observations. Number of observations and duration of their time intervals are the same as in the case when the coefficients of satellite deceleration were not among refined parameters. As the most probable values of the coefficients of deceleration we took averages over the three variants, adopting the maximum deviations from the averaged values as their errors. As seen from Table 6, it is only for Miranda that the coefficient of deceleration can be defined with an accuracy of 15 per cent. For the rest of the satellites, accuracy is much worse. However, calculated values can be used as estimations limiting ranges of possible values of the coefficients. For all five satellites the coefficients of deceleration turned out to be negative, which means that the satellites have negative deceleration. Tides in the interior of Uranus, which rotates faster than the satellites, transfer additional mechanical energy to them. As a result, the satellites decelerate their orbital motion, moving away from the planet. Taking into account the deceleration of the satellites should provide better agreement of the model of motion with the observations, i.e. we can produce more accurate ephemerides. Obviously, the satellite ephemeris that is fit to observations and where deceleration of satellites is taken into consideration would differ from that generated without taking into account this effect. This difference Figure 5. The differences between topocentric positions of the satellite U5 (Miranda) obtained with the ephemerides generated with both taking into account the quadratic term in longitude and without it. should be evaluated. With this in view, we calculated residuals of topocentric satellite positions for a series of instants of time using both ephemerides. In both cases, the same set of observations (30 139 observed satellite positions) was used. For the satellite U5 (Miranda), calculations were made at the time interval from 1948 to 2031 (see Fig. 5). For the remaining four satellites, such comparison was made at the time interval between 1788 and 2031 (see the results in Fig. 6). The figures show that the differences in satellite positions exceed the estimations of the ephemerides errors given above as well as surpass the level of accuracy of present-day observations. Therefore, generation of the ephemerides of the main Uranian satellites that are based on observations should take account of the effect of the secular deceleration of the satellites orbital motion. 6 CONCLUSION In this paper we presented the new model of motion of the five main Uranian satellites based on all available observations. This model is used to generate ephemerides of the satellites at the time interval from 1787 to 2031. As distinct from the previous ephemerides produced by other authors, our ephemerides have significantly greater