3.3.1 Introduction Uranus, the seventh planet of the Solar System, has 27 known moons. Uranus’s moons are divided into three groups: thirteen inner moons, five major moons, and nine irregular moons. The inner moons are small dark bodies that share common properties and origins with the Uranus’s rings, the five major moons (Miranda, Ariel, Umbriel, Titania, and Oberon) are massive enough to have achieved hydrostatic equilibrium. They range in diameter from 472 km for Miranda to 1578 km for Titania. While, Uranus’s irregular moons range in size from 120–200 km (Sycorax) to about 20 km (Trinculo).

It is worth note that “Margaret” is the only known irregular prograde moon of Uranus, and it currently has the most eccentric orbit of any moon in the Solar System, though Neptune’s moon Nereid has a higher mean eccentricity. As of 2008, Margaret’s eccentricity is 0.7979. 3.3.2 Analytical Theories of Uranian satellites Laskar (1986) presented a general analytical theory (GUST) of the main satellites of Uranus, taking into account the all secular and short-period perturbations. GUST, is a first approach to a general analytical theory of the Uranian satellites.

We mean by general that it includes all secular perturbations (secular mutual perturbations of the satellites) but also the short period perturbations which have not already been taken into account. Laskar and Jacobson (1987) fitted GUST86 theory to observations (Earth based observations and Voyager data) and calculated the masses of the satellites. The effects of mutual perturbations of the satellites and of J2 and J4 of Uranus are included. The analysis of the short-period terms in the solution of the Uranian satellites Umbriel, Titania and Oberon shows that the optical data from Voyager encounter should lead to a good determination of the masses of Titania and Oberon.

The disturbing function Ri for the satellite Si is given by: , (3.1) where, is the vector which links Uranus with the satellite , is the distance from to and the part of the disturbing function depending on Uranus harmonics . In this theory, secular terms are limited to the first degree of the first order system , (3.2) where denotes the secular part and the short period part. However, it still can be improved by computing the secular system of order two and a few more short period terms of order two. The precision of the theory is given by direct comparison with the numerical integration of the equations of motion. The maximum differences, is given in kilometers over 12 years for the main satellites as shown in Table 10.

Table 10 Precision of the theory over 12 years Satellite Differences (km) Miranda 10 Ariel 20 Umbriel 40 Titania 100 Oberon 100

Fig. 9 shows the comparison between GUST and numerical integration. Malhotra et al. (1989) introduced a secular perturbation theory of the Uranian satellites, derived from Fourier analysis of a long numerical integration of the equations of motion. In addition to the perturbations of the oblateness of Uranus, Malhotra et al. revised the theory of Laskar (1987) and Laskar & Jacobson (1987) by including the effects of near-resonance of the satellites Titania-Oberon and Umbriel-Titania. With the inclusion of the effects of such near-resonances, the maximum error in the secular frequencies is reduced from 16% to less than 3% for eccentricity, and less than 1% for inclination.

The osculating eccentricities of the satellites show large amplitude variations on both short (86-145 day) and long (120-980 year) time scales. Apostolos and Murray (1997) constructed a second order Laplace-Lagrange theory applied to the Uranian satellite system. The theory takes into account the oblateness of Uranus and near-resonance of satellites (e.g. Umbriel-Titania and Titania-Oberon), and developed in satellites’ masses up to second order. It is compared to the previous theory constructed by Malhotra et al. (1989).

Fig. 9: Precision of the theory (Laskar 1986)

Lainey (2008) developed a new dynamical model of the main Uranian satellites, based on numerical integration and fitted to astrometric observations. Old observations, as well as modern and Voyager observations have been included. The proposed model takes into account: (i) the Uranian gravity field up to degree 4, (ii) the perturbations of the Sun, (iii) the mass of each Uranian satellite, and (iv) the IAU2000 Uranian northern pole orientation. The integrator subroutine is from Everhart (1985) and called RA15. An error of more than 0.1 arcsec on the Uranian position is observed.

3.3.3 Outer satellites of Uranus It is interesting to recall that, until 1997, Uranus was the only giant planet without observable outer satellites. The system of Uranus currently numbers only nine outer satellites, with only one of them having a prograde motion. The semi-major axis of its orbit lies within the range of semi-major axes for the retrograde orbits (see Table 11).

The maximum eccentricities are approximately the same (0.70–0.85), while the maximum angles between the orbital and ecliptical planes are about 65o and 40o, respectively, for the prograde and retrograde orbits (Vashkov’yak & N. M. Teslenko 2008). Table 11. Characteristics of the family of evolving orbits for the outer Uranian satellites Types of orbits Number of satellites amin million km amax million km emin emax imin deg. imax deg. Prograde orbits 1 14.5 14.7 0.44 0.85 47 66 Retrograde orbits 8 4.3 20.8 0.07 0.70 139 172

3.4. Moons of Neptune 3.4.1 Introduction According to the recent space explorations, Neptune has 14 known moons. Of them are seven inner moons. Their orbits lie among several faint rings and partial rings composed of dust and ice particles. Outwards from Neptune, the inner moons are Naiad, Thalassa, Despina, Galatea, Larissa, S/2004N1 and Proteus. Next outward from Neptune, Triton is one of four moons in the solar system known to have an atmosphere.

Triton is Neptune’s largest moon, at 1,677 miles (2,700 km) in diameter. Because Triton moves backward (retrograde) in its orbit, astronomers think the icy moon formed in the frozen Kuiper belt, located in the outer solar system. Later, Neptune’s gravity captured Triton. After Triton comes Nereid. At 211 miles (340 km) in diameter, Nereid is one of the largest irregular moons in the solar system. Irregular moons have orbits that are highly inclined to the planet’s equator. Much further out than Nereid orbit five outer moons, Halimede, Sao, Laomedeia, Neso and Psmathe. These irregular moons travel in highly tilted orbits.

The following figure shows the inner satellites of Neptune, the orbits of the five outer satellites, circular orbit of Triton and the highly eccentric orbit of Nereid. It is shown that, Triton has a size slightly less than the Moon. Fig. 10 3.4.2 Analytical Theories Several analytical theories have constructed to study the effects of perturbing forces on the orbital motion of Neptune’s moons, most of them are devoted for the bizarre orbit of Nereid. Mignard (1975) constructed a satellite theory disturbed by the Sun with use of the Von Zeipel method and applied his theory to Nereid in (1981). He adopted an eccentric anomaly for the expressions of the solution of which method was proposed by Hori (1963).

In the elimination of the long periodic terms, he assumed the inclination is small and the solution was obtained up to the second order of the inclination. Oberti (1990) extended the theory of Mignard by including the perturbation from Triton with use of Deprit method (1969). Segerman and Richardson (1997) included the oblateness perturbation of Neptune with use of Deprit method. Several authors have dealt with the orbital determinations of Nereid (Rose 1974; Veillet 1982, 1988; Jacobson 1990, 1991). Rose fit van Biesbroeck’s (1951, 1957) observations, while Veillet used the theory of Mignard (1975). Jacobson fit the numerically integrated Neptunian satellite orbits (Nereid and Triton) to Earth-based astrometric observations and Voyager spacecraft observations.

Saad and Kinoshita (2000) constructed an analytical theory of motion of the second Neptunian satellite Nereid using Lie transformation approach advanced by Hori (1966). The main perturbing forces which coming from the solar influence are taken into account. The disturbing function is developed in powers of the ratio of the semimajor axes of the satellite and the Sun (6 10-3) and put in a closed form with respect to the eccentricity.

The theory includes secular perturbations up to the fourth order, short, intermediate and long period perturbations up to the third order. The osculating orbital elements which describe the orbital motion of Nereid are evaluated analytically. The comparison with the numerical integration of the equations of motion gives an accuracy on the level of 0.2 km in the semimajor axis, 10-7 in the eccentricity and 10-4 degree in the angular variables over a period of several hundred years 3.4.3 Hamiltonian of the Motion The Hamiltonian equation of the nonplanar case is given by , (3.3) where , (3.4) , (3.5)

(3.6) (3.7) , and n are the mean motions of the Sun and the satellites, are Delaunay’s elements and k defines the longitude of the Sun(which is moving in a Keplerian orbit). To reach our goal we applied a successive of canonical transformations on the Hamiltonian equations. At the first stage we eliminate the short-periodic terms and take into account the eccentric anomaly of Nereid u as an independent variable, since the high eccentric orbit of Nereid precludes replacing functions of the true anomaly by expansions involving the mean anomaly.

The analytical expressions of the new Hamiltonian and the determining functions are given in Saad & Kinoshita 2000). In the present study, the short-period is days, which describes the orbital revolution of Nereid around Neptune. Removal the intermediate term, k will be achieved by building another transformation. Here, the intermediate period k is about 165 years, which defines the orbital revolution of Neptune around the Sun. The Hamiltonian is also free from the node h* since the disturbing potential becomes axial symmetric.

After eliminating the long (intermediate) terms, the orbital elements are computed from . (3.8) The analytical expressions of the new Hamiltonian and determining function have evaluated and checked by two ways from the analytical point of view. The first one is satisfying d’Alembert characteristics. The second way, we put (inclination of Nereid) in the general formulae and got the analytical expressions of a fictitious Nereid (Saad & Kinoshita 1999). Up to this stage, the Hamiltonian system is still including the long terms g (~13000 years). We overcome this problem using, Jacobian elliptic functions (Kinoshita and Nakai 1991, 1999), and the mean elements of Nereid are evaluated. 3.4.4 The osculating orbital elements

The osculating orbital elements are a combination of secular, long-periodic and short-periodic variations in elements. In order to get the osculating orbital elements, and hence, evaluate ephemerides for Nereid, we have started with the mean elements (by solving the final Hamiltonian system in Jacobian elliptic functions). Then we substituted reversely in the formulae of long-periodic and short-periodic perturbations.

The above analytical expressions are implemented for digital computations by constructing a computational algorithm described by its purpose, input and its computational sequence. The final equations of the osculating elements have the forms (3.9) where, the subscripts long and sho have the meanings long-period and short-period perturbations respectively. For any element x, , defines the mean elements and the symbol refers to the variations of the elements. Tables 12 and 13 show the amplitudes and the accuracy in the osculating orbital elements for both short and long periodic perturbations respectively.

Table I2. Amplitudes of the osculating elements Elements Short-period Long-period Units Semi-major axis 747.989 1196.78 km eccentricity 0.0004 0.0115 rad Arg. of pericenter 0.006 0.7 deg. Inclination 0.0025 0.16 deg. Long. asc. node 0.01 0.17 deg. Mean anomaly 0.0325 2.0 deg.

Table 13. Accuracy of the osculating elements Elements Short-period Long-period Units Semi-major axis 0.2 0.2 km eccentricity

rad Arg. of pericenter

deg. Inclination

deg. Long. asc. node

deg. Mean anomaly

deg.

Fig. 11 shows the residuals in the osculating elements after making some corrections in mean motions of the elements , g and h using least-squares method. The global internal accuracy of the present theory is obtained by direct comparison with the numerical integration of the equations of motion. The maximum discrepancies reached 0.2 km in the semi-major axis, ~ 10-7 in the eccentricity and 4 10-4 degree in the angular variables over several hundred years.

The theory has not fitted to the observations. We intend to do that after including the perturbations of Triton and the oblateness of Neptune although the latter’s effect is very small. Then the integrations constants can have the real meaning. Fig. 11: Residuals in the osculating orbital elements of Nereid for long time interval. The semi-major axis is given in km, eccentricity in radians and the rest of elements are in degree (Saad & Kinoshita 2001).

Vashkov’yak and Teslenko (2010) in addition to the effects of the mother planet Neptune, they considered the influence of the attraction of the internal satellite Triton on the evolution of the orbit of the external satellite Nereid. The disturbing function of Triton is given by: , (3.8) where is the product of the gravitational constant and mass of Triton, and refer to the radius vectors of Triton and Nereid respectively and .

In spite of the fact that Nereid is classified as a distant satellite, the evolution of its orbit is presented as regular enough with practically secular variations of pericentric argument and node longitude, and with a conditionally periodic change of eccentricity and ecliptic inclinations. Perturbations from Triton are essentially smaller solar perturbations and are most noticeable during inclination change. The change of eccentricity is relatively insignificant and is about 0.02 for several thousand years. For other elements of the orbit, these perturbations are much less noticeable.