5 Numerical integration

5.1 Integration method

Numerical integration has been used very effectively in generating ephemerides of natural satellites. A numerical integration can provide a powerful way for checking the validity of the analytical theory before making any fit to real observations (see, for example, Sinclair & Taylor 1985; Harper et al. 1989). In this paper, we explore the use of numerical integration in producing an ephemeris of the five major Uranian satellites, which can also be used in our calibration calculation as an alternative to an analytical theory.

As concerns numerical integration methods, Hadjifotinou & Harper (1995) and Hadjifotinou & Ichtiaroglou (1997) studied the behaviour of two numerical integration methods, the 10th-order Gauss-Jackson backward difference and the 12th-order Runge-Kutta-Nyström method (RKN12(10)17M formulae of Brankin et al. 1989). They concluded that when using the first method, an instability occurs in the integration of the equations for the partial derivatives when the step-size is larger than 1/76 of the orbital period of the innermost satellite. In contrast, the RKN method is stable and can achieve very good accuracy with step-sizes much larger than the critical step-sizes of the Gauss-Jackson method. Therefore the RKN method is preferable than Gauss-Jackson.

 

 

Table 4: Statistics of O-C residuals, obtained from GUST86 and from numerical integration respectively, in arcseconds for the CCD astrometric observations for each of the Uranus satellites, Oberon excepted (used as the reference satellite). $\mu$ is the mean residual, $\sigma$ the standard deviation about the mean. Nu is the number of observations used. The rejection level that we used was $0\hbox{$.\!\!^{\prime\prime}$ }8$.

		GUST86				Numerical integration
		Position angle		Separation		Position angle		Separation
Satellites	Nu	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma$( $^{\prime\prime}$)	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma(\hbox{$^{\prime\prime}$ })$	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma$( $^{\prime\prime}$)	$\mu(\hbox{$^{\prime\prime}$ })$	$\sigma(\hbox{$^{\prime\prime}$ })$
Ariel	114	-0.0006$~\pm~$0.0036	0.0361	-0.0033$~\pm~$0.0043	0.0455	-0.0011$~\pm~$0.0034	0.0360	-0.0016$~\pm~$0.0043	0.0447
Umbriel	113	0.0022$~\pm~$0.0042	0.0429	-0.0047$~\pm~$0.0044	0.0464	0.0048$~\pm~$0.0041	0.0423	0.0054$~\pm~$0.0043	0.0462
Titania	122	-0.0101$~\pm~$0.0040	0.0424	-0.0050$~\pm~$0.0039	0.0425	-0.0122$~\pm~$0.0040	0.0427	0.0042$~\pm~$0.0038	0.0411
Miranda	83	0.0183$~\pm~$0.0091	0.0796	-0.0060$~\pm~$0.0093	0.0842	0.0195$~\pm~$0.0090	0.0798	-0.0039$~\pm~$0.0092	0.0836

5.2 Main parameters and reference system used

Taylor (1998) computed the ephemerides of the five major Uranian satellites by numerical integration using all available Earth-based observations. His excellent work provided the newest physical parameters for the integration of the Uranian satellite system, the most of which have been retained in our calculation. The main parameters that we used are given as follows:

Mass of Uranus = $4.36587\times10^{-5} ~{M}_\odot$
Mass of Miranda: $7.050\times10^{-7}$
Mass of Titania: $3.839\times10^{-5}$
Mass of Oberon: $3.230\times10^{-5}$
J₂ = 0.003365

Equatorial radius of Uranus = 26 200 km.
These parameters come from fitting to all of the data, including those from the spacecraft mission, and are taken from Taylor (1998).

Mass of Ariel: $1.558\times10^{-5}$
Mass of Umbriel: $1.3497\times10^{-5}$.
The masses of Ariel and Umbriel come from Jacobson et al. (1992). The masses of the satellites given above are all expressed in units of the mass of Uranus.

Coordinates of the pole of Uranus:

$\alpha_{\rm p} = 76\hbox{$.\!\!^\circ$ }5969$, $\delta_{\rm p} = 15\hbox{$.\!\!^\circ$ }1117$.
The reference is the Earth's Mean Equator and Equinox of B1950.

J₄ = -0.0000321.

Both the pole of Uranus and J₄ were taken from French et al. (1988).

The equatorial plane of Uranus is chosen as the reference plane for the numerical integration. The integration equations were formulated in Cartesian coordinates with the x-axis directed at the ascending node of the equator of Uranus w.r.t. Earth's equator of B1950.0 (JED 2433282.423), and the origin at the center of mass of Uranus. For simplification of reduction, no planetary and solar perturbations were included in our numerical integration. From trial computation we found no significant loss of accuracy by omitting them. The position of Uranus itself was given from the ephemerides DE406.

5.3 Fitting of the numerical integration

A numerical integration was iteratively fitted to the GUST86 theory over a span of 128 days in order to obtain a first set of initial positions and velocities at the epoch JED 2449948.5, which is close to the middle of the time span covered by the observations.

Next, the numerical integration was iteratively fitted to the observations. The resulting initial positions and velocities are given in Table 3.

The RKN integration method calculates the step-size adaptively, but an initial estimate is required to start the process. A value of 0.0625 days was used. In the fitting of integration to the observations, good convergence was obtained after only three iterations. The observed-minus-computed residuals of inter-satellite positions from numerical integration, in separation ($\Delta S$) and position angle ($S\Delta P$), are given also in Table 4. Oberon is excepted, as it was used as a reference. The rejection level that we used was $0\hbox{$.\!\!^{\prime\prime}$ }8$.