The determination of the orbit employs an algorithm which minimizes the sum of squares of the observed-minus-computed observation residuals. This procedure requires the formation of computed observables. In our approach, we attempt, where possible, to compute the values of the observables as actually reported rather than transform those values to a standard system (e.g. B1950 or J2000 system) as other authors have done.
Because the integration is performed using the DE403 planetary ephemeris and SAT077 satellite ephemeris for perturbing body positions, the Phoebe orbit is in fact generated in the J2000 reference frame of the International Earth Rotation Service (IERS). The observation processing software also refers the Earth orientation (observer location) to that same frame. The first step in computing the observables is the calculation of the natural direction from the observer to Phoebe in the IERS/J2000 system (see Murray 1983). For observations referred to the FK5/J2000 system, we form computed observables directly from the IERS/J2000 natural direction; the frame tie between the two systems is presumed to be much better than the error in the star catalogues used in the reduction of the observations. For observations referred to a mean equator of epoch system (this includes the FK4/B1950 system), we first precess the IERS/J2000 natural direction to the mean equator at the time of the observation with the IAU76 precession, we then precess it from that time to the mean equator of epoch with the Newcomb precession. In addition, we apply corrections for the FK4-FK5 equinox offset and the elliptic aberration. With the exception of those from Yerkes, all observations are available in a mean equator system.
The Yerkes observations, published as apparent positions, were originally
reduced with the aid of star catalogues (i.e., catalogue mean places of
stars). Before processing, we converted these apparent positions back to
mean-of-date positions by correcting for the circular part of the stellar
aberration and for the nutation. As we have found no record of the constants
of aberration used by the observers to construct their apparent positions, we
used a value of 20
47 (Paris conference 1911). The E.W. Woolard theory
(Explanatory Supplement 1961) provided the nutation corrections rather than
the 1980 IAU theory currently in use.
We should also comment that as an experiment we computed the observables corresponding to Yerkes observations directly as apparent positions in the J2000 system (see the 1987 Astronomical Almanac for the procedure). The residuals for these computed apparent positions against the published apparent positions were not significantly different from those obtained for the computed mean-of-date positions against the observations corrected to mean-of-date positions.
The Voyager observations are modelled in JPL's Optical Navigation Program (Owen & Vaughan 1991). This software package computes the pixel and line locations of the Phoebe and star images using a model of the Voyager camera, the trajectory of the Voyager spacecraft, the Phoebe orbit, and the catalogue positions of the stars. The spacecraft trajectory was the same one used in the SAT077 major satellite ephemeris development.
For each observation set, we used Householder transformations (Lawson & Hanson 1974) to pack the matrix of weighted observation partial derivatives and the weighted residual vector into an upper triangular square root information matrix and associated residual vector. This matrix and vector constitute the square root information array which is equivalent to the normal equations. Each column of the matrix and each element of the vector are associated with an epoch state vector component. Combining the separate information arrays via Householder transformations led to the square root information array for the complete data set. The solution for the state vector was generated and analyzed by means of singular value decomposition techniques (Lawson & Hanson 1974) applied to the composite square root information array.
The weight assigned to each observation set is based on our assessment of the
quality of the data in the set; numerically it is the reciprocal of our
assumed accuracy or standard error for the set. After discussions with the
observer, we selected accuracies of 05 for the 1992 and 1994 photographic
McDonald observations
and 04 for the 1996 CCD observations. The remainder of the accuracies we
determined through an iterative procedure in which we picked a set of weights,
fit the observations, computed a new orbit, and examined the statistics of the
residuals associated with that orbit. In general, for a particular observation
set we took the accuracy to be equal to the root-mean-square of the
residuals. However, we also imposed a lower limit of 05 on the accuracy of
the absolute positions. This limit follows from our assumption that the
recent McDonald observations are representative of the best photographic
absolute positions normally expected for a distant dim object such as Phoebe.
The limit mainly affects data sets with only one or two observations. For the
relative observations we imposed a lower limit of 02, a value typically
associated with photographic relative observations of the Saturnian satellites.
Most observation sets actually have two separate weights assigned: one for
right ascension, 
, or relative right ascension, 
, and
the other for declination, 
, or relative declination, 
.
The weights for the image locations in the Voyager data correspond to
accuracies of 0.5 pixel; the same weights were used during Voyager operations.
The publications of the Greenwich observations give positions of Saturn
which were reduced in the same manner as those of Phoebe. In the publications
these positions are compared to the tabular place of Saturn in the Nautical
Almanac in order to develop a set of corrections for the elimination of star
catalog and Saturn ephemeris errors. Rather than apply the published
corrections, we chose instead to determine our own corrections through direct
processing of the Saturn positions. We formed residuals against positions
computed from the DE403 and SAT077 ephemerides (the former gives the Saturnian
system barycenter position and the latter relates the planet position to the
barycenter). We then extended our processing to include the determination of
right ascension and declination biases which minimize these residuals. The
biases affect both the Phoebe and Saturn positions. The accuracy for
the Saturn positions was set at 05.