  The apparent close encounters of two satellites in the plane of the sky,
called mutual approximations, have been suggested as a different type of
astrometric observation to refine the moons' ephemerides. The main observables
are the central instants of the close encounters, which have the advantage of
being free of any scaling and orientation errors. However, no analytical
formulation is available yet for the partials of these central instants,
leaving numerical approaches or alternative observables (e.g. derivatives of
the apparent distance) as options. Filling that gap, this paper develops an
analytical method to include central instants as direct observables in the
ephemerides estimation and assesses the quality of the resulting solution. To
this end, we ran a covariance analysis to compare the estimated solutions
obtained with the two types of observables (central instants versus alternative
observables), using the Galilean moons of Jupiter as a test case. Alternative
observables can be equivalent to central instants in most cases. However,
obtaining consistent solutions between the two observable types requires to
accurately weigh each alternative observable individually, based on each mutual
approximation's characteristics. In that case, using central instants still
yields a small improvement of 10-20% of the formal errors in the radial and
normal directions (RSW frame), compared to the alternative observables'
solution. This improvement increases for mutual approximations with low impact
parameters and large impact velocities. Choosing between the two observables
thus requires careful assessment based on the characteristics of the available
observations. Using central instants over alternative observables ensures that
the state estimation fully benefits from the information encoded in mutual
approximations, which can be desirable for certain applications.
