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International cooperation and technology transfer
Mussio, Luigi

The Spacewise Approach to the Data Treatment for the
SAGE Mission Project
A.Albertella, F.Migliaccio, F.Sansó
Dip. IIAR, Sez. Rilevamento, Politecnico di Milano, Milano, Italy
After a call for proposals for small satellite missions from the Italian Space Agency (ASI), a group of Italian
research teams and industries led by the Politecnico di Milano proposed the concept of SAGE, a mission aimed
at determining the gravity field of the Earth by means of high-low SST, which means that the satellite orbit is
determined by GPS, while the non-gravitational perturbations are determined by a three-axes accelerometer.
This is basically the same concept of the CHAMP mission [Reigber et al., 1996]. SAGE underwent a Phase A
Study during the year 1998 [ASI, 1998]. In this framework, the task of the Politecnico di Milano group was to
analyze the data by means of the spacewise approach. The complete spacewise approach, besides studying new
simulations of the data to be expected from SAGE, requires to perform the inversion of Hill’s equations, to form
average values on a regular grid over the sphere and to recover the gravity field coefficients. The simulations are
requested in order to: assess the accuracy of the data obtained after the inversion of Hill’s equations introducing
a realistic measurement noise; formulate the overdetermined boundary value problem to be solved; determine
indices enabling to evaluate the performances of the solution.
1 The concept of space ac-
The proposal of the mission SAGE (Satellite Ac-
celerometry by Gravity field Exploration) consisted in
using a GPS receiver together with an accelerometer on
a low, polar orbit satellite. The accelerometer proof-
mass, positioned in the centre of mass of the satellite, is
subject to a purely gravitational acceleration g, while
the centre of mass of the satellite is subject (besides the
same acceleration g) also to all non-gravitational forces
which act on the surface, whose sum is /. Therefore
the accelerometer gives a direct measure of /.
The GPS tracking (aided by a SLR device) allows to
reconstruct (with very high relative precision between
two points along an orbit arc) the satellite trajectory
x(t). The difference between the “observed” orbit x(t)
and the orbit x(t) modelled by all available information
is due to the residual gravitational effects:
£(*) = x{t) - x(t) (1)
where £(£) is the orbit anomaly, equivalent to a “virtu
al” orbit ruled by the residual gravitational potential.
From £(£) it is possible, by differencing and smooth
ing, to obtain observed values of g along the orbit,
which can be integrated to give the harmonic coeffi
cients of the field, in the framework of an overdeter-
mined boundary value problem.
In particular, we write the equation of motion of the
satellite as [Bassanino et al., 1992]
x — VuoGD + V<5it(r) + (x) + / (x) (2)
considering the gravitational potential u(x) as the sum
of a reference potential uo(x) and a residual part du(x).
In this equation f g (x) represents the effects of the sun,
moon and tides (which can be modelled) and / (®)
represents the effect of the surface forces (mainly due
to the drag, which is measured by the accelerometer).
It must be remarked that f g and f ng can be computed
along the nominal orbit without significant errors.
The residual gravitational effects 5g along the orbit
can be obtained by inverting Hill’s equations, which
are written here under the hypotheses that £ is small
and that the orbit arc is circular:
Ca +2u£ r = Sg a
ir -2w{ a -3u 2 ( r = <5g r (3)
L +w 2 fo = Sg„
The indices a, r, o respectively denote the along track,
radial and orthogonal (out of plane) component.
This system is inadequate to produce realistic orbit
ephemerides; nevertheless it is useful because it can
certainly be used to produce simulations, to under
stand how well (3) can be inverted. The general so
lution of (3) can be written as
¿(i)=ifc(t)+S p (i) (4)