Capítulo 7: Actualización de los componentes de Dr.Web Enterprise
7.3. Actualizar los Agentes Dr.Web
For very precise orbital analysis other perturbations may be considered, whose indi-vidual contributions to the acceleration of the satellite are usually far below 10−9m/s2. These are, for example:
− friction caused by charged particles in the upper atmosphere,
− thermal radiation of the satellite,
− heating effects at shadow boundaries,
− electromagnetic interaction in the geomagnetic field, and
− influences of the inter-planetary dust.
A discussion and evaluation of such perturbing forces can be found in special literature, e.g. Ries (1997); Ries, Tapley (1999). For practical purposes in satellite geodesy these perturbations are mostly not considered.
It should be mentioned that so-called apparent forces arise if the equation of motion is formulated with respect to a moving reference system instead of a space-fixed one (Schneider, 1981, p. 30, Reigber, 1981). These are, if the moving system is non-uniformly rotating and/or accelerated as against the space-fixed system, the
centrifugal force Z = −md× (d× r),
gyro force T = −m∂d
∂t × r, Coriolis force C = −2md× ∂r
∂t ,
(3.146)
which an observer in a space-fixed system would not notice. From the equation of motion in the stationary system
md2r
dt2 = K(t, r, ˙r) (3.147)
follows the equation of motion in the moving system (Schneider, 1981):
m∂2r
∂t2 = K− md× (d× r) − m∂d
∂t × r− 2md× ∂r
∂t , (3.148) withKthe force in the rotating system,rthe position vector in the rotating system, anddthe rotation vector.
For an explicit computation of the apparent accelerating forces, using the expres-sion for the rotational vectordand the derivatives thereof in the moving system, see for example Reigber (1981).
Relativistic effects are, for most applications in satellite geodesy, smaller than the observation accuracy. In many cases they are cancelled by the observation technique, or they are modeled through other parameters. Insofar as relativistic effects are of importance, they will be discussed together with the particular satellite methods (e.g.
[7.4.1]). With respect to orbital dynamics it follows from general relativity that the orbital elements are subject to additional secular perturbations. These influences are much greater for the orbits of near-Earth satellites than for planets (cf. the relativis-tic perihelion rotation of Mercury). Cugusi, Proverbio (1978) give the appropriate formulas, and they find as mean values for satellites of geodetic interest:
10/year for ω, 0.2/year for ., and 0.2/year for M.
The correction to the acceleration of an artificial satellite, based on general relativity, is (McCarthy, 2000)
¨rrel= GM c2r3
4GM r − ˙r2
r + 4 (r · ˙r) ˙r
, (3.149)
with
c speed of light,
r satellite position vector,
˙r satellite velocity vector, and GM geocentric constant of gravitation.
The relativistic correction of the accelerations is in the order of 3· 10−10m/s2for GPS satellites and 1· 10−8m/s2for TOPEX/POSEIDON.
For some satellite systems particular, non-gravitational accelerations are generated from thrust or attitude control maneuvers. They have to be considered in orbital analyses. Thrust forces appear in connection with maneuvers for orbit corrections.
Attitude control systems change the satellite’s orientation in space. Cappelari et al.
(1976) or Montenbruck, Gill (2000, p. 104f) give formulas for the consideration of such effects.
In dynamical orbit determination it is not possible to model all perturbations per-fectly. This holds in particular for the non-conservative force models which are limited by uncertainties in the knowledge of platform orientation, material properties, and sur-face temperatures (Montenbruck, Gill, 2000). This is why empirical accelerations are employed to take account of this effect. In general the empirical forces are described by an equation of the following type:
¨rem= E(a0+ a1sinν + a2cosν), (3.150) were a0 is a constant acceleration bias,a1 anda2 are coefficients related to the fre-quency (e.g. one cycle per orbital revolution), and E is a matrix to transform the acceleration biases from the local orbital frame (radial, cross-track, and along-track) into the inertial system. For details see e.g. Montenbruck, Gill (2000, p. 112).
3.2.3.6 Resonances
Resonances occur when the period of a satellite revolution is an integer multiple of Earth’s rotation period. This leads to an amplification of certain non-zonal harmonics Snm, Cnm, resulting in much higher amplitudes in the element perturbation than nor-mal. In geometric terms, resonances appear when consecutive revolutions of a satellite are separated exactly by an interval which corresponds to the wave-length of the par-ticular harmonic coefficient. After a given number of revolutions, the sub-satellite orbit repeats, i.e. the satellite crosses the same regions and is subject to the same per-turbations. This causes an amplification of the initial perturbation and generates the resonance effect. Consequently, a satellite with≈m revolutions/day will be sensitive to resonant influences from the tesseral coefficientsCnm, Snm.
From a mathematical point of view, resonances develop when the denominator (3.118) in the perturbation equation (3.119) becomes very small:
˙ψnmpq = (n − 2p) ˙ω + (n − 2p + q) ˙M + m( ˙. − ˙R) ≈ 0. (3.151)
Satellite orbits can be explicitly selected to determine particular tesseral harmonics with high accuracy, using the resonance effect and equation (3.118). The corollary is that in orbit computation it is essential to know whether specific high order potential coefficients can give rise to large perturbations, caused by resonances. Low orbiting satellites, because of their frequent revolutions, are particularly affected by short wave structures of the geopotential. Resonances may be present also for Earth observation and remote sensing satellites, because of their dedicated orbital design with partic-ular repetition rates. GPS satellites experience resonance effects caused by Earth’s ellipticity (Delikaraoglou, 1989).
Insofar as different coefficients generate resonances of identical phase and ampli-tude, they cannot be separated, and only derived jointly from orbital analyses. The determination of such so-called lumped coefficients is treated for example by Klokoˇcník (1982). For a detailed discussion of resonances in high satellite orbits (GPS, geosyn-chronous) see e.g. Hugentobler (1998).