Courtes et longues priodes dans le mouvement orbital

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Theory of mean orbital motion current status and
future developments,
for applications in spatial dynamics.
Florent DELEFLIE Anne LEMAITRE
Observatoire de la Côte dAzur Grasse,
France FUNDP, Unité de Systèmes Dynamiques,
Belgium
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• Short and long periods a specific approach in
  orbital
• dynamics

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Equations of motion of an artificial satellite
Orbital elements
Sat
N ascending node T perigee Sat
satellite
r
longitude of node argument of
perigee true anomaly inclination
Perturbative equations
O
Many forces to be taken into account
gravitational ones, non gravitational
ones
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Orbital dynamics why and how ???
Definition simultaneous improvement of orbital
modelling which forces acting on a satellite ?
force modelling quantifying the corresponding
parameters Methods of celestial mechanics and
numerical analysis 1. Direct problem orbit
extrapolation integration of the equations of
dynamics on the basis of 1) a force model
2) given initial conditions 2. Inverse
problem orbit restitution adjustment of
parameters from tracking data (GPS, SLR)
Goal reaching a centimeter accuracy !
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Orbital dynamics over a long time scale why ?
Geodynamics Determination of parameters
producing small but cumulative effects
Effect over 20 years (on the node of LAGEOS-1)
100 m
Mission analysis studying the main
properties of an orbit
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Orbital dynamics over a long time scale how ?
Osculating Motion

Long Periods

Short Periods
Effects of and on the eccentricity of
STARLETTE
Goal Removal of all short periods included in
the osculating equations
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How does it work in practice a complementarity
between analytical and numerical approaches
Analytical step Averaging of the
osculating equations of motion (realized once for
all) Numerical integration of the averaged
equations (large integration step size)
Time (Days)
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• Short and long periods a specific approach in
  orbital
• dynamics
• 2. Building of the averaged equations of dynamics

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The analytic averaging approach
Idea
Method Removal of the mean anomaly from
osculating equations of motion Idea valid for
 small  perturbations (equations of Lagrange or
of Gauss) But not valid for the static central
gravity field and third body
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Static gravity field and third bodya specific
hamiltonian approach
Canonical transformation of the equations of
dynamics
Use of the algorithm of Deprit, 1969
Advantage high precision (coupling effects)
Hamiltonian (potential) developed in a power
series of a small parameter
Building n by n of the averaged Hamiltonian
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The result can easy be implemented in the CODIOR
softwarethanks to the MS software Claes et al.,
1988
Results implemented once for all
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Orbital model for long period studies
Goal reaching a centimeter accuracy !

• Gravitational Effects
• Gravity field of the central body (including time
  variability)
• Third body potentials (Moon, Sun,planets)
• Terrestrial tides
• Oceanic tides
• Relativistic corrections
• Non gravitational effects
• Atmospheric drag
• Radiation pressures
• Precession-nutation
• Yarkovsky effects
• or use of accelerometric data

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Tests for different kinds of orbits
LAGEOS-1 perturbed by J2 to J40, Moon and Sun,
terrestrial tides
Signal to be modelled
3 km
Residuals
1 cm
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• Short and long periods a specific approach in
  orbital
• dynamics
• 2. Building of the averaged equations of dynamics
• 3. Current applications

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Satellites around the Earth
Different sensibilities
A constellation to decorrelate geodynamical
parameters
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Example time variations of low degree zonals

• Link between time variations of orbital
  parameters
• and corresponding geodynamical parameters

(LAGEOS-1)
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• Short and long periods a specific approach in
  orbital
• dynamics
• 2. Building of the averaged equations of dynamics
• 3. Current applications
• 4. Next developments and applications

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Effects of tesseral coefficients on orbits
Short periodic effects (mean value 0)
Semi-major axis perturbed by tesserals of degree
3 (STARLETTE)
Except in case of resonance
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Modelling of resonances
Resonance angle
A frequence analysis is required (short periods
 become  long periods), as well as
regularization procedures
Resolution of the resonance equation
Same semi-analytical approach as for zonals
Actions for a long term validity
Small divisors precise determination of
amplitudes
Dynamical study of small but cumulative effects
High order Theory
Resonances with Moon and Sun have also to be
managed
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Applications
Long term evolution of the GALILEO
constellation (5rev/3d) Links with space
debris ??? Altimetric missions Others
Mapping of resonance areas
Evolution of the trajectory (with maneuvers how
minimizing them ?)
Evolution of the free trajectory (over centuries)
To be questionned
Which consequences for the repetitivity of the
trajectory on measurements ?
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• Short and long periods a specific approach in
  orbital
• dynamics
• 2. Building of the averaged equations of dynamics
• 3. Current applications
• 4. Next developments and applications
• 5. Conclusion

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Sum up of the theory of mean motionimplemented
in the CODIOR software
Methods of celestial mechanics (different kinds
of variables)
Numerical integration of a differential system
which has been averaged in an analytical way once
for all
Most of the perturbations modelled
Limits
Now...
A new challenge resonance modelling (fix and
varying altitudes)
For new applications of orbitography over long
periods of time
Global geodynamics (GRACE, GOCE), Mars (MGS,
MEX ???) GALILEO constellation, altimetric
missions Space debris