PreprocessingVariationalEquation
This program integrates an orbit dynamically using the given forces and setup the state transition matrix for each time step. These are the prerequisite for a least squares adjustement (e.g. gravity field determination) using the variational equation approach. The variational equations are computed arc wise as defined by inputfileOrbit. This means for each arc new initial state parameters are setup.
In a first step the forces acting on the satellite are evaluated at the apriori positions given by inputfileOrbit. Non-conservative forces like solar radiation pressure needs the orientation of the satellite (inputfileStarCamera) and additional a satellite macro model (satelliteModel) with the surface properties. Furthermore inputfileAccelerometer observations are also considered.
In a second step the accelerations are integrated twice to an dynamic orbit using a moving polynomial with the degree integrationDegree. The orbit is corrected to be self-consistent. This means the forces should be evaluated at the new integrated positions instead of the apriori ones. This correction is computed in a linear approximation using the gradient of the forces with respect to the positions (gradientfield). As this term is small generally only the largest force components has to be considered. A low degree spherical harmonic expansion of the static gravity field (about up to degree 5) is sufficient in almost all cases. In this step also the state transition matrix (the partial derivatices of the current state, position and velocity) with respect to the initial state is computed. The integrated orbit together with the state transitions are stored in outputfileVariational, the integrated orbit only in outputfileOrbit.
To improve the numerical stability a reference ellipse can be reduced beforehand using Enke's method (useEnke). Mathematically the result is the same, but as the large central term is removed before and restored afterwards more digits are available for the computation.
The integrated orbit should be fitted to observations afterwards by the programs PreprocessingVariationalEquationOrbitFit and/or PreprocessingVariationalEquationSstFit. They apply a least squares adjustment by estimating some satellite parameters (e.g. an accelerometer bias). If the fitted orbit is too far away from the original inputfileOrbit the linearization may not be accurate enough. In this case PreprocessingVariationalEquation should be run again with the fitted orbit as inputfileOrbit and introducing the estimatedParameters as additional forces.
| Name | Type | Annotation |
|---|---|---|
outputfileVariational | filename | approximate position and integrated state matrix |
outputfileOrbit | filename | integrated orbit |
inputfileSatelliteModel | filename | satellite macro model |
inputfileOrbit | filename | approximate position, used to evaluate the force |
inputfileStarCamera | filename | rotation from body frame to CRF |
inputfileAccelerometer | filename | non-gravitational forces in satellite reference frame |
forces | forces | |
estimatedParameters | sequence | satellite parameters e.g. from orbit fit |
parametrizationAcceleration | parametrizationAcceleration | orbit force parameters |
inputfileParameter | filename | estimated orbit force parameters |
earthRotation | earthRotation | |
ephemerides | ephemerides | |
gradientfield | gravityfield | low order field to estimate the change of the gravity by position adjustement |
integrationDegree | uint | integration of forces by polynomial approximation of degree n |
useEnke | sequence | integrate differential forces to an elliptical reference trajectory |
GM | double | geocentric gravitational constant used for elliptical reference orbit |