The SATAN Programs and Software Development

The SATAN package consists o f two main programs: ORBIT and RGODYN (Sinclair and Appleby, 1986). The first computes satellite position and velocity at specified dates by numerical integration o f the equations o f motion. The second determines corrections to the parameters that define the orbit by fitting it to laser observations.

Throughout this study several facilities were introduced into the programs:

- Solution for multiple drag coefficients

- Solution for orbital elements o f initial position and velocity instead of cartesian coordinates

- Possibility o f applying constraints to the solve-for parameters

- Bayesian least squares solution in addition to conventional least squares

- Processing o f PRARE range data

- Processing of altimeter data

3.2.1 Program ORBIT

Program ORBIT performs the numerical integration o f the equations o f motion (Sinclair, 1988):

r = f ( r, r . t ) = rg + rb + ra + rrp + rtides

(3.1)

where

- acceleration due to the Earth's gravitational field

Tb - acceleration due to third bodies (sun, moon and planets )

- acceleration due to atmospheric drag

^rp - acceleration due to radiation pressure %

^tides - acceleration due to solid earth and ocean tides

The integration method used is an eighth order Gauss-Jackson method with an iterative starting scheme. Given the position and velocity at an initial time ti, the starting schem e calculates the position and velocity for the next 8 points t2, . ..,

ig.

The step length is fixed for each integration. This limits the use o f the programs to satellites o f low eccentricity. For each satellite, the step length should be selected to give the required accuracy. It has to be short enough to cope with the short period perturbations caused by the high order tesseral harmonics o f the gravitational field.

The celestial reference frame used for the integration is the equator and equinox o f J2000. However, the acceleration due to the gravitational field has to be calculated in a terrestrial reference frame (the real equator o f the Earth, as defined by the geopotential coefficients). So, at every step o f integration, the satellite position is transformed to the terrestrial reference frame and the acceleration is evaluated. The acceleration is then transformed to the J2000 frame for the numerical integration. The rotation matrices used in the

transformation between the two system s are those o f the FK5 system: lA U (1976) precession, lA U (1980) nutation, and lA U (1982) sidereal time as described in section 2 .4 .3 .

The timescale used is an atomic timescale that is equal to UTC at the starting epoch o f integration (called ATC). So ATC is calculated by subtracting from TAI the leap seconds that occurred until the starting epoch, but if a leap second occurs during the span o f the integration, it will not be applied.

The program also calculates the partial derivatives o f satellite position and velocity with respect to the solve-for parameters that affect the orbit. These partial derivatives are necessary for subsequent use in RGODYN to fit the orbit to observations. In its original form the program included the calculation o f the partial derivatives with respect to the initial state position and velocity vectors, a drag coefficient and its rate o f change, solar radiation coefficient and the product CM. The main modifications introduced into this program were the implementation o f the solution for multiple drag coefficients and for a selected set of geopotential coefficients.

The force model used, described in section 3.3, is the recommended one in project MERIT standards (Melbourne et al., 1983), and includes effects due to :

• The Earth's gravitational field

• Gravitational attraction o f the sun, moon and the planets • Atmospheric drag

• Radiation pressure

• The solid Earth and ocean tides

3.2.2 Program RGODYN

For most o f the applications, the accuracy achieved when computing an orbit by simple integration o f the equations o f motion is not sufficiently good. This is due to the inabihty o f the force model to describe accurately the forces that act on a satellite. In practice, the satellite is tracked from a number o f stations on the Earth, and orbital parameters are estimated by fitting the orbit to observations.

In its initial form, RGODYN performed the adjustment o f the orbit to laser observations only. The processing of PRARE range and altimeter data has been added to the program, allowing any combination o f the three data types. The program takes each observation (O) in time, computes a corresponding quantity (C), according to initial conditions, and forms the "0-C" residuals. In addition, it computes the partial derivatives o f the observed quantity with respect to the solve-for parameters, and adds their contribution to

the normal equations. After all observations have been processed, the normal equations are solved to derive corrections to the initial parameters. In this way, an updated set of parameters is determined.

The least squares procedure implemented in RGODYN was the traditional method o f "observation equations" described in section 3.4.1. This algorithm assumes that there is no "a priori" information on the initial set o f parameters. When such information exists, it may be used to help to constrain the new solution. Hence, the Bayesian Least Squares technique described in section 3.4.3 was also implemented in the program.

The new formulae implemented in RGODYN for the partial derivatives o f PRARE Range and altimeter data with respect to satellite parameters are presented in section 3.5. The algorithms for the computation o f the computed (C) values are described in chapter 6 for PRARE and laser range data, and in chapter 7 for altimeter data.