Numerical Methods of Orbit Integration

The numerical methods are distinguished by their simplicity and universal applicability when compared with analytical methods. With the use of the modern computer tech-niques the numerical effort only plays a minor role. This is why numerical methods are now used almost exclusively for orbit computations in satellite geodesy.

One basic requirement for the numerical integration is a suitable orbit determina-tion method, for example a procedure named after Cowell or Encke (Noton, 1998). The method of Cowell (1910) was developed at the beginning of the last century and has been applied to the orbit determination of Halley’s comet and the moons of Jupiter.

With the availability of fast and efficient computers the method is now particularly suitable. The basic idea is that the equation of motion (3.97)

¨r = −GM r³ r + ks,

including all perturbations, is integrated stepwise. Equation (3.97) can be re-written in the form of two first order differential equations:

˙r = v, ˙v = k_s− GM

r³ r, (3.167)

and further broken down into vector components:

˙x = v_x, ˙v_x = k_sx− GM r³ x,

˙y = v_y, ˙v_y = k_sy−GM

r³ y, (3.168)

˙z = v_z, ˙v_z = k_sz−GM r³ z.

The perturbing accelerationk_s, following the equations of section [3.2.3], can as well be written in the form ˙r = v and ˙v for each particular perturbation effect, e.g. the gravitational attraction of the Moon or of the Sun. The satellite state (r, v) for the desired epoch is then calculated using suitable methods of numerical integration (see later).

The main advantage of Cowell’s method lies in its conceptual simplicity. The influences of all individual perturbations can be considered at the same time. One disadvantage is that smaller integration steps are required near a large attracting body, which leads to an increase of computation time and to an accumulation of round-off errors. In such cases some improvements can be made by formulating the equation (3.167) in spherical polar coordinates (r, θ, ). The integration steps can be larger because of much smaller coordinate variations. The equations of motion are then (cf.

Bate et al., 1971, p. 389):

¨r − r( ˙θ cos² + ˙²) = −GM r² r ¨θ cos  + 2˙r ˙θ cos  − 2r ˙θ ˙ sin  = 0

r ¨ + 2˙r ˙ + r ˙θ²sin cos  = 0.

(3.169)

true orbit (disturbed) t1

rectification

t0

initial epoch

new reference orbit

reference orbit (Kepler)

Figure 3.23. Numerical integration of the orbit, after Encke

In Cowell’s method the total force acting on the satellite is integrated, whereas in the method of Encke (1857) a reference orbit is introduced, and only the difference between the primary ac-celeration and all perturbing accelera-tions is subject to integration. Usually an osculating Kepler ellipse is chosen as a reference trajectory for the initial epocht0(Fig. 3.23). Hence, one portion of the total acceleration is separated from the numerical integration, and is given a simplified analytical solution. The os-culating orbit serves until the deviations from the true orbit become too large.

Then a new ellipse is introduced as a reference orbit for the new initial epocht1, with the true position and velocity vector valid fort0. This process is called rectification.

t0

Figure 3.24. Explanation of Encke’s method of orbit integration

Let r be the position vector of the true perturbed orbit and ρ the position vector of the osculating (reference) orbit for a particular epochτ = t − t0. Then we find for the true orbit

¨r +GM

r³ r = k_s, (3.170) and for the osculating orbit

¨ρ +GM ρ³ ρ = 0.

For the initial epocht0we have r(t0) = ρ(t0) and v(t0) = ˙ρ(t0).

Referring to Fig. 3.24, we introduce

the deviation vector δr between the reference orbit and the true orbit (we skip the parametert for the sake of simplicity):

δr = r − ρ, δ¨r = ¨r − ¨ρ. (3.171)

Substituting (3.170) into (3.171) yields δ¨r = k_s+

The deviation vectorδr can be calculated for arbitrary epochs δr(t0+ ,t) with nu-merical integration techniques. Equation (3.172), however, may lead to nunu-merical difficulties, because the expression(1 − ρ³/r³) nearly equals zero. These difficulties can be solved by substituting

2q = 1 − r²

ρ², (3.173)

and series expansions (Bate et al., 1971, p. 393). Withδr changing much more slowly thanr, Encke’s method usually requires fewer integration steps, and consequently less computer time than Cowell’s method (Taff, 1985, p. 393).

Generally speaking, Cowell’s method is adequate when the perturbing acceleration equals or exceeds the primary acceleration, and Encke’s method is more suitable when perturbing accelerations are small compared with the primary acceleration (Battin et al., 1978).

Many alternative orbit determination methods have been developed in celestial mechanics (see e.g Stumpff, 1959/1965/1974; Roy, 1978; Battin et al., 1978); none have particular advantages when compared with the two above-mentioned methods.

The numerical integration itself is realized with methods taken from approximation theory. Basically a polynomial has to be fitted to a limited series of consecutive points, in order to generate an additional point through extrapolation of the polynomial. This process is repeated at will. The polynomial coefficients are derived from the given points and their derivatives, based on the equation of motion. Different methods are used, depending on the number of points required, on the extrapolated values, on the smoothing process, and so on. They are generally subdivided into single-step and multi-step methods.

A well-known member of the family of single-step methods is the Runge–Kutta method. Here, a Taylor series of a certain order is used as an extrapolation function. A special feature of the single-step method is that only the last integration step is used, so the knowledge of the “history” of the function to be integrated is neglected. To avoid this multi-step solutions are usually applied in satellite geodesy. They are also called predictor-corrector methods.

The basic idea is to first predict a satellite position with a certain algorithm and then to correct that position. As a first step a predicted valueX_n+1is calculated from X_n. It is then substituted in the differential equation of the process in order to obtain with ˙X_n+1a corrected value ofX_n+1. The process can be iterated, until the result does not change. Many predictor-corrector formulas may be found in the literature. The formulas of Adam–Moulton or Gauss–Jackson are frequently used. In principle these are filter techniques; this is also why the Kalman filter is now used more and more (Battin et al., 1978). Herer and v are state vectors with a relevant variance-covariance matrix, see e.g. Egge (1985).

Two main error sources have to be considered when numerical integration methods are applied. These are round-off errors and truncation errors. The round-off errors depend on the numerical representation accuracy in the computer being used. In order to limit these influences, rather large step-sizes are an advantage in the integration.

Truncation errors develop when the last terms of a series expansion, which is used for the integration, are cut off. The errors can be minimized with small step-sizes. These two conflicting requirements have to be fulfilled with appropriate compromises.

Except for these deficiencies, the numerical integration methods can be considered as rigorous orbit integration methods, without approximations. The only disadvantage is that many, unwanted, intermediate satellite positions have to be calculated before the final solution is obtained.

For a detailed treatment of current numerical integration solutions, including exer-cises, see Montenbruck, Gill (2000). A short review is given by Beutler et al. (1998).