Representation of the Perturbed Orbital Motion .1 Osculating and Mean Orbital Elements.1Osculating and Mean Orbital Elements

The requirement for orbital elements to be time-dependent leads to the concept of osculating orbital elements. Let a satellite, whose motion is described through equation (3.97), have the position vector r(t_k) at time t = t_k and the velocity vector ˙r(t_k).

Suppose that all perturbationskscould be removed at this particular epochtk, then the further motion of the satellite would be in an undisturbed Keplerian orbit, governed by the initial conditionsr(t_k) and ˙r(t_k). This orbit is called an osculating (lat. osculare = to kiss, to embrace) or instantaneous orbit, because it coincides with the true, disturbed, orbital path at epocht_k, when the initial parameters are equal.

In reality the perturbing forces do not disappear; this is why the satellite is located on a different osculating orbit for each particular epoch. The true satellite orbit can be regarded to be the envelope of all successive osculating orbits with the osculating elementsa(t_k), e(t_k), . . . M(t_k). With t_kas a time parameter, continuously increasing, the perturbed satellite motion can be interpreted to be a Keplerian motion with time-variable elements

a(t), e(t), i(t), ω(t), .(t), M(t).

For applications in satellite geodesy, with Earth’s gravitation as the primary force, the osculating elements serve very well for orbit approximation because they change

slowly. This is because the acceleration of the central body exceeds the remaining perturbing accelerations at least by a factor 10³ (cf. [3.2.4], [12.2]). It is therefore possible to approximate, for the use of orbit predictions, the orbital elements by a power series in time differences, (t − t0), witht0being a mean epoch:

a_i(t) = a_i(t0) + ˙a_i(t − t0) + ¨a_i(t − t0)²+ · · · (i = 1, . . . , 6). (3.100) The “history” of an osculating elementa_i(t) is represented as the sum of long- and short-periodic terms:

a_i(t) = a_i(t) + ,a_i(t). (3.101) a_i(t) contains the sum of low frequency, secular and constant parts, ,a_i(t) represents the high-frequency oscillations. The termsa_i(t) are also called mean elements. Thus, mean elements can be considered as osculating elements with vanishing periodic terms.

3.2.1.2 Lagrange’s Perturbation Equations

We now have to establish a relation between the acting perturbing forces and the time dependent variations of the orbital elements. The appropriate basic equations were formulated by Lagrange (1736–1813). The explicit derivations can be found in textbooks on celestial mechanics (e.g. Brouwer, Clemence, 1961; Taff, 1985), also Kaula (1966).

Following equations (3.61) and (3.78) the total energy of the satellite motion is determined by

E_M = v²

2 − GM

r = −GM 2a .

The negative term of the total energy, GM/2a, is also named the force function F . With the potentialV as the negative value of the potential energy and the symbol T for the kinetic energy we find the following form of the force function (e.g. Kaula, 1966):

F = V − T . (3.102)

In a non-central force field

V = GM r + R, F = GM

r + R − T = GM

2a + R. (3.103)

The functionR contains all components of V excluding the central term ^GM_r , and is called the disturbing function or disturbing potential.

For the sake of completeness an alternative form of the equation of motion (3.97) in a non-central force field is given

¨r = grad V = ∇V. (3.104)

This form will be used later in [3.2.2.3].

With Lagrange’s perturbation equations a relationship between the disturbing potential R and the variation of the orbital elements is established, e.g. Brouwer, Clemence (1961, p. 284), Kaula (1966, p. 29):

da

In order to avoid singularities, alternative forms of perturbation equations are available, e.g. Brouwer, Clemence (1961, p. 287), Taff (1985, p. 308ff). In particular for orbits with small eccentricity (ω indeterminate) or with small inclination (. indeterminate) other forms are preferable.

For the canonical set of Hill’s elements (3.94) the following relations hold with the geocentric gravitational constantGM = µ:

d ˙r

The analytical integration of the perturbation equations (3.96) or (3.106) requires that the disturbing potentialR is written as a function of the orbital elements. With the derivatives at hand, the integration can then be executed, cf. [3.3.2.2]. So long asR only depends on Earth’s anomalous gravity field, the relation between the coefficients of the potential expansion and the orbit perturbations can be formulated. This aspect is

used when Earth’s gravity field is derived from an analysis of perturbed satellite orbits [12.2].

It becomes evident that the implication of particular perturbations on satellite orbits can at best be investigated via the analytical solution of the perturbation equations.

3.2.1.3 Gaussian Form of Perturbation Equation

In some cases it is useful to formulate the disturbing accelerations directly at the satellite in componential form, instead of using partial derivatives of the disturbing

Z

Figure 3.13. Gaussian form of perturbing forces potential in the elements. This is, for

ex-ample, true for orbits with large eccen-tricities, where series expansions would require many terms in e. The formu-las of type (3.105) are also less suitable for numerical treatment. An appropri-ate alternative form was developed by Gauss. According to Fig. 3.13 the per-turbing forces at the satellite are resolved into three mutually perpendicular com-ponents

K1 perpendicular to the orbital plane, positive toward the north pole,

K2 perpendicular to the radius vector in the orbital plane, positive in the direction of increasing longitude, and

K3 in the direction of the radius vector, positive in the direction of increasing radial distance.

The corresponding perturbation equations are, Brouwer, Clemence (1961, p. 301), Arnold (1970, p. 28), Taff (1985, p. 314), Beutler et al. (1998, p. 61f):

da

dω

Equations (3.108) are convenient in that they allow the influences of the components K1,K2,K3to be separately discussed. We see immediately that onlyK1is capable of changing the orientation of the orbital plane (elements. and i). A change of the semi-major axis a can be achieved by K2 fore  1, i.e. by the component in the direction of satellite motion. This is of importance for satellite maneuvers. Note that in the literature the symbolsW, S, R are also used instead of K1,K2,K3.