Numerical calculations revealed major problems with the stability of the P3-D orbits . Only when an analytic model of the perturbations  was available, the detailed dependencies of the perturbations from the orbital parameters and the special properties of the P3-D orbits could be understood . One special result of these investigations is presented in the following.
The gravitational field of the Earth, which is not quite that of a point-like mass, causes a rotation of the plane of the satellite orbit around the rotational axis of the Earth as well as a rotation of the line of apsides of the orbital ellipsis. Table I contains the formulae for these two rotations. As variables we encounter the semi-major axis a, the eccentricity e and the inclination i, see for example . In addition we use the designations according to Delaunay for the argument of perigee g and for the longitude of node h (right ascension of the ascending node = RAAN). The rotation of the orbital plane - that is the rate of change of the longitude of node - is proportional to the negative cosine of inclination. The rotation of the line of apsides - that is the rate of change of the argument of perigee - is positive at inclinations below the so-called "critical" inclination of 63.4° and negative for higher inclinations, corresponding to the expression "5 cos²(i) - 1". These perturbations depend on the size of the semi-major axis with the power -7/2.
Table I: Rates of change of argument of perigee dg/dt and longitude of node dh/dt, caused by the oblateness of the Earth.
Table II shows the (simplified) equations for long-term changes of eccentricity and inclination due to the gravitational perturbations by the Moon and the Sun . Long-term means here, that all terms with periods of less than one year have been omitted. The simplification concerns the influence of the longitude of the node of the Moon's orbit, which is not very big anyway. These perturbations depend on the size of the semi-major axis with the power +3/2.
Table II: Long-term rates of change of eccentricity de/dt and inclination di/dt due to the influences of Moon and Sun.
If we now look at the ratio of the perturbances caused by Moon and Sun to those caused by the oblateness of the Earth, we see an increase with the fifth power of the size of the semi-major axis. Concerning our satellites, the influence of Moon and Sun for the P3-D orbit is about three times greater than for the orbits of OSCAR 10 or AO-13.
If we analyse the equations of table II in detail, we see the rates of change of eccentricity and inclination disappear, when g equals a multiple of 90° and h equals a multiple of 180°. For polar orbits (i=90°) this is valid also when g and h equals a multiple of 90°. For all other values of g and h we can expect changes of e and i with time. The only exceptions are perfectly circular orbits (e=0), which will remain circular - but only in theory.
During the planned mission period of 20 years the node of the P3-D orbit will cover one complete orbit or even more. The knowledge of the disturbances during the whole period is therefore essential.
The dependencies of the perturbations of e and i from h and g for the nominal values of the P3-D orbit are shown in figures 1 and 2. The values for the rates of change are given in units per day. If we convert to the values per year, we get a maximum change of eccentricity of 0.02 (this corresponds in our example to a reduction of the height of perigee from 4000 km to 3300 km), and a maximum change of inclination of 1.5°. For a perigee at 270° the perturbations of the inclination are comparatively small - an advantage realised by the Molniya, Taiga and M-HEO orbits [4,6].
Figure 1: Long-term rates of change of eccentricity de/dt (per day) in dependency of argument of perigee g and longitude of node h for P3-D orbits (a/R=5.054, e=0.678, i=63.43°).
Figure 2: Long-term rates of change of inclination di/dt (degrees per day) in dependency of argument of perigee g and longitude of node h for P3-D orbits (a/R=5.054, e=0.678, i=63.43°).
Furthermore there are similar - and even longer - equations than those of table II for the perturbations of g and h by the Moon and the Sun . For g the perturbations in dependency from h are shown in figure 3.
Figure 3: Long-term rates of change of argument of perigee dg/dt (degrees per day) in dependency of argument of perigee and longitude of node h for P3-D orbits (a/R=5.054, e=0.678, i=63.43°).
2 g, h ± 2 g, 2 h ± 2 g.
The terms for perturbations of inclination contain in addition sin-functions of the angles
h and 2 h.
If one of these angles remains constant or nearly constant and does not belong to the special values mentioned above (multiples of 90° or 180°, respectively), the sin-function yields a constant non-zero value. In this case eccentricity and inclination will drift in just one direction. If these angles change slowly, also eccentricity and inclination will follow the same rhythm. We can speek of a resonance. It is interesting here, that not only the sum or difference of two frequencies can cause a resonance, but also the constancy of one angle, corresponding to a frequency of 0.
This situation applies especially to the planned argument of perigee of 225°, because sin(2 x 225°) = sin(90°) = +1, and therefore the corresponding perturbation term will reach its maximum. The P3-D orbit is at a maximum of a resonance.
For AO-13 the term with the angle h + 2 g is the greatest one. This term causes a variation of the height of perigee of about 5000 km (3000 miles) within a period of 40 years. There is another smaller variation - caused by the term with the angle 2 g and within a period of between 7 and 12 years . Both variations together are causing the early return of AO-13 by end of 1996.
Figure 4 shows three P3-D orbits, all of them starting with a node at 0°, an argument of perigee of 225°, and an eccentricity of 0.7 . The initial inclination of the central orbit has been selected so that the argument of perigee will stay as long as possible within a useful range (225 ± 15°). The other two orbits (dashed lines) are differing in the initial inclination by ±1°. As long as the argument of perigee is near 225°, the eccentricity will grow continuously and the inclination will decrease continuously. With the selected initial values and without any further corrections we can expect a life-time of 5 to 6 years. With a higher initial eccentricity an even shorter life-time of the satellite will result.
Figure 4: Evolution of the orbital parameters for 3 different initial inclinations (68 ± 1°) and initial values g=225° and h=0°.
However, the gravitational perturbations of the two orbits are completely different. Since sin(2 x 315°) = sin(270°) = -1 , the corresponding perturbation terms have the sign inverted, and the evolution of eccentricity and inclination is also inverted. Figure 5 shows three orbits starting with a node at 0°, an argument of perigee of 315°, and an eccentricity of 0.7 . The initial inclination of the central orbit has been selected - as above - so that the argument of perigee remains in the useful range as long as possible. The initial inclination of the other two orbits (dashed lines) differ by ± 1°.
Figure 5: Evolution of the orbital parameters for 3 different inclinations (62 ± 1°) and initial values g=315° and h=0°.
We encounter here a possibility to extend the life-time of the satellite without any corrections to 20 years or even longer. This advantage can be achieved only with a continuously increasing height of perigee. Within 20 years the height of perigee will increase from 3500 km to 16000 km. Since the semi-major axis will remain constant, the height of apogee will decrease correspondingly. This might be a desirable effect, since the decrease of efficiency of the sun panels and the corresponding decrease of transmitter power - will be offset, at least partially, by the decreasing path loss.
For highly-elliptic orbits the possibilities of disposal are different. For example OSCAR 10, with a relatively small inclination of about 26° and therefore at a - not very effective - 5:3 resonance of nodal to apsidal rotation, will maintain its orbit for many more years and even centuries. On the contrary AO-13, very near a 2:1 resonance which causes extreme perturbations to eccentricity, will return to the Earth's surface around the end of 1996, due to gravitational effects.
From the examples presented in the previous chapter we can see that for P3-D with an argument of perigee of 225° the fate will be very similar to that of AO-13. On the other hand an argument of perigee of 315° will yield a more and more circular orbit, which will remain fairly stable for at least 20 years. The final fate for such an orbit cannot be predicted generally, it has to be calculated for each specific case.
Figure 6: Evolution of orbital parameters for 3 different inclinations (59 ± 1°) and initial values g=315° and h=90°.
The original German version of this article has been published in AMSAT-DL Journal, Vol. 21, Nr. 4, Dec. 1994.