ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: the analytical solution and the real solution that increases with time. This effect was observed for all the solutions studied for this work. Figure 3. Long-term error evolution in the radial distance, the latitude, and the longitude for a frozen sun-synchronous orbit. 6.3.2 Example: Molniya orbit As an example of this formulation to eccentric orbits, we select a typical Molniya orbit with initial elements: a = 26 600.0 km, e = 0.74, , , , and . Figs 4 and 5 present the results of applying this formulation to this particular initial conditions. From these figures, it can be seen that, even for very eccentric orbits, the formulation along with the perturbation theory used provides a very good approximation to the problem even for a first-order solution. In particular, the order of magnitude of the error in latitude and longitude remains the same as in the near-circular case, while the radial distance experiences an increase in its error due to the larger variation of this variable during the dynamic. Therefore, this example shows that this formulation can be applied without any problem even to high eccentric orbits. Figure 4. Evolution of radial distance, latitude, and longitude for a Molniya orbit. Figure 5. Evolution of error in radial distance, latitude, and longitude for a Molniya orbit. Moreover, and as done in the previous example, we can study the performance of a long-term propagation of this orbit. Fig. 6 shows the evolution of the error in radial distance, latitude, and longitude of the orbit as a function of the number of orbital periods that the orbit is propagated. In this figure, it can be observed the increase in error as the propagation time increases due to the approximation performed in the frequencies of the motion using this first-order solution. Nevertheless, even with this increase in error, there is an angular error smaller than , and a radial error of 0.5 km after 100 orbital revolutions. These results represent a relative error of 10 −5 after this long propagation when compared to absolute value of their variables, which shows the good performance of the methodology even for high eccentric orbits. Additionally, and if we compare this result with the long-term errors obtained from the sun-synchronous orbit (Fig. 3 ), we observe a decrease in the absolute accuracy due to the higher variation of the variables. Nevertheless, the relative error is still in the same order of magnitude. Figure 6. Long-term error evolution in the radial distance, the latitude, and the longitude for a Molniya orbit. 6.3.3 Example: hyperbolic orbit For the final example, we select an hyperbolic orbit to show that this formulation can also be successfully applied to this kind of trajectories. To this end, an orbit with the following initial orbital elements is used: a = −35 000.0 km, e = 1.2, , , , and . Figs 7 and 8 show the solution for these initial conditions. Note that, since the orbit is hyperbolic, only a section of the orbit was