Parameterization of highly eccentric orbit
Abstract
In this report, we present a new way to parameterize the Highly Eccentric Orbit (HEO). The new parameterization is introduced to overcome the difficulty in propagating the HEO in restricted scenarios. By restricted scenarios, we mean the space debris where the HEO objects are numerous or the onboard environment where the hardware capability is limited.In the restricted scenarios, it is time-consuming to use the numerical propagation since the large eccentricity requires the propagator to adaptively change the step size to obtain a good balance between accuracy and speed. Even so, propagation at the perigee is inevitably slow, not to mention that frequent adapting the step size itself comes at the cost of efficiency. The storage for HEO ephemeris is also a concern. Due to the fast motion at perigee, the ephemeris has to be saved at a reasonably small step size at perigee.Analytical solution is another popular option in such scenarios, among which SGP is always used by many agencies. However, the analytical approach faces a critical problem with HEO, where the large eccentricity renders the expansion (with respect to the mean anomaly) highly inaccurate or even invalid. Many approximations applied to nearly circular orbits do not hold either in these circumstances and the neglected higher-order terms (with respect to the eccentricity) have to be reconsidered. All these restrictions compromise the current analytical solutions in the application of HEO.There have been some earlier researches, providing alternative presentations of the orbits of the space objects, including HEO objects or debris. One effective approach is to follow the broadcast ephemeris in satellite navigation (GPS, Galileo, BeiDou) that uses certain numbers of parameters to fit the propagated orbit based on known variation pattern of the orbital elements. These parameters, together with predetermined model, are used to compute the orbit or position at real time. This approach always involves frequency analysis or Fourier analysis, to find the contributing periods. The limit of this approach on HEO is that, since the HEO objects move non-uniformly and it is not easy to model the orbit variation with simple combinations of various frequencies. To model the orbit using combinations of cos(ω_kt) and sin(ω_kt) could require a large number of terms.In our approach, we start modeling the orbit from the underlying dynamics. Recognizing that the orbital element actually varies with respect to the true anomaly f, instead of the mean anomaly (or equivalently the time). The orbit variation is modeled in the form of summation of cos(kf) and sin(kf), together with other possible terms including linear or quadratic terms and/or monthly or semimonthly periods. The tricky part of this approach is about how to solve for the true anomaly when recovering the orbital elements from the fitting parameters. We use a recurrence approach to determine the true anomaly at the beginning of the orbit calculation and with the determined f time series, the other elements can be determined as well.Preliminary tests show that it is easy to obtain high accuracy with this method, without having to include too many frequencies in the model. This new parameterization is feasible in terms of both the orbit accuracy and the data storage of fitting parameters.
- Publication:
-
42nd COSPAR Scientific Assembly
- Pub Date:
- July 2018
- Bibcode:
- 2018cosp...42E3333T