A modified classical method for preliminary orbitdetermination is presented. In our proposal, the spread of theobservations is considerably wider than in the original method, aswell as the order of convergence of the iterative scheme involved. The numerical approach is made by using matricial weight functions,which will lead us to a class of iterative methods with a sixthlocal order of convergence. This is a process widely used in thedesign of iterative methods for solving nonlinear scalar equations,but rarely employed in vectorial cases. The numerical tests confirmthe theoretical results, and the analysis of the dynamics of theproblem shows the stability of the proposed schemes.
The first step in orbit determination methods is to obtain preliminary orbits, as the motion analyzed is under the premises of the two bodies problem. It is possible to set a two-dimensional coordinate system (see Figure 1), where the X axis points to the perigee of the orbit, the closest point of the elliptical orbit to the focus and center of the system, the Earth. In this picture, the true anomaly and the eccentric anomaly can be observed. In order to place this orbit in the celestial sphere and determine completely the position of a body in the orbit, some elements (called orbital or keplerian elements) must be determined. These orbital elements are as follows.(i) (right ascension of the ascending node): defined as the equatorial angle between the Vernal point and the ascending node ; it orients the orbit in the equatorial plane.(ii) (argument of the perigee): defined as the angle of the orbital plane, centered at the focus, between the ascending node and the perigee of the orbit; it orients the orbit in its plane.(iii) (inclination): dihedral angle between the equatorial and the orbital planes.(iv) (semimajor axis): which sets the size of the orbit.(v) (eccentricity): which gives the shape of the ellipse.(vi) (perigee epoch): time for the passing of the object over the perigee, to determine a reference origin in time. It can be denoted by a exact date, in Julian days, or by the amount of time ago the object was over the perigee.
Two reference orbits have been used in the test for the preliminary orbit determination. The first can be found in , and the second one is a commercial real orbit called Tundra. As the orbital elements of each one are known, the vector positions (measured in Earth radius) at the instants and have been recalculated with 500 exact digits. These vector positions arefor Reference Orbit I andfor Tundra Orbit. Then, our aim is to gain from these positions the orbital elements showed in Tables 1 and 2 with a precision as high as possible by means of proposed iterative schemes.
For the representation of the convergence basins of our procedures and classical methods, we have used the software described in . We draw a mesh with two thousand points per axis; each point of the mesh is a different initial estimation which we introduce in each procedure. If the method reaches the final solution in less than five hundred iterations, this point is drawn in orange. The color will be more intense when the number of iterations is lower. Otherwise, if the method arrives at the maximum of iterations, the point will be drawn in black. In each axis, we will represent each of the variables with which we work. The ratio sector triangle is represented in the abscissas and the difference of eccentric anomalies in the ordinates. In addition, we will use the reference orbit I, which is defined in Table 1, and the solution of the nonlinear system is in this case around (1, 0.1). For this reason, we choose as the region of representation.
The reliability of the uncertainty characterization, also known as uncertainty realism, is of the uttermost importance for Space Situational Awareness (SSA) services. Among the different sources of uncertainty related to the orbits of Resident Space Objects (RSOs), the uncertainty of dynamic models is one of the most relevant ones, although it is not always included in orbit determination processes. A classical approach to account for these sources of uncertainty is the consider parameters theory, which consists in including parameters in the underlying dynamical models whose variance aims to represent the uncertainty of the system. However, realistic variances of these consider parameters are not known a-priori. This work presents a method to infer the variance of the consider parameters, based on the distribution of the Mahalanobis distance of the orbital differences between predicted and estimated orbits, which theoretically shall follow a \(\chi ^2\) distribution under Gaussian assumption. This paper presents results in a simulated scenario focusing on Geostationary (GEO) regimes. The effectiveness and traceability of the uncertainty sources is assessed via covariance realism metrics.
In an operational environment, simple and robust techniques are required to improve covariance realism since, as previously discussed, the nominal covariance determination methods tend to provide optimistic results. The most applied solutions are:
Scaling techniques which inflate the covariance by certain factors. Some authors propose the computation of such scaling based on increasing the initial position uncertainty to match the velocity error  whereas other options explore the usage of the Mahalanobis distance of the orbital differences to find the scale factor . However, a common drawback of artificially increasing the covariance is that the physical meaning of the correction is lost, not being able to understand the contributions of each source of uncertainty. These sort of methods are used nowadays in operation centers such as Space Operations Center (CSpOC) .
In the work at hand, the covariance determination methodology is applied to GEO RSOs, continuing the efforts of previous studies that applied the proposed methodology to Low Earth Orbit (LEO) RSOs for drag and range bias uncertainty . The realism of the determined covariance matrices remains as the cornerstone of this study, and thus specific covariance realism metrics such as the covariance containment are analyzed. For a geostationary orbit, two of the main sources of uncertainty that come into play are related to the Solar Radiation Pressure (SRP) and the time bias of the sensors, which are analysed in the presented work.
Operational reference orbit: We propose an alternative orbit to use for the computation of the orbital differences of Eq. (14) other than the aforementioned reference orbit. The operational reference orbit consists in the output of an OD whose determination arc includes the propagation epochs of interest for each Monte Carlo iteration. In other words, if we want to analyse the current orbit between \(t_0\)+7 days and \(t_0\)+21 days, we can use as reference the output of another OD whose arc ranges from \(t_0\) days up to \(t_0\)+28 (see Fig. 1). The reason behind this choice is that, for most SST operational environments, precise orbits whose ephemeris can be assumed perfect are not commonly accessible. For instance, external sources of precise GNSS orbit data is not available when the targets are non-collaborative objects such as space debris. It is important to remark that, the estimation corresponding to this operational reference orbit has certain noise-only covariance due to the measurements uncertainty. Thus, it is necessary to include its noise-only covariance inside the Mahalanobis distance computation of Eq. (14).
The Mahalanobis distances can be calculated once all the necessary data for all the orbits is available at the desired analysis epochs. Then, the proposed covariance determination methodology can be applied. This process is summarised in Algorithm 1.
The results shown in this work indicate that the presented covariance determination methodology is capable of accurately capturing the model error present within the dynamics of an RSO on a simulated SST scenario. It has been successfully applied to GEO, considering SRP and sensor time bias uncertainty. We have tested the robustness of the model by enforcing large perturbation levels, while also ensuring that sensitivity to lower values is maintained. The deviation between the uncertainty introduced as perturbation inputs and the determined consider parameters has remained lower than a 11% throughout the development of this work. Additionally, successful results have been obtained when estimated orbits are used as reference to compute the Mahalanobis distance, indicating the operational suitability of the methodology for operational scenarios. Relevant metrics for covariance realism assessment such as the covariance containment tests have shown that the proposed methodology is able to determine a realistic covariance, applicable to the complete propagation region of interest in GEO and without over-sizing.
The rise/set problem may be defined as the process of determining the times at which a satellite rises and sets with respect to a ground location. The easiest solution uses a numerical method to determine visibility periods for the site and satellite by evaluating UK position vectors of each. It advances vectors by a small time increment, Δt, and checks visibility at each step. A drawback to this method is computation time, especially when modeling many perturbations and processing several satellites. Escobal ,  proposed a faster method to solve the rise/set problem by developing a closed-form solution for unrestricted visibility periods about an oblate Earth. He assumes infinite range, azimuth, and elevation visibility for the site. Escobal transforms the geometry for the satellite and tracking station into a single transcendental equation for time as a function of eccentric anomaly. He then uses numerical methods to find the rise and set anomalies, if they exist. Lawton  has developed another method to solve for satellite-satellite and satellite-ground station visibility periods for vehicles in circular or near circular orbits by approximating the visibility function, by a Fourier series. 2b1af7f3a8