A Ptolemaic Approach Improving the Conjunction Analysis Pipeline for Leo

Abstract

Low-Earth-Orbit (LEO) region congestion is becoming one of the big issues of the modern space era. To avoid the Kessler syndrome, now more than ever it is needed to improve awareness about space traffic, and upgrade the entire monitoring process. Extensive literature is available covering the topics of orbital conjunction filtering techniques and computation of the Minimum Orbital Intersection Distance (MOID). The present paper investigates Funding and/or Conflicts of interests/Conflict of interest. An alternative filtering method exploits the near-circularity of certain orbits (a condition often verified in LEO), to improve conjunction analysis performance. Elliptical orbits are reshaped through an auxiliary deferent model, inspired by C. Ptolemy’s orbital theory, replacing the real motion along conjunction analysis. To recover satellites’ averaged mean orbital elements, CelesTrack LEO catalogue was considered and propagated. Based on averaged parameters, off-centric circular orbits are considered instead of elliptical ones. The resulting deferents (off-centric circles) are not far from osculating orbits due to LEOs low eccentricities, becoming the basis for the conjunction analysis algorithm. The algorithm is conceived as a sequence of pre-filters and a final MOID computation. Performances are inspected through an all-vs-all analysis, taking as reference a combination of Hoots’ and Gronchi’s algorithms. This method achieves good performance as compared with these traditional benchmarks. Adopting this approach could reduce the time needed for a preliminary conjunction inspection during the first phases of the Collision Avoidance (CA) process, especially in LEO, where pre-filtering aims to reduce the number of orbit couples where precise MOID computation is needed.

Appendices

Appendix A: Test-4 Description

Test-4 setup is better described here.

The following steps allow computing the two crossing points required for Test-4, between the Def-2 and Def-1’s plane.

• 1. Since Def-2 has been translated in perifocal RF of Def-1, the two points we are looking for belong exactly to the nodal line of Def-2. Moreover $\Omega _{2}$ is simply the angle from the perigee of Def-1 up to the line of crossing between the planes of orbits. Crossing points are represented in Fig. 8 (down): L and M. Where C is the centre of Def-1.

• 2. Exploiting previous deductions, two useful triangles could be identified (Fig. 8) on the plane of Def-1: the first one for the point M ($\hat{\text {Earth-C-M}}$), and the second for point L ($\hat{\text {Earth-C-L}}$). What is needed to set up Test-4 are distances d1 and d2 (represented in Fig. 8), from the centre of Def-1. Once obtained, it will be straightforward to compare these distances with $R_{1}$, to verify conjunctions at these two locations.

• 3. In order to obtain d1 and d2, the two previous triangles are solved. Since it is already known the common side $c_{1}$ (the ellipse parameter of Def-1), at least another side and one angle are needed. Angles between $c_{1}$ and the two sides $r_{2_right}$ and $r_{2_left}$ (distances of L and M from Earth) could be easily computed with $\Omega$ of Def-2, assuming point (1).

  $$\begin{aligned} \begin{aligned}&\hat{{\text {(C-Earth-M)}}}=180^\circ \, -\, \Omega _{2}\\&\hat{{\text {(C-Earth-L)}}}=\Omega _{2}\\ \end{aligned} \end{aligned}$$

  (4)

• 4. Distances from Earth of points L and M are computed instead considering the geometrical definition of $\omega$ for Def-2. The procedure is the one represented in Fig. 8b. In this representation Def-1 is the green one, while Def-2 is the red one; L and M are the two big white points. The two distances are computed by recovering eccentric anomalies from true anomalies (the one represented in Fig. 8).

• 5. Through a well-known theorem, knowing two sides of a triangle and the angle between them the geometry can be solved. In one step, more d1 and d2 are computed.

Once distances d1 and d2 are computed, the aim of this test is to verify if these two points come closer than D to Def-1. This could be verified through the following inequalities:

$$\begin{aligned} \begin{aligned}&|R_{1}-d_{1}|> D\\&|R_{2}-d_{2}| > D\\ \end{aligned} \end{aligned}$$

(5)

In case inequalities are both verified, any conjunction is possible at these two points. From the sign of the two differences, we can understand if L and M stay inside or outside Def-1.

Appendix B: Test-5 Description

In this appendix, a more in-depth description of Test-5 is provided.

Since Def-1 lays completely on the horizontal x–y plane of the main RF (due to the rotation performed, in its perifocal RF), the relative distance between deferents can be computed as the sum of the squares between distance from $C_{1}$ (planar distance) and elevation (out-of-plane distance) from the Def-1 plane, for all points on Def-2. Once the minimum distance is found, it is possible to state if any conjunction will happen or not: if the minimum distance is bigger than D no conjunctions are possible, otherwise at least one is present. The relative distance between deferents is reshaped as a function of the eccentric anomaly E of Def-2 ($E_{2}$): this is called F, the function to be minimized. This is defined here:

$$\begin{aligned} \begin{aligned}&F = f(E_{2})\\&F = \sqrt{(|\vec {P}_{Def-2} -\vec {C_{1}}|-a_{1})^{2} + z_{P}^{2} }\\ \end{aligned} \end{aligned}$$

(6)

where $\vec {P}$ is the position vector of a generic point on Def-2 and $z_{P}$ is its elevation from horizontal plane (Def-1’s plane). And $\vec {C_{1}}$ is the position vector of the Def-1 centre. The F function is simply the planar distance from Def-1 together with the vertical distance from the plane of Def-1. Adding the two contributions, the resulting distance constitutes the relative distance between deferents.

The passages to obtain F are reported in the following equation:

$$\begin{aligned} \begin{aligned}&\vec {P}_{\text {perifocal2}}(E_{2})= \begin{bmatrix} (a_{2}\,\textrm{cos}(E_{2})-a_{2}\,e_{2})\\ (a_{2}\,\textrm{sin}(E_{2}))\\ 0 \end{bmatrix}&\\&\vec {P}_{\text {perifocal1}}=\bar{\bar{Q_{xX}}} \,\,\vec {P}_{\text {perifocal2}}\\&\\&z_{P}(E_{2})=\vec {P}_{\text {perifocal1}}(3)\\&\\&F(E_{2}) = \sqrt{ (\sqrt{(\vec {P}_{\text {perifocal1}}(1)+c_{1})^{2} + \vec {P}_{\text {perifocal1}}(2)^{2} } - a1)^2 + z_{P}^{2} } \end{aligned} \end{aligned}$$

(7)

$\bar{\bar{Q_{xX}}}$ is the matrix providing transformation from perifocal to equatorial RF with angles of Def-2 relative to Def-1, this matrix is not a function of E. $\vec {P}_{\text {perifocal1}}$ is the position vector on Def-2 in perifocal frame of Def-1 at the anomaly selected. The function F provides the relative distance between the two deferents. Once minimised this function, it is sufficient to compare its minimum with the threshold D and verify if at least one conjunction is present or not.

This function is minimised through the algorithm described before. The characteristics of this function are the following:

• single variable function;

• non-linear function;

• F presents probably more than one local minima.

Along the conjunction analysis, in Test-5 both in-plane and out-of-plane thresholds are involved. The minimum found is first checked for the out-of-plane distance and in case it is smaller, it is checked for the in-plane too.