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On a set of J\(_2\) equinoctial orbital elements and their use for uncertainty propagation

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Abstract

This paper defines a set of six non-singular orbital elements designed specifically for the characterization of uncertainty in the state of a resident space object in circular or elliptic orbit and demonstrates their use for uncertainty propagation in the context of the perturbed two-body problem of orbital mechanics. As evidenced by the time evolution of the Cramér–von Mises test statistic, representation of the orbital state probability density function in J\(_2\)EqOE yields less nonlinear uncertainty propagation and provides covariance and uncertainty realism for much longer periods of time than what is possible using Cartesian coordinates or even equinoctial orbital elements, without an appreciable increase in computational cost.

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Notes

  1. Covariance realism is the proper characterization of the covariance in the state of a system under Gaussian assumptions. It implies that the estimate of the mean is the true mean (i.e., the estimate is unbiased) and that the covariance possesses the right size, shape, and orientation (i.e., consistency). Uncertainty realism relaxes the Gaussian assumptions. Covariance realism is a necessary but not sufficient condition for achieving uncertainty realism. An extensive study on covariance and uncertainty realism in the SSA domain is provided in the report of Poore et al. (2016).

  2. More than 25% of cataloged space objects have eccentricities less than \(10^{-3}\) and nearly 2% have eccentricities less than \(10^{-4}\) (Space-Track.Org 2020).

  3. Lyddane’s algorithm contains an intrinsic singularity because the final output is the Keplerian elements.

  4. Brouwer’s algorithm contains explicit singularities for zero eccentricity and zero inclination.

  5. Suppose \(\theta _1\) and \(\theta _2\) are two angles equivalent up to integer \(2\pi \) shifts. Let \(b_{cut} = \theta _1 - \pi \), and redefine \(\theta _2\) according to \(\theta _2 = \theta _2 + 2\pi \lceil (b_{cut} - \theta _2)/(2\pi )\rceil \). Then, \(\theta _1\) and \(\theta _2\) can be safely added or subtracted.

  6. The geocoefficients are taken from the 2008 Earth gravity model (Pavlis et al. 2008).

  7. High-precision ephemerides (DE405) from the Jet Propulsion Laboratory’s HORIZONS system are used for determining the position of the Sun and Moon.

  8. Note that the orbital propagations themselves are performed in Cartesian space; however, the PDF need not be represented in Cartesian coordinates.

  9. The Mahalanobis distance is given by

    $$\begin{aligned} {\mathcal {M}}^{(i)}(\varvec{x}^{(i)}; \varvec{\mu }, \mathbf {P}) = (\varvec{x}^{(i)} - \varvec{\mu })^{T} \mathbf {P}^{-1} (\varvec{x}^{(i)} - \varvec{\mu }), \end{aligned}$$
    (13)

    where \(\mu \) is the n-dimensional state estimate, and \(\mathbf {P}\) the n by n covariance matrix. The expected value of \({\mathcal {M}}\) is n, and it follows that \({\mathcal {M}} \sim \chi ^{2}(n)\), i.e., \({\mathcal {M}}\) is chi-squared distributed with n degrees of freedom.

  10. The Cramér–von Mises test statistic is given by

    $$\begin{aligned} Q = \frac{1}{12k} + \sum _{i=1}^{k} \left[ \frac{2i-1}{2k} - F(y^{(i)}) \right] ^{2}, \end{aligned}$$
    (14)

    where the \(y^{(i)}\), \(i = 1, \ldots , k\), are the observed samples in increasing order, and F is the cumulative distribution function of the target distribution. Further details are given in Horwood et al. (2014b).

  11. Conceivably, the tests described in this section could be performed in other systems of Cartesian coordinates, such as the perifocal PQW coordinates or the RSW or NTW satellite coordinate systems (Vallado 2013). However, because any two Cartesian frames are related by an affine transformation (i.e., a rotation and translation), the same results would be obtained if one used a different Cartesian coordinate system.

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Acknowledgements

The authors thank Navraj Singh and Alex Ferris for assistance with the simulation studies. This work was funded, in part, by a Phase II STTR from the Air Force Office of Scientific Research (FA9550-12-C-0034) and by a Phase I and a Phase II SBIR from the Air Force Research Laboratory Space Vehicles Directorate (FA9453-15-M-0482, FA8650-19-C-9205). This work was cleared for public release (case number AFMC-2019-0691).

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Authors and Affiliations

  1. Numerica Corporation, Fort Collins, CO, USA

    Jeffrey M. Aristoff & Joshua T. Horwood

  2. Texas A&M University, College Station, TX, USA

    Kyle T. Alfriend

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  1. Jeffrey M. Aristoff
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Correspondence to Joshua T. Horwood.

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Appendix: Numerical example

Appendix: Numerical example

To assist the reader in an implementation of the J\(_2\)EqOE to EqOE conversion (and its inverse), we provide details of how to recover the Keplerian orbital elements for the LEO example presented in Table 1 following all of the intermediate steps described in Sect. 2.

The following values are assumed for the gravitational parameter \(\mu \), Earth radius R, and \(J_2\) coefficient:

$$\begin{aligned} \mu&= 3.986004415 \text {e+05}\, \text {km}^{3}/\text {s}^{2} \\ R&= 6.378137 \text {e+03}\, \text {km} \\ J_{2}&= 1.0826261738522 \text {e-03} \end{aligned}$$

The input J\(_2\)EqOE are the following:

$$\begin{aligned} n''&= 1.0487861550167\mathrm{e} {-}{03}\, \text {s}^{-1}&\qquad p''&= 6.6371635585001\mathrm{e} {-}{01} \\ h''&= 1.4260897234681\mathrm{e} {-}{03}&\qquad q''&= -3.2380704393120\mathrm{e} {-}{01} \\ k''&= -9.2912382513865\mathrm{e} {-}{03}&\qquad \ell ''&= 4.8735182761240\mathrm{e} {+}{00} \end{aligned}$$

In what follows, all angular quantities have implied units of radians.

1.1 J\(_2\)EqOE to J\(_2\)IOE Conversion

$$\begin{aligned} I''_{1}&= 7.1294270251742 \text {e+03}\, \text {km}&\qquad I''_{4}&= 5.3390774990364\mathrm{e} {-}{01} \\ I''_{2}&= 8.9420945754925\mathrm{e} {-}{03}&\qquad I''_{5}&= -2.6047736914253\mathrm{e} {-}{01} \\ I''_{3}&= -2.8982382142187\mathrm{e} {-}{03}&\qquad I''_{6}&= 4.8735182761240 \text {e+00} \end{aligned}$$

1.2 J\(_2\)IOE to IOE Conversion: Step 1

$$\begin{aligned} a''&= 7.1294270251742 \text {e+03}\, \text {km}&\qquad h''&= 2.0246926216394 \text {e+00} \\ e''&= 9.4000446883730\mathrm{e} {-}{03}&\qquad g''&= 9.6460100038126\mathrm{e} {-}{01} \\ I''&= 1.2721904014791 \text {e+00}&\qquad l''&= 1.8842246541034 \text {e+00} \end{aligned}$$

1.3 J\(_2\)IOE to IOE Conversion: Step 2

$$\begin{aligned} k&= 2.2020958264503 \text {e+04}\, \text {km}^{2}&\qquad \gamma&= 4.3323841440301\mathrm{e} {-}{04} \\ \eta&= 9.9995581860393\mathrm{e} {-}{01}&\qquad \gamma '&= 4.3331498717247\mathrm{e} {-}{04} \\ \theta&= 2.9418810951484\mathrm{e} {-}{01} \end{aligned}$$

1.4 J\(_2\)IOE to IOE Conversion: Step 3

$$\begin{aligned} E''&= 1.8931405532060 \text {e+00} \end{aligned}$$

1.5 J\(_2\)IOE to IOE Conversion: Step 4

$$\begin{aligned} r''&= 7.1506573807183 \text {e+03}\, \text {km}&\qquad f''&= 1.9020433345198 \text {e+00} \end{aligned}$$

1.6 J\(_2\)IOE to IOE Conversion: Step 5

$$\begin{aligned} a&= 7.1366000000000 \text {e+03}\, \text {km}&\qquad \delta h&= -1.1070091328349\mathrm{e} {-} {04} \end{aligned}$$

1.7 J\(_2\)IOE to IOE Conversion: Step 6

$$\begin{aligned} l + g + h&= 4.8729592715682 \text {e+00} \end{aligned}$$

1.8 J\(_2\)IOE to IOE Conversion: Step 7

$$\begin{aligned} \nu _{1}&= -9.7268781528180 \mathrm{e}{-}{01}&\qquad \delta e&= 8.1222972039185\mathrm{e} {-}{05} \\ \nu _{2}&= -9.5884220957866\mathrm{e} {-}{01}&\qquad e'' \delta l&= -4.0701787616281\mathrm{e} {-}{04} \\ \nu _{3}&= -9.6354316647622\mathrm{e} {-}{01}&\\ \nu _{4}&= 1.9910139555114 \mathrm{e}{+00} \end{aligned}$$

1.9 J\(_2\)IOE to IOE Conversion: Step 8

$$\begin{aligned} \delta I&= 1.5460976151790 \mathrm{e}{-}04&\qquad \sin \left( \tfrac{1}{2}I''\right) \delta h&= -6.5762859846059\mathrm{e} {-}{05} \end{aligned}$$

1.10 J\(_2\)IOE to IOE Conversion: Step 9

$$\begin{aligned} I_{1}&= 7.1366000000000 \text {e+03}\, \text {km}&\qquad I_{4}&= 5.3399247411729\mathrm{e} {-}{01} \\ I_{2}&= 9.1448530009497\mathrm{e} {-}{03}&\qquad I_{5}&= -2.6044553167591\mathrm{e} {-}{01} \\ I_{3}&= -2.5360921889622\mathrm{e} {-}{03}&\qquad I_{6}&= 4.8729592715682 \text {e+00} \end{aligned}$$

1.11 IOE to EqOE Conversion

$$\begin{aligned} a&= 7.1366000000000 \text {e+03}\, \text {km}&\qquad p&= 6.6385958338730\mathrm{e} {-}{01} \\ h&= 1.0413786122339\mathrm{e} {-}{03}&\qquad q&= -3.2378595304974\mathrm{e} {-}{01} \\ k&= -9.4326894672662\mathrm{e} {-}{03}&\qquad \ell&= 4.8729592715682 \text {e+00} \end{aligned}$$

1.12 IOE to Keplerian Orbital Elements Conversion (Optional)

$$\begin{aligned} a&= 7.1366000000000 \text {e+03}\, \text {km}&\qquad \varOmega&= 2.0245819323134 \text {e+00} \\ e&= 9.4900000000000\mathrm{e} {-}{03}&\qquad \omega&= 1.0070549784028 \text {e+00} \\ i&= 1.2723450247039 \text {e+00}&\qquad M&= 1.8413223608519 \text {e+00} \end{aligned}$$

These Keplerian orbital elements are precisely those for the LEO object in Table 1. Note that the angular quantities \((i, \varOmega , \omega , M)\) in the above are given in radians whereas the table converts them to degrees.

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Aristoff, J.M., Horwood, J.T. & Alfriend, K.T. On a set of J\(_2\) equinoctial orbital elements and their use for uncertainty propagation. Celest Mech Dyn Astr 133, 9 (2021). https://doi.org/10.1007/s10569-021-10004-0

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