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On the nodal distance between two Keplerian trajectories with a common focus

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Abstract

We study the possible values of the nodal distance \(\delta _\mathrm{nod}\) between two non-coplanar Keplerian trajectories \(\mathcal{A}, \mathcal{A}'\) with a common focus. In particular, given \(\mathcal{A}'\) and assuming it is bounded, we compute optimal lower and upper bounds for \(\delta _\mathrm{nod}\) as functions of a selected pair of orbital elements of \(\mathcal{A}\), when the other elements can vary. This work arises in the attempt to extend to the elliptic case the optimal estimates for the orbit distance given in Gronchi and Valsecchi (Mon Not R Astron Soc 29(3):2687–2699, 2013) in case of a circular trajectory \(\mathcal{A}'\). These estimates are relevant to understand the observability of celestial bodies moving (approximately) along \(\mathcal{A}\) when the observer trajectory is (close to) \(\mathcal{A}'\).

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Notes

  1. Minimal Orbital Intersection Distance.

  2. Defined by assigning an orientation to both trajectories.

  3. \((I,\Omega ,\omega )\) and \((I',\Omega ',\omega ')\) are, respectively, inclination, longitude of the ascending node, argument of pericenter of \(\mathcal{A}\) and \(\mathcal{A}'\).

  4. We admit infinite values for the considered functions, e.g., \(\ell ^\omega _\mathrm{int}(q,0) = -\,\infty \).

  5. Here \(\ell ^{\,e}_\mathrm{int}(q,1)=-\,\infty \), and \(u^{\,e}_\mathrm{link}(q,1)=Q'-q\).

  6. Here we state the result presented in Gronchi and Valsecchi (2013) with a formula that is not singular for \(e=1\).

  7. We discard the solution giving a value of \(\xi '\) which is \(<-1\).

  8. We discard the solution giving a negative value of e.

  9. https://newton.spacedys.com/neodys.

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Acknowledgements

Part of this work has been done during a visiting period of G. F. Gronchi at the Institut de mécanique céleste et de calcul des éphémérides (IMCCE), Observatoire de Paris. The same author also acknowledges the Project MIUR-PRIN 20178CJA2B titled “New frontiers of Celestial Mechanics: theory and applications”. We thank the anonymous referees for their suggestions, allowing us to improve the paper.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Giovanni Federico Gronchi

  2. Départment de Mathématiques d’Orsay, Université Paris-Saclay, 91405, Orsay, France

    Laurent Niederman

  3. Observatoire de Paris, 77 Avenue Denfert-Rochereau, 75014, Paris, France

    Laurent Niederman

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  1. Giovanni Federico Gronchi
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  2. Laurent Niederman
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Correspondence to Giovanni Federico Gronchi.

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Gronchi, G.F., Niederman, L. On the nodal distance between two Keplerian trajectories with a common focus. Celest Mech Dyn Astr 132, 5 (2020). https://doi.org/10.1007/s10569-019-9944-y

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