# A lower bound of the distance between two elliptic orbits

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## Abstract

We obtain a lower bound of the distance function (MOID) between two noncoplanar bounded Keplerian orbits (either circular or elliptic) with a common focus. This lower bound is positive and vanishes if and only if the orbits intersect. It is expressed explicitly, using only elementary functions of orbital elements, and allows us to significantly increase the speed of processing for large asteroid catalogs. Benchmarks confirm high practical benefits of the lower bound constructed.

## Keywords

Elliptic orbits MOID Linking coefficient Distance function Catalogs Asteroids and comets Near-Earth asteroids Space debris Close encounters Collisions## Notes

### Acknowledgements

We are grateful to Professor K. V. Kholshevnikov for the statement of the problem, for important remarks and for his help in preparing the manuscript. We also express our gratitude to A. Ravsky for valuable discussion, as well as anonymous reviewers, whose constructive and valuable comments greatly helped the authors to improve the manuscript. All calculations made in the work were conducted by means of the equipment of the Computing Centre of Research Park of Saint Petersburg State University. This work is supported by the Russian Science Foundation Grant No. 18-12-00050.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflicts of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

### Informed consent

This research did not involve human participants.

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