A Population Dynamics Approach to the Distribution of Space Debris in Low Earth Orbit

Abstract

The presence of the debris in the Earth’s orbit poses a significant risk to human activity in outer space. This debris population continues to grow due to ground launches, the loss of external parts from space ships, and uncontrollable collisions between objects. A computationally feasible continuum model for the growth of the debris population and its spatial distribution is therefore critical. Here we propose a diffusion-collision model for the evolution of the debris density in the low-Earth orbit and its dependence on the ground-launch policy. We parametrize this model and test it against data from publicly available object catalogs to examine timescales for the uncontrolled growth. Finally, we consider sensible launch policies and cleanup strategies and how they reduce the future risk of collisions with active satellites or space ships.

Keywords

Population dynamics / Space debris / Kessler syndrome / Diffusion / Low-Earth orbit

References

[1.] ApolloSat.com.: U.S. Satellite Destroyed in Space Collision (2022). Access at https://apollosat.com/space-collision/. Accessed 1 Aug 2021

[2.] Bonnal C, McKnight D, Phipps C, Dupont C, Missonnier S, Lequette L, Merle M, Rommelaere S. Just in time collision avoidance—a review. Acta Astronaut., 2020, 170: 637-651,

[3.] Brouwer D. Solution of the problem of artificial satellite theory without drag. Astron. J., 1959, 64: 378-396,

[4.] Carnegie Endowment for International Peace: The dangerous fallout of Russia’s anti-satellite missile test (2022). Access at https://carnegieendowment.org. Accessed 20 June 2022

[5.] Gast, R.: Ein All für alle (Die Zeit, 07-23-2022, in German) (2022)

[6.] Hoots FR. Reformulation of the Brouwer geopotential theory for improved computational efficiency. Celest. Mech., 1981, 24: 367-375,

[7.] Johnston, E.: List of satellites in geostationary orbit (2022). Access at https://www.satsig.net/sslist.htm. Accessed 20 May 2022

[8.] Kessler DJ, Cour-Palais BG. Collision frequency of artificial satellites: the creation of a debris belt. J. Geophys. Res., 1978, 83: 2637-2646,

[9.] Klima R, Bloembergen D, Savani R, Tuyls K, Wittig A, Sapera A, Izzo D. Space debris removal: learning to cooperate and the price of anarchy. Front. Robot. AI, 2018, 5: 54,

[10.] Lemaitre A, Hubaux C. Celletti A, Locatelli U, Ruggeri T, Strickland E. Space debris long term dynamics. Mathematical Models and Methods for Planet Earth, 2013 Cham Springer 113-121

[11.] Liou J-C. An active debris removal parametric study for LEO environment remediation. Adv. Space Res., 2011, 47: 1865-1876,

[12.] Liou J-C, Hall DT, Krisko PH, Opiela JN. LEGEND—a three-dimensional LEO-to-GEO debris evolutionary model. Adv. Space Res., 2004, 34: 981-986,

[13.] Lyddane RH. Small eccentricities or inclinations in the Brouwer theory of the artificial satellite. Astron. J., 1963, 68: 555-558,

[14.] McInnes CR. An analytical model for the catastrophic production of orbital debris. ESA J., 1993, 17: 293-305

[15.] NASA: LEGEND: 3D/OD evolutionary model (2022). orbitaldebris.jsc.nasa.gov. Accessed 20 Nov 2022

[16.] Nikolaev, S., Phillion, D., Springer, H.K., deVries, W., Jiang, M., Pertica, A., Henderson, J., Horsley, M., Olivier, S.: Brute force modeling of the Kessler syndrome. In: Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS), Maui, HI, September 2012 (2012)

[17.] Okubo A. . Diffusion and Ecological Problems: Mathematical Models, 1980 Berlin Springer

[18.] Reynolds RC, Eichler P. A comparison of debris environment projections using the EVOLVE and CHAIN models. Adv. Space Res., 1995, 16: 127-135,

[19.] Riesing, K., Kahoy, K.: Orbit determination from two-line element sets for ISS-deployed CubeSats. In: 29th Annual AIAA/USA Conference on Small Satellites (2015)

[20.] Rossi, A.: Population models of space debris. In: Knežević, Z., Milani, A. (eds.) Dynamics of Populations of Planetary Systems, vol. 197 of Proceedings IAU Colloquium, pp. 427–438. International Astronomical Union, Paris (2005)

[21.] Space.com: A rogue 3-ton piece rocket debris just collided with the moon (2022). Access at https://www.space.com/rogue-rocket-stage-hit-moon-today Accessed 20 June 2022

[22.] Space.com: Starlink satellites: everything you need to know about the controversial internet megaconstellation (2022). Access at https://www.space.com/spacex-starlink-satellites.html. Accessed 10 Dec 2022

[23.] Takahashi K, Charles C, Boswell RW, Ando A. Demonstrating a new technology for space debris removal using a bidirectional plasma thruster. Sci. Rep., 2018, 8: 14417,

[24.] US Space Command: Space-track.org (2010). Space-Track.org. Accessed 8 Aug 2022

[25.] Zhang B, Wang Z, Zhang Y. Discrete evolution model based on mean spatial density for space debris environment. Astrophys. Space Sci., 2019, 364: 70,

Funding

University of Wisconsin-Milwaukee Graduate School