# The Random Walk of Cars and Their Collision Probabilities with Planets

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Setup and Yarkovsky Effect

`REBOUND`integrator package [9] to query JPL’s NASA Horizons database for the initial ephemerides of all Solar System planets and the Tesla. As initial conditions, we use the NASA JPL solution #7 for the Tesla, generated on February 15th. We start the integrations at a time at which the Tesla is not expected to make any more course corrections. The integrations were carried out in the centre-of-mass frame of the Solar System and we use the high order Gauß-Radau

`IAS15`integrator [10]. This integrator uses an adaptive timestep and can handle frequent close encounters with high accuracy. The error in the conservation of energy is close to the double floating point precision limit. Using an even higher precision integrator would not further improve the results because, as we show below, nearby trajectories diverge due to close encounters, i.e., not due to numerical precision.

## 3. Evolution over the Next Few Hundred Years

## 4. Long-Term Evolution

## 5. Collision Probabilities

## 6. Conclusions

## Author Contributions

## Acknowledgments

`Jupyter`[22],

`iPython`[23], and

`matplotlib`[24].

## Conflicts of Interest

## References

- Farinella, P.; Froeschlé, C.; Froeschlé, C.; Gonczi, R.; Hahn, G.; Morbidelli, A.; Valsecchi, G.B. Asteroids falling into the Sun. Nature
**1994**, 371, 314–317. [Google Scholar] [CrossRef] - Granvik, M.; Morbidelli, A.; Jedicke, R.; Bolin, B.; Bottke, W.F.; Beshore, E.; Vokrouhlický, D.; Delbò, M.; Michel, P. Super-catastrophic disruption of asteroids at small perihelion distances. Nature
**2016**, 530, 303–306. [Google Scholar] [CrossRef] [PubMed] - Bottke, W.F.; Morbidelli, A.; Jedicke, R.; Petit, J.M.; Levison, H.F.; Michel, P.; Metcalfe, T.S. Debiased Orbital and Absolute Magnitude Distribution of the Near-Earth Objects. Icarus
**2002**, 156, 399–433. [Google Scholar] [CrossRef] - Gladman, B.J.; Migliorini, F.; Morbidelli, A.; Zappala, V.; Michel, P.; Cellino, A.; Froeschle, C.; Levison, H.F.; Bailey, M.; Duncan, M. Dynamical lifetimes of objects injected into asteroid belt resonances. Science
**1997**, 277, 197–201. [Google Scholar] [CrossRef] - Gladman, B.J.; Burns, J.A.; Duncan, M.; Lee, P.; Levison, H.F. The exchange of impact ejecta between terrestrial planets. Science
**1996**, 271, 1387–1392. [Google Scholar] [CrossRef] - Bottke, W.; Vokrouhlický, D.; Marchi, S.; Swindle, T.; Scott, E.; Weirich, J.; Levison, H. Dating the Moon-forming impact event with asteroidal meteorites. Science
**2015**, 348, 321–323. [Google Scholar] [CrossRef] [PubMed] - Farnocchia, D.; Chesley, S.R.; Milani, A.; Gronchi, G.F.; Chodas, P.W. Orbits, Long-Term Predictions, Impact Monitoring. In Asteroids IV; University of Arizona Press: Tucson, AR, USA, 2015; pp. 815–834. [Google Scholar]
- Milani, A.; Chesley, S.R.; Chodas, P.W.; Valsecchi, G.B. Asteroid Close Approaches: Analysis and Potential Impact Detection. In Asteroids III; University of Arizona Press: Tucson, AR, USA, 2002; pp. 55–69. [Google Scholar]
- Rein, H.; Liu, S.F. REBOUND: An open-source multi-purpose N-body code for collisional dynamics. A&A
**2012**, 537, A128. [Google Scholar] - Rein, H.; Spiegel, D.S. IAS15: A fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits. MNRAS
**2015**, 446, 1424–1437. [Google Scholar] [CrossRef] - Nobili, A.; Roxburgh, I.W. Simulation of general relativistic corrections in long term numerical integrations of planetary orbits. In Relativity in Celestial Mechanics and Astrometry. High Precision Dynamical Theories and Observational Verifications; Kovalevsky, J., Brumberg, V.A., Eds.; D. Reidel Publishing Co.: Dordrecht, The Netherlands, 1986; Volume 114, pp. 105–110. [Google Scholar]
- Laskar, J.; Gastineau, M. Existence of collisional trajectories of Mercury, Mars and Venus with the Earth. Nature
**2009**, 459, 817–819. [Google Scholar] [CrossRef] [PubMed] - Vokrouhlický, D.; Bottke, W.F.; Chesley, S.R.; Scheeres, D.J.; Statler, T.S. The Yarkovsky and YORP Effects. In Asteroids IV; University of Arizona Press: Tucson, AR, USA, 2015; pp. 509–531. [Google Scholar]
- Mommert, M.; Hora, J.L.; Farnocchia, D.; Chesley, S.R.; Vokrouhlický, D.; Trilling, D.E.; Mueller, M.; Harris, A.W.; Smith, H.A.; Fazio, G.G. Constraining the Physical Properties of Near-Earth Object 2009 BD. ApJ
**2014**, 786, 148. [Google Scholar] [CrossRef] - Bottke, W.F.; Vokrouhlický, D.; Rubincam, D.P.; Nesvorný, D. The Yarkovsky and YORP effects: Implications for asteroid dynamics. Annu. Rev. Earth Planet. Sci.
**2006**, 34, 157–191. [Google Scholar] [CrossRef] - Farnocchia, D.; Chesley, S.; Vokrouhlický, D.; Milani, A.; Spoto, F.; Bottke, W. Near Earth asteroids with measurable Yarkovsky effect. Icarus
**2013**, 224, 1–13. [Google Scholar] [CrossRef] - Greenstreet, S.; Ngo, H.; Gladman, B. The orbital distribution of Near-Earth Objects inside Earth’s orbit. Icarus
**2012**, 217, 355–366. [Google Scholar] [CrossRef] - Milani, A.; Carpino, M.; Hahn, G.; Nobili, A.M. Dynamics of planet-crossing asteroids—Classes of orbital behavior. Icarus
**1989**, 78, 212–269. [Google Scholar] [CrossRef] - Froeschle, C.; Hahn, G.; Gonczi, R.; Morbidelli, A.; Farinella, P. Secular resonances and the dynamics of Mars-crossing and Near-Earth asteroids. Icarus
**1995**, 117, 45–61. [Google Scholar] [CrossRef] - Michel, P.; Froeschlé, C. The Location of Linear Secular Resonances for Semimajor Axes Smaller Than 2 AU. Icarus
**1997**, 128, 230–240. [Google Scholar] [CrossRef] - Michel, P.; Thomas, F. The Kozai resonance for near-Earth asteroids with semimajor axes smaller than 2AU. A&A
**1996**, 307, 310. [Google Scholar] - Kluyver, T.; Ragan-Kelley, B.; Pérez, F.; Granger, B.; Bussonnier, M.; Frederic, J.; Kelley, K.; Hamrick, J.; Grout, J.; Corlay, S.; et al. Jupyter Notebooks—A publishing format for reproducible computational workflows. In Positioning and Power in Academic Publishing: Players, Agents and Agendas; IOS Press: Amsterdam, The Netherlands, 2016; p. 87. [Google Scholar]
- Pérez, F.; Granger, B.E. IPython: A System for Interactive Scientific Computing. Comput. Sci. Eng.
**2007**, 9, 21–29. [Google Scholar] [CrossRef] - Hunter, J.D. Matplotlib: A 2D graphics environment. Comput. Sci. Eng.
**2007**, 9, 90–95. [Google Scholar] [CrossRef]

**Figure 1.**The short-term orbital evolution of 48 realizations of the Tesla, initially perturbed by ${10}^{-6}$, over the next 1000 years. The top, middle, and bottom plots show the semi-major axis, eccentricity and minimum close approach distance to Earth for all realizations. The orbits diverge after a few deep encounter in the first 100 years.

**Figure 2.**Orbit of the Tesla, together with Earth and Mars. The Tesla’s initial orbit is nearly tangent to that of the Earth. This makes close encounters within a Hill sphere of the Earth possible over a range of longitudes, highlighted in yellow.

**Figure 3.**Long-term orbital evolution of the Tesla, showing the semi-major axis and eccentricity of 240 realizations. The star shows the initial orbit. The curves indicate the set of orbits having aphelion or perihelion that intersects the orbit of Mercury, Venus, Earth, or Mars. Close encounters with planets are only possible between the aphelion and perihelion lines. Some orbits temporarily decouple from these zones by effects of weak mean-motion resonances with terrestrial planets, visiting low-eccentricity states before again reaching the planet-crossing region.

**Figure 4.**Long-term evolution of the Tesla’s semi-major axis, eccentricity, and inclination as a function of time. The orbital elements undergo a random walk. On a long timescale, the median eccentricity and inclination of the clone orbits increase due to effects of mean-motion and secular resonances. This slightly lowers their impact probability on planets.

**Figure 5.**Top panel: Fraction of Tesla’s realizations surviving in our simulation as a function of time (240 clones initially). Bottom panel: The observed frequency of the Tesla having physical collisions with Solar system planets and the Sun. For all planets not shown, no collision was observed in our simulations with the 240 realizations.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rein, H.; Tamayo, D.; Vokrouhlický, D.
The Random Walk of Cars and Their Collision Probabilities with Planets. *Aerospace* **2018**, *5*, 57.
https://doi.org/10.3390/aerospace5020057

**AMA Style**

Rein H, Tamayo D, Vokrouhlický D.
The Random Walk of Cars and Their Collision Probabilities with Planets. *Aerospace*. 2018; 5(2):57.
https://doi.org/10.3390/aerospace5020057

**Chicago/Turabian Style**

Rein, Hanno, Daniel Tamayo, and David Vokrouhlický.
2018. "The Random Walk of Cars and Their Collision Probabilities with Planets" *Aerospace* 5, no. 2: 57.
https://doi.org/10.3390/aerospace5020057