^{*}

Edited by: Lorenzo Iorio, Ministry of Education, Universities and Research, Italy

Reviewed by: Áron László Süli, Eötvös Loránd University, Hungary; Bojan Novakovic, University of Belgrade, Serbia

This article was submitted to Fundamental Astronomy, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this work we describe a genetic algorithm which is used in order to study orbits of minor bodies in the frames of close encounters. We find that the algorithm in combination with standard orbital numerical integrators can be used as a good proxy for finding typical orbits of minor bodies in close encounters with planets and even their moons, saving a lot of computational time compared to long-term orbital numerical integrations. Here, we study close encounters of Centaurs with Callisto and Ganymede in particular. We also perform n-body numerical simulations for comparison. We find typical impact velocities to be between υ_{rel} = 20[υ_{esc}] and υ_{rel} = 30[υ_{esc}] for Ganymede and between υ_{rel} = 25[υ_{esc}] and υ_{rel} = 35[υ_{esc}] for Callisto.

Jupiter's large icy moons such as Ganymede and Callisto show countless impact craters across their surface. Studying these craters gives deep insights into the impactors as well as the moons themselves. This is the first approach in the frame of future works of studying collisions with the outermost moon Callisto. We are especially interested in the Valhalla crater system, Callisto's biggest crater. This impact structure measures several hundred of kilometers in diameter and shows some extraordinary features such as an extensive ring system in the outskirts of the crater (Greeley et al.,

In this first work we focus on the use of a genetic algorithm^{1}^{2}^{7} (Volk and Malhotra, ^{9} (Di Sisto and Brunini,

Measuring close encounters or even collisions between minor and major bodies in the context of n-body simulations is computationally demanding. Typically one has to constrain the parameter space of the minor bodies to selected regions within the Solar System (e.g., Kuiper belt objects or specific families of objects). We use a genetic algorithm to find asteroids of the Centaur type family which are likely to have a close encounter with the Jovian moons Ganymede and Callisto. Centaurs mainly origin from TNOs and in particular from the Kuiper Belt (Galiazzo et al.,

Parameter space for random initial conditions of the test particles.

4.95 | 30.33 | |

0 | 0.99 | |

0 | 180 | |

ω (deg) | 0 | 360 |

Ω (deg) | 0 | 360 |

0 | 360 |

A genetic algorithm (Turing,

Genetic n-body algorithm.

We use the code to find objects which are likely to collide with either Ganymede or Callisto within a certain time interval. The following list gives an overview how the GA and the n-body code are associated with each other. In analogy to section 2.1, we link the terms between the genetic algorithm and the actual problem:

Population: Centaur type asteroids with random initial conditions

DNA: initial Keplerian orbital elements (

Generation: numerical simulation of the system via an n-body method

Evolution: iterative process of consecutive simulations

Fitness: score of each test particle at each simulation given by the fitness function

Crossover: combination of initial orbital elements of parent particles given by the crossover function

Child: a test particle with a new set of initial orbital elements

Mutation: small, random variation of initial orbital elements

The so-called fitness function is the function to be optimized. We use a fitness function of _{rel} being the minimum relative distance between a particle and the corresponding moon during each generation. The squared distance was found to be useful if the fitness is used as a probability distribution to sample quite fit parents for crossover, ensuring a high fraction of fit parents and thus increasing the overall performance of the GA. Note that one can use other fitness functions to study completely different problems. One only has to find a quantity to score objects which show a specific behavior to obtain a population which develops that behavior. The fitness of each body is measured throughout the simulation. If a close approach with the moon occurs, a measurement is taken and the algorithm starts a completely new evolution process. If no close approach happens during the simulation, unfit objects are replaced by new children which are created by objects with a high fitness score. The initial orbital elements _{e} = ζ ·σ_{e}). The initial mutation scaling is set to ζ = 0.1 for all evolutions. At this point it should be noted that the functions for fitness, crossover and mutation are empirical functions found to yield good results (high performance due to learning curves with steep slopes) for this specific problem. Mathematical formulations of the functions we use for fitness, crossover and mutation are given in the Appendix.

The size of the population is _{pop} = 30 and the total number of bodies in each simulation is _{tot} = 38, including the Sun, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. The mass of Mercury was added to the Sun. The orbital elements for the massive bodies are obtained from the JPL HORIZONS system. We perform the simulations in the Jovian-centric system, as we expect only negligible changes of the results because the systematic error produced by the GA is significantly larger than the error made by not using a barycentric frame of reference. We use the Lie-Series n-body integrator as described in Hanslmeier and Dvorak (^{−11}. Note that we do not explicitly include Ganymede and Callisto in the simulations due to a performance gain. In contrast to symplectic n-body integrators, the stepsize of the Lie-integrator is limited by the minimum relative distance between any of the objects,
_{G} is the Gaussian gravitational constant and ε is the numerical accuracy. The quantities ν and

We set the maximum simulation time _{max} = 165yr in order to allow at least a full orbital period for each individual. The maximum number of generations _{max} = 250. A new iteration is started either if there is no measurement (i.e., close encounter with either Ganymede or Callisto) within these _{max} iterations or as soon as such a close encounter is recorded. Finally, for a quick check of the results with the GA method, we also check its output with the ones from full integrations of Centaurs (through all their lifetime) taking data kindly provided by Galiazzo et al. (

After about 3300 CPU hours, total number of 531 and 625 measurements was obtained for Ganymede and Callisto, respectively. Each measurement corresponds to an individual evolution process of the GA and represents an individual close encounter scenario which contains all relevant information about positions, velocities, orbital elements, etc. about the bodies at both the time of closest approach and at the beginning of the respective simulation.

Figure

The low-velocity class (

The intermediate-velocity class A (

The intermediate-velocity class B (

The high-velocity class (

Left: Ganymede, Right: Callisto. The histograms show the relative velocities between the test particles and the corresponding moon at closest approach in units of the escape velocity. The four colorized components represent the possible geometrical encounter scenarios. For example, the orange component

The terms

Mean and standard deviations of close encounter velocities for the four classes, as well as their numbercounts.

_{esc}) |
_{esc}) |
||||
---|---|---|---|---|---|

Class 1, |
15.03 | 4.55 | 15.36 | 5.17 | 78 |

Class 2, |
24.38 | 3.09 | 26.00 | 5.33 | 110 |

Class 3, |
22.39 | 5.59 | 23.45 | 6.68 | 150 |

Class 4, |
39.29 | 3.84 | 40.65 | 8.48 | 193 |

Class 1, |
11.63 | 3.40 | 18.44 | 5.96 | 111 |

Class 2, |
19.41 | 3.57 | 29.53 | 7.91 | 156 |

Class 3, |
23.85 | 3.59 | 36.84 | 8.40 | 147 |

Class 4, |
34.96 | 3.12 | 52.86 | 10.45 | 211 |

The most probable close encounter velocities can be seen between _{rel} = 20[_{esc}] and _{rel} = 30[_{esc}] for Ganymede and between _{rel} = 25[_{esc}] and _{rel} = 35[_{esc}] for Callisto.

Further observations can be obtained from the results:

The overall form of the relative velocity histograms can be reproduced by overlapping Gaussian distributions which are represented by the four classes.

The classes are overlapping stronger for Ganymede than for Callisto, even swapping places (in velocity) when comparing class 2 and class 3 for Ganymede.

There is a clear trend favouring retrograde encounters (for both Jupiter and the respective moon), with most close encounters being

The first row in Figure

Left column: Ganymede, right column: Callisto. The top row shows the semi-major axes at closest approach. The middle row shows the initial eccentricity of the last generation vs. the eccentricity at the measurement. The bottom row shows the respective inclinations. Each datapoint corresponds to an independent evolution.

Several other obervations can be obtained from the datasets:

The needed number of generations for taking the measurement grows steadily until about

The most dramatic changes of initial orbits happen during the first few tens of generations.

It is easier for the GA to find intersection orbits after a short simulation time. Therefore, the time of measurement peaks toward low values with a mean simulation time of _{mean} = 45.5yr.

The intersection probability is approximately twice as high for Callisto because her Hill radius is larger than Ganymede's. For statistical reasons, we therefore assigned more computation time to Ganymede.

Two actual collisions are measured for Ganymede (with impact velocities of _{rel} = 34.6 km/s and _{rel} = 42.2 km/s, respectively), none for Callisto.

We take the orbital evolution of Centaurs to do a comparison between the predicted Centaurs' orbits at close encounters and the Jovian moons. We integrate forward a sub-sample of the Centaurs for 30 Myrs and check all the close encounters with Jupiter, using the Lie-integrator. This study considers only close encounters up to a distance of ^{3}

We take 319 Centaurs with 15 clones in each interval ranging over 5AU in semi-major axis (for a total of 5104 bodies)^{4}

From the evolution of 5104 objects, a total number of 292 measurements was obtained for Ganymede close encounters. From our sample of Centaurs we find that ~ 22.6% can have close encounters with Jupiter. As the percentage of Centaurs which can cross Ganymede orbits is about 20.1% (8.7% with

Comparison between measurements of GA and Lie integrations of evoluted orbits for Ganymede close encounters. For the GA, all measurements are taken within Ganymede's Hill sphere. For the Lie integrations, the measurements are taken as described in the text.

Figure

The velocities of the four classes as described in section 3.1 can be explained by geometrical considerations: Class 1 encounters happen if the particles experience deceleration from the Jupiter flyby and additionally “lose” relative velocity due to the parallel direction of flight compared to the respective moon. The intermediate classes 2 and 3 either experience deceleration or acceleration from the Jupiter flyby and additionally “gain” or “lose” relative velocity due to the antiparallel or parallel direction of flight compared to the respective moon. Both acceleration by Jupiter and a “gain” of relative velocity is true for class 4.

Comparing the numbercounts of the classes, a correlation exists with higher fractions of both types of retrograde encounters being more probable. The higher fractions of retrograde encounters can be described by a simple geometrical effect: While a particle moves within the torus-shaped Hill region (the complete area which is accessed by the moon's Hill sphere during a full orbital period), the probability for a retrograde measurement is much higher than a prograde one since the typical relative velocity between the particle and the moon is high. This finding supports the observation of the heavily cratered front-side of the rotationally bound moons. For example, many large crater systems on Callisto (namely Valhalla, Asgard, Adlinda, Utgard, etc.) are located at the front-side.

Comparing class 2 and class 3, it seems that the influence caused by the flyby at Jupiter and the parallel or antiparallel direction of flight are similar in strength, producing similar numbercounts for these classes. More discreet or weaker effects like the selection of the parameter space and the behavior of the GA may influence the shape of the curves as well as the numbercounts for the classes. However, the overall shape of the individual components can be interpreted as Gaussian distributions, covering a large interval for possible close encounter velocities.

Recalling the distributions of semi-major axes in Figure _{max} = 165yr indicates that these drastic changes are caused by a single or only a handful of close encounters with Jupiter. The stronger scattering toward higher values of the semi-major axes and eccentricities for Ganymede is intuitive, because particles experience a stronger acceleration by Jupiter during the flyby. The data implies that more than half of all particles even undergo the transition from eccentric to hyperbolic trajectories during the simulations (taking into account we only considered elliptic orbits initially). The comparison between the final semi-major axes indicates that the large peak (at Callisto) vanishes at closer distances to Jupiter (toward Ganymede) and gets scattered over a wider range at typically higher values.

For the inclination, the GA has a selection effect for the sector

The comparison with the numerical integations (see Figure _{max} = 165yr, which do not allow for modeling the long-term behavior of chaotic orbits. However, the GA still yields a good coverage of parameter space, even with this short simulation time. Interestingly, the GA also finds orbits which lie far beyond the initial parameter space of semi-major axes given in Table

For future work with the GA we may include the simulation time and further properties of the orbits in the fitness function in order to avoid too short simulations and make high inclinations less likely, for example. We may also include the TNO region as an important extension to the parameter space.

The GA can easily be adapted in order to efficiently measure actual collisions with any given object. In this work we measure only two collisions with Ganymede because as soon as the first particle has its closest approach within the Hill sphere, a completely new evolution process is started.

In summary, the results from the GA are not fully consistent compared to the classical approach because the underlying principles are different. GAs in general tend to find all possible solutions to a given problem rather than only the realistic, physical ones. However, in our implementation this effect can be overcome by optimizing the GA either to avoid clearly non-physical solutions or to enhance realistic solutions by increasing the simulation time, enlarging the parameter space and refining the functions for fitness, crossover and mutation.

The comparison with the Lie integrations reveal that the genetic n-body algorithm yield both a high number of physical as well as non-physical results. For example, an unrealistic high number of retrograde orbits are found while using a realistic probability distribution for the random initial inclinations. This is clearly a selection effect of the GA, as it finds that the probability for measuring retrograde encounters is significantly higher than for prograde ones. Several reasons such as a too short simulation time, too powerful crossover- and mutation functions or the choice of hyperparameters (such as the population size, number of generations, etc.) can be responsible for causing this high fraction non-physical results. However, it is expected that this high fraction can be significantly reduced by applying one or more of the following improvements:

A higher simulation time for each generation enables more dynamical effects in general. Therefore, the GA will also tend to produce more physically motivated results.

The fitness function can be refined to avoid solutions which are clearly non-physical. For example, one may introduce additional terms which depend on the inclination, the time of measurement, etc.

Since crossover tends to find orbits within the initial random parameter space, this parameter space should be large enough, e.g., 50% larger compared to the parameter space of interest. Regions outside the inital random parameter space are only accessible via mutation.

We found the mutation function to have a significant effect on the behavior of the GA's learning curve (generation vs. mean fitness). An optimized mutation function can boost the overall performance of the GA drastically, enabling the use of a higher simulation time, a larger population, obtaining more measurements, etc. with the same computation resources.

A larger population is able to cover the parameter space more homogeneously and may reveal further close encounter families with low probabilities.

Like in this work, the non-physical results can be efficiently filtered by comparing the results with classical approaches.

However, the GA also yields useable results, especially on the small-scale close encounter dynamics. We find it to be an efficient tool to get a rough idea of the underlying dynamics of the problem and the expected families of solutions before investigating into more detailed analysis with classical approaches. The GA supports the use of existing approaches rather than replacing them. In this work, the GA efficiently finds all possible close encounter geometries even beyond the initial parameter space with low computational effort. The measurements cover all major areas of the parameter spaces in semi-major axis, eccentricity and inclination. Even a weak correlation in the density distributions for the

Apart from the behavior of the GA itself, relevant information is obtained from the measurements:

The four classes, which are motivated by geometrical considerations, can be distinguished well in the datasets. The classes allow for a more detailed and structured way for analysing close encounter events in the Jovian System.

One may distinguish between different impact scenarios depending on the impact velocity. For example, there is no need for analysing retrograde collisions if the particle is classified as class 1 and vice-versa for prograde collisions and class 4.

There are significantly more retrograde than prograde encounters for both moons. This fact is supported by the heavily cratered front-side of the rotationally bound moons.

As shown in Figure

Moreover, the distributions of semi-major axes reveal a double-peak structure for Ganymede in contrast to a single-peak structure for Callisto. This can be explained by a stronger scattering of the classes

PW wrote the code for the genetic algorithm and the transformations for the osculating elements. He wrote about 70% of the paper, especially the part inherent to the genetic algorithm, the core of this paper. MG has contributed performing the full orbital numerical part section, selecting the initial populations for this study, he suggested how to consider the close encounters with the Jupiter moons and he edited several parts of the paper for quality improvements in any section of the paper, apart the description of the genetic algorithm. He wrote about 15% of the paper. TM helped in editing the quality of the paper and wrote about 15% of the paper. He also helped in the description of the genetic algorithm and performing the Kolmogorov-Smirnov test.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors wish to thank the three referees for their comments which helped to significantly improve the manuscript. The authors acknowledge support by the FWF Austrian Science Fund projects S11603-N16 (PW and TM) and P23810-N16 (MG), respectively.

The Supplementary Material for this article can be found online at:

^{1}Herein “GA”.

^{2}Data taken from JPL Small-Body Database Search Engine

^{3}We assume the semi-major axes of the moons as a proxy for the distance, neglecting their small eccetricities.

^{4}The first region is the one with a semi-major axis between 5 and 10 AU. The second region is between 10 and 15 AU and congruently for the other regions up to 30 AU.