# Capture in Regime of a Trapped Motion with Further Inelastic Collision for Finite-Sized Asteroid in ER3BP

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{Sun}and m

_{p}(m

_{p}< M

_{Sun}) revolve around their barycenter on Keplerian orbits with low eccentricities. A smaller body (projectile for attacking a large asteroid) is supposed to be a solid, almost symmetric ellipsoid, having the gravitational potential of the MacCullagh type. Our aim is to develop a previously introduced solving procedure and to investigate the updated dynamics of the projectile captured to a trapped dynamical resonance, thereby having the inelastic collision of a small projectile orbiting on quasi-stable elliptic orbits around the large asteroid, m

_{p}.

## 1. Introduction

_{Sun}and large asteroid m

_{p}(or minor planet)—in such a way that small asteroid m will be captured into further inelastic collision with the bigger ones, m

_{p}, both moving on quasi-stable elliptic orbits.

_{Sun}and m

_{p}, m

_{p}< M

_{Sun}, which are orbiting around their barycenter in mutual Keplerian motion) with the help of clear mathematical formulae by the system of two coupled ordinary differential equations of the second order. We restrict our current study by using the final form of the approximated scaled equations derived in work [43].

## 2. Convenient form of Equations of Motion for Further Solving Procedure

## 3. Graphical Plots for Numerical Solutions

- First, we choose from the very beginning the set of initial data with zero values;
- Then, we initiate the numerical solving of system (1) using the Runge–Kutta fourth-order method with step 0.001 starting from the abovementioned set of initial conditions;
- In the case where we obtain the required positive result (which is associated with recognizing the dynamical features of the projectile’s trapped motion when approaching the large asteroid during such celestial motion in ER3PB), the algorithm will be stopped;
- In the case where we obtain a negative result, initial data should be shifted by step (+/−) 0.001 with respect to the absolute magnitudes of the initial conditions. Hereafter, the algorithm should work during the next cycles of numerical calculating up to reaching the positive result.

_{0}= −1, ${(\dot{x})}_{0}$ = −0.53, y

_{0}= 0.167, ${(\dot{y})}_{0}$ = −0.57.

## 4. Discussion and Conclusions

_{Sun}and m

_{p}(m

_{p}< M

_{Sun}), are revolving around their barycenter on Kepler orbits with low eccentricities. A smaller body (a projectile attacking a large asteroid) is supposed to be a solid, almost symmetric ellipsoid, having the gravitational potential of the MacCullagh type.

_{p}.

_{p}or circa 6.4 km, in the case of Ceres) if we wish to exclude the direct impact of the projectile to the minor primary’s surface. In any case, our calculations are valid for a restricted part of the projectile’s trajectory, where such a small mass m (projectile) will be moving inside the sphere of effective attraction, approaching the minor primary m

_{p}inside the so-called Hill sphere [6] (but beyond Hill spheres of other planets of the Solar system):

_{p}(1 − e²)} = 1. Thus, we have for the case of Ceres, ${r}_{H}\cong {\left(\frac{\mu}{3}\right)}^{\frac{1}{3}},\mu \cong 4.7\xb7{10}^{-10}\to {r}_{2}<{r}_{H}\cong 5.38\xb7{10}^{-4}$; this means that in the case of Ceres, the distance from the small asteroid to this minor planet should be less than ${r}_{2}<{r}_{H}\cong 5.38\xb7{10}^{-4}(2.7653\xb71a.e.)$ ≅ 222.6⋅10

^{3}km. Taking into account that the average velocity of a small asteroid (projectile) is circa 12–15 km/s, this means that such a projectile will be moving more than 4 h in the Hill sphere of a minor planet before reaching the minor primary (as the final target of the projectile’s trajectory). Despite the fact that the aforementioned distance within the Hill sphere is located sufficiently close to the secondary planet (minor planet), it is very important to describe correctly the final approaching motion of the small asteroid (projectile) to the target minor planet or large asteroid using the approximation of ER3BP (elliptic restricted three-body problem).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

H | absolute magnitude, a measure of the luminosity of asteroid on an inverse logarithmic magnitudes scale, dimensionless |

p | geometric albedo, dimensionless |

m | mass of finite-sized small asteroid or projectile (of ellipsoidal form), kg |

m_{p} | mass of large asteroid or minor planet (here, Ceres), kg |

M_{Sun} | mass of Sun, kg |

$\overrightarrow{r}$ = {x, y} | coordinates of the scaled, pulsating, planar coordinate system, dimensionless |

e | eccentricity, dimensionless |

f | the true anomaly (in radians), rad |

h | the extent of deviation of projectile from symmetrical form of ellipsoid of rotation, dimensionless |

r_{1} | distance of projectile m from M_{Sun}, dimensionless |

r_{2} | distance of projectile m from Ceres, dimensionless |

R_{p} | radius of minor planet or large asteroid (here, Ceres), dimensionless |

a_{p} | semimajor axis of elliptic orbits of the rotating primaries around their barycenter, dimensionless (here, {a_{p} (1 − e^{2})} = 1) |

r_{H} | radius of Hill sphere for large asteroid (here, Ceres), dimensionless |

Greek symbols | |

μ | the ratio of the mass (mass-parameter), dimensionless |

Subscripts | |

1, 2 | components of distances of projectile from each of primaries with mass M_{Sun} and m_{planet}, accordingly |

## Appendix A1. Estimation of Absolute Magnitudes for Parameter h

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**Figure 1.**Numerical solution for coordinate x(f) using Equation (1) (parameter h = $4\xb7{10}^{-16}$ as previously shown in [43]).

**Figure 2.**Numerical solution for coordinate y(f) using Equation (1) (parameter h = $4\xb7{10}^{-16}$ as previously shown in [43]).

**Figure 3.**Numerical solution for distance r

_{1}(f) using Equation (1) of projectile m from Sun (parameter h = $4\xb7{10}^{-16}$ as previously shown in [43]), ${r}_{1}^{2}={(x-\mu )}^{2}+{y}^{2}$.

**Figure 4.**Numerical solution for distance r

_{2}(f) using Equation (1) of projectile m from Ceres (parameter h = $4\xb7{10}^{-16}$ as previously shown in [43]), ${r}_{2}^{2}={(x-\mu +1)}^{2}+{y}^{2}$.

**Figure 8.**Numerical solution for distance r

_{2}(f) using Equation (1), min. distance of close approach to Ceres, r

_{2}min ≅ 0.00011 (at f = 2.343).

Name | Approx. Mass (×10^{18} kg) | Approx. Proportion of All Asteroids |
---|---|---|

1. Ceres | 938.4 | 31% |

2. Vesta | 259.1 | 8.6% |

3. Pallas | 204 | 6.7% |

4. Hygiea | 87 | 3.7% |

5. Interamnia | 35 | 1.3% |

6. Eunomia | 30 | 1.1% |

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**MDPI and ACS Style**

Ershkov, S.; Leshchenko, D.; Rachinskaya, A.
Capture in Regime of a Trapped Motion with Further Inelastic Collision for *Finite-Sized* Asteroid in ER3BP. *Symmetry* **2022**, *14*, 1548.
https://doi.org/10.3390/sym14081548

**AMA Style**

Ershkov S, Leshchenko D, Rachinskaya A.
Capture in Regime of a Trapped Motion with Further Inelastic Collision for *Finite-Sized* Asteroid in ER3BP. *Symmetry*. 2022; 14(8):1548.
https://doi.org/10.3390/sym14081548

**Chicago/Turabian Style**

Ershkov, Sergey, Dmytro Leshchenko, and Alla Rachinskaya.
2022. "Capture in Regime of a Trapped Motion with Further Inelastic Collision for *Finite-Sized* Asteroid in ER3BP" *Symmetry* 14, no. 8: 1548.
https://doi.org/10.3390/sym14081548