# Asteroids and Their Mathematical Methods

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Classification and Exploration of Small Celestial Bodies

#### 2.1. Overview of Small Celestial Bodies

#### 2.2. Exploration of Small Celestial Bodies

## 3. Research on Small Celestial Body Gravitational Field Environments and Orbital Mechanics

#### 3.1. Research on the Gravitational Field Model of Small Celestial Bodies

_{1}> R

_{2}. Yu [96] gave the gravitational radii of 23 small celestial bodies in his doctoral dissertation. Table 4 lists the radius range of the small celestial body’s gravitational action, calculated according to Formulas (2) and (3). It is not difficult to see that the radius ${R}_{1}$ of the gravitational influence sphere is about two orders of magnitude larger than the radius ${R}_{2}$ at the gravitational neutralization point.

^{−11}m

^{3}·kg

^{−1}·s

^{−2}; $r$ is the position vector of the particle; $r,\varphi ,\lambda $ are the three components of the vector in spherical coordinates; M

_{A}is the mass of the small celestial body; P

_{lm}is the associative Legendre polynomial; r

_{e}is the radius of the Brillouin sphere, reflecting the range of convergence of the series, that is, the applicable range of Formula (4); and C

_{lm}and S

_{lm}are spherical harmonic coefficients, reflecting the shape irregularity and inhomogeneity of internal mass distribution [101]. The advantage of this method is that the gravitational potential can be given analytically. It is convenient for obtaining theoretical solutions through analytical methods. In addition, once the spherical harmonic function coefficient is obtained, it can be directly substituted in the subsequent numerical calculation, which is convenient to use, especially for inversion calculation through flight data [102]. During the orbiting of Ceres by the Dawn probe, Takahashi et al. [103,104] used the spherical harmonic model to estimate the precise gravitational field of Ceres and iteratively iterated the known spherical harmonics to give the direction of its principal axis. The main limitations are that the model cannot be applied to the region located within the Brillouin sphere. The reason for this is that the series does not converge [105]. The truncation error in calculations may lead to large errors in the obtained gravitational field model in some cases [106].

_{1}, λ

_{2}, λ

_{3}are the ellipsoid coordinate components of the vector $r$. λ

_{e}is the parameter of the Brillouin ellipsoid, which reflects the range of convergence of the series, that is, the range of use of Formula (5). F

_{lm}is the Lamé equation canonical solution, and α

_{lm}is the ellipsoid harmonic coefficient [105]. A conversion method between spherical harmonics and ellipsoidal harmonics was proposed by Dechambre et al. [109] to simplify the solution process. The ellipsoid harmonic function model expands the convergence region of small celestial bodies while still retaining the characteristics of the spherical harmonic function model for easy calculation [110].

_{i}. Then, the gravitational potential can be expressed as

_{3}and J

_{33}terms in the spherical harmonic model. Then, Werner et al. [117] sorted out the previous work, taking (4769) Castalia as an example, and introduced the modeling method of the polyhedral gravitational field in detail. Mirtich [118] also applied Gauss’s theorem and Green’s formula to replace the integral by summation and calculated the center of mass, the moment of inertia, the product of inertia, and other physical quantities of a homogeneous polyhedron.

_{e}, E

_{e}, F

_{f}are the quantities related to the edge and the side, and θ

_{f}is the solid angle formed by the points at the side f and r. Its specific calculation formula is

#### 3.2. Research on Orbital Dynamics near Small Celestial Bodies

_{20}and C

_{30}terms on the frozen orbits. Antreasian et al. [135] and Scheeres et al. [136] successively used the second-order quadratic gravitational field model and the average method to analyze the motions near (433) Eros and found a family of retrograde periodic orbits, which were used for the Shoemaker mission. Shang et al. [137] investigated various periodic orbits near non-principal-axis rotation asteroids.

_{20}and C

_{22}terms on the energy and angular momentum of the particle motion and numerically calculated the stable and unstable orbital regions in the parameter space. They further studied the orbital dynamics considering the non-spherical gravity of small celestial bodies, solar radiation pressure, and solar gravity and used the averaging method to find the frozen orbits near small celestial bodies. The property is closely related to the area-to-mass ratio of the spacecraft and the distance from the small celestial body to the Sun. Because the search for periodic orbits is very complicated, it is generally necessary to use symmetry for analysis and research, but the gravitational field of small irregular celestial bodies does not have this feature [128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143]. Yu et al. [144] proposed a global search method for 3D periodic orbit families in the vicinity of irregular small bodies by using the polyhedral model and the hierarchical grid method. The 29 periodic orbits are given by taking asteroid (216) Kleopatra as an example. The periodic orbits are classified into seven types according to the orbits of the four-dimensional symplectic manifold by calculating the eigenvalues of the monodromy matrix of periodic orbits [145]. Topological types, the bifurcation phenomenon, and the stability of periodic orbits with the continuation of energy are studied. Jiang et al. pointed out that periodic orbits move in a six-dimensional symplectic manifold and that its manifold structure is different from that of the four-dimensional case and re-classified the periodic orbits near small irregular celestial bodies into 13 topological types [132]. Applying this theory, Yu et al. [146] found periodic orbit families belonging to different topological types near (243) Ida and the bifurcation behavior in the continuation process in the periodical orbit search and continuation near (243) Ida. Jiang’s theory provides a powerful tool for follow-up research to better understand the type and stability of periodic orbits near irregular small celestial bodies from the topological structure. Non-equatorial equilibrium points near an asteroid with gravitational orbit-attitude coupling perturbation were analyzed in reference [147]. Li et al. [148] calculated the geophysical environments and periodic orbits near 2016 HO3 by using different shape models.

#### 3.3. Research on Surface Motion Dynamics and the Capillary Phenomenon of Small Celestial Bodies

#### 3.4. Dynamic Characteristics under Varying Parameters

^{2}[204], and the rotation rate of the small celestial body (1620) Geographos can be accelerated by $1.15\times {10}^{-8}$ rad/d

^{2}[205]. Numerical experiments show that the YORP effect can lead to the disintegration of rubble-like celestial bodies and the formation of small moons. The YORP effect may also indirectly affect the distribution and topological characteristics of the relative equilibrium points in the gravitational field. The shapes of asteroids may also be deformed as landslides and mass shedding occur. Similar to Comet Shoemaker-Levy 9, small rubble-like bodies can change their topography dramatically as they approach a planet, and some even disintegrate. Holsapple et al. [206,207] established the mechanical mechanism of tidal deformation caused by the influence of nearby larger celestial bodies. The variations in the shape, spin, and state during the slowly increasing angular momentum of rubble-pile, self-gravitating, homogeneous ellipsoidal bodies were investigated in reference [208,209]. Zhang et al. [210] found that three typical tidal response outcomes may appear on rubble piles, namely, deformation, scattering, and destruction. During the long-term evolution of small celestial bodies, their density, rotational speed, shape, and internal structure may change. The disintegration and gravitational aggregation of asteroids will also affect their internal structure, average density, and shape. These factors lead to changes in the parameters of small celestial bodies. If the parameters of the primary in binary asteroids with a large-scale ratio vary, it is bound to have an impact on the movement of the small moon. In addition, changes in parameters will also affect the movement of dust, particles, and gravel in the gravitational field of small celestial bodies.

## 4. Summary and Future Development

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The geometric shape of (101955) Bennu based on the polyhedron model [119].

Type | Semi-Major Axis a | Perihelion Distance q | Aphelion Distance Q |
---|---|---|---|

Atira | a < 1 AU | - | Q < 0.983 AU |

Aten | a < 1 AU | - | Q > 0.983 AU |

Apollo | a > 1 AU | q < 1.017 AU | - |

Amor | a > 1 AU | 1.017 AU < q < 1.3 AU | - |

Group | Type | Criterion | Representative |
---|---|---|---|

C | B | The general properties are the same as the C type, but the ultraviolet absorption below 0.5 μm is smaller, and the slight blueness is more obvious than the redness in the spectrum. The albedo also tends to be greater than the darker C-type. | (2) Pallas |

C | There is moderate absorption at UV wavelengths of 0.4–0.5 μm, and there are no obvious features but slight reddening at longer wavelengths. There is a mineral feature indicative of hydration known as water absorption around the wavelength of 3 μm. | (10) Hygiea | |

F | Similar to B-type asteroids but lacks water absorption features indicative of hydrated minerals around wavelengths around 3 μm and differs from the B-type in the low-wavelength UV portion below 0.4 μm. | (704) Interamnia | |

G | Similar to C-type asteroids but has strong absorption characteristics for ultraviolet wavelengths below 0.5 μm. There may also be absorption properties around 0.7 μm, implying the presence of layered silicate minerals such as clay and mica. | (1) Ceres | |

S | A | Significant olivine features at a 1 μm wavelength and strong reddening at wavelengths below 0.7 μm. | (446) Aeternitas |

K | There is moderate reddening at wavelengths below 0.75 μm and slight bluing at wavelengths above 0.75 μm. | (221) Eos | |

L | There is strong reddening at wavelengths below 0.75 μm, and the spectrum is flat at wavelengths above 0.75 μm. Compared with the K type, the redness is more obvious in the visible band, and the spectrum in the infrared band is more gentle. | (83) Beatrix | |

Q | There are prominent features of olivine and pyroxene in the 1 μm band, and their spectral changes indicate the possible presence of metallic substances. There is an absorption spectrum at 0.7 μm. | (1862) Apollo | |

R | There are distinct olivine and pyroxene features at 1 μm and 2 μm. The spectrum is strongly reddened at wavelengths below 0.7 μm. | (349) Dembowska | |

S | There is moderate spectral variation at wavelengths shorter than 0.7 μm and moderate spectral absorption at 1 μm and 2 μm wavelengths. There is also a shallow but broad spectral absorption around 0.63 µm. | (3) Juno | |

X | E | The albedo is greater than 0.3, the spectrum is flat and reddish, and there are no obvious features. | (44) Nysa |

M | The albedo is between 0.1 and 0.2, there are subtle spectral absorption lines in the bands above 0.75 μm and below 0.55 μm, and the overall spectrum is flat and slightly reddened, lacking obvious features. | (16) Psyche | |

P | The albedo is less than 0.1, and the color is redder than that of the S-type asteroid, but it is not reflected in the spectral properties. | Sylvia | |

Not grouped | D | Very low albedo and featureless, light red electromagnetic spectrum. | (624) Hektor |

O | Strong spectral absorption in the band above 0.75 μm | (3628) Božněmcová | |

T | The spectrum is moderately reddened, darker, and has moderate spectral absorption in the band below 0.85 μm. | (114) Kassandra | |

V | There is strong spectral absorption in the bands above 0.75 μm and 1 μm and strong reddening in the bands below 0.7 μm. | (4) Vesta |

Spacecraft | Agency | Start Date | Asteroid | Mission Type |
---|---|---|---|---|

International Comet Explorer | NASA ESA | 1982 | 21P/Giacobini-Zinner | Fly-by |

Vega 1/2 | IKI | 1984 | 1P/Halley | Fly-by |

Pioneer/Comet | JAXA | 1985 | 1P/Halley | Fly-by |

Giotto | ESA | 1985 | 1P/Halley 26P/Grigg–Skjellerup | Fly-by |

Galileo | NASA | 1989 | (951) Gaspra (243) Ida | Fly-by |

Near-Shoemaker | NASA | 1996 | (253) Mathilde (433) Eros | Fly-by/Orbiting/Landing |

Cassini-Huygens | NASA | 1997 | (2685) Masursky | Fly-by |

Deep Space 1 | NASA | 1998 | (9969) Braille (19P/Borrelly) | Fly-by/Orbiting |

Stardust | NASA | 1999 | (5535) Annefrank 81P/Wild 2 9P/Tempel 1 | Fly-by |

Comet Nucleus Tourer (Failed) | NASA | 2002 | 2P/Encke 73P/Schwassmann-Wachmann 3 6P/d’Arrest | Fly-by |

Hayabusa | JAXA | 2003 | (25143) Itokawa | Orbiting/Landing/Sample return |

Rosetta | ESA | 2004 | (2867) Steins (21) Lutetia 67P/Churyumov-Gerasimenko | Orbiting/Landing/Sample return |

Deep Impact/EPOXI | NASA | 2005 | 9P/Tempel 1 103P/Hartley 2 | Impact/ Fly-by |

New Horizons | NASA | 2006 | 132524 APL (134340) Pluto 2014 MU69 | Fly-by |

Dawn | NASA | 2007 | (4) Vesta (1) Ceres | Orbiting |

Chang’e 2 | CNSA | 2010 | (4179) Toutatis | Fly-by |

Hayabusa 2 | JAXA | 2014 | (162173) Ryugu | Orbiting/Landing/Sample return |

OSIRIS-REx | NASA | 2016 | (101955) Bennu | Orbiting/Landing/Sample return |

Don Quixote (in progress) | ESA | - | 2003 SM84 | Fly-by/Impact |

Double Asteroid Redirection Test (DART) | NASA | 2021 | (65803) Didymos | Impact |

Lucy | NASA | 2021 | 15094 Polymele 21900 Orus | Fly-by |

Psyche mission | NASA | 2022 | (16) Psyche | Orbiting |

Tianwen 2 (in progress) | CNSA | 2025 | (469219) Kamo‘oalewa (2016 HO3) 311P/PanSTARRS | Orbiting/Landing/Sample return Orbiting |

Asteroids | MA/MS | D/AU | R_{1}/km | R_{2}/km |
---|---|---|---|---|

(216) Kleopatra | 2.33 × 10^{−12} | [2.09, 3.49] | [6.97 × 10^{3}, 1.16 × 10^{4}] | [4.78 × 10^{2}, 7.99 × 10^{2}] |

(243) Ida | 2.11 × 10^{−14} | [2.74, 2.98] | [1.39 × 10^{3}, 1.51 × 10^{3}] | [5.97 × 10, 6.50 × 10] |

(433) Eros | 3.36 × 10^{−15} | [1.13, 1.78] | [2.75 × 10^{2}, 4.34 × 10^{2}] | [9.83 × 10^{0}, 1.55 × 10] |

(1620) Geographos | 1.30 × 10^{−17} | [0.83, 1.66] | [2.19 × 10, 4.38 × 10] | [4.49 × 10^{−1}, 8.98 × 10^{−1}] |

(6489) Golevka | 1.06 × 10^{−19} | [0.99, 4.02] | [3.81 × 10^{0}, 1.55 × 10] | [4.83 × 10^{−2}, 1.96 × 10^{−1}] |

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**MDPI and ACS Style**

Jiang, Y.; Ni, Y.; Baoyin, H.; Li, J.; Liu, Y.
Asteroids and Their Mathematical Methods. *Mathematics* **2022**, *10*, 2897.
https://doi.org/10.3390/math10162897

**AMA Style**

Jiang Y, Ni Y, Baoyin H, Li J, Liu Y.
Asteroids and Their Mathematical Methods. *Mathematics*. 2022; 10(16):2897.
https://doi.org/10.3390/math10162897

**Chicago/Turabian Style**

Jiang, Yu, Yanshuo Ni, Hexi Baoyin, Junfeng Li, and Yongjie Liu.
2022. "Asteroids and Their Mathematical Methods" *Mathematics* 10, no. 16: 2897.
https://doi.org/10.3390/math10162897