Dynamical configurations of celestial systems comprised of multiple irregular bodies

Abstract

This manuscript considers the main features of the nonlinear dynamics of multiple irregular celestial body systems. The gravitational potential, static electric potential, and magnetic potential are considered. Based on the three established potentials, we show that three conservative values exist for this system, including a Jacobi integral. The equilibrium conditions for the system are derived and their stability analyzed. The equilibrium conditions of a celestial system comprised of $n$ irregular bodies are reduced to $12n - 9$ equations. The dynamical results are applied to simulate the motion of multiple-asteroid systems. The simulation is useful for the study of the stability of multiple irregular celestial body systems and for the design of spacecraft orbits to triple-asteroid systems discovered in the solar system. The dynamical configurations of the five triple-asteroid systems 45 Eugenia, 87 Sylvia, 93 Minerva, 216 Kleopatra, and 136617 1994CC, and the six-body system 134340 Pluto are calculated and analyzed.

Appendices

Appendix A: The static electric potential and magnetic potential

1.1 A.1 The potential and dynamic equation relative to the inertial space

In this section, we present the static electric potential and magnetic potential of multiple irregular bodies, as well as the dynamic equation relative to the inertial space.

The total static electric potential energy can be written as

$$\begin{aligned} U_{\mathit{se}} =& - \sum_{k = 1}^{n - 1} \sum _{j = k + 1}^{n} \frac{1}{4\pi \varepsilon_{0}} \\ &{}\times\int_{\beta_{k}} \int_{\beta_{j}} \frac{\rho_{\mathit{se}} ( \mathbf{D}_{k} ) \rho_{\mathit{se}} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) }{ \Vert \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{j} \mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} \Vert }, \end{aligned}$$

(A.1)

and the total magnetic potential energy can be written as

$$\begin{aligned} U_{m} =& - \sum_{k = 1}^{n - 1} \sum _{j = k + 1}^{n} \frac{\mu_{0}}{4 \pi } \\ &{}\times \int_{\beta_{k}} \int_{\beta_{j}} \frac{\mathbf{J} ( \mathbf{D}_{k} ) \cdot \mathbf{J} ( \mathbf{D} _{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) }{\Vert \mathbf{A}_{k}\mathbf{D} _{k} - \mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} \Vert }. \end{aligned}$$

(A.2)

Thus, the total energy of the system becomes

$$\begin{aligned} H =& T + U_{g} + U_{e} + U_{m} \\ =& \frac{1}{2}\sum_{k = 1}^{n} \bigl( m _{k}\Vert \dot{\mathbf{r}}_{k} \Vert ^{2} + \langle \varPhi _{k},\mathbf{I}_{k}\varPhi_{k} \rangle \bigr) \\ &{}- \sum_{k = 1}^{n - 1} \sum_{j = k + 1}^{n} \int_{\beta_{k}} \int_{\beta_{j}} \biggl[ G\rho_{g} ( \mathbf{D}_{k} ) \rho _{g} ( \mathbf{D}_{j} ) \\ &{} + \frac{1}{4\pi \varepsilon_{0}}\rho_{\mathit{se}} ( \mathbf{D}_{k} ) \rho_{\mathit{se}} ( \mathbf{D}_{j} ) + \frac{\mu_{0}}{4\pi } \mathbf{J} ( \mathbf{D}_{k} ) \cdot \mathbf{J} ( \mathbf{D}_{j} ) \biggr] \\ &{}\times\frac{dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) }{\Vert \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{j} \mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} \Vert } \end{aligned}$$

(A.3)

The total static electric force acting on $\beta_{k}$ can be written as

$$\begin{aligned} \mathbf{f}_{\mathit{se}}^{k} =& - \frac{1}{4\pi \varepsilon_{0}} \\ &{}\times \sum _{j = 1,j \ne k}^{n} \int_{\beta_{k}} \int_{\beta_{j}} \frac{ ( \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{j} \mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} ) }{ \Vert \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{j} \mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} \Vert ^{3}} \\ &{}\times\rho_{\mathit{se}} ( \mathbf{D}_{k} ) \rho_{\mathit{se}} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) , \end{aligned}$$

(A.4)

and the total magnetic force acting on $\beta_{k}$ can be written as

$$\begin{aligned} \mathbf{f}_{m}^{k} =& - \frac{\mu_{0}}{4\pi } \\ &{}\times \sum _{j = 1,j \ne k} ^{n} \int_{\beta_{k}} \int_{\beta_{j}} \frac{ ( \mathbf{A} _{k}\mathbf{D}_{k} - \mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} ) }{\Vert \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{j} \mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} \Vert ^{3}} \\ &{}\times\mathbf{J} ( \mathbf{D}_{k} ) \cdot \mathbf{J} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) . \end{aligned}$$

(A.5)

The resultant static electric torque acting on $\beta_{k}$ can be expressed as

$$\begin{aligned} \mathbf{n}_{\mathit{se}}^{k} =& \frac{1}{4\pi \varepsilon_{0}} \\ &{}\times \sum _{j = 1,j \ne k}^{n} \int_{\beta_{k}} \int_{\beta_{j}} \frac{ ( \mathbf{A}_{k}\mathbf{D}_{k} + \mathbf{r}_{k} ) \times ( \mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{r} _{j} ) }{\Vert \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} \Vert ^{3}} \\ &{}\times\rho_{\mathit{se}} ( \mathbf{D}_{k} ) \rho_{\mathit{se}} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) , \end{aligned}$$

(A.6)

and the resultant magnetic torque acting on $\beta_{k}$ can be expressed as

$$\begin{aligned} \mathbf{n}_{m}^{k} =& \frac{\mu_{0}}{4\pi } \\ &{}\times \sum _{j = 1,j \ne k} ^{n} \int_{\beta_{k}} \int_{\beta_{j}} \frac{ ( \mathbf{A} _{k}\mathbf{D}_{k} + \mathbf{r}_{k} ) \times ( \mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{r}_{j} ) }{ \Vert \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{j} \mathbf{D}_{j} + \mathbf{r}_{k} - \mathbf{r}_{j} \Vert ^{3}} \\ &{}\times \mathbf{J} ( \mathbf{D}_{k} ) \cdot \mathbf{J} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) . \end{aligned}$$

(A.7)

Then the dynamical equation relative to the inertial space is now

$$\{\begin{array}{l}{\dot{\mathbf{p}}}_k={\mathbf{f}}_g^k+{\mathbf{f}}_{\mathit{se}}^k+{\mathbf{f}}_m^k,\\ {\dot{\mathbf{r}}}_k=\frac{{\mathbf{p}}_k}{m_k},\\ {\dot{\mathbf{K}}}_k={\mathbf{n}}_g^k+{\mathbf{n}}_{\mathit{se}}^k+{\mathbf{n}}_m^k,\\ {\dot{\mathbf{A}}}_k={\stackrel{⌢}{\mathit{\psi }}}_k{\mathbf{A}}_k,\end{array}k=1,2,\dots ,n.$$

(A.8)

The three conservative values of the dynamics equation are

$$ \begin{aligned}[c] &H = T + U_{g} + U_{\mathit{se}} + U_{m}, \\ &\mathbf{p} = \sum_{k = 1}^{n} \mathbf{p}_{k},\\ &\mathbf{K} = \sum_{k = 1}^{n} \mathbf{K}_{k}. \end{aligned} $$

(A.9)

1.2 A.2 The total static electric potential energy and the total magnetic potential energy in the body-fixed frame of $\beta_{n}$

The total static electric potential energy and the total magnetic potential energy can be written relative to the body-fixed frame of $\beta_{n}$ as follows:

$$ \left \{ \textstyle\begin{array}{rcl} \displaystyle U_{\mathit{se}} &=& \displaystyle - \sum_{k = 1}^{n - 1} \sum_{j = k + 1}^{n} \frac{1}{4\pi \varepsilon_{0}}\\ &&\displaystyle{}\times\int_{\beta_{k}} \int_{\beta_{j}} \frac{\rho_{\mathit{se}} ( \mathbf{D}_{k} ) \rho_{\mathit{se}} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) }{ \Vert \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{n}^{T}\mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{R}_{kn} - \mathbf{R}_{jn} \Vert }, \\ \displaystyle U_{m} &=& \displaystyle - \sum_{k = 1}^{n - 1} \sum_{j = k + 1}^{n} \frac{\mu_{0}}{4 \pi }\\ \displaystyle &&\displaystyle{}\times\int_{\beta_{k}} \int_{\beta_{j}} \frac{\mathbf{J} ( \mathbf{D}_{k} ) \cdot \mathbf{J} ( \mathbf{D} _{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) }{\Vert \mathbf{A}_{n}^{T} \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{n}^{T} \mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{R}_{kn} - \mathbf{R}_{jn} \Vert }. \end{array}\displaystyle \right. $$

(A.10)

If we denote $U = U_{g} + U_{\mathit{se}} + U_{m}$, the dynamical equation of the system is also Eq. (13). The torque is

$$ \left\{ \textstyle\begin{array}{l} \boldsymbol{\mu }^{k} = \boldsymbol{\mu }_{g}^{k} + \boldsymbol{\mu }_{e}^{k} + \boldsymbol{\mu }_{m}^{k},\quad k = 1,2, \ldots ,n - 1, \\ \boldsymbol{\mu }^{n} = \boldsymbol{\mu }_{g}^{n} + \boldsymbol{\mu } _{\mathit{se}}^{n} + \boldsymbol{\mu }_{m}^{n}, \end{array}\displaystyle \right. $$

(A.11)

where

$$ \left\{ \textstyle\begin{array}{rcl} \displaystyle \boldsymbol{\mu }_{\mathit{se}}^{k} &=& \displaystyle - \frac{1}{4\pi \varepsilon_{0}}\\ &&{}\times\displaystyle\sum_{k = 1}^{n - 1} \sum_{j = k + 1}^{n} \int_{\beta_{k}} \int_{\beta_{j}} \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k}\\ &&\displaystyle{} \times \frac{ ( \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{n}^{T}\mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{R}_{kn} - \mathbf{R}_{jn} ) }{\Vert \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{n}^{T}\mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{R}_{kn} - \mathbf{R}_{jn} \Vert ^{3}} \\ &&\displaystyle{} \times \rho_{\mathit{se}} ( \mathbf{D}_{k} ) \rho_{\mathit{se}} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ) \\ \displaystyle \boldsymbol{\mu }_{m}^{k} &=& \displaystyle - \frac{\mu_{0}}{4\pi }\\ &&{}\times \displaystyle\sum_{k = 1}^{n - 1} \sum_{j = k + 1}^{n} \int_{\beta_{k}} \int_{\beta_{j}} \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k} \\ &&\displaystyle{} \times \frac{ ( \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{n}^{T}\mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{R}_{kn} - \mathbf{R}_{jn} ) }{\Vert \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{A}_{n}^{T}\mathbf{A}_{j}\mathbf{D}_{j} + \mathbf{R}_{kn} - \mathbf{R}_{jn} \Vert ^{3}}\\ &&\displaystyle{}\times\mathbf{J} ( \mathbf{D}_{k} ) \cdot \mathbf{J} (\mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV (\mathbf{D}_{k}) \end{array}\displaystyle \right. $$

(A.12)

and

$$ \left\{ \textstyle\begin{array}{rcl} \displaystyle \boldsymbol{\mu }_{\mathit{se}}^{n} &=& \displaystyle\frac{1}{4\pi \varepsilon_{0}}\sum_{k = 1} ^{n - 1} \sum_{j = k + 1}^{n} \int_{\beta_{k}} \int_{\beta_{j}} \mathbf{D}_{n} \\ &&\displaystyle{} \times \frac{ ( \mathbf{A}_{n}^{T} \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{D}_{n} + \mathbf{R}_{kn} ) }{\Vert \mathbf{A}_{n}^{T} \mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{D}_{n} + \mathbf{R}_{kn} \Vert ^{3}}\\ &&\displaystyle{} \times \rho_{\mathit{se}} ( \mathbf{D}_{k} ) \rho_{\mathit{se}} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ), \\ \displaystyle \boldsymbol{\mu }_{m}^{n} &=& \displaystyle\frac{\mu_{0}}{4\pi } \sum_{k = 1}^{n - 1} \sum_{j = k + 1}^{n} \int_{\beta_{k}} \int_{\beta_{j}} \mathbf{D} _{n}\\ &&\displaystyle{} \times \frac{ ( \mathbf{A}_{n}^{T}\mathbf{A}_{k} \mathbf{D}_{k} - \mathbf{D}_{n} + \mathbf{R}_{kn} ) }{ \Vert \mathbf{A}_{n}^{T}\mathbf{A}_{k}\mathbf{D}_{k} - \mathbf{D}_{n} + \mathbf{R}_{kn} \Vert ^{3}}\\ &&\displaystyle{} \times \mathbf{J} ( \mathbf{D}_{k} ) \cdot \mathbf{J} ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{j} ) dV ( \mathbf{D}_{k} ). \end{array}\displaystyle \right. $$

(A.13)

Careful analysis of the motion of multiple-asteroid systems indicates that the static electric force and the magnetic force acting between each asteroid are negligible. Therefore,

$$U = U_{g}\quad \mbox{and}\quad \left\{\textstyle\begin{array}{l@{\quad }l} \boldsymbol{\mu }^{k} = \boldsymbol{\mu }_{g}^{k}, & k = 1,2, \ldots ,n - 1\\ \boldsymbol{\mu }^{n} = \boldsymbol{\mu }_{g}^{n} \end{array}\displaystyle \right. $$

are established for multiple-asteroid systems in the Solar system in Eqs. (13) and (15).

Appendix B: Initial parameters

Table 1 Initial conditions for calculation of dynamical configurations

Table 2 Initial conditions for calculation of the dynamical configurations of the six-body system (134340) Pluto