The linear stability of the post-Newtonian triangular equilibrium in the three-body problem

Abstract

Continuing a work initiated in an earlier publication (Yamada et al. in Phys Rev D 91:124016, 2015), we reexamine the linear stability of the triangular solution in the relativistic three-body problem for general masses by the standard linear algebraic analysis. In this paper, we start with the Einstein–Infeld–Hoffmann form of equations of motion for N-body systems in the uniformly rotating frame. As an extension of the previous work, we consider general perturbations to the equilibrium, i.e., we take account of perturbations orthogonal to the orbital plane, as well as perturbations lying on it. It is found that the orthogonal perturbations depend on each other by the first post-Newtonian (1PN) three-body interactions, though these are independent of the lying ones likewise the Newtonian case. We also show that the orthogonal perturbations do not affect the condition of stability. This is because these do not grow with time, but always precess with two frequency modes, namely, the same with the orbital frequency and the slightly different one due to the 1PN effect. The condition of stability, which is identical to that obtained by the previous work (Yamada et al. 2015) and is valid for the general perturbations, is obtained from the lying perturbations.

Appendices

Appendix A: Derivation of EIH equations of motion in uniformly rotating frame

The EIH equation of motion for N-body systems is given by (Misner et al. 1973; Landau and Lifshitz 1962; Will 1993; Asada et al. 2011)

$$\begin{aligned} \frac{\hbox {d}^2\varvec{r}_K}{\hbox {d} t^2} =&\, \sum _{A\ne K}\frac{Gm_A}{r_{AK}^3}\varvec{r}_{AK} \biggl [1 - 4\sum _{B\ne K}\frac{Gm_B}{c^2r_{BK}} - \sum _{C\ne A}\frac{Gm_C}{c^2r_{CA}}\left( 1-\frac{\varvec{r}_{AK}\cdot \varvec{r}_{CA}}{2r^2_{CA}}\right) + \frac{v_K^2}{c^2} + 2\frac{v_A^2}{c^2} \nonumber \\&- \,4\frac{\varvec{v}_A\cdot \varvec{v}_K}{c^2} - \frac{3}{2}\left( \frac{\varvec{v}_A}{c}\cdot \varvec{x}_{AK}\right) ^2 \biggr ] - \sum _{A\ne K}\frac{Gm_A}{c^2r^2_{AK}} \varvec{x}_{AK}\cdot \left( 3\frac{\varvec{v}_A}{c} -4\frac{\varvec{v}_K}{c} \right) \left( \frac{\varvec{v}_A}{c} - \frac{\varvec{v}_K}{c}\right) \nonumber \\&+ \,\frac{7}{2}\sum _{A\ne K}\sum _{C\ne A}\frac{G^2m_Am_C}{c^2r_{AK}r^3_{CA}} \varvec{r}_{CA}, \end{aligned}$$

(61)

where $\varvec{r}_I$, $\varvec{v}_I$, and $m_I$ are the position, the velocity, and the mass of Ith particle, respectively. $\varvec{r}_{I J}\equiv \varvec{r}_I-\varvec{r}_J$, $r_{I J}\equiv |\varvec{r}_{I J}|$, $\varvec{x}_{I J}\equiv \varvec{r}_{I J}/r_{I J}$, $v_I \equiv |\varvec{v}_I|$. For the above equation, we consider the linear transformation of the function t. In general, a linear transformation from $\mathbb {R}^n$ to $\mathbb {R}^n$ is given by

$$\begin{aligned} \varvec{r}^{\prime }=R\varvec{r}, \end{aligned}$$

(62)

where $\varvec{r}, \varvec{r}^{\prime }\in \mathbb {R}^n$, R is an $n\times n$ matrix. If R is a one to one and onto mappings, the linear mapping of the first order and second order time derivatives of $\varvec{r}$ are calculated as

$$\begin{aligned}&\displaystyle R\frac{\hbox {d}\varvec{r}}{\hbox {d} t} = \frac{\hbox {d}\varvec{r}^{\prime }}{\hbox {d} t} + S\varvec{r}^{\prime }, \end{aligned}$$

(63)

$$\begin{aligned}&\displaystyle R\frac{\hbox {d}^2\varvec{r}}{\hbox {d} t^2} = \frac{\hbox {d}^2\varvec{r}^{\prime }}{\hbox {d} t^2} + 2S\frac{\hbox {d}\varvec{r}^{\prime }}{\hbox {d} t} + S^2\varvec{r}^{\prime } + \frac{\hbox {d}S}{\hbox {d} t}\varvec{r}^{\prime }, \end{aligned}$$

(64)

where $S \equiv -(\hbox {d}R)/(\hbox {d} t)R^{-1}$.

If R is a rotation matrix, R is the orthogonal matrix and the determinant of R is 1. Therefore, the transposition of R is consistent with the inverse of R, namely ${}^t R=R^{-1}$, and S becomes skew-symmetric. Thus, for a three-dimensional transformation, we can set S as

$$\begin{aligned} S=\begin{pmatrix} 0 &{} -w_3 &{} w_2\\ w_3 &{} 0 &{} -w_1\\ -w_2 &{} w_1 &{} 0 \end{pmatrix}, \end{aligned}$$

(65)

then, for all vectors $\varvec{v}\in \mathbb {R}^{3}$, it satisfies the relation $S\varvec{v}=\varvec{\varOmega }\times \varvec{v}$, where $\varvec{\varOmega }=(w_1, w_2, w_3)$.

The EIH equation in a uniformly rotating frame of constant angular velocity $\varvec{\varOmega }$ can be expressed as

$$\begin{aligned} \frac{\hbox {d}^2 \varvec{r}_K^{\prime }}{\hbox {d} t^2} =&\, \sum _{A \ne K} \frac{G m_A}{(r_{K A}^{\prime })^3} \varvec{r}_{A K}^{\prime } - 2 (\varvec{\varOmega } \times \varvec{v}_K^{\prime }) - (\varvec{\varOmega } \cdot \varvec{r}_K^{\prime }) \varvec{\varOmega } + \varOmega ^2 \varvec{r}_K^{\prime } \nonumber \\&+\, \sum _{A \ne K} \frac{G m_A}{(r_{K A}^{\prime })^3}\varvec{r}_{A K}^{\prime } \left[ - 4 \sum _{B \ne K} \frac{G m_B}{c^2 r_{K B}^{\prime }} - \sum _{C \ne A} \frac{G m_C}{c^2 r_{A C}^{\prime }} \left( 1 + \frac{\varvec{r}_{A K}^{\prime } \cdot \varvec{r}_{A C}^{\prime }}{2 (r_{C A}^{\prime })^2} \right) \right. \nonumber \\&+\, \left( \frac{\varvec{v}_K^{\prime } + (\varvec{\varOmega } \times \varvec{r}_K^{\prime })}{c} \right) ^2 + 2 \left( \frac{\varvec{v}_A^{\prime } + (\varvec{\varOmega } \times \varvec{r}_A^{\prime })}{c} \right) ^2 \nonumber \\&\left. -\, 4 \left( \frac{\varvec{v}_K^{\prime } + (\varvec{\varOmega } \times \varvec{r}_K^{\prime })}{c} \right) \cdot \left( \frac{\varvec{v}_A^{\prime } + (\varvec{\varOmega } \times \varvec{r}_A^{\prime })}{c} \right) - \frac{3}{2} \left\{ \left( \frac{\varvec{v}_A^{\prime }+ (\varvec{\varOmega } \times \varvec{r}_A^{\prime })}{c} \right) \cdot \varvec{x}_{A K}^{\prime } \right\} ^2 \right] \nonumber \\&-\, \sum _{A \ne K} \frac{G m_A}{c^2 (r_{K A}^{\prime })^2} \left[ \varvec{x}_{A K}^{\prime } \cdot \left( \frac{4 [\varvec{v}_K^{\prime } + (\varvec{\varOmega } \times \varvec{r}_K^{\prime })] - 3 [\varvec{v}_A^{\prime } + (\varvec{\varOmega } \times \varvec{r}_A^{\prime })]}{c} \right) \right] \nonumber \\&\times \, \left( \frac{[\varvec{v}_K^{\prime } + (\varvec{\varOmega } \times \varvec{r}_K^{\prime })] - [\varvec{v}_A^{\prime } + (\varvec{\varOmega } \times \varvec{r}_A^{\prime })]}{c} \right) \nonumber \\&+\, \frac{7}{2} \sum _{A \ne K} \sum _{C \ne A} \frac{G m_A}{c^2 r_{K A}^{\prime }} \frac{G m_C}{(r_{A C}^{\prime })^3} \varvec{r}_{C A}^{\prime }, \end{aligned}$$

(66)

using the relations

$$\begin{aligned} \varvec{r}_{I J}\cdot \varvec{r}_{MN}&\,= \varvec{r}_{I J}^{\prime }\cdot \varvec{r}_{MN}^{\prime }, \end{aligned}$$

(67)

$$\begin{aligned} \varvec{v}_I\cdot \varvec{v}_J&\,= (\varvec{v}_I^{\prime } + \varvec{\varOmega }\times \varvec{r}_I^{\prime })\cdot (\varvec{v}_J^{\prime } + \varvec{\varOmega }\times \varvec{r}_J^{\prime }),\end{aligned}$$

(68)

$$\begin{aligned} \varvec{v}_I\cdot \varvec{r}_{J K}&\,= (\varvec{r}_I^{\prime } + \varvec{\varOmega }\times \varvec{r}_{I}^{\prime })\cdot \varvec{r}_{J K}^{\prime }, \end{aligned}$$

(69)

and where we set $\varOmega \equiv |\varvec{\varOmega }|$.

Appendix B: Transformation to Routh’s variables

Eliminating $\xi _{23}$ and $\eta _{23}$ from Eq. (25), the perturbed equations of motion for $\xi _{12}$, $\eta _{12}$, $\xi _{31}$, $\eta _{31}$ become

$$\begin{aligned} 0 =&\, \left( D^2+\frac{9 \nu _3}{4}-3\right) \xi _{12} +\left( \frac{3 \sqrt{3} \nu _3}{4}-2 D\right) \eta _{12} -\frac{3}{4} \sqrt{3} \nu _3 \eta _{31} -\frac{9}{4} \nu _3 \xi _{31} \nonumber \\&+\, \varepsilon \left[ \left( \frac{1}{8} \sqrt{3} D \nu _3 (-9 \nu _2 \nu _3+21 \nu _2+4 \nu _3-8) \right. \right. \nonumber \\&\left. +\, \frac{1}{32} \left[ -22 \nu _2^2 (9 \nu _3+4)+\nu _2 \left( -126 \nu _3^2+136 \nu _3+88\right) -108 \nu _3^3+270 \nu _3^2-537 \nu _3+540\right] \right) \xi _{12} \nonumber \\&+\, \left( \frac{1}{8} D \left[ -2 \nu _2^2 (9 \nu _3+17)+\nu _2 \left( -9 \nu _3^2-9 \nu _3+34\right) +10 \nu _3^2+2 \nu _3+45\right] \right. \nonumber \\&\left. -\, \frac{1}{32} \sqrt{3} \nu _3 \left[ 30 \nu _2^2+\nu _2 (66 \nu _3+56)+84 \nu _3^2-66 \nu _3+127\right] \right) \eta _{12} \nonumber \\&+\, \left( \frac{1}{32} \nu _3 \left[ 126 \nu _2^2+2 \nu _2 (27 \nu _3-76)+144 \nu _3^2-322 \nu _3+553\right] \right. \nonumber \\&\left. -\,\frac{1}{8} \sqrt{3} D \nu _3 \left[ \nu _2 (9 \nu _3+7)+9 \nu _3^2-12 \nu _3-1\right] \right) \xi _{31} \nonumber \\&+\, \left( \frac{1}{8} D \nu _3 \left[ -\nu _2 (9 \nu _3+7)+9 \nu _3^2-32 \nu _3+11\right] \right. \nonumber \\&\left. \left. +\, \frac{1}{32} \sqrt{3} \nu _3 \left[ 30 \nu _2^2+\nu _2 (66 \nu _3+56)+84 \nu _3^2-66 \nu _3+127\right] \right) \eta _{31} \right] , \end{aligned}$$

(70)

$$\begin{aligned} 0 =&\, \left( D^2-\frac{9 \nu _3}{4}\right) \eta _{12} +\left( 2 D+\frac{3 \sqrt{3} \nu _3}{4}\right) \xi _{12} + \frac{9}{4} \nu _3 \eta _{31} -\frac{3}{4} \sqrt{3} \nu _3 \xi _{31} \nonumber \\&+\, \varepsilon \left[ \left( \frac{1}{8} D \left[ -6 \nu _2^2 (3 \nu _3+5)+\nu _2 \left( -9 \nu _3^2-13 \nu _3+30\right) -2 \nu _3^2+14 \nu _3-61\right] \right. \right. \nonumber \\&\left. -\,\frac{1}{32} \sqrt{3} \nu _3 \left[ 54 \nu _2^2+6 \nu _2 (3 \nu _3+8)+36 \nu _3^2+6 \nu _3+103\right] \right) \xi _{12} \nonumber \\&+\, \left( \frac{1}{8} \sqrt{3} D \nu _3 [\nu _2 (9 \nu _3-25)-4 \nu _3+8] \right. \nonumber \\&\left. +\,\frac{3}{32} \nu _3 \left[ 66 \nu _2^2+6 \nu _2 (7 \nu _3-8)+36 \nu _3^2-66 \nu _3+155\right] \right) \eta _{12} \nonumber \\&+\, \left( \frac{1}{8} D \nu _3 \left( -9 \nu _2 \nu _3+\nu _2+9 \nu _3^2-34 \nu _3+13\right) \right. \nonumber \\&\left. + \,\frac{1}{32} \sqrt{3} \nu _3 \left[ 54 \nu _2^2+6 \nu _2 (15 \nu _3-8)+72 \nu _3^2-78 \nu _3+151\right] \right) \xi _{31} \nonumber \\&+\, \left( \frac{1}{8} \sqrt{3} D \nu _3 \left[ \nu _2 (9 \nu _3-1)+9 \nu _3^2-18 \nu _3+5\right] \right. \nonumber \\&\left. \left. -\, \frac{3}{32} \nu _3 \left[ 66 \nu _2^2+6 \nu _2 (7 \nu _3-8)+36 \nu _3^2-66 \nu _3+155\right] \right) \eta _{31} \right] , \end{aligned}$$

(71)

$$\begin{aligned} 0 =&\, \left( D^2+\frac{9 \nu _2}{4}-3\right) \xi _{31} +\left( -2 D-\frac{3 \sqrt{3} \nu _2}{4}\right) \eta _{31} + \frac{3}{4} \sqrt{3} \nu _2 \eta _{12} -\frac{9}{4} \nu _2 \xi _{12} \nonumber \\&+\, \varepsilon \left[ \left( \frac{1}{8} \sqrt{3} D \nu _2 \left[ 9 \nu _2^2+3 \nu _2 (3 \nu _3-4)+7 \nu _3-1\right] \right. \right. \nonumber \\&\left. +\, \frac{1}{32} \nu _2 \left[ 144 \nu _2^2+\nu _2 (54 \nu _3-322)+126 \nu _3^2-152 \nu _3+553\right] \right) \xi _{12} \nonumber \\&+\, \left( \frac{1}{8} D \nu _2 \left[ 9 \nu _2^2-\nu _2 (9 \nu _3+32)-7 \nu _3+11\right] \right. \nonumber \\&\left. -\, \frac{1}{32} \sqrt{3} \nu _2 \left[ 84 \nu _2^2+66 \nu _2 (\nu _3-1)+30 \nu _3^2+56 \nu _3+127\right] \right) \eta _{12} \nonumber \\&+\, \left( \frac{1}{8} \sqrt{3} D \nu _2 [\nu _2 (9 \nu _3-4)-21 \nu _3+8] \right. \nonumber \\&\left. + \,\frac{1}{32} \left[ -108 \nu _2^3-18 \nu _2^2 (7 \nu _3-15)+\nu _2 \left( -198 \nu _3^2+136 \nu _3-537\right) -88 \nu _3^2+88 \nu _3+540\right] \right) \xi _{31} \nonumber \\&+\, \left( \frac{1}{8} D \left[ \nu _2^2 (10-9 \nu _3)+\nu _2 \left( -18 \nu _3^2-9 \nu _3+2\right) -34 \nu _3^2+34 \nu _3+45\right] \right. \nonumber \\&\left. \left. +\, \frac{1}{32} \sqrt{3} \nu _2 \left[ 84 \nu _2^2+66 \nu _2 (\nu _3-1)+30 \nu _3^2+56 \nu _3+127\right] \right) \eta _{31} \right] , \end{aligned}$$

(72)

$$\begin{aligned} 0 =&\, \left( D^2-\frac{9 \nu _2}{4}\right) \eta _{31} +\left( 2 D-\frac{3 \sqrt{3} \nu _2}{4}\right) \xi _{31} + \frac{9}{4} \nu _2 \eta _{12} +\frac{3}{4} \sqrt{3} \nu _2 \xi _{12} \nonumber \\&+\, \varepsilon \left[ \left( \frac{1}{8} D \nu _2 \left[ 9 \nu _2^2-\nu _2 (9 \nu _3+34)+\nu _3+13\right] \right. \right. \nonumber \\&\left. -\, \frac{1}{32} \sqrt{3} \nu _2 \left[ 72 \nu _2^2+\nu _2 (90 \nu _3-78)+54 \nu _3^2-48 \nu _3+151\right] \right) \xi _{12} \nonumber \\&+\, \left( \frac{1}{8} \sqrt{3} D \nu _2 \left[ -9 \nu _2^2-9 \nu _2 (\nu _3-2)+\nu _3-5\right] \right. \nonumber \\&\left. -\, \frac{3}{32} \nu _2 \left[ 36 \nu _2^2+6 \nu _2 (7 \nu _3-11)+66 \nu _3^2-48 \nu _3+155\right] \right) \eta _{12} \nonumber \\&+\, \left( \frac{1}{8} D \left[ \nu _2^2 (-(9 \nu _3+2))+\nu _2 \left( -18 \nu _3^2-13 \nu _3+14\right) -30 \nu _3^2+30 \nu _3-61\right] \right. \nonumber \\&\left. + \,\frac{1}{32} \sqrt{3} \nu _2 \left[ 36 \nu _2^2+6 \nu _2 (3 \nu _3+1)+54 \nu _3^2+48 \nu _3+103\right] \right) \xi _{31} \nonumber \\&+\, \left( \frac{1}{8} \sqrt{3} D \nu _2 [\nu _2 (4-9 \nu _3)+25 \nu _3-8] \right. \nonumber \\&\left. \left. +\, \frac{3}{32} \nu _2 \left[ 36 \nu _2^2+6 \nu _2 (7 \nu _3-11)+66 \nu _3^2-48 \nu _3+155\right] \right) \eta _{31} \right] . \end{aligned}$$

(73)

Let us seek the relations between $(\xi _{I J}, \eta _{I J})$ and Routh’s variables. First, since $\chi _{12}$ is a perturbation to $r_{12}$, we obtain the relation

$$\begin{aligned} \ell (1 + \rho _{12}) (1 + \xi _{12}) = \ell (1 + \rho _{12} + \chi _{12}). \end{aligned}$$

(74)

Therefore,

$$\begin{aligned} \chi _{12} = (1 + \rho _{12}) \xi _{12}. \end{aligned}$$

(75)

In the same way, we obtain the relation for X as

$$\begin{aligned} X = (1 + \rho _{31}) \xi _{31} - (1 + \rho _{12}) \xi _{12}. \end{aligned}$$

(76)

Next, let us define the projection of vectors onto the orbital plane as

$$\begin{aligned} \bar{\varvec{A}} \equiv \varvec{A} - (\varvec{A} \cdot \varvec{z}) \varvec{z}. \end{aligned}$$

(77)

Using this, $\sigma$ is expressed as a perturbation to the angle between $\bar{\varvec{r}}_{12}$ and $\bar{\varvec{r}}_{12} + \bar{\delta \varvec{r}}_{12}$, so that,

$$\begin{aligned} \sin \sigma = \frac{\left| \bar{\varvec{r}}_{12} \times ( \bar{\varvec{r}}_{12} + \bar{\delta \varvec{r}}_{12}) \right| }{r_{12}^2 (1 + \xi _{12})}. \end{aligned}$$

(78)

Solving this for $\sigma$ to the 1PN order, we obtain

$$\begin{aligned} \sigma = \eta _{12}. \end{aligned}$$

(79)

Finally, since $\psi _{23}$ is a perturbation in the opposite angle of $r_{23}$, we obtain

$$\begin{aligned} \cos \left( \frac{\pi }{3} + \sqrt{3} \rho _{23} + \psi _{23} \right) = - \frac{(\bar{\varvec{r}}_{31} + \bar{\delta \varvec{r}}_{31}) \cdot ( \bar{\varvec{r}}_{12} + \bar{\delta \varvec{r}}_{12})}{r_{31} r_{12} (1 + \xi _{31}) (1 + \xi _{12})}. \end{aligned}$$

(80)

This leads to

$$\begin{aligned} \psi _{23} = \eta _{31} - \eta _{12}, \end{aligned}$$

(81)

at the 1PN order. Using these relations, we obtain the perturbed equations of motion (39)–(42).

Appendix C: The components of the coefficient matrix of the perturbation equation

The components of the coefficient matrix $\mathcal {N}$ in Eq. (43) are written as

$$\begin{aligned}&\mathcal {N}_{11} = \,\frac{\varepsilon }{8} \sqrt{3} \nu _3 (9 \nu _3-7) (2 \nu _2+\nu _3-1), \end{aligned}$$

(82)

$$\begin{aligned}&\mathcal {N}_{12} =\, \frac{\varepsilon }{8} \sqrt{3} \nu _3 \left[ \nu _2 (9 \nu _3+7) +9 \nu _3^2-12 \nu _3-1\right] ,\end{aligned}$$

(83)

$$\begin{aligned}&\mathcal {N}_{13} =\, -\frac{\varepsilon }{8} \nu _3 \left[ -\nu _2 (9 \nu _3+7)+9 \nu _3^2-32 \nu _3+11\right] ,\end{aligned}$$

(84)

$$\begin{aligned} \mathcal {N}_{14} =&\, 2-\frac{\varepsilon }{24} \left[ -6 \nu _2^2 (9 \nu _3+19) -6 \nu _2 \left( 9 \nu _3^2+10 \nu _3-19\right) +27 \nu _3^3-60 \nu _3^2+63 \nu _3\right. \nonumber \\&\left. +\,125\right] ,\end{aligned}$$

(85)

$$\begin{aligned} \mathcal {N}_{15} =&\, 3 - \frac{\varepsilon }{32}\left[ -11 \nu _2^2 (9 \nu _3+8) + \nu _2 \left( -72 \nu _3^2 -34 \nu _3+88\right) +63 \nu _3^3-34 \nu _3^2+16 \nu _3 \right. \nonumber \\&\left. +\,540\right] ,\end{aligned}$$

(86)

$$\begin{aligned} \mathcal {N}_{16} =&\, \frac{9}{4}\nu _3 - \frac{\varepsilon }{32} \nu _3 \left[ 99 \nu _2^2+2 \nu _2 (27 \nu _3-85) +171 \nu _3^2-304 \nu _3+553\right] ,\end{aligned}$$

(87)

$$\begin{aligned} \mathcal {N}_{17} =&\, \frac{3}{4}\sqrt{3}\nu _3 - \frac{\varepsilon }{32} \sqrt{3} \nu _3\left[ 24 \nu _2^2 +\nu _2 (60 \nu _3+62)+87 \nu _3^2-54 \nu _3+122\right] ,\end{aligned}$$

(88)

$$\begin{aligned} \mathcal {N}_{21} =&\, - \frac{\varepsilon }{8} \sqrt{3}\nu _2 (9 \nu _2-7) (\nu _2+2 \nu _3-1) -\frac{\varepsilon }{8} \sqrt{3} \nu _3 (9 \nu _3-7) (2 \nu _2+\nu _3-1),\end{aligned}$$

(89)

$$\begin{aligned} \mathcal {N}_{22} =&\, -\frac{\varepsilon }{8} \sqrt{3}\nu _2 \left[ \nu _2 (9 \nu _3-4)-21 \nu _3+8 \right] - \frac{\varepsilon }{8} \sqrt{3} \nu _3 \left[ \nu _2 (9 \nu _3+7) +9 \nu _3^2-12 \nu _3-1\right] ,\end{aligned}$$

(90)

$$\begin{aligned} \mathcal {N}_{23} =&\, 2- \frac{\varepsilon }{24}\left[ -9 \nu _2^2 (3 \nu _3-4)+3\nu _2 \left( -18 \nu _3^2-13 \nu _3+10\right) -114 \nu _3^2+114 \nu _3+125\right] \nonumber \\&+\, \frac{\varepsilon }{8} \nu _3 \left[ -\nu _2 (9 \nu _3+7)+9 \nu _3^2-32 \nu _3+11\right] ,\end{aligned}$$

(91)

$$\begin{aligned} \mathcal {N}_{24} =&\, - \frac{\varepsilon }{24}\left[ 27 \nu _2^3-6 \nu _2^2 (9 \nu _3+10)+3\nu _2 \left( -18 \nu _3^2-20\nu _3+21\right) -114 \nu _3^2+114 \nu _3 +125\right] \nonumber \\&+ \frac{\varepsilon }{24} \left[ -6 \nu _2^2 (9 \nu _3+19) -6 \nu _2 \left( 9 \nu _3^2+10 \nu _3-19\right) +27 \nu _3^3-60 \nu _3^2 +63 \nu _3+125\right] ,\end{aligned}$$

(92)

$$\begin{aligned} \mathcal {N}_{25} =&\, - \frac{\varepsilon }{32}\left[ 63 \nu _2^3-2 \nu _2^2 (36 \nu _3+17)+\nu _2 \left( -99 \nu _3^2-34 \nu _3+16\right) -88 \nu _3^2+88 \nu _3 +540\right] \nonumber \\&+ \frac{\varepsilon }{32} \left[ -11 \nu _2^2 (9 \nu _3+8) + \nu _2 \left( -72 \nu _3^2 -34 \nu _3+88\right) +63 \nu _3^3-34 \nu _3^2 \right. \nonumber \\&\left. +\,16 \nu _3+540\right] ,\end{aligned}$$

(93)

$$\begin{aligned} \mathcal {N}_{26} =&\, -\frac{9\nu _2}{4}+3 - \frac{\varepsilon }{32}\left[ -108 \nu _2^3-18 \nu _2^2 (7 \nu _3-15)+\nu _2 \left( -198 \nu _3^2+136 \nu _3-537\right) \right. \nonumber \\&\left. -\,88 \nu _3^2+88 \nu _3+540\right] - \frac{9}{4}\nu _3 + \frac{\varepsilon }{32} \nu _3 \left[ 99 \nu _2^2+2 \nu _2 (27 \nu _3-85) +171 \nu _3^2 \right. \nonumber \\&\left. -\,304 \nu _3+553\right] ,\end{aligned}$$

(94)

$$\begin{aligned} \mathcal {N}_{27} =&\, \frac{3\sqrt{3}\nu _2}{4} - \frac{\varepsilon }{32} \sqrt{3} \nu _2 \left[ 87 \nu _2^2+6 \nu _2 (10 \nu _3-9) +24 \nu _3^2+62 \nu _3+122\right] - \frac{3}{4}\sqrt{3}\nu _3 \nonumber \\&+ \,\frac{\varepsilon }{32} \sqrt{3} \nu _3\left[ 24 \nu _2^2 +\nu _2 (60 \nu _3+62)+87 \nu _3^2-54 \nu _3+122\right] ,\end{aligned}$$

(95)

$$\begin{aligned} \mathcal {N}_{31} =&\, - \frac{\varepsilon }{24} \left[ 27 \nu _2^3-54 \nu _2^2 (\nu _3+2) -\nu _2 \left( 54\nu _3^2+42\nu _3-101\right) -90 \nu _3^2+80 \nu _3-183 \right. \nonumber \\&\left. -\,2\nu _1(3\nu _2-6\nu _3+5)\right] + \frac{\varepsilon }{24}\left[ -6 \nu _2^2 (9 \nu _3+17) - 6\nu _2\left( 9\nu _3^2 + 8\nu _3 - 17\right) \right. \nonumber \\&\left. +\, 27\nu _3^3 - 102\nu _3^2 + 105\nu _3-193\right] ,\end{aligned}$$

(96)

$$\begin{aligned} \mathcal {N}_{32} =&\, 2 - \frac{\varepsilon }{24} \left[ -3(9 \nu _3+2) \nu _2^2 -\nu _2 \left( 54 \nu _3^2+45\nu _3-62\right) -90 \nu _3^2+80 \nu _3-183 \right. \nonumber \\&\left. -\,2\nu _1(3\nu _2-6\nu _3+5)\right] +\frac{\varepsilon }{8}\nu _3 \left( -9 \nu _2 \nu _3+\nu _2+9 \nu _3^2-34 \nu _3 +13\right) ,\end{aligned}$$

(97)

$$\begin{aligned} \mathcal {N}_{33} =&\, -\frac{\varepsilon }{8} \sqrt{3} \nu _2 \left[ \nu _2 (4-9 \nu _3)+25 \nu _3-8\right] +\frac{\varepsilon }{8} \sqrt{3}\nu _3 \left[ \nu _2 (9 \nu _3-1)+9 \nu _3^2-18 \nu _3 \right. \nonumber \\&\left. +\,5\right] ,\end{aligned}$$

(98)

$$\begin{aligned} \mathcal {N}_{34} =&\, \frac{\varepsilon }{8} \sqrt{3} \nu _2 (9 \nu _2-13) (\nu _2+2 \nu _3-1) + \frac{\varepsilon }{8} \sqrt{3} \nu _3 (9 \nu _3-13) (2 \nu _2+\nu _3-1),\end{aligned}$$

(99)

$$\begin{aligned} \mathcal {N}_{35} =&\, \frac{3\varepsilon }{32} \sqrt{3} \nu _2 \left[ 15 \nu _2^2+\nu _2 (24 \nu _3-26) -3 \nu _3^2-34 \nu _3+16\right] \nonumber \\&+ \,\frac{3\varepsilon }{32} \sqrt{3} \nu _3[-3 \nu _2^2+\nu _2 (24 \nu _3-34) +15 \nu _3^2-26\nu _3+16],\end{aligned}$$

(100)

$$\begin{aligned} \mathcal {N}_{36} =&\, \frac{3\sqrt{3}\nu _2}{4} - \frac{\varepsilon }{32} \sqrt{3} \nu _2 \left[ 36 \nu _2^2+ \nu _2 (21 \nu _3-4) +54 \nu _3^2+53 \nu _3+103 \right. \nonumber \\&\left. +\, \nu _1(3\nu _2-6\nu _3+5)\right] - \frac{3}{4}\sqrt{3}\nu _3 + \frac{3\sqrt{3}\varepsilon }{32}\nu _3\left[ 17 \nu _2^2 + 4 \nu _2 (8 \nu _3-5) + 26 \nu _3^2 \right. \nonumber \\&\left. -\,28 \nu _3 + 52\right] ,\end{aligned}$$

(101)

$$\begin{aligned} \mathcal {N}_{37} =&\, \frac{9\nu _2}{4} - \frac{3\varepsilon }{32} \nu _2 \left[ 39\nu _2^2+\nu _2 (39\nu _3-64)+60 \nu _3^2 -37 \nu _3+150 \right. \nonumber \\&\left. +\,\nu _1(3\nu _2-6\nu _3+5)\right] + \frac{9}{4}\nu _3 - \frac{3\varepsilon }{32} \nu _3 \left[ 66\nu _2^2 + 6\nu _2(7 \nu _3-8) +36 \nu _3^2-66 \nu _3 \right. \nonumber \\&\left. +\,155\right] ,\end{aligned}$$

(102)

$$\begin{aligned} \mathcal {N}_{41} =&\, - 2- \frac{\varepsilon }{24}\left[ -6 \nu _2^2 (9 \nu _3+17) - 6\nu _2\left( 9\nu _3^2 + 8\nu _3 - 17\right) + 27\nu _3^3 - 102\nu _3^2 \right. \nonumber \\&\left. +\, 105\nu _3-193\right] ,\end{aligned}$$

(103)

$$\begin{aligned}&\mathcal {N}_{42} =\, - \frac{\varepsilon }{8}\nu _3 \left( -9 \nu _2 \nu _3+\nu _2+9 \nu _3^2-34 \nu _3 +13\right) ,\end{aligned}$$

(104)

$$\begin{aligned}&\mathcal {N}_{43} =\, - \frac{\varepsilon }{8} \sqrt{3}\nu _3 \left[ \nu _2 (9 \nu _3-1)+9 \nu _3^2-18 \nu _3+5\right] ,\end{aligned}$$

(105)

$$\begin{aligned}&\mathcal {N}_{44} =\, - \frac{\varepsilon }{8} \sqrt{3} \nu _3 (9 \nu _3-13) (2 \nu _2+\nu _3-1),\end{aligned}$$

(106)

$$\begin{aligned}&\mathcal {N}_{45} =\, - \frac{3\varepsilon }{32} \sqrt{3} \nu _3\left[ -3 \nu _2^2+\nu _2 (24 \nu _3-34) +15 \nu _3^2-26\nu _3+16\right] ,\end{aligned}$$

(107)

$$\begin{aligned}&\mathcal {N}_{46} =\, \frac{3}{4}\sqrt{3}\nu _3 -\frac{3\sqrt{3}\varepsilon }{32}\nu _3\left[ 17 \nu _2^2 + 4 \nu _2 (8 \nu _3-5) + 26 \nu _3^2-28 \nu _3 + 52\right] ,\end{aligned}$$

(108)

$$\begin{aligned}&\mathcal {N}_{47} =\, - \frac{9}{4}\nu _3 + \frac{3\varepsilon }{32} \nu _3 \left[ 66\nu _2^2 + 6\nu _2(7 \nu _3-8) +36 \nu _3^2-66 \nu _3+155\right] , \end{aligned}$$

(109)

$\mathcal {N}_{51} = \mathcal {N}_{62} = \mathcal {N}_{73} = 1$, and the others are 0.

Appendix D: Relation between perturbations to size and orbital frequency

Equation (59) can be rewritten as

$$\begin{aligned} \varvec{\chi } =&\,QE_1Q^{-1}\varvec{\chi }_0 + e^{\omega _{N}t\lambda _{1+}}QE_2Q^{-1}\varvec{\chi }_0 + e^{\omega _{N}t\lambda _{1-}}QE_3Q^{-1}\varvec{\chi }_0 + e^{\omega _{N}t\lambda _{2+}}QE_4Q^{-1}\varvec{\chi }_0 \nonumber \\&+\, e^{\omega _{N}t\lambda _{2-}}QE_5Q^{-1}\varvec{\chi }_0 + e^{\omega _{N}t\lambda _{3+}}QE_6Q^{-1}\varvec{\chi }_0 + e^{\omega _{N}t\lambda _{3-}}QE_7Q^{-1}\varvec{\chi }_0, \end{aligned}$$

(110)

where $E_i$ is a $7 \times 7$ matrix with components

$$\begin{aligned} E_i = \left\{ \begin{array}{cl} 1 &{} (\text {for the } ii\ \text {component}),\\ 0 &{} (\text {for the others}). \end{array} \right. \end{aligned}$$

(111)

The regular matrix Q is given by

$$\begin{aligned} Q=(\varvec{v}\,\, \varvec{v}_{1+}\,\, \varvec{v}_{1-}\,\, \varvec{v}_{2+}\,\, \varvec{v}_{2-}\,\, \varvec{v}_{3+}\,\, \varvec{v}_{3-}), \end{aligned}$$

(112)

where $\varvec{v}_a$ are the eigenvectors corresponding to the eigenvalues $\lambda _a$, and $\varvec{v}$ is the eigenvector corresponding to the zero eigenvalue. Therefore, we have

$$\begin{aligned} \left\{ \begin{array}{l} QE_1 =(\varvec{v}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}),\\ QE_2 =(\varvec{0}\,\,\varvec{v}_{1+}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}),\\ QE_3 =(\varvec{0}\,\,\varvec{0}\,\,\varvec{v}_{1-}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}),\\ QE_4 =(\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{v}_{2+}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}),\\ QE_5 =(\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{v}_{2-}\,\,\varvec{0}\,\,\varvec{0}),\\ QE_6 =(\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{v}_{3+}\,\,\varvec{0}),\\ QE_7 =(\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{0}\,\,\varvec{v}_{3-}), \end{array} \right. \end{aligned}$$

(113)

and Eq. (110) is calculated as

$$\begin{aligned} \varvec{\chi } =&\, c_1\varvec{v} + e^{\omega _{N}t\lambda _{1+}}c_2\varvec{v}_{1+} + e^{\omega _{N}t\lambda _{1-}}c_3\varvec{v}_{1-} \nonumber \\&+\, e^{\omega _{N}t\lambda _{2+}}c_4\varvec{v}_{2+} + e^{\omega _{N}t\lambda _{2-}}c_5\varvec{v}_{2-} + e^{\omega _{N}t\lambda _{3+}}c_6\varvec{v}_{3+} + e^{\omega _{N}t\lambda _{3-}}c_7\varvec{v}_{3-}, \end{aligned}$$

(114)

where $\varvec{c}=(c_1, c_2, \ldots , c_7)\equiv Q^{-1}\varvec{\chi }_0$. By setting the coefficients as

$$\begin{aligned} c_1\varvec{v}&\,\equiv (\ldots , C_{41}, C_{11}, C_{21}, C_{31}), \end{aligned}$$

(115)

$$\begin{aligned} c_2\varvec{v}_{1+}&\,\equiv (\ldots , C_{42}, C_{12}, C_{22}, C_{32}), \end{aligned}$$

(116)

$$\begin{aligned} c_3\varvec{v}_{1-}&\,\equiv (\ldots , C_{43}, C_{13}, C_{23}, C_{33}), \end{aligned}$$

(117)

$$\begin{aligned} c_4\varvec{v}_{2+}&\,\equiv (\ldots , C_{44}, C_{14}, C_{24}, C_{34}), \end{aligned}$$

(118)

$$\begin{aligned} c_5\varvec{v}_{2-}&\,\equiv (\ldots , C_{45}, C_{15}, C_{25}, C_{35}), \end{aligned}$$

(119)

$$\begin{aligned} c_6\varvec{v}_{3+}&\,\equiv (\ldots , C_{46}, C_{16}, C_{26}, C_{36}), \end{aligned}$$

(120)

$$\begin{aligned} c_7\varvec{v}_{3-}&\,\equiv (\ldots , C_{47}, C_{17}, C_{27}, C_{37}), \end{aligned}$$

(121)

we obtain the solution of Eq. (60).

In order to consider $C_{41}$, which corresponds to the unique secular term, therefore, let us focus on the zero eigenvector $\varvec{v}$. By the definition of the zero eigenvector, we have the equation

$$\begin{aligned} \mathcal {N}\varvec{v}=\varvec{0}. \end{aligned}$$

(122)

By straightforward calculations, we obtain the equations for the components of $\varvec{v}$ as

$$\begin{aligned}&\displaystyle c_1 \varvec{v} \,= (0, 0, 0, C_{41}, C_{11}, C_{21}, C_{31}), \end{aligned}$$

(123)

$$\begin{aligned}&\displaystyle C_{21} \,= 0, \end{aligned}$$

(124)

$$\begin{aligned}&\displaystyle C_{31} \,= - \frac{\sqrt{3}}{24} \left( -10\nu _1^2+5\nu _2^2+5\nu _3^2+4\nu _2\nu _3-2\nu _1\nu _2 - 2\nu _1\nu _3 \right) \varepsilon C_{11}, \end{aligned}$$

(125)

$$\begin{aligned}&\displaystyle C_{41}\,= - \frac{3}{2} \left[ 1 - \frac{5}{48} (29 - 14 V) \varepsilon \right] C_{11}. \end{aligned}$$

(126)

Therefore, $C_{41}$ is not independent of $C_{11}$, and hence, the change in the size corresponding to $C_{11}$ leads to the change in the orbital frequency $C_{41}$ regarding the energy and angular momentum changes.

In fact, Eq. (126) can be derived from Eqs. (18)–(20). Let us consider a perturbation $\ell \rightarrow \ell (1 + x)$ in Eq. (19). This leads to the perturbation of $\varepsilon$ as

$$\begin{aligned} \varepsilon \rightarrow \varepsilon (1 - x), \end{aligned}$$

(127)

at the leading order. Therefore, the orbital frequency is perturbed as

$$\begin{aligned} \omega \rightarrow \omega _{N} \left[ 1 + \tilde{\omega }_{PN} - \frac{3}{2} \left( 1 + \frac{5}{3} \tilde{\omega }_{PN} \right) x \right] . \end{aligned}$$

(128)

Replacing $x \rightarrow C_{11}$ in the last term, we obtain the perturbations to the orbital frequency equivalent to Eq. (126).