Long-term dynamics beyond Neptune: secular models to study the regular motions

Abstract

Two semi-analytical one-degree-of-freedom secular models are presented for the motion of small bodies beyond Neptune. A special attention is given to trajectories entirely exterior to the planetary orbits. The first one is the well-known non-resonant model of Kozai (Astron J 67:591, 1962) adapted to the transneptunian region. Contrary to previous papers, the dynamics is fully characterized with respect to the fixed parameters. A maximum perihelion excursion possible of 16.4 AU is determined. The second model handles the occurrence of a mean-motion resonance with one of the planets. In that case, the one-degree-of-freedom integrable approximation is obtained by postulating the adiabatic invariance, and is much more general and accurate than previous secular models found in the literature. It brings out in a plain way the possibility of perihelion oscillations with a very high amplitude. Such a model could thus be used in future studies to deeper explore that kind of motion. For complex resonant orbits (especially of type 1 : k), a segmented secular description is necessary since the trajectories are only “integrable by parts”. The two models are applied to the Solar System but the notations are kept general so that it could be used for any quasi-circular and coplanar planetary system.

Appendix: Expansion of the secular non-resonant Hamiltonian

We present here some hints about the construction of the secular analytical non-resonant Hamiltonian function. With the planetary model chosen (Eq. 6), the angle $\psi _i$ is simply defined by:

$$\begin{aligned} \cos \psi _i =\alpha \cos \lambda _i + \beta \sin \lambda _i \end{aligned}$$

(56)

where:

$$\begin{aligned} \left\{ \begin{aligned} \alpha&= \cos (\omega +\nu )\cos {\varOmega }-\sin (\omega +\nu )\sin {\varOmega }\cos I\\ \beta&= \cos (\omega +\nu )\sin {\varOmega }+\sin (\omega +\nu )\cos {\varOmega }\cos I \end{aligned} \right. \end{aligned}$$

(57)

Then, the multiple average of $\varepsilon {\mathcal {H}}_1$ is computed in two steps, beginning with the integration over $\lambda _1,\lambda _2\ldots \lambda _N$. As the indirect part vanishes, we have, using the Legendre development (9):

$$\begin{aligned} \frac{1}{(2\pi )^N}\int _{0}^{2\pi } \int _{0}^{2\pi } \cdots \int _{0}^{2\pi }\varepsilon {\mathcal {H}}_1\ \ {\mathrm {d}}\lambda _1{\mathrm {d}}\lambda _2\ldots {\mathrm {d}}\lambda _N = -\frac{1}{r}\,\sum _{n=0}^{+\infty }\left( \sum _{i=1}^{N}\mu _i\left( \frac{a_i}{r}\right) ^{2n}\right) P_{2n}(\chi ) \end{aligned}$$

(58)

where $\chi$ is defined as:

$$\begin{aligned} \left\{ \begin{array}{ll} \chi ^0 &{}= 1 \\ \chi ^{2k} &{}= \frac{1\times 3\times 5\times \cdots \times (2k-1)}{2\times 4\times 6\times \cdots \times 2k}(\alpha ^2+\beta ^2)^{k},\quad k = 1,2,3\ldots \end{array} \right. \end{aligned}$$

(59)

One can see that the odd terms have disappeared. The average over l is less straightforward, since each polynomial 2n of (58) requires the computation of $n+1$ integrals of the form:

$$\begin{aligned} \frac{1}{2\pi }\int _{0}^{2\pi }\frac{(\alpha ^2+\beta ^2)^k}{r^{2n+1}}\,{\mathrm {d}}l = \frac{1}{a^2\sqrt{1-e^2}}\ \frac{1}{2\pi }\int _{0}^{2\pi }\frac{(\alpha ^2+\beta ^2)^k}{r^{2n-1}}\,{\mathrm {d}}\nu ,\quad k = 0,1,2\ldots n \end{aligned}$$

(60)

where $\alpha$, $\beta$ and r are functions of the true anomaly $\nu$. The general form of the result is presented in Eqs. (12) and (13), and the first terms are the following

$n = 1$	$n = 2$	$n = 3$
$\alpha _1 = 1/8$	$\alpha _2 = 9/1024$	$\alpha _3 = 25/65536$
$P_1^0(x) = 1$	$P_2^0(x) = 2+3\,x^2$	$P_3^0(x) = 2\,(8+40\,x^2+15\,x^4)$
$Q_1^0(x) = -1+3\,x^2$	$Q_2^0(x) = 3-30\,x^2+35\,x^4$	$Q_3^0(x) = -5+105\,x^2-315\,x^4+231\,x^6$
	$P_2^1(x) = 10$	$P_3^1(x) = 210\,(2+x^2)$
	$Q_2^1(x) = -1+7\,x^2$	$Q_3^1(x) = 1-18\,x^2+33\,x^4$
		$P_3^2(x) = 63$
		$Q_3^2(x) = -1+11\,x^2$

$n = 4$	$n = 5$
$\alpha _4 = 245/33554432$	$\alpha _5 = 567/4294967296$
$P_4^0(x) = 5\,(16+168\,x^2+210\,x^4+35\,x^6)$	$P_5^0(x) = 14\,(128+2304\,x^2+6048\,x^4+3360\,x^6+315\,x^8)$
$Q_4^0(x) = 35-1260\,x^2+6930\,x^4-12012\,x^6+6435\,x^8$	$Q_5^0(x) = -63+3465\,x^2-30030\,x^4+90090\,x^6-109395\,x^8+46189\,x^{10}$
$P_4^1(x) = 630\,(48+80\,x^2+15\,x^4)$	$P_5^1(x) = 9240\,(32+112\,x^2+70\,x^4+7\,x^6)$
$Q_4^1(x) = -1+33\,x^2-143\,x^4+143\,x^6$	$Q_5^1(x) = 7-364\,x^2+2730\,x^4-6188\,x^6+4199\,x^8$
$P_4^2(x) = 1386\,(10+3\,x^2)$	$P_5^2(x) = 240240\,(8+8\,x^2+x^4)$
$Q_4^2(x) = 1-26\,x^2+65\,x^4$	$Q_5^2(x) = -1+45\,x^2-255\,x^4+323\,x^6$
$P_4^3(x) = 858$	$P_5^3(x) = 8580\,(14+3\,x^2)$
$Q_4^3(x) = -1+15\,x^2$	$Q_5^3(x) = 3-102\,x^2+323\,x^4$
	$P_5^4(x) = 12155$
	$Q_5^4(x) = -1+19\,x^2$

$n = 6$
$\alpha _6 = 7623/549755813888$
$P_6^0(x) = 14\,(256+7040\,x^2+31690\,x^4+36960\,x^6+11550\,x^8+693\,x^{10})$
$Q_6^0(x) = 231-18018\,x^2+225225\,x^4-1021020\,x^6+2078505\,x^8-1939938\,x^{10}+676039\,x^{12}$
$P_6^1(x) = 60060\,(128+768\,x^2+1008\,x^4+336\,x^2+21\,x^8)$
$Q_6^1(x) = -3+225\,x^2-2550\,x^4+9690\,x^6-14535\,x^8+7429\,x^{10}$
$P_6^2(x) = 90090\,(80+168\,x^2+70\,x^4+5\,x^6)$
$Q_6^2(x) = 5-340\,x^2+3230\,x^4-9044\,x^6+7429\,x^8$
$P_6^3(x) = 12155\,(224+160\,x^2+15\,x^4)$
$Q_6^3(x) = -5+285\,x^2-1995\,x^4+3059\,x^6$
$P_6^4(x) = 230945\,(6+x^2)$
$Q_6^4(x) = 1-42\,x^2+161\,x^4$
$P_6^5(x) = 29393$
$Q_6^5(x) = -1+23\,x^2$