Secular structure of 1:2 and 1:3 mean motion resonances with Neptune

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Table 1

Proper frequencies of major planets in references (Nobili et al. 1989; Zhou et al. 2020) and in this paper.

Frequency	Nobili 1989	Zhou 2020	This paper
𝑔₅	0.03285	0.03263	0.03262
𝑔₆	0.21794	0.21353	0.21737
𝑔₇	0.02382	0.02384	0.02385
𝑔₈	0.00519	0.00522	0.00519
s₅	0.00008	0.00008	<0.00029
s₆	0.20328	0.20355	0.20305
s₇	0.02309	0.02325	0.02296
s₈	0.00534	0.00536	0.00537

2N:1U	2.3606	2.3238	2.35524
5S:2J	N.A.	N.A.	11.33012

Notes. The frequencies are given in 10⁻⁴ 2π yr⁻¹. The symbols 𝑔_5,…₈ and s₅,…_,8 stand for the precessions of perihelion and ascending node for planets Jupiter, Saturn, Uranus, and Neptune, respectively. The 2N:1U and 5S:2J are for the quasi MMRs between two pairs of planets.

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