The dynamical history of the Kepler-221 planet system

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Table 3

Parameter setups in different simulation stages corresponding to figures from Figures 2–6.

Simulation stage	Figures	Phase(s)	Mass model	Parameter^(a)	Value/range
`complete`	2	all	M2a	m_d₂/m_d₁	1.02
				t_a,I	1.4 × 10⁴ yr
				t_{a, I}/t_e,I	250
				t_d	5 × 10³ yr
				Δa/a	0.7%
				Q_phy,all	4.6
				t_a0,III^b	1.9 × 10⁸ yr
				t_{Γ, III}^b	7.5 × 10⁶ yr

`before planet merging`	2	I, II	M2a, M2d	m_d₂/m_d₁	[1, 1.04]
				t_{a, I}	[10³, 10⁶] yr
				t_{a, I}/t_{e, I}	[150, 250]
				t_d	[50,10⁴] yr
				Δa/a	[0.5%, 1.2%]

`3BR reformation`	3, 4	III	M2a, M2d	Q_{phy, c}	[3,10]
				Q_phy,/Q_{phy, b}	[0.01, 1]
				t_{a0, III}^b	[6 × 10⁷, 2 × 10⁸] yr
				t_{Γ, III}^b	[2.5 × 10⁶, 7.5 × 10⁶] yr

`orbital expansion`	2, 5, 6	IV	M1, M2a, M2d	P_d/P_c	[1.712, 1.728]
				t_a0,^b	[4.6 × 10⁹, 2.3 × 10¹⁰] yr
				t_{Γ, IV}^b	[4.6 × 10⁸, 1.4 × 10⁹] yr

Notes. ^(a) Here the parameters denote −m_d2/m_d1: the mass ratio between planet d₂ and d₁ before merging, t_{a, I}: semi-major axis damping timescale in the disk phase, t_{a, I}/t_{e, I}: the ratio between semi-major axis damping and eccentricity damping timescale in disk phase, t_d: the disk dispersal timescale, Δa/a: the instantaneous kick applied to the semi-major axis of planet d2 to break the resonances, Q_{phy, x}: the effective tidal damping parameter for planet x (see Section 2.3), t_{a 0, III}^b, t_{a0, IV}^b: the equivalent semi-major axis migration timescale due to torque on planet b in Phase III and Phase IV, t_{Γ, II}^b, t_{Γ, IV}^b: the exponential decay timescale of the torque on planet b in Phase III and Phase IV, P_d/P_c: the period ratio of planet c and d after planet merging.

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