Non-linear three-mode coupling of gravity modes in rotating slowly pulsating B stars - Stationary solutions and modeling potential

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Volume 687 (July 2024)

A&A, 687 (2024) A265

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Table 4.

Linear co-rotating-frame frequencies Ω_φ, largest absolute spin parameters |s_φ| (of the triad modes), radial orders n_φ and linear driving or linear damping rates γ_φ of g modes φ in all identified isolated (g) mode triads with ordering numbers (k₁, k₂, k₃) = (0, 0, 0) and azimuthal orders (m₁, m₂, m₃) = (2, 1, 1) for the models in Table 2 that yield stable and valid stationary solutions.

Model	(n₁, n₂, n₃)	δω (d⁻¹)	γ_⊞ (10⁻³ d⁻¹)	(Ω₁, Ω₂, Ω₃) (d⁻¹)	(γ₁, γ₂, γ₃) (10⁻⁵ d⁻¹)	\|s\|_max
Fiducial	(−51, −38, −90)	0.00145	−0.32249	(3.23, 2.32, 0.91)	(0.39, −0.30, −32.3)	2.20
ΔX_c, 1	(−20, −13, −58)	−0.00197	−2.0248	(6.92, 5.74, 1.18)	(0.18, −0.16, −202.5)	2.71
	(−21, −14, −54)	−0.00528	−1.657	(6.60, 5.33, 1.28)	(0.57, −0.17, −166.1)	2.51
	(−22, −15, −53)	0.00370	−1.536	(6.29, 4.99, 1.30)	(1.06, −0.19, −154.4)	2.46
	(−39, −40, −40)	0.00303	−0.350	(3.53, 1.76, 1.76)	(11.2, −23.1, −23.1)	1.81
ΔX_c, 2_(a)	(−10, −7, −22)	−0.01510	−1.025	(12.0, 9.16, 2.88)	(0.06, −0.08, −102.4)	1.98
	(−14, −10, −26)	0.03121	−3.252	(8.85, 6.41, 2.41)	(4.19, −0.08, −329.3)	2.36
ΔX_c, 2_(b)	(−10, −6, −36)	−0.01614	−7.765	(11.26, 9.65, 1.63)	(0.07, −0.05, −776.5)	10.5
	(−12, −8, −23)	−0.00844	−1.565	(9.62, 7.08, 2.54)	(0.94, −0.12, −157.3)	6.71
	(−17, −12, −30)	−0.00723	−5.144	(6.80, 4.86, 1.95)	(8.45, −0.08, −522.8)	8.74
	(−18, −13, −31)	0.03334	−5.825	(6.42, 4.49, 1.89)	(7.03, −0.19, −589.3)	9.02
ΔM_ini, 1	(−50, −48, −56)	−0.00083	−0.507	(2.37, 1.29, 1.09)	(15.5, −13.3, −52.9)	1.59
	(−40, −30, −70)	0.00836	−1.429	(2.98, 2.12, 0.85)	(6.65, −0.62, −148.9)	2.03
$Δ X_{c} \| M_{ini}$ $\Delta \mathrm{X_{\mathrm{c}}}\|\rm M_{\mathrm{ini}}$	(−16, −14, −22)	0.00130	−0.120	(6.19, 3.81, 2.38)	(0.01, −0.56, −11.4)	1.17
	(−17, −13, −29)	0.00830	−0.915	(5.88, 4.08, 1.79)	(0.16, −0.57, −91.1)	1.56
	(−23, −17, −41)	0.01453	−4.495	(4.37, 3.12, 1.23)	(5.33, −1.15, −453.6)	2.26
ΔM_ini, 2	(−31, −23, −56)	0.00283	−2.018	(3.03, 2.18, 0.85)	(2.30, −2.14, −202.0)	1.80
	(−32, −24, −58)	0.00631	−2.176	(2.92, 2.10, 0.82)	(2.43, −2.42, −217.6)	1.86
	(−35, −26, −62)	−0.00274	−2.915	(2.68, 1.93, 0.76)	(7.54, −3.13, −295.9)	2.01
	(−36, −27, −63)	0.00546	−2.734	(2.61, 1.86, 0.75)	(2.02, −3.20, −272.2)	2.04
	(−38, −28, −67)	−0.00575	−2.724	(2.47, 1.78, 0.70)	(0.24, −4.27, −268.3)	2.18

Notes.

The effective triad damping γ_⊞ and the triad detuning δω are also listed. The mode triad detuning δω is equal to ΔΩ^l because we set 𝔍 = 1 in Eq. (34). We compute the largest absolute spin parameter as ${| s |}_{\max} = max (| s_{φ} |) \equiv max (| 2 Ω / Ω_{φ} |) \forall φ \in 〚 3 〛$ $\lvert s\rvert_{\mathrm{max}} = \max\left(\lvert s_\varphi\rvert\right) \equiv \max\left(\lvert 2\,\Omega / \Omega_\varphi\rvert\right)\,\forall\varphi\in{\llbracket3\rrbracket}$ .

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