Table 2.
Radial order ranges {nu} (u ∈ ⟦3⟧) of the (k, m)=(0, 2) parent and (k, m)=(0, 1) daughter modes computed with GYRE and based on the MESA models listed in Table 1, angular rotation rates Ω expressed in percentages of the Roche critical rotation rate ΩRoche and in d−1, model radius R, and the order of magnitude of the bounds for the ranges of ϑ2 and ϑ3 (denoted by ϑ2, 3), q1 and the quality factors Qφ (φ ∈ ⟦3⟧).
| Model | {n1} | {n2, 3} | Ω/ΩRoche (%) | ⌈ϑ2, 3⌋ | ⌈q1⌋ | ⌈Qφ⌋ | Ω (d−1) | R (R⊙) |
|---|---|---|---|---|---|---|---|---|
| Fiducial | ⟦70 , 50⟧ | ⟦120 , 70⟧ ∪ ⟦38 , 10⟧ | 20 | [102, 10−2] | [106, 100] | [107, 102] | 0.9995 | 5.1874 |
| ΔXc, 1 | ⟦40 , 20⟧ | ⟦70 , 40⟧ ∪ ⟦15 , 3⟧ | 20 | [103, 10−4] | [107, 101] | [108, 102] | 1.6006 | 3.7900 |
| ΔXc, 2(a) | ⟦20 , 10⟧ | ⟦47 , 17⟧ ∪ ⟦10 , 3⟧ | 20 | [104, 10−4] | [107, 102] | [108, 102] | 2.8419 | 2.5848 |
| ΔXc, 2(b) | ⟦19 , 10⟧ | ⟦48 , 3⟧ | 60 | [104, 10−4] | [107, 101] | [108, 102] | 8.5256 | 2.5848 |
| ΔMini, 1 | ⟦55 , 33⟧ | ⟦120 , 45⟧ ∪ ⟦30 , 5⟧ | 20 | [102, 10−3] | [106, 10−1] | [107, 102] | 0.8628 | 6.5483 |
| ΔXc|Mini | ⟦26 , 16⟧ | ⟦55 , 5⟧ | 20 | [104, 10−3] | [108, 100] | [107, 102] | 1.3904 | 4.7645 |
| ΔMini, 2 | ⟦39 , 30⟧ | ⟦85 , 5⟧ | 20 | [103, 10−2] | [106, 100] | [106, 102] | 0.7611 | 7.8322 |
Notes. We define the radial order ranges using ⟦u , h⟧ ≡ { x∈ℕ0 | −u≤x≤−h }, and we define the range of the dimensionless parameter a as ⌈a⌋≡[𝒪(sup|a|) , 𝒪(inf|a|)]. Mode triads that consist of modes whose radial orders lie within the radial order ranges quoted in this table satisfy the necessary conditions for parametric resonance: γ1 > 0 , γ2 < 0 and γ3 < 0. We consider the ΔXc, 2 model at two rotation rates, labeled with the subscripts (a) and (b). Gaps in the radial order ranges {n2, 3} indicate the presence of linearly excited (k, m)=(0, 1) modes. Such linearly excited modes cannot act as potential daughter modes in parametric resonances with the (k, m)=(0, 2) parent modes and are therefore not considered in this work. Note that the κ mechanism excites a strongly different set of potential (k, m)=(0, 2) parent modes for each of the different models.
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