Table 1
Model parameters for pure helium models.
| M ⋆ | L ⋆ | Γe | log (Tc) | log (T ⋆ ) | Rc | Re | R ⋆ | ΔR | ΔR | ΔM | ρmin | fρ | Q | Wc | We | Pe |
| [M⊙] | [L⊙] | [K] | [K] | [R⊙] | [R⊙] | [R⊙] | [Rc] | [Re] | [M⊙] |
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| X = 0.00, D = 1 | ||||||||||||||||
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| 23.7 | 5.813 | 0.414 | 5.11 | 4.61 | 1.6 | 14.6 | 16.5 | 7.99 | 0.89 | 1.9 × 10-6 | 4.4 × 10-11 | 2.4 | 30.50 | 0.89 | 7.99 | 6.8 × 105 |
| 23.5 | 5.807 | 0.412 | 5.10 | 4.68 | 1.7 | 10.5 | 11.4 | 5.35 | 0.84 | 4.6 × 10-7 | 4.6 × 10-11 | 2.1 | 25.27 | 0.84 | 5.35 | 6.8 × 105 |
| 23.0 | 5.792 | 0.406 | 5.10 | 4.80 | 1.6 | 6.3 | 6.6 | 2.88 | 0.74 | 5.7 × 10-8 | 5.4 × 10-11 | 2.1 | 21.90 | 0.74 | 2.88 | 6.9 × 105 |
| 22.0 | 5.761 | 0.396 | 5.11 | 4.92 | 1.6 | 3.6 | 3.7 | 1.35 | 0.57 | 6.6 × 10-9 | 7.3 × 10-11 | 2.0 | 16.61 | 0.57 | 1.35 | 7.1 × 105 |
| 21.0 | 5.729 | 0.385 | 5.11 | 4.98 | 1.5 | 2.7 | 2.7 | 0.79 | 0.44 | 2.0 × 10-9 | 9.9 × 10-11 | 2.0 | 12.69 | 0.44 | 0.79 | 7.1 × 105 |
| 20.0 | 5.694 | 0.373 | 5.11 | 5.01 | 1.4 | 2.2 | 2.2 | 0.52 | 0.34 | 9.6 × 10-10 | 1.4 × 10-10 | 2.1 | 10.28 | 0.34 | 0.52 | 7.4 × 105 |
| 19.0 | 5.658 | 0.361 | 5.11 | 5.04 | 1.4 | 1.9 | 1.9 | 0.36 | 0.26 | 5.3 × 10-10 | 1.9 × 10-10 | 2.1 | 7.80 | 0.26 | 0.36 | 6.9 × 105 |
| 18.0 | 5.618 | 0.348 | 5.11 | 5.06 | 1.3 | 1.6 | 1.7 | 0.25 | 0.20 | 3.6 × 10-10 | 2.7 × 10-10 | 2.1 | 5.90 | 0.20 | 0.25 | 6.9 × 105 |
| 17.0 | 5.576 | 0.335 | 5.11 | 5.07 | 1.3 | 1.5 | 1.5 | 0.19 | 0.16 | 2.7 × 10-10 | 3.9 × 10-10 | 2.3 | 4.81 | 0.16 | 0.19 | 7.0 × 105 |
| 16.0 | 5.531 | 0.320 | 5.10 | 5.07 | 1.2 | 1.4 | 1.4 | 0.14 | 0.12 | 2.1 × 10-10 | 5.5 × 10-10 | 2.3 | 3.76 | 0.12 | 0.14 | 6.8 × 105 |
| 15.0 | 5.482 | 0.305 | 5.10 | 5.08 | 1.2 | – | 1.3 | – | – | – | – | – | 3.10 | – | – | – |
| 14.0 | 5.429 | 0.289 | 5.10 | 5.08 | 1.1 | – | 1.2 | – | – | – | – | – | 2.31 | – | – | – |
| 12.0 | 5.307 | 0.255 | 5.09 | 5.07 | 1.0 | – | 1.1 | – | – | – | – | – | 1.27 | – | – | – |
| 10.0 | 5.155 | 0.216 | 5.07 | 5.06 | 0.9 | – | 0.9 | – | – | – | – | – | 0.69 | – | – | – |
| 8.0 | 4.959 | 0.172 | 5.05 | 5.04 | 0.8 | – | 0.8 | – | – | – | – | – | 0.36 | – | – | – |
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| X = 0.00, D = 4 | ||||||||||||||||
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| 18.5 | 5.638 | 0.355 | 5.10 | 4.69 | 1.4 | 8.1 | 9.2 | 4.82 | 0.83 | 2.0 × 10-7 | 5.8 × 10-11 | 1.9 | 21.09 | 0.83 | 4.82 | 6.4 × 105 |
| 18.0 | 5.618 | 0.348 | 5.10 | 4.81 | 1.4 | 4.8 | 5.1 | 2.51 | 0.72 | 2.2 × 10-8 | 6.9 × 10-11 | 1.9 | 18.09 | 0.72 | 2.51 | 6.3 × 105 |
| 17.0 | 5.576 | 0.335 | 5.10 | 4.93 | 1.3 | 2.8 | 2.9 | 1.14 | 0.53 | 2.6 × 10-9 | 9.8 × 10-11 | 1.9 | 13.21 | 0.53 | 1.14 | 6.2 × 105 |
| 16.0 | 5.531 | 0.320 | 5.10 | 4.98 | 1.2 | 2.0 | 2.1 | 0.65 | 0.40 | 7.7 × 10-10 | 1.4 × 10-10 | 1.9 | 9.86 | 0.40 | 0.65 | 5.9 × 105 |
| 15.0 | 5.482 | 0.305 | 5.10 | 5.02 | 1.2 | 1.7 | 1.7 | 0.42 | 0.29 | 3.6 × 10-10 | 2.0 × 10-10 | 2.0 | 7.44 | 0.29 | 0.42 | 5.6 × 105 |
| 14.0 | 5.429 | 0.289 | 5.09 | 5.04 | 1.1 | – | 1.5 | – | – | – | – | – | 6.14 | – | – | – |
| 13.0 | 5.371 | 0.273 | 5.09 | 5.05 | 1.1 | – | 1.3 | – | – | – | – | – | 4.45 | – | – | – |
| 12.0 | 5.307 | 0.255 | 5.08 | 5.05 | 1.0 | – | 1.2 | – | – | – | – | – | 3.01 | – | – | – |
| 10.0 | 5.155 | 0.216 | 5.07 | 5.05 | 0.9 | – | 1.0 | – | – | – | – | – | 1.43 | – | – | – |
| 8.0 | 4.959 | 0.172 | 5.05 | 5.04 | 0.8 | – | 0.8 | – | – | – | – | – | 0.69 | – | – | – |
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| X = 0.00, D = 10 | ||||||||||||||||
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| 15.5 | 5.507 | 0.313 | 5.10 | 4.66 | 1.2 | 7.3 | 8.9 | 5.10 | 0.84 | 1.5 × 10-7 | 6.7 × 10-11 | 2.2 | 23.29 | 0.84 | 5.10 | 5.6 × 105 |
| 15.0 | 5.482 | 0.305 | 5.09 | 4.80 | 1.2 | 4.2 | 4.6 | 2.52 | 0.72 | 1.3 × 10-8 | 8.1 × 10-11 | 2.0 | 18.22 | 0.72 | 2.52 | 5.4 × 105 |
| 14.0 | 5.429 | 0.289 | 5.09 | 4.92 | 1.1 | – | 2.5 | – | – | – | – | – | 14.52 | – | – | – |
| 13.0 | 5.371 | 0.273 | 5.09 | 4.98 | 1.1 | – | 1.8 | – | – | – | – | – | 10.24 | – | – | – |
| 12.0 | 5.307 | 0.255 | 5.08 | 5.01 | 1.0 | – | 1.5 | – | – | – | – | – | 6.97 | – | – | – |
| 10.0 | 5.155 | 0.216 | 5.07 | 5.03 | 0.9 | – | 1.1 | – | – | – | – | – | 3.07 | – | – | – |
| 8.0 | 4.959 | 0.172 | 5.05 | 5.03 | 0.8 | – | 0.9 | – | – | – | – | – | 1.33 | – | – | – |
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| X = 0.00, D = 16 | ||||||||||||||||
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| 14.0 | 5.429 | 0.289 | 5.08 | 4.65 | 1.2 | 5.9 | 8.5 | 3.97 | 0.80 | 1.0 × 10-7 | 7.4 × 10-11 | 1.5 | 14.59 | 0.80 | 3.97 | 4.7 × 105 |
| 13.5 | 5.401 | 0.281 | 5.08 | 4.80 | 1.2 | 3.5 | 4.3 | 1.97 | 0.66 | 8.1 × 10-8 | 8.9 × 10-11 | 1.4 | 10.99 | 0.66 | 1.97 | 4.0 × 105 |
| 13.0 | 5.371 | 0.273 | 4.88 | 4.87 | 2.8 | – | 3.0 | – | – | – | – | – | 0.56 | – | – | – |
| 12.0 | 5.307 | 0.255 | 5.08 | 4.95 | 1.0 | – | 1.9 | – | – | – | – | – | 10.49 | – | – | – |
| 10.0 | 5.155 | 0.216 | 5.07 | 5.01 | 0.9 | – | 1.2 | – | – | – | – | – | 4.75 | – | – | – |
| 8.0 | 4.959 | 0.172 | 5.05 | 5.02 | 0.8 | – | 0.9 | – | – | – | – | – | 2.06 | – | – | – |
Notes. Given are: the stellar mass M⋆, the stellar luminosity L⋆, the classical Eddington parameter Γe (Eq. (6)), the effective temperature Tc related to the core radius Rc, the effective temperature T⋆ related to the stellar surface radius R⋆, the inner core radius Rc (at the bottom of the inflated envelope), the outer envelope radius Re, the stellar surface radius R⋆, the envelope extension ΔR in units of Rc, and Re, the mass Δ M of the outer shell (with Re < r < R⋆), the “minimum density” ρmin at the point of the lowest gas pressure within the inflated envelope, the ratio 
(with 
from Eqs. (18), (19)), the ratio Q according to Eq. (27), the parameters Wc and We according to Eqs. (19), (21), and the total pressure Pe at the outer envelope radius Re. The values given in this table are inferred numerically from our computational models, and may be compared to the analytical results from Sect. 3.
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