Stability and Pulsations of Wolf - Rayet Stars

Transcription

1 Stability and Pulsations of Wolf - Rayet Stars Talk presented on the conference Hydrogen - Deficient Stars Tübingen, September 20, 2007 Wolfgang Glatzel Institut für Astrophysik, Universität Göttingen

2 in collaboration with S. Wende S. Schuh S. Chernigovski M. Grott M. Kiriakidis

3 Outline Linear stability of Wolf - Rayet Stars Strange modes, instability Dependence on stellar parameters WR 123: Observation and interpretation Final result of strange mode instabilities in Wolf - Rayet stars (see also the poster by Wende et al.) Nonlinear pulsations Caveats Energies, Acoustic luminosities Pulsationally driven winds (?)

4 Linear analysis Models: generalized main sequences, parameter: mass Standard linear nonadiabatic stability analysis growth rates in the dynamical regime

5 He-ZAMS (X, Y, Z) = (0, 0.98, 0.02) First Prev Next Last Go Back Full Screen Close Quit

6 Occurrence of additional modes = strange modes Strange modes come in pairs with approximately complex conjugate symmetry Associated with instabilities having growth rates in the dynamical regime NAR approximation: heat capacity of the stellar envelope artificially set to zero

7 He-ZAMS (X, Y, Z) = (0, 0.98, 0.02) NAR approximation First Prev Next Last Go Back Full Screen Close Quit

8 Strange modes still exist in the NAR approximation Consequences of vanishing heat capacity Matter cannot store heat, luminosity perturbation vanishes Thermal modes cannot exist, strange modes are acoustic modes Thermodynamics irrelevant, mechanical system System is time reversible, eigenvalues come in complex conjugate pairs Instability mechanisms of Carnot type (ε, κ mechanism) excluded

9 Model for mechanism of strange mode instability based on considering limit of vanishing gas pressure in the NAR approximation Local analysis (ω: frequency, k: wavenumber) provides dispersion relation ω 2 = ikgκ ρ g: gravity; κ ρ : logarithmic derivative of opacity with respect to density Conditions for the occurrence of strange mode instabilities: dominant radiation pressure and κ ρ 0. κ T is essentially irrelevant. Differential work integral has oscillatory character. Interpretation of differential work integral: caveat!!

10 Differential work integral. Model: He-ZAMS (X, Y, Z) = (0, 0.98, 0.02); M = 18.25M First Prev Next Last Go Back Full Screen Close Quit

11 Dependence on chemical composition: ZAMS with (X, Y, Z) = (0.2, 0.78, 0.02) Similar result for WC and WO composition

12 Dependence on chemical composition: ZAMS with (X, Y, Z) = (0, 0.999, 0.001) Indirect effect: Reduced opacity decreases fraction of radiation pressure

13 Dependence on κ T : ZAMS with (X, Y, Z) = (0, 0.98, 0.02) κ T is artificially set to zero First Prev Next Last Go Back Full Screen Close Quit

14 Dependence on the harmonic degree of the perturbation Model (ZAMS): (X, Y, Z) = (0, 0.98, 0.02); M = 16M First Prev Next Last Go Back Full Screen Close Quit

15 WR 123 MOST Photometry (Lefèvre et al., 2005): Light variation with a period of 9.8h Strange mode periods for generalized ZAMS models: below 1h Driving by strange mode instability rejected Dorfi, Gautschy & Saio (2006): Discrepancy removed by adopting correct radius for WR 123 Radius of WR 123: 15.3R (Crowther 1997); ZAMS - radii: 1R Period - density relation: P R 3/2 Variety of hydrodynamical models of strange mode driven pulsations for WR 123 with periods in the observed range

16 Townsend & MacDonald (2006): Instability of l=1,2 g-modes driven by the κ - mechanism due to an opacity bump around log T 6.25 in models appropriate for WR 123 Period of unstable modes depend on mass fraction of hydrogen: Hydrogen - depleted models consistent with observations both with respect to chemical composition and periods Strange mode driven pulsations (Dorfi et al. 2006) are attributed to a for WR 123 unrealistically high hydrogen mass fraction of X = 0.35 and therefore rejected - see, however below Normalized growth rates of g-modes at least 3 orders of magnitude below that of strange modes

17 Nonlinear analysis Direct numerical simulation of the evolution of strange mode instabilities from hydrostatic equilibrium through the linear range of exponential growth into the nonlinear regime Only subphotospheric layers are considered Initial models: Hydrostatic envelopes with prescribed chemical composition, mass and effective temperature. Luminosity is taken from the mass - luminosity relation of the corresponding generalized ZAMS. So far only 1D simulations in spherical geometry available Requirement: without external perturbation the code has to pick up the correct predetermined unstable modes from numerical noise in hydrostatic equilibrium Linear regime may be used for validation

18 Evolution of the photospheric velocity Initial model: (X, Y, Z) = (0, 0.98, 0.02); M = 13.04M ; T eff = 90000K First Prev Next Last Go Back Full Screen Close Quit

19 Nonlinear regime: Variation of bolometric luminosity (red) and effective temperature (green) Initial model: (X, Y, Z) = (0, 0.98, 0.02); M = 13.04M ; T eff = 90000K First Prev Next Last Go Back Full Screen Close Quit

20 Snapshot in the nonlinear regime Initial model: (X, Y, Z) = (0, 0.98, 0.02); M = 13.04M ; T eff = 90000K First Prev Next Last Go Back Full Screen Close Quit

21 Evolution of the effective temperature Initial models: (X, Y, Z) = (0, 0.98, 0.02); M = 13.04M ; T eff = 120 kk, 90 kk and 60 kk All models reach a similar final mean effective temperature around 30 kk

22 Nonlinear regime: Variation of bolometric luminosity Initial models: (X, Y, Z) = (0, 0.98, 0.02); M = 13.04M ; T eff = 120 kk (bottom panel), 90 kk (middle panel) and 60 kk (top panel) First Prev Next Last Go Back Full Screen Close Quit

23 HRD: Initial and final mean effective temperatures (X, Y, Z) = (0, 0.98, 0.02) First Prev Next Last Go Back Full Screen Close Quit

24 Scatter plot of the opacity in the outer part of the envelope for various time dependent models Outer boundaries (final mean effective temperatures) are only found where the opacity increases with temperature

25 Final result of strange mode instabilities in Wolf - Rayet stars: Finite amplitude pulsations Shock waves inflate the envelope: Nonlinear periods are considerably higher than the linear ones. P 10h can be achieved even with X = 0. Existence of three unique attracting final mean effective temperatures (opacity effect?) Velocity amplitudes O 100 km/sec (up to v esc /2)

26 Caveats Outer boundary of the model does not correspond to the physical boundary of the system Boundary conditions ambiguous Optically thin regime disregarded, restriction to subphotospheric levels Spherical symmetry Resolution of shock waves requires artificial viscosity

27 Energy balance (hydrostatic values subtracted): gravitational (red) and internal (green) energies, time integral of the luminosity (blue) Initial model: (X, Y, Z) = (0, 0.98, 0.02); M = 17M ; T eff = 60kK Final period: 11.24h

28 Energy balance (hydrostatic values subtracted): kinetic energy (red), sum of previous terms (green) Initial model: (X, Y, Z) = (0, 0.98, 0.02); M = 17M ; T eff = 60kK

29 Energy balance: time integral of the acoustic luminosity Initial model: (X, Y, Z) = (0, 0.98, 0.02); M = 17M ; T eff = 60kK Mean slope corresponds to the mechanical luminosity generated by the instabilities

30 Error in the energy balance Initial model: (X, Y, Z) = (0, 0.98, 0.02); M = 17M ; T eff = 60kK

31 Determination of acoustic luminosities requires fully conservative numerical schemes Acoustic luminosities depend on numerical viscosity Determined values are to be regarded as lower limits Maximum values: L acoustic ergs/sec corresponding to 10 6 M /y (using L acoustic 1 2Mv esc) 2 Are winds in Wolf - Rayet stars driven by strange mode pulsations? M