The instability strip and pulsation features of postAGB star models
T. Aikawa
Tohoku gakuin University, Sendai 9813193, Japan
Received 27 July 2009 / Accepted 15 January 2010
Abstract
Aims. The instability strip and characteristics of the
nonlinear behavior of the radial pulsation are examined for the
observed photometric variability of postAGB stars.
Methods. A linear stability analysis for radial pulsation modes is performed in the domain
,
,
assuming a total mass
and
for postAGB stars. Radial modes up to the 5th overtone are considered,
since we expect that there are some strange modes amongst highorder
modes. Nonlinear simulations are tried for all the models, which have
at least one pulsationally unstable mode.
Results. Pulsationally unstable modes are found in a large
region of the domain, and these modes may be responsible for the
observed variability. Nonlinear simulations on models that have
unstable modes with higher overtones reveal the variability through
small peaktopeak variations. These variations are characteristics of
the observed variations for some postAGB stars located in the bluer
region of the classical instability strip.
Key words: stars: oscillations  stars: AGB and postAGB  hydrodynamics
1 Introduction
PostAGB stars are luminous objects that evolve on a fast track to hotter effective temperatures at roughly constant luminosity after AGB evolution by a very strong mass loss. In the course of such evolution they cross the classical instability strip extended to higher luminosity. Photometric variability observed in some postAGB stars thus are believed to be caused by stellar pulsation. However, the behavior of pulsation in the postAGB stars may be quite different from those in classical Cepheid variables, since the postAGB stars are low mass stars but they have high luminosities.
In this study, we first summarize the behavior of the photometric variability for a selected sample of postAGB stars and postAGB candidates. Then we construct the envelope models for linear analysis and nonlinear simulation of radial pulsation, intending to explain the observed photometric variability due to radial pulsation of postAGB envelopes.
Pulsation is sometimes a key to understanding the nature of postAGB stars (Córsico et al. 2007) and postAGB related stars, such as FG Sagittae (Whitney 1978; Jeffery & Schönberner 2006). For a specific star, HD 56126, Jeannin et al. (1996, 1997) and Barthès et al. (2000) reported the results of nonlinear simulation of radial pulsation for comparison with the observed light variation. This study was the first attempt at direct comparison of the variability in postAGB stars with theoretical nonlinear pulsation models.
Kiss et al. (2007) studied the instability strip of postAGB stars based on 30 known and candidate binary postAGB stars. They confirmed the instability strip; besides this, on the bluer side of the instability strip, they found some variable stars where the peaktopeak lightcurve amplitudes are relatively small (less than 0.5 mag) compared with the major stars in the strip. They found a few similar stars at redder side, too.
Schmidt (2001) presents a table of spectroscopic parameters ( and g) of selected postAGB stars based on the literature. We start our discussion with postAGB stars in the table to confirm the existence of the region for the small amplitude variables that are bluer than the classical instability strip, as mentioned by Kiss et al. (2007) and to define the extension of the region of smallamplitude variables beyond the classical instability strip, though this table is preliminary (cf. Szczerba et al. 2007). We first searched in the literature for the photometric variability of the postAGB stars in the table. We summarize the result in Fig. 1. It is pointed out that the photometric variability in the literatures is mainly reported for the stars in the late type region of the HR diagram, and variable stars with small amplitude (the peaktopeak variation less than 0.5) are located in a hotter part of the region. Pulsation stars classified as RV Tau stars are located in the main part of the region.
Figure 1: The features of photometric variability of postAGB on the . Filled circles indicate stars with peaktopeak variation greater than 0.5 mag, empty circles correspond to stars with no remarks on photometric variability, and filled triangles depict variable stars of the peaktopeak variation less than 0.5. HD 46703 (see text) is marked with a filled square. 

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In summary, the stars that show photometric variability are confined to the late type region where is less than 3.90. The smallamplitude variations are observed in a bluer region of the main domain. In Fig. 1, in the smallamplitude region, there are several stars (i.e., IRAS Z02229+6208, IRAS 04296+3429, ROA 24, AFGL 2688) that have no remarks on the variability and that might also have smallamplitude variations.
Probably in the table we have missed some interesting variable stars in the earlytype region classified as luminous hothelium stars (Saio and Jeffery 1988). The cause of variability is known as irongroup bump in the opacity (Saio 1995; Fadeyev & LynasGray 1996), and the stars are definitely related to postAGB stars.
In this paper, we assume that the variability of the postAGB stars is caused by radial pulsation of the envelope. In particular, we stress that the smallamplitude variability may be explained by the strange mode in radial pulsation. For examining the amplitude of model pulsation, we perform nonlinear simulation to get the pulsation behavior in the nonlinear regime. We first examine the linear stability of radial modes, and then the nonlinear simulations are performed for the models with pulsationally unstable modes. The nonlinear simulation, as well as linear analysis, are performed with a hydrodynamic code with the variable Eddington factor approximation of radiative transfer (Aikawa 2008).
In this paper we have entirely ignored convection effects. For a static point of view, convection acts an energy transporter, so the radiative transfer becomes less effective than a purelly radiative envelope. It is expected that convection may stabilize pulsation. But there must be a dynamic coupling between convection and pulsation. It is thus desirable to include convection, although Stothers (2003) studies the effects of convection on the stability of luminous envelopes qualitatively.
2 Hydrostatic models
We first construct the envelope models according to observed
effective temperature(
)
and surface gravity(). We
adopt 0.6 and
as the total mass of postAGB stars.
According to the hydrogenrich envelope mass vs. the effective
temperature relationship by Paczynski (1971) and
Schönberner (1981, 1983),
Blöcker (1995) and Frankowski (2003), the envelope mass in the range of observed effective temperature is about
,
so the total mass should
be equal to the core mass. To convert
to the luminosity, we use the following equation
with using
and
are 5777 K and 4.4378 for the
Sun, respectively (Allen 2000):
(1) 
The hydrostatic equilibrium envelope models were constructed as described by Aikawa (2008). In this method the variable Eddington factor approximation is applied to radiative hydrodynamics instead of the diffusion approximation. Chemical abundances are assumed as X=0.700, Y=0.298, Z=0.002. The metal abundance ratio is a simplified version of the solar abundance ratio. We used OP opacity (Seaton 1992; Seaton et al. 1994) for the Roseland mean and Planck mean opacities. For regions of low density and low temperature, the opacities supplied by Alexander & Ferguson (1994) are used. The equations of radiative hydrodynamics are reduced to finite difference equations, and there are about 200 zones for the difference equations. Equations of the hydrostatic equilibrium and equation of radiative transfer are solved iteratively until obtaining a consistent solution. We denote the Eddington factor in the hydrostatic equilibrium as . The hydrostatic equations are integrated inward from the surface, defined as the radius of the optical depth , until the ratio of the radius (R) of the bottom to one at the surface () becomes less than 0.05. For the mass , it is 0.005 at most. We assume that radial pulsation is confined to the region defined by the two radii.
3 Linear analysis
The linear nonadiabatic eigenvalue problem for radial modes was solved by using as the Eddington factor. While sometimes the Eddington factor estimated by the linear perturbed quantities may yield quantitatively different results on the growth rates for strange modes (Zalewski 1992), we used as the Eddington factor for linear analysis for simplicity. Since we are interested in the strange mode in postAGB models, higher modes up to the fifth overtone were examined.
Since the present models have strong nonadiabatic effects, the periods and the growth rates deviate strongly from those in the adiabatic approximation (Wood 1976). We thus followed Aikawa (2008) to get the periods and the growth rates of radial pulsation at the fully nonadiabatic condition. Starting in the adiabatic condition, the periods and the growth rates were calculated stepbystep by increasing the effects of nonadiabaticity. The increment of the effect was carefully set to ensure the succession in the order of modes from those of an adiabatic case where we may know the complete set of the eigenvalue problem. Thus, we may label the order of modes in fully nonadiabatic approximation with the correspondig modes in adiabatic condition, and so sometimes the order of modes in the fully nonadiabatic approximation is not the decrease in the periods, which is a definition of the order of modes in adiabatic condition(see, Fig. 2). The periods of strange modes that appear as unstable highorder modes with high growth rates seriously drift among those of other modes. Probably this comes from the strange modes being caused mainly by the hydrogen ionization zone at outer envelopes (Aikawa & Sreenivasan 1996), so those modes are strongly affected by nonadiabatic effects.
Figure 2: The results of linear analysis for model sequences for fixed effective temperatures ( for . The ordinate corresponds to of the models, and the abscissa corresponds to the linear pulsation periods in a logarithmic scale. Linear nonadiabatic analysis is applied to radial modes up to the 5th overtone mode. The numbers at left of each curve denote the radial order of the modes (0: the fundamental, 1: the 1st overtone ...). Pulsationally stable modes are indicated by a hollow circle, and unstable modes are marked by a filled circle. 

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Figure 3: Same as Fig. 2, but for the case of . 

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The results are summarized in Fig. 2 for 0.6 . For lower effective temperature models, such as , the fundamental and/or the 1stovertone modes are pulsationally unstable. For higher effective temperature models, however, these loworder modes become stable but highorder modes become unstable. This tendency first appears in the models of middle values. Finally, all models are stable in the loworder modes, but unstable in highorder modes at higher effective temperature (for instance, ). For , all the modes up to the 5th overtone are stable in the range of relevant to the observed postAGB. The results are summarized in Fig. 3 for 0.8 . The characteristics of instability are almost the same as in the case of 0.6 . For , all the modes are also stable.
For both the masses of and , there are pulsationally unstable modes of high overtone in the model sequences of higher effective temperatures ( ). Aikawa (1991, 1993) showed that these pulsationally unstable highorder modes have the characteristics of the strange mode. In some models, only these high overtone modes are unstable, so these strange modes should be responsible for the photometric variability, if radial modes are assumed to be the cause of that.
Figure 4: The instability strip resulting from the linear analysis for . Models that are pulsationally stable up to the 5th overtone modes are shown by circles, models where the fundamental and/or the 1stovertone modes are unstable are displayed with filled circles, and models where only higher modes than 1stovertone mode are unstable are depicted with filled triangles. 

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Figure 5: Same as in Fig. 4, but for . 

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In Figs. 4 and 5, we show summary maps of the instability strip of the radial modes for and . For both the masses, there is a instability region that comes from high overtone modes (probably strange modes) at a hotter regime than in the case of the fundamental and 1stovertone modes.
The instability of the fundamental, and the 1stovertone at higher at and 3.70 comes from the model location just inside the blue edge of the classical instability strip extended to the high luminosity of lowmass stars (for instance W Verginis stars, Bridger 1985). On the other hand, the instability due to the fundamental and the 1stovertone modes at lower are those found by Saio et al. (1984) and may be related to the opacitymodified Eddington limit (Asplund 1998).
4 Nonlinear simulation
Nonlinear simulations for models with pulsationally unstable modes were performed with the variable Eddington factor, (Aikawa 2008), snapshot values of the Eddington factor at each time step. The simulation of each model was started from the hydrostatic equilibrium except for the velocity perturbation. The velocity profile was the form of the eigenfunction and the amplitude of the velocity perturbation at the surface is 1 km s^{1}, which is quite small compared to one expected to the limit cycles of classical Cepheids.
Besides not taking convection into account, the main ambiguities of
nonlinear modeling come from the artificial viscosity to stabilize
calculations and the boundary condition at the surface. For the
artificial viscosity, we
used the formula of the von NeumannRichtmyer type with a cutoff
(Stellingwerf 1975a,b), i.e.,
(2) 
where P_{i} and V_{i} are ordinary pressure and specific volume at the cell between i and i+1 shells, respectively, and u_{i}s are velocities of the shell at i. The constants of C_{Q} = 4 and are used.
Figure 6: The photospheric light curves predicted by nonlinear models for and . The ordinate corresponds to the time in unit of days, and the abscissa corresponds to the photometric magnitude. The values of the surface gravity are indicated for each curve. The scale of the magnitude is valid for each curve that is shifted vertically for clarity. 

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We assumed the perfect reflection of the wave at the surface as one of the necessary boundary
condition at the outer boundary:
(3) 
The pressure at the outermost cell was assumed to be equal to the radiative pressure evaluated by other quantities at the cells. This condition would be invalid for models in which the velocities of some outer shells exceed their escape velocities, and the wave may propagate beyond the outer boundary.
The resulting finite difference equations are time dependent and solved by the fully implicit scheme (Castor et al. 1977). Simulation for each model was followed enough to confirm that the pulsation is settled into a stationary state. The results are depicted as light curves at the model photosphere in Figs. 68 for .
Figure 7: Same as in Fig. 6 but for and . 

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Figure 8: Same as in Fig. 6 but for . The upper scale is applied to the upper four curves, and the lower scale is valid for the lower three curves. Only the transient light curve is plotted for . 

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We notice that the amplitudes of nonlinear pulsation in a stationary state are quite small in higher effective temperature models (for instance ). These amplitudes are much small than those observed in postAGB variables. The lower the models effective temperature, the larger the amplitudes. For , the amplitudes are comparable to those observed, though the pulsation driving modes are still highorder modes and probably strange modes. For even lower temperature, there are large amplitude pulsations at a higher region due to loworder modes(mainly the fundamental mode), as expected, though there are smallamplitude pulsation caused by highorder modes at lower region. It is known that pulsation amplitudes in stars in the instability strip of lowmass supergiant stars classified as W Virginis stars are quite large with propagating shock wave features (Davis 1972). The fundamental pulsation at higher is the same. We also notice that stationary pulsations at lower models are irregular with small amplitudes, though those at higher are regular. This feature of nonlinear pulsation may be useful for comparing to observed features.
Figures 911 show the results for . For a wide range of the HR diagram the pulsation behavior for a parameter set with and is essentially the same as in the model of the same parameter of . For , quite regular pulsations are realized but these are too small amplitudes again. For lower temperatures, such as , the amplitudes of regular pulsation become larger, and irregular pulsation is also realized for lower gravity models. For even lower temperature, regular welldeveloped pulsation sets in for higher gravity and irregular pulsation for lower gravity (see, Fig. 11).
Figure 9: Same as in Fig. 6, but for of . 

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Figure 10: Same as in Fig. 6, but for and . 

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Figure 11: Same as in Fig. 6, but for and . The upper scale is applied to the upper four curves, and the lower scale is valid for the lower three curves. 

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It is very impressive that the variation amplitudes due to the instability of highorder modes are quite small compared with those from the instability of loworder modes. The highorder modes that are responsible to the variation are strange modes, and they have pulsation periods close to those of strongly damped modes, so nonlinear coupling between the strange modes and the very strong damped modes should be strong. This may suppress the amplitude of highorder modes drastically.
It is interesting to see the transition from regular pulsation to irregular pulsation as seen in Figs. 711 at low values of . The period doubling (Buchler et al. 1987) and the intermittency (Aikawa 1987) have been found in pulsation models as the transition routes from regular to irregular pulsation. To see such a transition in the present model sequence, more fine tuning of the model parameters would be necessary.
5 Discussion and concluding remarks
The instability strip for postAGB pulsation was examined. On the bluer side of the classical instability strip of loworder pulsation modes, there is a significant size of the instability domain of highorder modes. The region may be responsible for the photometric variability of some postAGB stars with small peaktopeak variations.
For the models with the parameters of the bluer instability region, nonlinear simulations performed with a hydrodynamic code with the variable Eddington approximation of radiative transfer reveal small peaktopeak variations like those observed in some of postAGB stars. The time scale of the variations in general is comparatively shorter than that observed (for instance, Hrivnaket al. 2001). However, Bond et al. (1984) describe the variation of HD 46703 (which is included in Fig. 1) as brightness variations of 0.1 mag on the time scale of a week. The pulsation behavior of our models may be a perfect match to this star.
To get the stationary pulsation state of nonlinear models, we met some difficulties on numerical treatments during the nonlinear simulation. Starting with the initial conditions as mentioned before, we could run the models for a while, but the convergence of implicit calculation became very poor, and so we could not continue the simulation to confirm stationary state of nonlinear pulsation. Those are models with higher , lower (for instance, , ), and lower (for instance ). The first may be related to the opacitymodified Eddington limit (Asplund 1998). The region where the opacitymodified Eddington limit realized so the radiation pressure is dominant becomes close to the dynamical instability. Tuchman et al. (1979) show that pulsation becomes violent and mass loss may occur in the model envelope close to the dynamical instability. We confirmed that the velocity of the outer shell reached the escape velocity during pulsation. We need a pulsation code that includes running waves for these stars. The second is the models in the redder region of the instability strip and nonlinear amplitudes became too large to continue the simulation, so our analysis should include some effects of convection and/or pulsationinduced turbulence to get the stationary state of moderate nonlinear amplitudes. There have been some attempts as nonlinear pulsation related to the region (Fadeyev & Tutukov 1981; Fadeyev 1982; Fadeyev & Muthsam 1990).
RV Tau stars are mainly located in the main part of the instability strip of postAGB stars. Pulsation behavior of the models for parameters corresponding to that region should reproduce the photometric variations of RV Tau stars that are supposed to be postAGB stars. AC Her and R Sct are ones that have been observed well and monitored for a long time. Kolláth (1990) and Kolláth et al. (1998) compiled data of the photometric variation of these two stars and tried to characterize the longmonitored pulsation. At the same time, they mention that the behavior of the variations of these two stars is quite different. The difference may come from the their locations located at different parts of the instability strip, because the effective temperatures of R Sct is estimated as 4500 K, while it is 5900 K for AC Her. This difference may cause the quite large difference in photometric variation, because AC Her may be located at the edge of the instability bay with a small amplitude pulsation, while R Sct is located well within the main instability strip. It is interesting to compare the models with observations using these well monitored data. We will discuss this problem in a future paper.
AcknowledgementsPart of this work was supported by the Japanese GrantinAid for Scientific Research of the Ministry of Education, Culture, Sports, Science, and Technology, project number 14540229.
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All Figures
Figure 1: The features of photometric variability of postAGB on the . Filled circles indicate stars with peaktopeak variation greater than 0.5 mag, empty circles correspond to stars with no remarks on photometric variability, and filled triangles depict variable stars of the peaktopeak variation less than 0.5. HD 46703 (see text) is marked with a filled square. 

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In the text 
Figure 2: The results of linear analysis for model sequences for fixed effective temperatures ( for . The ordinate corresponds to of the models, and the abscissa corresponds to the linear pulsation periods in a logarithmic scale. Linear nonadiabatic analysis is applied to radial modes up to the 5th overtone mode. The numbers at left of each curve denote the radial order of the modes (0: the fundamental, 1: the 1st overtone ...). Pulsationally stable modes are indicated by a hollow circle, and unstable modes are marked by a filled circle. 

Open with DEXTER  
In the text 
Figure 3: Same as Fig. 2, but for the case of . 

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In the text 
Figure 4: The instability strip resulting from the linear analysis for . Models that are pulsationally stable up to the 5th overtone modes are shown by circles, models where the fundamental and/or the 1stovertone modes are unstable are displayed with filled circles, and models where only higher modes than 1stovertone mode are unstable are depicted with filled triangles. 

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In the text 
Figure 5: Same as in Fig. 4, but for . 

Open with DEXTER  
In the text 
Figure 6: The photospheric light curves predicted by nonlinear models for and . The ordinate corresponds to the time in unit of days, and the abscissa corresponds to the photometric magnitude. The values of the surface gravity are indicated for each curve. The scale of the magnitude is valid for each curve that is shifted vertically for clarity. 

Open with DEXTER  
In the text 
Figure 7: Same as in Fig. 6 but for and . 

Open with DEXTER  
In the text 
Figure 8: Same as in Fig. 6 but for . The upper scale is applied to the upper four curves, and the lower scale is valid for the lower three curves. Only the transient light curve is plotted for . 

Open with DEXTER  
In the text 
Figure 9: Same as in Fig. 6, but for of . 

Open with DEXTER  
In the text 
Figure 10: Same as in Fig. 6, but for and . 

Open with DEXTER  
In the text 
Figure 11: Same as in Fig. 6, but for and . The upper scale is applied to the upper four curves, and the lower scale is valid for the lower three curves. 

Open with DEXTER  
In the text 
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