Time-dependent Theory Of Convection Research Articles

Convection Theory and Relevant Problems in Stellar Structure, Evolution, and Pulsational Stability I Convection Theory and Structure of Convection Zone and Stellar Evolution

A non-local and time-dependent theory of convection was briefly described. This theory was used to calculate the structure of solar convection zones, the evolution of massive stars, lithium depletion in the atmosphere of the Sun and late-type dwarfs, and stellar oscillations (in Part Ⅱ). The results show that: 1) the theoretical turbulent velocity and temperature fields in the atmosphere and the thermal structure of the convective envelope of the Sun agree with the observations and inferences from helioseismic inversion very well. 2) The so-called semi-convection contradiction in the evolutionary calculations of massive stars was removed automatically, as predicted by us. The theoretical evolution tracks of massive stars run at higher luminosity and the main sequence band becomes noticeably wider in comparison with those calculated using the local mixing-length theory (MLT). This means that the evolutionary mass for a given luminosity was overestimated and the width of the main sequence band was underestimated by the local MLT, which may be part of the reason for the contradiction between the evolutionary and pulsational masses of Cepheid variables and the contradiction between theoretical and observed distributions of luminous stars in the H-R diagram. 3) The predicted lithium depletion, in general, agrees well with the observation of the Sun and Galactic open clusters of different ages. 4) Our theoretical results for non-adiabatic oscillations are in good agreement with the observed mode instability from classic variables of high-luminosity red giants. Almost all the instability strips of the classical pulsating variables (including the Cepheid, δ Scuti, γ Doradus, βCephei, and SPB strips) were reproduced (Part Ⅱ).

Convection Theory and Related Problems in Stellar Structure, Evolution, and Pulsational Stability II. Turbulent Convection and Pulsational Stability of Stars

Using our non-local and time-dependent theory of convection and a fixed set of convective parameters (C1, C2/C1 , C3)= (0.70, 0.50, 3.0) calibrated against the Sun, the linear non-adiabatic oscillations for evolutionary models with masses 1–20 M⊙ are calculated. The results show that almost all the classical instability strips can be reproduced. The theoretical instability strips of δ Scuti and γ Doradusvariables agree well with Kepler spacecraft observations. There is no essential difference in the excitation mechanism for δ Scuti and γ Doradus stars. They are excited by the combined effects of the radiative κ-mechanism and coupling between convection and oscillations. They represent two subgroups of a broader type of δ Scuti and γ Doradus stars, located in the lower part of the Cepheid instability strip. δ Scuti is the p-mode subgroup and γ Doradus is the g-mode subgroup. The luminous variable red giants observed by MACHO and OGLE are low-order radial pulsators among low-mass red giant and asymptotic giant branch stars. The excitation and damping mechanism of oscillations for low-temperature stars is studied in detail. Convective flux and turbulent viscosity are consistent damping mechanisms. The damping effect of the convective enthalpy flux is inversely proportional to the frequency of the modes, so it plays an important role in stabilizing the low-order modes and defining the red edge of the Cepheid instability strip. The damping effect of turbulent viscosity reaches its maximum at 3ωτc/16∼1, where τc is the dynamic time scale of turbulent convection and ω is the angular frequency of the modes. Turbulent viscosity is the main damping mechanism for stabilizing the high-order modes of low-temperature variables. The turbulent pressure is, in general, an excitation mechanism; it reaches maximum at 3ωτc/4∼1, and it plays an important role for the excitation of red variables. Convection is not, in fact, a pure damping effect for stellar oscillations. The relative contributions of turbulent pressure, turbulent viscosity, and convective enthalpy flux for excitation and damping effects change with stellar parameters (mass, luminosity, effective temperature) and with the radial order and spherical harmonic degree of the oscillation mode; therefore, the combined effect of convection is sometimes damping, and sometimes the excitation of oscillations. Our research shows that, for low-luminosity red giants, the low-order modes are pulsationally stable, while the intermediate- and high-order modes are unstable. Toward higher luminosity, the range of unstable modes shifts gradually toward the lower order. All of the intermediate- and high-order modes become stable, and a few low-order modes become unstable for high-luminosity red giants. They show the typical pulsational characteristics of Mira-like variables. The variable red giants are, at least for the high-luminosity RGs, self-excited. For red giants, the frequency of the maximally unstable modes predicted by our theory is similar to that given by the semi-empirical scaling relation.

Turbulent convection and pulsation stability of stars – III. Non-adiabatic oscillations of red giants

We have computed linear non-adiabatic oscillations of luminous red giants using a non-local and anisotropic time-dependent theory of convection. The results show that low-order radial modes can be self-excited. Their excitation is the result of radiation and the coupling between convection and oscillations. Turbulent pressure has important effects on the excitation of oscillations in red variables.

Turbulent convection and pulsation stability of stars – II. Theoretical instability strip forδScuti andγDoradus stars

By using a non-local and time-dependent convection theory, we have calculated radial and low-degree non-radial oscillations for stellar evolutionary models with $M=1.4$--3.0\,$\mathrm{M}_\odot$. The results of our study predict theoretical instability strips for $\delta$ Scuti and $\gamma$ Doradus stars, which overlap with each other. The strip of $\gamma$ Doradus is slightly redder in colour than that of $\delta$ Scuti. We have paid great attention to the excitation and stabilization mechanisms for these two types of oscillations, and we conclude that radiative $\kappa$ mechanism plays a major role in the excitation of warm $\delta$ Scuti and $\gamma$ Doradus stars, while the coupling between convection and oscillations is responsible for excitation and stabilization in cool stars. Generally speaking, turbulent pressure is an excitation of oscillations, especially in cool $\delta$ Scuti and $\gamma$ Doradus stars and all cool Cepheid- and Mira-like stars. Turbulent thermal convection, on the other hand, is a damping mechanism against oscillations that actually plays the major role in giving rise to the red edge of the instability strip. Our study shows that oscillations of $\delta$ Scuti and $\gamma$ Doradus stars are both due to the combination of $\kappa$ mechanism and the coupling between convection and oscillations, and they belong to the same class of variables at the low-luminosity part of the Cepheid instability strip. Within the $\delta$ Scuti--$\gamma$ Doradus instability strip, most of the pulsating variables are very likely hybrids that are excited in both p and g modes.

Dependence of Pulsation Stability on the Helium Abundance of Stellar Convective Envelope

By using a non-local and time-dependent theory of convection, the linear non-adiabatic oscillations of radial and low-degree non-radial F-p8 modes are calculated for the stellar evolutionary models in the mass range of 1.4∼3.0 Mʘ, with three helium abundances of convective envelope (Y =0.28, 0.13, 0.00), and the solar metal abundance (Z =0.02). The numerical results show that the red edge of theoretical δ Scuti instability strip almost does not change with the helium abundance. With the decrease of the helium abundance, the blue edge of the δ Scuti instability strip moves toward the direction of low temperatures, the high-temperature stars on the hot side of instability strip become more stable, while the low-temperature stars on the cold side of instability strip become more instable. It seems to be impossible to explain the non-variable stars in the δ Scuti instability strip by using the diffusion of helium. However, the ratios of non-variable stars to variable stars in the hot and cold sides of the δ Scuti strip may be taken as the observational evidence of helium diffusion.

Solar-like and Mira-like oscillations of stars — A uniform excitation mechanism

Using a non-local and time-dependent theory of convection, we have calculated the linear non-adiabatic oscillations of the radial and low-degree F-p39 modes for evolutionary models from the main sequence to the asymptotic giant branch for stars with solar abundance (X = 0.70, Z = 0.02) in the mass range of 0.6–3.0 M⊙. The results show that low luminosity cool stars tend to be solar-like oscillators, whose low-order modes are stable, but intermediate and high order p-modes are pulsationally unstable; their unstable modes have a wide range in frequency and small values for amplitude growth rates. For stars with increasing luminosity and therefore lower temperature, the unstable modes shift towards lower orders, the corresponding range of frequency decreases, and the amplitude growth rate increases. High luminosity red giant stars behave like typical Mira-like oscillators. The effects of the coupling between convection and oscillations on pulsational instability have been carefully analyzed in this work. Our research shows that convection does not simply act as a damping mechanism for oscillations, and the complex nature of the coupling between convection and oscillations makes turbulent convection sometimes behave as damping, and sometimes as excitation. Such a picture can not only naturally account for the red edge of the instability strip, but also the solar-like oscillations in low luminosity red stars and Mira-like ones in high luminosity red giants.

A Theoretical Probe for Excitation Mechanisms of Solar-like and Mira-like Oscillations of Stars

The linear stability analysis of the radial and non-radial oscillations for the evolutionary model of a star with the mass of 0.6∼3 M8 has been per- formed by using the nonlocal and time-dependent convection theory. The results show that the unstable low-temperature stars on the right side of the instabil- ity strip in the HR diagram can be divided into two groups. One is of the stars of solar-like oscillations, composed of the main-sequence dwarfs, subgiants, and the red giants with low- and intermediate-luminosity, which are unstable in the intermediate- and high-order (nr ≥ 12) p-modes, but stable in the low- order (nr ≤ 5) p-modes. Another is of the Mira-like stars, composed of the luminous red giants and AGB stars, which are just contrary to the solar-like stars, unstable in the low-order (nr ≤ 5) p-modes, but stable in the intermediate- and high-order (nr ≥ 12) p-modes. On the red edge of Cepheid (δ Scuti) insta- bility strip, the oscillations of solar-like and Mira-like stars can be explained uniformly by the coupling between convection and oscillations (CCO). For the low-temperature stars on the right side of the instability strip, the CCO is the dominant excitation and damping mechanism for the low- and intermediate-order p-modes, and the stochastic excitation of turbulence becomes important only for the high-order p-modes of solar-like oscillations.

A Theoretical Probe for Excitation Mechanisms of Sun-like and Mira-like Oscillations of Stars

The linear nonadiabatic oscillations for evolutionary models of 0.6- 3M8 stars are calculated by using a nonlocal and time-dependent convection theory. The results show that in the HR diagram the pulsation-unstable low- temperature stars on the right side of instability strip can be divided into two groups. One group indicates the Sun-like oscillation stars composed of the main- sequence dwarfs, sub-giants and red giants (RGs) of low and intermediate lu- minosities, which are unstable in the intermediate- and high-order (n≥12) p- modes, and stable in the low-order (n≤5) p-modes. Another group indicates the Mira-like stars composed of the bright RGs and asymptotic giant branch (AGB) stars, which are just contrary to Sun-like stars, unstable in low-order (n≤5) p-modes and stable in the intermediate- and high-order (n≥12) p-modes. The oscillations for the red edge of Cepheid (δ Scuti) instability strip, Sun-like and Mira-like stars can be explained uniformly by the coupling between convection and oscillation (CCO). For the low-temperature stars on the right side of in- stability strip, CCO is the dominant excitation and damping mechanism of the oscillations of low- and intermediate-order p-modes, and the turbulent stochas- tic excitation becomes important only for the high-order p-modes of Sun-like oscillations.

Unstable Non-radial Modes of Upper Main-sequence Stars†

Using a non-local time-dependent theory of convection, the linear stability is analyzed for low-degree (l = 1 − 4) g39-p4 modes of the evolutionary sequences of stars with masses from 3M_ to 30M_. The influence of convection on the pulsation stability of the non-radial modes is studied. The results show that this influence is not negligible. However, if the details of the single star and the single mode are not concerned, the location and scope of instability strip are not affected significantly by convection.

The planetary system host HR 8799: on its λ Bootis nature

HR\,8799 is a $\lambda$ Bootis, $\gamma$ Doradus star hosting a planetary system and a debris disk with two rings. This makes this system a very interesting target for asteroseismic studies. This work is devoted to the determination of the internal metallicity of this star, linked with its $\lambda$ Bootis nature (i.e., solar surface abundances of light elements, and subsolar surface abundances of heavy elements), taking advantage of its $\gamma$ Doradus pulsations. This is the most accurate way to obtain this information, and this is the first time such a study is performed for a planetary-system-host star. We have used the equilibrium code CESAM and the non-adiabatic pulsational code GraCo. We have applied the Frequency Ratio Method (FRM) and the Time Dependent Convection theory (TDC) to estimate the mode identification, the Brunt-Va\"is\"al\"a frequency integral and the mode instability, making the selection of the possible models. When the non-seismological constraints (i.e its position in the HR diagram) are used, the solar abundance models are discarded. This result contradicts one of the main hypothesis for explaining the $\lambda$ Bootis nature, namely the accretion/diffusion of gas by a star with solar abundance. Therefore, according to these results, a revision of this hypothesis is needed. The inclusion of accurate internal chemical mixing processes seems to be necessary to explain the peculiar abundances observed in the surface of stars with internal subsolar metallicities. The use of the asteroseismological constraints, like those provided by the FRM or the instability analysis, provides a very accurate determination of the physical characteristics of HR 8799. However, a dependence of the results on the inclination angle $i$ still remains. The determination of this angle, more accurate multicolour photometric observations, and high resolution spectroscopy can definitively fix the mass and metallicity of this star.

Non-adiabatic oscillations of the low- and intermediate-degree modes of the Sun

Using a non-local time-dependent theory of convection, the linear stability is analysed for low- and intermediate-degree (l = 1-25) g4-p39 modes of the Sun. The results show that all the p modes with periods from similar to 3 to similar to 16 min are pulsationally unstable, while all the g, f and p modes with periods longer than similar to 16 min and the high-frequency p modes with periods shorter than similar to 3 min are stable except the low-degree (l = 1-5) p1 modes. The pulsation amplitude growth rates depend on only the frequency and almost do not depend on l. They achieve the maximum at nu similar to 3700 mu Hz (P similar to 270 s). The effects of radiation and convection on the stability are analysed in detail. The coupling between convection and oscillations plays a key role for stabilization of low-frequency f and p modes and excitation of intermediate- and high-frequency p modes. We propose that the solar 5-min oscillations are not excited by a single mechanism, but are resulted from the combined effects of the 'regular' coupling between convection and oscillations and the turbulent stochastic excitation. The coupling between convection and oscillations is the dominant excitation and damping mechanism for the low- and intermediate-frequency modes, and the turbulent stochastic excitation is the main excitation mechanism for high-frequency p modes.

On the Excitation Mechanism of Solar Five-minute Oscillations

The excitation mechanism of solar five-minute oscillations is studied in the present paper. We calculated the non-adiabatic oscillations of low- and intermediate-degree ( l = 1 − 25) g4-p39 modes for the Sun. Both the thermodynamic and dynamic couplings are taken into account by using our non-local and time-dependent theory of convection. The results show that all the lowfrequencyf- and p-modes with periods P > 5 .4 min are pulsationally unstable, while the coupling between convection and oscillations is neglected. However, when the convection coupling is taken into account, all the g- and low-frequency f- and p-modes with periods longer than ∼16 minutes (except the low-degree p1-modes) and the high frequency p-modes with periods shorter than ∼3 minutes become stable, and the intermediate-frequency p-modes with period from ∼3 to ∼16 minutes are pulsationally unstable. The pulsation amplitude growth rates depend only on the frequency and almost do not depend on l. They achieve the maximum at ν ∼ 3700 μHz (or P ∼ 270 sec). The coupling between convection and oscillations plays a key role for stabilization of low-frequency f- and p-modes and excitation of intermediate-frequency p-modes. We propose that the solar 5-minute oscillations are not caused by any single excitation mechanism, but they are resulted from the combined effect of “regular” coupling between convection and oscillations and turbulent stochastic excitation. For low- and intermediatefrequency p-modes, the coupling between convection and oscillations dominates; while for high-frequency modes, stochastic excitation dominates.

Non-adiabatic oscillations of red giants

Abstract Using our non-local time-dependent theory of convection, the linear non-adiabatic oscillations of 10 evolutionary model series with masses of 1–3 M⊙ are calculated. The results show that there is a red giant instability strip in the lower temperature side of the Hertzsprung–Russell diagram which goes along the sequences of the red giant branch and the asymptotic giant branch. For red giants of lower luminosities, pulsation instability is found at high order overtones; the lower order modes from the fundamental to the second overtone are stable. Towards higher luminosity and lower effective temperature, instability moves to lower order modes, and the amplitude growth rate of oscillations also grows. At the high luminosity end of the strip, the fundamental and the first overtone become unstable, while all the modes above the fourth order become stable. The excitation mechanisms have been studied in detail. It is found that turbulent pressure plays a key role for excitation of red variables. The frozen convection approximation is unavailable for the low temperature stars with extended convective envelopes. In any case, this approximation can explain neither the red edge of the Cepheid instability strip, nor the blue edge of the pulsating red giant instability strip. An analytic expression of a pulsation constant as a function of stellar mass, luminosity and effective temperature is presented from this work.

An asteroseismic Main Sequence model for the δ Scuti star 44 Tau

Antoci et al. (2007) have found that 44 Tau oscillates with 13 independent frequencies with amplitudes higher than the observational limit of 0.6 mmag. The observations were made in the two Stromgren filters (v,y) which allow good discrimination of the spherical harmonic of the corresponding non-radial mode, as it was shown by Garrido et al. (1990). The star has Teff = 6900 ± 150K ; log g = 3.6 ± 0.05 (log L/L = 1.3 ± 0.2) and solar ([M/H] = 0.0 ± 0.1) composition. Different diagnostic diagrams for amplitude ratios and phase differences have been calculated using the “time-dependent convection (TDC)” described by Grigahcene et al. (2005) and Dupret et al. (2004) together with atmospheric parameters taken from Heiter et al. (2002) and Barban et al. (2003). All of these point to the tentative modal identification depicted in Table 1. The ratio f1/f5 = 0.7698± 0.0005 indicates that these two frequencies correspond to the radial fundamental and first overtone. We can then use the Petersen diagrams to fix the evolutionary status of 44 Tau. Rotation is very low and has not been taken into account (see Suarez et al. 2006). Models with 1.85 M and solar composition do not fit the frequency ratio at the required period. We have tried different masses, compositions and overshooting parameters and the only reasonable combinations allow masses from 1.90 to 2.00 M and main sequence models (i.e. hydrogen not yet exhausted in the core). Work is in progress to find the best model (in a least squares sense) but the closest theoretical frequencies to those observed which agree with the mode identification indicated in Table 1 are for a model with 1.94 M , αov = 0.3, [M/H] = 0.10 and an age of 1130 Myr where the “avoiding crossing” phenomenon is present. The predicted range of instability depends on the α parameter: low values represent better the excited range of this star when time dependent convection theory is used. In conclusion, the observed frequencies of 44 Tau can be theoretically explained by a Main Sequence model of 1.94 M with a small metal overabundance and a moderate overshooting at an effective temperature 150 K lower than indicated by the photometric calibration. All the proposed frequencies fit the following constraints: colour identification, instability range, photometric visibility and period ratios for radial modes (Petersen diagrams). Besides, if our identification is correct, then values up to 2 are visible in this star, at the above mentioned photometric precision, and then three more new frequencies are predicted to be unstable in the observed range which could be detectable from new ground based observations.

Time Dependent Convectionvs. Frozen Convection Approximations

The approximation known as Frozen Convection has been widely used in the context of stellar oscillations due to the lack of a Time Dependent Convection theory. This approximation supposes to neglect the interaction between convection and oscillations and there are different ways to do it. The choice can affect theoretical predictions concerning the non-adiabatic treatment of the stellar oscillations. In fact, fictitious excitation or damping of the pulsation modes can be produced depending on the approximation used.

Convection-pulsation interaction: theories and applications

The problem of the interaction between convection and pulsations is considered. A precise modeling of this interaction is required for the seismic study of stars with non-negligible convective envelope. After a short introduction to the Time-Dependent Convection theory of Gabriel (1996) and Grigahc ene et al. (2005), we give a summary of the main results obtained with this treatment for Sct, Dor and solar-like stars.

Small amplitude variable red giants

AbstractSmall amplitude variable red giants were discovered long time ago (Eggen 1969, Henry et al. 2000, Jorison et al. 1997). The revolutionary turning point for the study of this type of variable was due to the work of Wood (2000) and Soszyński et al. (2004) based on the very large stellar sample of the Magellanic clouds MACHO project and OGLE photometric data. By using our non-local time-dependent theory of convection (Xiong 1989), we carried out a linear stability survey for red giants with initial mass of M = 1 − 3M⊙. Explanations for the observed variabilities and analysis of mode identifications in red giant stars are presented in this work.

Inclusion of a time-dependent, non-local convection theory in a stellar evolution code

AbstractThe non-local, time-dependent convection theory of Kuhfuß (1986) in both its one- and three-equation form has been implemented in the Garching stellar evolution code. We present details of the implementation and the difficulties encountered. Specific test cases have been calculated, among them a 5 M⊙ star and the Sun. These cases point out deficits of the theory. In particular, the assumption of an isotropic velocity field leads to too extensive overshooting and has to be modified at convective boundaries. Some encouraging aspects are indicated as well.

Non-linear pulsation models of red giants

There are currently unresolved differences between fundamental mode red giant pulsation models and the large amplitude pulsation behaviour of real stars. In particular, current models tend to be driven too hard and they consequently have excessive amplitudes. This may be a result of the convection energy transport approximations used. We have developed a new pulsation code which includes a time-dependent convection theory, turbulent pressure and turbulent viscosity. Exploratory calculations performed using this code show that the turbulent pressure and turbulent energy transport seem to have little effect on pulsation properties such as velocity, radius and light variation. All calculations also reveal the presence of sharp luminosity spikes that are not observed in real stars. The calculations indicate that the inclusion of a turbulent viscosity in the models may be what is required to bring the model pulsation amplitudes down into agreement with observations. Calculations for a high mass and luminosity model show a double-peaked light curve comparable to that observed for high-luminosity Large Magellanic Cloud Miras.

On the Red Edge of the δ Scuti Instability Strip

The delta Scuti star catalogue is used to derive the observational locations of such stars on the HR diagram. The theoretical and observational instability strips are compared to check the theoretical red edge obtained by considering non-local time-dependent convection theory. The observational instability strip almost overlaps with the theoretical one, but the observed blue and red envelopes are hotter than the theoretical edges. The distribution of delta Scuti stars in the pulsation strip is not uniform.