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The Demarcation Point from E-AGB Stars to TP-AGB Stars in HR Diagram for Medium-mass Stars
ELSEVIER
CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy and Astrophysics 36 (2012) 12–26
The Demarcation Point from E-AGB Stars to TP-AGB Stars in HR Diagram for Medium-mass Stars† HONG Ya-fang JIANG Su-yun College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004 Abstract Via a study of the evolutionary tracks of 3∼10 M stars on the Hertzsprung-Russell diagram, the variations of the energy, density, temperature at the peak of helium-shell burning, ratio of surface luminosity of helium shell to stellar surface luminosity as well as the stellar radius are analyzed. Then the demarcation point of medium-mass stars in the evolution from early AGB stars to thermally pulsing AGB stars on the HR diagram is determined, and for 119 carbon stars our analysis agrees rather well with observation. At the same time the following is suggested. After arriving at this demarcation point in stellar evolution, in the formula of the loss of stellar wind material it is probably needed to introduce a quantity which is not concerned with the surface luminosity, but it dominates the formation of super stellar wind. On this basis and via the analysis of the structure and evolution of 5 M stars as well as the rate of mass loss of stellar wind, it is found that the effect of turbulent pressure on the mass loss of stellar wind in the stage of thermally pulsing AGB stars is rather great, hence the turbulent pressure of thermally pulsing AGB stars cannot be overlooked. Furthermore, the physical factors which possibly affect the matter loss of the stellar winds of thermally pulsing AGB stars are suggested. Key words: stars: evolution, AGB stars, HR diagram, mass loss, turbulence
Received 2010–11–19; revised version 2011–01–12 A translation of Acta Astron. Sin. Vol. 52, No. 4, pp. 275–287, 2011 [email protected]
0275-1062/11/$-see front front mattermatter © 2012 Elsevier All rights reserved. c 2012 B.V. 0275-1062/01/$-see Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2011.12.005 PII:
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1. INTRODUCTION After the termination of central helium burning, the stars with medium masses may form the electron-degenerate C-O cores. The outer layers of their shell-sources expand rapidly, and this makes the stars to evolve toward red supergiants. Their track of evolution moves along an asymptotic giant branch, and their stage of AGB stars commences. As stars with small and medium masses, the AGB stars exhibit rather high luminosities in the entire process of evolution from the main sequence in company with nuclear reactions to the last stage. Therefore they are often taken to be an important means for the research of extragalactic systems[1−4] . At the same time, before becoming white dwarfs, in the form of super stellar winds more than 70% of matter is thrown out[5] . In this way the planetary nebulae are formed, and furthermore they expand and become interplanetary medium. This possesses unnegligible influence on the composition and origin of the elements in interstellar medium. In order to systematically understand the AGB stars, Iben et al.[6] divided AGB stars into two classes, i.e., the early AGB (E-AGB) stars and thermally pulsing AGB (TP-AGB) stars. Because of the periodic pulsations of the luminosity, temperature and other quantities of TP-AGB stars, the third dredge-up effect and the accompanying super stellar wind in this process are more noticeable. The concept of super stellar wind was first proposed by Renzini[7,8] . As noticed by him, in order to produce typical planetary nebulae, the rate of loss of matter for the AGB stars in their late stage should attain 3 × 10−5 M� · yr−1 . But it is evident that such a rate of loss cannot be attained by the formula of stellar wind proposed by Reimers[9] in 1975 and widely used nowadays. Subsequently, via the analysis of the model of the cores of planetary nebulae, Wood et al.[10] argued that all the AGB stars with the masses of central cores 1.4 M� > Mc ≥ 0.86 M� can form planetary nebulae by ejection of matter via surface stellar wind. The existence of super stellar wind has also been further verified by observations. Knapp et al.[11] and Wood et al.[12] discovered that in many AGB stars there exists the persistent loss of matter by super stellar winds. The super stellar winds are found very frequently, so this shows that planetary nebulae are not formed by intermittent ejections[13,14] . However, until present the mechanism of formation of super stellar wind is still not very clear. The main theory is the dust-driven stellar wind based on the mechanism of radiation pressure[15−19] . Iben et al.[6] suggested that the pulsations of shell may enhance the matter loss of stellar wind, and the outward migration of carbon element may also have certain influence on the rate of matter loss of stellar wind. This plays certain role of guidance for the establishment of the formula of matter loss of stellar wind. For instance, in the formula of matter loss of super stellar wind, Marigo et al.[20] introduced the period of pulsations P0 and the abundance ratio C/O in stellar surface. Afterwards in 1990 Dominik et al.[21] at first connected the mass loss with the formation and growth of dust. Then the related semi-empirical formulae of matter loss of stellar wind are established[22−25] . Together with the development of the theory of pulsating stellar atmospheres and the concerned hydrodynamical and phenomenological research, the observational data related with the matter loss of stellar winds are gradually improved. The fitting of the formula of matter loss in super stellar winds of AGB stars is incessantly carried on. Based on the hypothesis of the mechanism of radiation pressure as well as the observational summary of the lg M˙ − P0 relation (M˙ is the rate of mass loss), in 1993 Vassiliadis et al.[29] suggested
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that before P0 =500 d the matter loss of stellar wind grows in the form of the exponential function of the period of pulsations and that after this it becomes a constant which is several times larger than 10−5 M� . Therefore the function of pulsation period P0 is included in the formula of matter loss of AGB stellar winds, and P0 =500 d is taken to be the demarcation point. Thus the formula of the mass loss of AGB stellar winds is treated in segments. For the stars with M ≤ 3 M� , the observational results can be well fitted with theory. But for stars with larger masses, the fitting of observational results with theory is not satisfactory enough. Afterwards Blocker [30] also carried out the research concerned with the mass loss of AGB stellar winds. He also proposed the idea of demarcation point. Taking into consideration that for the stars with initial masses larger than 2 M� at least before P0 =100 d the mass losses of stellar winds are rather small, so the minimum value P0 =100 d is taken as the demarcation point. And on the basis of the work of Reimers [9] , the formula of the mass loss of stellar wind is verified. Analogously, in 2005 Bergeat et al.[25] chose the effective temperature Teff = 2900 K to be the demarcation point. The suggestion of demarcation point leads to the convenience for the fitting of the formula of the mass loss of super stellar winds in the stage of TP-AGB stars. However, all these are based on an empirical summary and have no theoretical basis. In 2009 Iwamoto [31] made a study on the metal-poor AGB stars with medium and small masses. He traced the path of evolution of AGB stars on the HR diagram from main-sequence stars to AGB stars. In the diagram it can be clearly seen that in the late period, due to the decline of luminosity there appears a peak in the track. After passing through the peak there occur the periodic pulsations of luminosity. This may be taken as the “minipulses” suggested by Becker et al. [32] . In this paper we like to use the improved Kippenhahn stellar structure and evolution program and calculate the stellar evolution from the main sequence for the stars with 3∼10 M� (with metal abundance Z=0.02). Then we get a part of the evolution track on the HR diagram after passing through the peak. Then again we make a further analysis of the peak value and get the demarcation point where the E-AGB stars begin to evolve on the HR diagram to the TP-AGB stars. In the second part of this paper, the parameters of the stellar model used in computation, the formula of stellar wind and the treatment of turbulent pressure are introduced. The third part describes the results of computation. In Section 3.1 and via theoretical analysis, the demarcation point where stars enter the stage of TP-AGB stars on the HR diagram is located. Section 3.2 clarifies that the turbulent pressure has an unnegligible influence on the stellar winds of TP-AGB stars. Section 3.3 qualitatively analyzes the possible method of revision of the formula of matter loss of stellar winds of TP-AGB stars. The fourth part gives some conclusions about the determination of the demarcation point and its application. 2. MODEL In this article the improved Kippenhahn stellar structure and evolution program is adopted, and the evolutions of stars with the metal abundance of 0.02 as well as initial masses of 3 M� , 4 M� , 5 M� , 8 M� and 10 M� from main-sequence stars to AGB stars are calculated. For the computation of convection zone the theory of local mixing length[33−35] is adopted, and the mixing parameter is taken to be α=1.8. The convective overshooting is not taken into consideration.
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2.1 Stellar Wind The way of 3-stage treatment of the formula of mass loss of stellar wind is adopted. And the effective stellar temperature is denoted by Teff , the stellar surface luminosity by L, the stellar mass by M and the radius of star by R. (1) The first stage For the stage from the beginning of central hydrogen burning to the end of central hydrogen burning, we adopt the semi-empirical formula given by de Jager et al.[36] : lg M˙ = 1.64 lg L − 1.61 lg Teff + 0.16 lg M − 7.93 .
(1)
(2) The second stage From the end of central hydrogen burning to the early period of AGB stars, the empirical formula of Reimers [9] can be used. M˙ r represents the rate of mass loss of star obtained with Reimers’ empirical formula[9] . Now we have LR η M˙ r = 4 × 10−13 M
0.3 < η < 3.
(2)
For the calculation of the mass loss of stellar wind, the values of η taken for stars of various masses are slightly different. In this work, for stars of 3∼8 M� we take η=1.0, and for stars of 10 M� we take η=1.5. (3) The third stage For the post-AGB stars, if Eq.(2) is still utilized, then in the late period of evolution the AGB stars may produce enormous stellar winds. But this does not well agree with observation[11], so Eq.(1) has to be revised. Now we adopt the theory of Blocker[30] to make the necessary revision of the formula of stellar wind after the beginning of thermal pulses. We let (3) M˙ = 4.83 × 10−9 M 2.7 L2.7 M˙ r .
As for from what time the formula of stellar wind can be used to treat the third stage, Blocker[30] made observational fitting and proposed his suggestion. On the basis of the model defined by Ostlie et al.[37] in 1986, i.e., by defining a periodic function P0 which is concerned with thermal pulsations, he let lg(P0 /d) = −1.92 − 0.73 lg M + 1.86 lg R .
(4)
Then he suggested that for the stars with primary masses larger than 2 M� and at least before P0 =100 d the rate of matter loss of stellar wind is comparatively small, so after P0 >100 d the formula of mass loss of stellar wind for the third stage can be used. In the above formulae, L, R and M are in the units of L� , R� and M� , respectively. And the unit of the rate of mass loss M˙ is M� · yr−1 . 2.2 Turbulent Pressure The turbulent pressure Pt is a function of density ρ and average velocity of convective element v¯. Its expression[38−41] is taken to be: υ2 . Pt = ρ¯
(5)
In this paper the effect of turbulent pressure on stellar evolution, especially in the stage of TP-AGB stars, will be adequately considered. Now in combination with the treatment
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of turbulent pressure proposed by Jiang Su-yun et al.[38,39] and Demarque et al.[42] and under the premise of ignoring the effects of magnetic pressure and rotation we like to make an adequate revision of the formula of static equilibrium. During the treatment of the convective region, we let (6) ∇(P + Pt ) = −ρg , P = Pg + PR ,
(7)
in which PR is the radiation pressure, Pg is the gas pressure and g is the gravitational acceleration. Herein we may consider the influence of turbulent pressure on the stellar evolution in various stages. 3. RESULTS AND ANALYSIS 3.1 Starting Point of TP-AGB Stars on HR Diagram 3.1.1 A detailed analysis of 5 M stars Fig.1 shows the track of evolution of 5 M stars on the HR diagram. In this figure we can see an astonishing result. After arriving at the stage of red supergiants the curve on the HR diagram deflects to the right and lower side. In order to study this situation, we present the diagram of the relation between the internal temperature and density (Fig.2) as well as the variations of various quantities in the process of He-shell burning (Fig.3).
Fig. 1 Evolutionary track of 5 M stars on HR diagram
It may be seen in Fig.2 that at this time the star is already situated in the electrondegenerate state of C-O core and enters the stage of evolution of AGB stars. In order to know in which layer of the stage of AGB stars it is situated, we like to analyze the various quantities of state of its shell-burning. Table 1 lists a part of the physical quantities at some points near the peak of the HR diagram (lg Teff ≈ 3.482). The physical quantities are the age, temperature at the peak of helium-shell burning lg Tr , density ρr , energy �r , luminosity on the outer border of shell
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lg(Lr /L ), effective temperature of the star lg Teff , surface luminosity lg(L/L ) and stellar radius R. From Fig.4 we may have an objective understanding on the time variation of the radius of a 5 M star. Table 1
A part of parameters at some points around the peak of evolutionary track of 5 M� star on HR diagram
Age(yr) lg Tr (K) ρr (g·cm−3 ) εr (erg · g−1 · s−1 ) lg(Lr /L� ) lg(L/L� ) lg Teff (K) R(R� ) 99565752 8.242 1745.82 722769.8 3.869 3.864 3.498 284 99576520 8.243 1678.80 783429.6 3.885 3.879 3.487 303 99579884 8.245 1706.08 801678.1 3.890 3.881 3.485 307 99580579 8.244 1690.44 809095.9 3.891 3.882 3.484 308 99581697 8.245 1702.15 814704.3 3.892 3.883 3.482 311 99582498 8.245 1694.34 822242.6 3.893 3.863 3.466 329 99582606 8.245 1702.16 822242.6 3.893 3.841 3.450 345
From Fig.3, Fig.4 and Table 1 it can be found that approximately from lg Teff =3.482 the surface luminosity of star attains its peak value, then it begins to decline. At the same time it may be seen that at this time the energy released by the helium shell-source in unit time rapidly increases and the rate of rise of temperature rapidly grows. At the same time the corresponding density quickly declines. The luminosity of shell-source exceeds the total luminosity of star and the amount of excess increases with time. It is peculiar that after passing through the vicinity of peak in the HR diagram the stellar radius increasingly expands.
Fig. 2 Variation of temperature with density in the center of a 5 M� star
So we may assume that after a 5 M star evolves to the place of the peak lgTeff = 3.482 in the HR diagram, the star undergoes the transition from the stage of E-AGB stars to that of TP-AGB stars.
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Fig. 3 Time variations of (a) energy production rate, (b) density, (c) temperature and (d) ratio of luminosity on outer border of helium shell to that of stellar surface
Fig. 4 Stellar radius as a function of time for 5 M stars
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3.1.2 A brief analysis of the stars with medium masses The primary masses of stars play an unnegligible role in the evolution of TP-AGB stars. In 2005 Herwig[43] made a classification of AGB stars. They are divided into three classes, i.e., super AGB stars, large-mass AGB stars and low-mass AGB stars. The primary mass of a super AGB star is M ≥ 8 M . Whether the stars with primary masses larger than 10 M can enter the stage of TP-AGB stars is to a certain degree concerned with the magnitude of the matter loss of stellar wind. The stars with masses smaller than 4 M are called as low-mass AGB stars. In order to ascertain the applicability of the demarcation point where the E-AGB stars enter the evolutionary track of TP-AGB stars on the HR diagram, we like to analyze the evolutionary tracks of 3 M , 5 M , 8 M and 10 M stars on the HR diagram (Fig.5) as well as their relations of central temperature with density (Fig.6).
Fig. 5 Evolutionary tracks in HR diagram for stars with different masses
It may be seen in Fig.5 that the evolutionary tracks of all the stars with medium masses in the HR diagram exhibit evident peaks, i.e., the demarcation points where E-AGB stars become TP-AGB stars. Besides, all the effective temperatures which correspond to the peaks of stars with various masses are almost the same, and their luminosities increase with the growth of their primary masses. In combination with Table 2 and under the error tolerance, we can make the following summary. (1) The effective temperature corresponding to the peak is lgTeff = 3.48±0.005. Namely, Teff = 3020 ± 35 K. With the increase of primary mass, it exhibits progressive decrease. (2) The luminosity range of the peak value is 2950∼23550 L , and it increases with the growth of primary mass. (3) The period function P0 corresponding to the peak value increases with the growth of the primary mass. According to the models calculated by Cristallo et al.[44] and Iwamoto[31], we like to make the corresponding and necessary supplement: The difference of metal abundance can affect the magnitude of Teff of the peak and it decreases with the increase of metal abundance. Iben et al.[6] believe that when the stars become AGB stars their C-O cores are already in the electron-degenerate state, so they suggest that the upper limit of the primary masses
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of AGB stars is approximately 8∼9 M� . However, as illustrated in Fig.5, the stars with primary masses 10 M� can evolve to be AGB stars, and in the vicinity of Teff = 3020 ± 35 K they enter the stage of TP-AGB stars. It may be seen in Fig.6 that at this time their C-O cores are not degenerate, and this just agrees with the characters of super AGB stars[43] .
Fig. 6 Central temperature versus density for stars with different masses
Vassiliadis et al.[29] and Blecker[30] revised the stellar wind formula for TP-AGB stars, and for this a key point is that the time of entrance into the stage of super stellar wind is estimated beforehand by through the period function P0 . As shown by Table 2, with the increase of the primary masses of stars, when the E-AGB stars are converted into TP-AGB stars, the magnitude of P0 increases in rather large scale. Therefore, the revision of stellar wind with P0 as the demarcation point has a certain limit of applicability. The revision of the super stellar wind formula made by Bergeat et al.[25] in 2005 was carried out via the empirical formula for the rates of mass loss of 119 carbon-rich giants[45−47] . They proposed Teff =2900 K as the demarcation point, and this differs not much from the demarcation point Teff = 3020 ± 35 K suggested by us according to the evolutionary track on the HR diagram. In order to be cautious, we analyzed the 119 stars listed by them as the objects of our research, and discovered that only V623 Cas (Teff =3360 K), V614 Mon (Teff =3320 K) and TX Psc (Teff =3125 K) exceed rather far the demarcation point proposed by us. Besides, the deviations of six stars are smaller than 20 K. Then all the other giants agree with the requirements of the demarcation point suggested by us in theory. In consideration of the possible influence of metal abundance on this demarcation point, we tentatively believe that the demarcation point yielded by our theoretical analysis does not contradict the revision suggested by Bergeat et al.[25] via their summary of observations. Thus we can affirm the following. When stars evolve and for the first time arrive at Teff = 3020 ± 35 K, they enter the stage of TP-AGB stars. For the stars with 3∼8 M� (Z=0.02), the range of luminosities at the demarcation point is 2050∼23550 L� . The characteristics of this criterion are that it is simple and easy to be observed.
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Table 2 Some physical quantities of stars with various masses at the peak (A) and right bottom (B) of HR diagram Mi (M� ) 3 4 5 8 10
Teff A lg(LA /L� ) (K) 103.485 3.471 103.484 3.700 103.482 3.883 103.476 4.204 103.476 4.372
˙A P0 A M A M (d) (M� ) (M� ·yr−1 ) 100 2.92 10−7.10 131 3.90 10−6.48 171 4.82 10−5.97 259 7.58 10−5.26 345 8.38 10−4.47
Teff B lg(LB /L� ) (K) 103.2147 2.935 103.2382 3.251 103.2563 3.475 103.2817 3.876 103.3038 4.089
˙B P0 B M B M (d) (M� ) (M� ·yr−1 ) 321 2.92 10−8.73 417 3.90 10−7.85 495 4.82 10−7.21 675 7.58 10−6.20 823 8.32 10−5.30
3.2 Influence of Turbulent Pressure on Stellar Evolution Nowadays the mechanism of production of super stellar wind is still not very clear, so all the studies of the mass loss of stellar wind are made by some empirical or semiempirical formulae. Most of the semi-empirical formulae are based on the theory of dustdriven stellar wind due to radiation pressure on dust. After the effect of turbulent convection has been successfully introduced into the theory of pulsating variables[48] and on the basis of consideration of the effect of pulsations on the matter loss of stellar wind[6] , it is recognized that the turbulent pressure is the possible physical cause of the production of stellar wind[40] . After making deepgoing researches on turbulent pressure, References [38,39,49-51] proposed that in the stages of red giants and AGB stars the turbulent pressure in the stellar surface districts in comparison with the total pressure attains 0.3, so the turbulent pressure cannot be ignored. After this, Denmarque et al.[42] introduced the turbulent pressure into the solar research and carried out respective modellings and analyses for the three stages of the sun, i.e., the zero-age main sequence stage, present stage and subgiant stage. Via the verification with helioseismology it is found that the introduction of turbulent pressure can make the theory to be closer to observational results. Moreover, when the stars evolve to the stage of subgiants, the effect of turbulent pressure becomes more important. Owing to the consideration of the mechanism of matter loss of stellar wind, we like to carry out adequate analysis and discussion of the turbulent pressure. Fig.7 demonstrates the influence of turbulent pressure on the evolutionary tracks of 5 M stars in the HR diagram. It can be seen in the figure that the two tracks almost coincide with each other, and they slightly differ with each other only in the stages of central helium burning and AGB stars. These two stages are just characterized by the situation that the outer layers of stars are in the state of convection. This illustrates that when the outer layers of stars are convective regions, the effect of turbulent pressure is rather obvious. Fig.8 shows the evolutions of the central temperature and density for the 5 M stars in the two cases of ignorance and consideration of turbulent pressure. From this figure it can be seen that before degeneration (i.e., before the stage of AGB stars) the two curves overlap well with each other. This implies that before the stage of AGB stars the influence of turbulence on the stellar centers is almost null. The reason may be understood as follows. The closer to the stellar center, the closer the gradient of convective temperature to the gradient of adiabatic temperature, and at the same time the smaller the influence of turbulent pressure. However, in the stage of AGB stars and for the central part of stars (the central helium cores have been situated in the electron-degenerate state) it can be seen that these two curves evidently differ from each other, and the turbulent pressure should not be the direct reason. Considering that the curves near the peaks (Teff ≈ 3030 K) of curves in the HR
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diagram of Fig.5 evidently differ from each other, so this may be due to the intensification of the effect of turbulent pressure on the evolution of stars which enter the stage of AGB stars (see Fig.9).
Fig. 7 Evolutionary tracks of 5 M stars in HR diagram
Fig. 8 Relation between central temperatures and densities of 5 M stars
Fig.9 shows the time variations of mass and rate of mass loss in the process of stellar evolution. From this figure it may be evidently seen that after the point C (about 9.6×107 yr) the turbulent pressure in the outer convective layers of stars cannot be overlooked. Under the condition of considering the effect of turbulent pressure, the loss of stellar wind matter is evidently larger than that with the ignorance of turbulent pressure. Yet in combination with Table 1 it can be found that the point C is close to the peak in the HR diagram (Teff ≈ 3030 K). Namely, it is close to the turning point from the stage of E-AGB stars to the stage of TP-AGB stars.
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Hence we may assume that because of entering into the stage of TP-AGB stars the turbulent pressure produces rather large influence on the production of super stellar winds and on the evolution of stars. Therefore, when stars evolve to TP-AGB stars, the turbulent pressure cannot be neglected.
Fig. 9 Variations of mass (left) and mass-loss rate (right) with time for 5 M stars
3.3 A Qualitative Discussion of the Mass Losses of TP-AGB Stars The evolution of 5 M� stars without consideration of turbulence is taken as an example. Fig. 10 shows the time variations of the pulsation period P0 and the rate of mass loss in the process of evolution of 5 M� stars. As demonstrated by this figure, after the stellar evolution has entered the stage of TP-AGB stars, P0 is far longer than 100 days, but the loss of mass can not attain the observed value, i.e., 10−5 ∼ 10−4 M� . Namely, this does not agree with the primary aim of the revision of the formula of mass loss of stellar wind proposed by Blocker[30] . As for the reason, it should be the large-scale decrease of surface luminosity.
Fig. 10 Pulsation period (left) and mass-loss rate (right) stars as functions of time for 5 M
Most of the existing formulae of mass loss of stellar wind are concerned with the surface luminosity, and they all have their own proportional relations with it. For example, the empirical formulae proposed by Reimers[9] , Lamers[52], Nieuwenhuij-zen et al.[53] are based on the functional relations of L, R and M . But the empirical formula proposed by de Jager
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et al.[36] is merely the functional relation of L and Teff . Instead Schroder et al.[54] used L, R, Teff and M as the variables. Because of the reason that the independent variables L and T can be used to express R, we may tentatively think that if these empirical or semiempirical formulae are used for the stage of AGB stars, then the questions brought forth by the oscillations of surface luminosity in the stage of AGB stars can hardly be avoided. For example, Blocker[30] used the correction expression (3) to make analyses for the 3 M , 4 M , 5 M and 7 stars. Then it is known that in the period of pulsations, a rather large rate of mass loss can be attained in a short time. However, because the time of persistence of mass loss is rather short (this should be concerned with the oscillations of surface luminosity), it seems to be insufficient to eject the due amount of mass before entering the stage of white dwarfs. Moreover, the persistent super stellar winds are frequently observed[11,12], and this clarifies that the mass loss of stellar winds of TP-AGB stars is not produced by intermittent ejections[13,14] . Therefore, for fitting the formula of stellar wind mass losses of TP-AGB stars, one has to avoid the large-scale intermittent variations of the magnitudes of stellar winds caused by the pulsations of surface luminosity. From Fig.1 and in consideration of the characteristics of TP-AGB stars, the following may be inferred. After a star evolves to be a thermally pulsing star its surface luminosity may exhibit large-scale oscillations. In the revision of the formula of matter loss of stellar wind, it is expected that some quantities are not affected by the surface luminosity and are directly added in. Otherwise, when the surface luminosity exhibits some influence, another quantity is introduced. Analogous ideas have been practised for many times. For instance, in their studies of super stellar wind Wood et al.[12] , Whitelock et al.[55] and Vassiliades et al.[29] directly introduced the period function P0 . After a star evolves to be a TP-AGB star, this quantity may have large influence on the matter loss of stellar wind, and it may even dominate the formation of the super stellar wind in the late period. For the AGB stars with M ≤ 3 M the theoretical values can effectively agree with observational ones. For massive stars the observational values are smaller than theoretical ones. Jiang Su-yun et al.[49] studied the outer convective regions of AGB stars and discovered that due to the effect of turbulent pressure there appears the dynamical instability of the inversion of density gradient of the outer stellar envelope close to surface region, which possibly causes the formation of super stellar wind. By considering the influence of turbulent pressure shown in Fig.9 on the matter loss of stellar wind and our analysis of the evolutionary tracks of stars with masses in the range 10 M ≥ M ≥ 3 M , maybe, after they evolve to the demarcation point (Teff = 3020 ± 35 K) between the E-AGB stars and TP-AGB stars, the consideration of turbulent pressure can be made together with the revision of the formula of matter loss of stellar wind.
4. CONCLUSIONS (1) In this paper, via a study of the evolutionary tracks of the stars with metal abundance Z=0.02 and medium masses 10 M ≥ M ≥ 3 M in the HR-diagram, the internal structure and the variations of quantities of the helium-burning shell are analyzed. The demarcation point on the HR diagram when the stars evolve to the stage of TP-AGB stars is determined. Namely, when the evolutionary track arrives at the vicinity of the peak value (Teff = 3020 ±
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35 K), the stars evolve into the stage of TP-AGB stars. For the stars with 10 M ≥ M ≥ 3 M , the range of luminosity at the demarcation point is 2950∼23550 L . Thereby the location of TP-AGB stars on the HR diagram is determined. (2) The following is qualitatively proposed. For an AGB star with primary mass M ≥ 3 M , after arriving at Teff = 3020± 35 K for the first time, because of the entrance into the stage of TP-AGB stars in stellar evolution, the formula of the matter loss of stellar wind should be revised. We may directly introduce a quantity which is not concerned with the surface luminosity, and make it dominate the formation of super stellar wind in the stage of TP-AGB stars. Moreover, via the analysis of the introduced turbulent pressure it is found that on the matter loss of stellar wind in the stage of TP-AGB stars the turbulent pressure exhibits rather large influence. Hence, it is proposed that the influence of turbulent pressure may be considered in the formula revision. References 1
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CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy and Astrophysics 36 (2012) 12–26
The Demarcation Point from E-AGB Stars to TP-AGB Stars in HR Diagram for Medium-mass Stars† HONG Ya-fang JIANG Su-yun College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004 Abstract Via a study of the evolutionary tracks of 3∼10 M stars on the Hertzsprung-Russell diagram, the variations of the energy, density, temperature at the peak of helium-shell burning, ratio of surface luminosity of helium shell to stellar surface luminosity as well as the stellar radius are analyzed. Then the demarcation point of medium-mass stars in the evolution from early AGB stars to thermally pulsing AGB stars on the HR diagram is determined, and for 119 carbon stars our analysis agrees rather well with observation. At the same time the following is suggested. After arriving at this demarcation point in stellar evolution, in the formula of the loss of stellar wind material it is probably needed to introduce a quantity which is not concerned with the surface luminosity, but it dominates the formation of super stellar wind. On this basis and via the analysis of the structure and evolution of 5 M stars as well as the rate of mass loss of stellar wind, it is found that the effect of turbulent pressure on the mass loss of stellar wind in the stage of thermally pulsing AGB stars is rather great, hence the turbulent pressure of thermally pulsing AGB stars cannot be overlooked. Furthermore, the physical factors which possibly affect the matter loss of the stellar winds of thermally pulsing AGB stars are suggested. Key words: stars: evolution, AGB stars, HR diagram, mass loss, turbulence
Received 2010–11–19; revised version 2011–01–12 A translation of Acta Astron. Sin. Vol. 52, No. 4, pp. 275–287, 2011 [email protected]
0275-1062/11/$-see front front mattermatter © 2012 Elsevier All rights reserved. c 2012 B.V. 0275-1062/01/$-see Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2011.12.005 PII:
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1. INTRODUCTION After the termination of central helium burning, the stars with medium masses may form the electron-degenerate C-O cores. The outer layers of their shell-sources expand rapidly, and this makes the stars to evolve toward red supergiants. Their track of evolution moves along an asymptotic giant branch, and their stage of AGB stars commences. As stars with small and medium masses, the AGB stars exhibit rather high luminosities in the entire process of evolution from the main sequence in company with nuclear reactions to the last stage. Therefore they are often taken to be an important means for the research of extragalactic systems[1−4] . At the same time, before becoming white dwarfs, in the form of super stellar winds more than 70% of matter is thrown out[5] . In this way the planetary nebulae are formed, and furthermore they expand and become interplanetary medium. This possesses unnegligible influence on the composition and origin of the elements in interstellar medium. In order to systematically understand the AGB stars, Iben et al.[6] divided AGB stars into two classes, i.e., the early AGB (E-AGB) stars and thermally pulsing AGB (TP-AGB) stars. Because of the periodic pulsations of the luminosity, temperature and other quantities of TP-AGB stars, the third dredge-up effect and the accompanying super stellar wind in this process are more noticeable. The concept of super stellar wind was first proposed by Renzini[7,8] . As noticed by him, in order to produce typical planetary nebulae, the rate of loss of matter for the AGB stars in their late stage should attain 3 × 10−5 M� · yr−1 . But it is evident that such a rate of loss cannot be attained by the formula of stellar wind proposed by Reimers[9] in 1975 and widely used nowadays. Subsequently, via the analysis of the model of the cores of planetary nebulae, Wood et al.[10] argued that all the AGB stars with the masses of central cores 1.4 M� > Mc ≥ 0.86 M� can form planetary nebulae by ejection of matter via surface stellar wind. The existence of super stellar wind has also been further verified by observations. Knapp et al.[11] and Wood et al.[12] discovered that in many AGB stars there exists the persistent loss of matter by super stellar winds. The super stellar winds are found very frequently, so this shows that planetary nebulae are not formed by intermittent ejections[13,14] . However, until present the mechanism of formation of super stellar wind is still not very clear. The main theory is the dust-driven stellar wind based on the mechanism of radiation pressure[15−19] . Iben et al.[6] suggested that the pulsations of shell may enhance the matter loss of stellar wind, and the outward migration of carbon element may also have certain influence on the rate of matter loss of stellar wind. This plays certain role of guidance for the establishment of the formula of matter loss of stellar wind. For instance, in the formula of matter loss of super stellar wind, Marigo et al.[20] introduced the period of pulsations P0 and the abundance ratio C/O in stellar surface. Afterwards in 1990 Dominik et al.[21] at first connected the mass loss with the formation and growth of dust. Then the related semi-empirical formulae of matter loss of stellar wind are established[22−25] . Together with the development of the theory of pulsating stellar atmospheres and the concerned hydrodynamical and phenomenological research, the observational data related with the matter loss of stellar winds are gradually improved. The fitting of the formula of matter loss in super stellar winds of AGB stars is incessantly carried on. Based on the hypothesis of the mechanism of radiation pressure as well as the observational summary of the lg M˙ − P0 relation (M˙ is the rate of mass loss), in 1993 Vassiliadis et al.[29] suggested
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that before P0 =500 d the matter loss of stellar wind grows in the form of the exponential function of the period of pulsations and that after this it becomes a constant which is several times larger than 10−5 M� . Therefore the function of pulsation period P0 is included in the formula of matter loss of AGB stellar winds, and P0 =500 d is taken to be the demarcation point. Thus the formula of the mass loss of AGB stellar winds is treated in segments. For the stars with M ≤ 3 M� , the observational results can be well fitted with theory. But for stars with larger masses, the fitting of observational results with theory is not satisfactory enough. Afterwards Blocker [30] also carried out the research concerned with the mass loss of AGB stellar winds. He also proposed the idea of demarcation point. Taking into consideration that for the stars with initial masses larger than 2 M� at least before P0 =100 d the mass losses of stellar winds are rather small, so the minimum value P0 =100 d is taken as the demarcation point. And on the basis of the work of Reimers [9] , the formula of the mass loss of stellar wind is verified. Analogously, in 2005 Bergeat et al.[25] chose the effective temperature Teff = 2900 K to be the demarcation point. The suggestion of demarcation point leads to the convenience for the fitting of the formula of the mass loss of super stellar winds in the stage of TP-AGB stars. However, all these are based on an empirical summary and have no theoretical basis. In 2009 Iwamoto [31] made a study on the metal-poor AGB stars with medium and small masses. He traced the path of evolution of AGB stars on the HR diagram from main-sequence stars to AGB stars. In the diagram it can be clearly seen that in the late period, due to the decline of luminosity there appears a peak in the track. After passing through the peak there occur the periodic pulsations of luminosity. This may be taken as the “minipulses” suggested by Becker et al. [32] . In this paper we like to use the improved Kippenhahn stellar structure and evolution program and calculate the stellar evolution from the main sequence for the stars with 3∼10 M� (with metal abundance Z=0.02). Then we get a part of the evolution track on the HR diagram after passing through the peak. Then again we make a further analysis of the peak value and get the demarcation point where the E-AGB stars begin to evolve on the HR diagram to the TP-AGB stars. In the second part of this paper, the parameters of the stellar model used in computation, the formula of stellar wind and the treatment of turbulent pressure are introduced. The third part describes the results of computation. In Section 3.1 and via theoretical analysis, the demarcation point where stars enter the stage of TP-AGB stars on the HR diagram is located. Section 3.2 clarifies that the turbulent pressure has an unnegligible influence on the stellar winds of TP-AGB stars. Section 3.3 qualitatively analyzes the possible method of revision of the formula of matter loss of stellar winds of TP-AGB stars. The fourth part gives some conclusions about the determination of the demarcation point and its application. 2. MODEL In this article the improved Kippenhahn stellar structure and evolution program is adopted, and the evolutions of stars with the metal abundance of 0.02 as well as initial masses of 3 M� , 4 M� , 5 M� , 8 M� and 10 M� from main-sequence stars to AGB stars are calculated. For the computation of convection zone the theory of local mixing length[33−35] is adopted, and the mixing parameter is taken to be α=1.8. The convective overshooting is not taken into consideration.
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2.1 Stellar Wind The way of 3-stage treatment of the formula of mass loss of stellar wind is adopted. And the effective stellar temperature is denoted by Teff , the stellar surface luminosity by L, the stellar mass by M and the radius of star by R. (1) The first stage For the stage from the beginning of central hydrogen burning to the end of central hydrogen burning, we adopt the semi-empirical formula given by de Jager et al.[36] : lg M˙ = 1.64 lg L − 1.61 lg Teff + 0.16 lg M − 7.93 .
(1)
(2) The second stage From the end of central hydrogen burning to the early period of AGB stars, the empirical formula of Reimers [9] can be used. M˙ r represents the rate of mass loss of star obtained with Reimers’ empirical formula[9] . Now we have LR η M˙ r = 4 × 10−13 M
0.3 < η < 3.
(2)
For the calculation of the mass loss of stellar wind, the values of η taken for stars of various masses are slightly different. In this work, for stars of 3∼8 M� we take η=1.0, and for stars of 10 M� we take η=1.5. (3) The third stage For the post-AGB stars, if Eq.(2) is still utilized, then in the late period of evolution the AGB stars may produce enormous stellar winds. But this does not well agree with observation[11], so Eq.(1) has to be revised. Now we adopt the theory of Blocker[30] to make the necessary revision of the formula of stellar wind after the beginning of thermal pulses. We let (3) M˙ = 4.83 × 10−9 M 2.7 L2.7 M˙ r .
As for from what time the formula of stellar wind can be used to treat the third stage, Blocker[30] made observational fitting and proposed his suggestion. On the basis of the model defined by Ostlie et al.[37] in 1986, i.e., by defining a periodic function P0 which is concerned with thermal pulsations, he let lg(P0 /d) = −1.92 − 0.73 lg M + 1.86 lg R .
(4)
Then he suggested that for the stars with primary masses larger than 2 M� and at least before P0 =100 d the rate of matter loss of stellar wind is comparatively small, so after P0 >100 d the formula of mass loss of stellar wind for the third stage can be used. In the above formulae, L, R and M are in the units of L� , R� and M� , respectively. And the unit of the rate of mass loss M˙ is M� · yr−1 . 2.2 Turbulent Pressure The turbulent pressure Pt is a function of density ρ and average velocity of convective element v¯. Its expression[38−41] is taken to be: υ2 . Pt = ρ¯
(5)
In this paper the effect of turbulent pressure on stellar evolution, especially in the stage of TP-AGB stars, will be adequately considered. Now in combination with the treatment
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of turbulent pressure proposed by Jiang Su-yun et al.[38,39] and Demarque et al.[42] and under the premise of ignoring the effects of magnetic pressure and rotation we like to make an adequate revision of the formula of static equilibrium. During the treatment of the convective region, we let (6) ∇(P + Pt ) = −ρg , P = Pg + PR ,
(7)
in which PR is the radiation pressure, Pg is the gas pressure and g is the gravitational acceleration. Herein we may consider the influence of turbulent pressure on the stellar evolution in various stages. 3. RESULTS AND ANALYSIS 3.1 Starting Point of TP-AGB Stars on HR Diagram 3.1.1 A detailed analysis of 5 M stars Fig.1 shows the track of evolution of 5 M stars on the HR diagram. In this figure we can see an astonishing result. After arriving at the stage of red supergiants the curve on the HR diagram deflects to the right and lower side. In order to study this situation, we present the diagram of the relation between the internal temperature and density (Fig.2) as well as the variations of various quantities in the process of He-shell burning (Fig.3).
Fig. 1 Evolutionary track of 5 M stars on HR diagram
It may be seen in Fig.2 that at this time the star is already situated in the electrondegenerate state of C-O core and enters the stage of evolution of AGB stars. In order to know in which layer of the stage of AGB stars it is situated, we like to analyze the various quantities of state of its shell-burning. Table 1 lists a part of the physical quantities at some points near the peak of the HR diagram (lg Teff ≈ 3.482). The physical quantities are the age, temperature at the peak of helium-shell burning lg Tr , density ρr , energy �r , luminosity on the outer border of shell
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lg(Lr /L ), effective temperature of the star lg Teff , surface luminosity lg(L/L ) and stellar radius R. From Fig.4 we may have an objective understanding on the time variation of the radius of a 5 M star. Table 1
A part of parameters at some points around the peak of evolutionary track of 5 M� star on HR diagram
Age(yr) lg Tr (K) ρr (g·cm−3 ) εr (erg · g−1 · s−1 ) lg(Lr /L� ) lg(L/L� ) lg Teff (K) R(R� ) 99565752 8.242 1745.82 722769.8 3.869 3.864 3.498 284 99576520 8.243 1678.80 783429.6 3.885 3.879 3.487 303 99579884 8.245 1706.08 801678.1 3.890 3.881 3.485 307 99580579 8.244 1690.44 809095.9 3.891 3.882 3.484 308 99581697 8.245 1702.15 814704.3 3.892 3.883 3.482 311 99582498 8.245 1694.34 822242.6 3.893 3.863 3.466 329 99582606 8.245 1702.16 822242.6 3.893 3.841 3.450 345
From Fig.3, Fig.4 and Table 1 it can be found that approximately from lg Teff =3.482 the surface luminosity of star attains its peak value, then it begins to decline. At the same time it may be seen that at this time the energy released by the helium shell-source in unit time rapidly increases and the rate of rise of temperature rapidly grows. At the same time the corresponding density quickly declines. The luminosity of shell-source exceeds the total luminosity of star and the amount of excess increases with time. It is peculiar that after passing through the vicinity of peak in the HR diagram the stellar radius increasingly expands.
Fig. 2 Variation of temperature with density in the center of a 5 M� star
So we may assume that after a 5 M star evolves to the place of the peak lgTeff = 3.482 in the HR diagram, the star undergoes the transition from the stage of E-AGB stars to that of TP-AGB stars.
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Fig. 3 Time variations of (a) energy production rate, (b) density, (c) temperature and (d) ratio of luminosity on outer border of helium shell to that of stellar surface
Fig. 4 Stellar radius as a function of time for 5 M stars
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3.1.2 A brief analysis of the stars with medium masses The primary masses of stars play an unnegligible role in the evolution of TP-AGB stars. In 2005 Herwig[43] made a classification of AGB stars. They are divided into three classes, i.e., super AGB stars, large-mass AGB stars and low-mass AGB stars. The primary mass of a super AGB star is M ≥ 8 M . Whether the stars with primary masses larger than 10 M can enter the stage of TP-AGB stars is to a certain degree concerned with the magnitude of the matter loss of stellar wind. The stars with masses smaller than 4 M are called as low-mass AGB stars. In order to ascertain the applicability of the demarcation point where the E-AGB stars enter the evolutionary track of TP-AGB stars on the HR diagram, we like to analyze the evolutionary tracks of 3 M , 5 M , 8 M and 10 M stars on the HR diagram (Fig.5) as well as their relations of central temperature with density (Fig.6).
Fig. 5 Evolutionary tracks in HR diagram for stars with different masses
It may be seen in Fig.5 that the evolutionary tracks of all the stars with medium masses in the HR diagram exhibit evident peaks, i.e., the demarcation points where E-AGB stars become TP-AGB stars. Besides, all the effective temperatures which correspond to the peaks of stars with various masses are almost the same, and their luminosities increase with the growth of their primary masses. In combination with Table 2 and under the error tolerance, we can make the following summary. (1) The effective temperature corresponding to the peak is lgTeff = 3.48±0.005. Namely, Teff = 3020 ± 35 K. With the increase of primary mass, it exhibits progressive decrease. (2) The luminosity range of the peak value is 2950∼23550 L , and it increases with the growth of primary mass. (3) The period function P0 corresponding to the peak value increases with the growth of the primary mass. According to the models calculated by Cristallo et al.[44] and Iwamoto[31], we like to make the corresponding and necessary supplement: The difference of metal abundance can affect the magnitude of Teff of the peak and it decreases with the increase of metal abundance. Iben et al.[6] believe that when the stars become AGB stars their C-O cores are already in the electron-degenerate state, so they suggest that the upper limit of the primary masses
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of AGB stars is approximately 8∼9 M� . However, as illustrated in Fig.5, the stars with primary masses 10 M� can evolve to be AGB stars, and in the vicinity of Teff = 3020 ± 35 K they enter the stage of TP-AGB stars. It may be seen in Fig.6 that at this time their C-O cores are not degenerate, and this just agrees with the characters of super AGB stars[43] .
Fig. 6 Central temperature versus density for stars with different masses
Vassiliadis et al.[29] and Blecker[30] revised the stellar wind formula for TP-AGB stars, and for this a key point is that the time of entrance into the stage of super stellar wind is estimated beforehand by through the period function P0 . As shown by Table 2, with the increase of the primary masses of stars, when the E-AGB stars are converted into TP-AGB stars, the magnitude of P0 increases in rather large scale. Therefore, the revision of stellar wind with P0 as the demarcation point has a certain limit of applicability. The revision of the super stellar wind formula made by Bergeat et al.[25] in 2005 was carried out via the empirical formula for the rates of mass loss of 119 carbon-rich giants[45−47] . They proposed Teff =2900 K as the demarcation point, and this differs not much from the demarcation point Teff = 3020 ± 35 K suggested by us according to the evolutionary track on the HR diagram. In order to be cautious, we analyzed the 119 stars listed by them as the objects of our research, and discovered that only V623 Cas (Teff =3360 K), V614 Mon (Teff =3320 K) and TX Psc (Teff =3125 K) exceed rather far the demarcation point proposed by us. Besides, the deviations of six stars are smaller than 20 K. Then all the other giants agree with the requirements of the demarcation point suggested by us in theory. In consideration of the possible influence of metal abundance on this demarcation point, we tentatively believe that the demarcation point yielded by our theoretical analysis does not contradict the revision suggested by Bergeat et al.[25] via their summary of observations. Thus we can affirm the following. When stars evolve and for the first time arrive at Teff = 3020 ± 35 K, they enter the stage of TP-AGB stars. For the stars with 3∼8 M� (Z=0.02), the range of luminosities at the demarcation point is 2050∼23550 L� . The characteristics of this criterion are that it is simple and easy to be observed.
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Table 2 Some physical quantities of stars with various masses at the peak (A) and right bottom (B) of HR diagram Mi (M� ) 3 4 5 8 10
Teff A lg(LA /L� ) (K) 103.485 3.471 103.484 3.700 103.482 3.883 103.476 4.204 103.476 4.372
˙A P0 A M A M (d) (M� ) (M� ·yr−1 ) 100 2.92 10−7.10 131 3.90 10−6.48 171 4.82 10−5.97 259 7.58 10−5.26 345 8.38 10−4.47
Teff B lg(LB /L� ) (K) 103.2147 2.935 103.2382 3.251 103.2563 3.475 103.2817 3.876 103.3038 4.089
˙B P0 B M B M (d) (M� ) (M� ·yr−1 ) 321 2.92 10−8.73 417 3.90 10−7.85 495 4.82 10−7.21 675 7.58 10−6.20 823 8.32 10−5.30
3.2 Influence of Turbulent Pressure on Stellar Evolution Nowadays the mechanism of production of super stellar wind is still not very clear, so all the studies of the mass loss of stellar wind are made by some empirical or semiempirical formulae. Most of the semi-empirical formulae are based on the theory of dustdriven stellar wind due to radiation pressure on dust. After the effect of turbulent convection has been successfully introduced into the theory of pulsating variables[48] and on the basis of consideration of the effect of pulsations on the matter loss of stellar wind[6] , it is recognized that the turbulent pressure is the possible physical cause of the production of stellar wind[40] . After making deepgoing researches on turbulent pressure, References [38,39,49-51] proposed that in the stages of red giants and AGB stars the turbulent pressure in the stellar surface districts in comparison with the total pressure attains 0.3, so the turbulent pressure cannot be ignored. After this, Denmarque et al.[42] introduced the turbulent pressure into the solar research and carried out respective modellings and analyses for the three stages of the sun, i.e., the zero-age main sequence stage, present stage and subgiant stage. Via the verification with helioseismology it is found that the introduction of turbulent pressure can make the theory to be closer to observational results. Moreover, when the stars evolve to the stage of subgiants, the effect of turbulent pressure becomes more important. Owing to the consideration of the mechanism of matter loss of stellar wind, we like to carry out adequate analysis and discussion of the turbulent pressure. Fig.7 demonstrates the influence of turbulent pressure on the evolutionary tracks of 5 M stars in the HR diagram. It can be seen in the figure that the two tracks almost coincide with each other, and they slightly differ with each other only in the stages of central helium burning and AGB stars. These two stages are just characterized by the situation that the outer layers of stars are in the state of convection. This illustrates that when the outer layers of stars are convective regions, the effect of turbulent pressure is rather obvious. Fig.8 shows the evolutions of the central temperature and density for the 5 M stars in the two cases of ignorance and consideration of turbulent pressure. From this figure it can be seen that before degeneration (i.e., before the stage of AGB stars) the two curves overlap well with each other. This implies that before the stage of AGB stars the influence of turbulence on the stellar centers is almost null. The reason may be understood as follows. The closer to the stellar center, the closer the gradient of convective temperature to the gradient of adiabatic temperature, and at the same time the smaller the influence of turbulent pressure. However, in the stage of AGB stars and for the central part of stars (the central helium cores have been situated in the electron-degenerate state) it can be seen that these two curves evidently differ from each other, and the turbulent pressure should not be the direct reason. Considering that the curves near the peaks (Teff ≈ 3030 K) of curves in the HR
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diagram of Fig.5 evidently differ from each other, so this may be due to the intensification of the effect of turbulent pressure on the evolution of stars which enter the stage of AGB stars (see Fig.9).
Fig. 7 Evolutionary tracks of 5 M stars in HR diagram
Fig. 8 Relation between central temperatures and densities of 5 M stars
Fig.9 shows the time variations of mass and rate of mass loss in the process of stellar evolution. From this figure it may be evidently seen that after the point C (about 9.6×107 yr) the turbulent pressure in the outer convective layers of stars cannot be overlooked. Under the condition of considering the effect of turbulent pressure, the loss of stellar wind matter is evidently larger than that with the ignorance of turbulent pressure. Yet in combination with Table 1 it can be found that the point C is close to the peak in the HR diagram (Teff ≈ 3030 K). Namely, it is close to the turning point from the stage of E-AGB stars to the stage of TP-AGB stars.
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Hence we may assume that because of entering into the stage of TP-AGB stars the turbulent pressure produces rather large influence on the production of super stellar winds and on the evolution of stars. Therefore, when stars evolve to TP-AGB stars, the turbulent pressure cannot be neglected.
Fig. 9 Variations of mass (left) and mass-loss rate (right) with time for 5 M stars
3.3 A Qualitative Discussion of the Mass Losses of TP-AGB Stars The evolution of 5 M� stars without consideration of turbulence is taken as an example. Fig. 10 shows the time variations of the pulsation period P0 and the rate of mass loss in the process of evolution of 5 M� stars. As demonstrated by this figure, after the stellar evolution has entered the stage of TP-AGB stars, P0 is far longer than 100 days, but the loss of mass can not attain the observed value, i.e., 10−5 ∼ 10−4 M� . Namely, this does not agree with the primary aim of the revision of the formula of mass loss of stellar wind proposed by Blocker[30] . As for the reason, it should be the large-scale decrease of surface luminosity.
Fig. 10 Pulsation period (left) and mass-loss rate (right) stars as functions of time for 5 M
Most of the existing formulae of mass loss of stellar wind are concerned with the surface luminosity, and they all have their own proportional relations with it. For example, the empirical formulae proposed by Reimers[9] , Lamers[52], Nieuwenhuij-zen et al.[53] are based on the functional relations of L, R and M . But the empirical formula proposed by de Jager
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et al.[36] is merely the functional relation of L and Teff . Instead Schroder et al.[54] used L, R, Teff and M as the variables. Because of the reason that the independent variables L and T can be used to express R, we may tentatively think that if these empirical or semiempirical formulae are used for the stage of AGB stars, then the questions brought forth by the oscillations of surface luminosity in the stage of AGB stars can hardly be avoided. For example, Blocker[30] used the correction expression (3) to make analyses for the 3 M , 4 M , 5 M and 7 stars. Then it is known that in the period of pulsations, a rather large rate of mass loss can be attained in a short time. However, because the time of persistence of mass loss is rather short (this should be concerned with the oscillations of surface luminosity), it seems to be insufficient to eject the due amount of mass before entering the stage of white dwarfs. Moreover, the persistent super stellar winds are frequently observed[11,12], and this clarifies that the mass loss of stellar winds of TP-AGB stars is not produced by intermittent ejections[13,14] . Therefore, for fitting the formula of stellar wind mass losses of TP-AGB stars, one has to avoid the large-scale intermittent variations of the magnitudes of stellar winds caused by the pulsations of surface luminosity. From Fig.1 and in consideration of the characteristics of TP-AGB stars, the following may be inferred. After a star evolves to be a thermally pulsing star its surface luminosity may exhibit large-scale oscillations. In the revision of the formula of matter loss of stellar wind, it is expected that some quantities are not affected by the surface luminosity and are directly added in. Otherwise, when the surface luminosity exhibits some influence, another quantity is introduced. Analogous ideas have been practised for many times. For instance, in their studies of super stellar wind Wood et al.[12] , Whitelock et al.[55] and Vassiliades et al.[29] directly introduced the period function P0 . After a star evolves to be a TP-AGB star, this quantity may have large influence on the matter loss of stellar wind, and it may even dominate the formation of the super stellar wind in the late period. For the AGB stars with M ≤ 3 M the theoretical values can effectively agree with observational ones. For massive stars the observational values are smaller than theoretical ones. Jiang Su-yun et al.[49] studied the outer convective regions of AGB stars and discovered that due to the effect of turbulent pressure there appears the dynamical instability of the inversion of density gradient of the outer stellar envelope close to surface region, which possibly causes the formation of super stellar wind. By considering the influence of turbulent pressure shown in Fig.9 on the matter loss of stellar wind and our analysis of the evolutionary tracks of stars with masses in the range 10 M ≥ M ≥ 3 M , maybe, after they evolve to the demarcation point (Teff = 3020 ± 35 K) between the E-AGB stars and TP-AGB stars, the consideration of turbulent pressure can be made together with the revision of the formula of matter loss of stellar wind.
4. CONCLUSIONS (1) In this paper, via a study of the evolutionary tracks of the stars with metal abundance Z=0.02 and medium masses 10 M ≥ M ≥ 3 M in the HR-diagram, the internal structure and the variations of quantities of the helium-burning shell are analyzed. The demarcation point on the HR diagram when the stars evolve to the stage of TP-AGB stars is determined. Namely, when the evolutionary track arrives at the vicinity of the peak value (Teff = 3020 ±
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35 K), the stars evolve into the stage of TP-AGB stars. For the stars with 10 M ≥ M ≥ 3 M , the range of luminosity at the demarcation point is 2950∼23550 L . Thereby the location of TP-AGB stars on the HR diagram is determined. (2) The following is qualitatively proposed. For an AGB star with primary mass M ≥ 3 M , after arriving at Teff = 3020± 35 K for the first time, because of the entrance into the stage of TP-AGB stars in stellar evolution, the formula of the matter loss of stellar wind should be revised. We may directly introduce a quantity which is not concerned with the surface luminosity, and make it dominate the formation of super stellar wind in the stage of TP-AGB stars. Moreover, via the analysis of the introduced turbulent pressure it is found that on the matter loss of stellar wind in the stage of TP-AGB stars the turbulent pressure exhibits rather large influence. Hence, it is proposed that the influence of turbulent pressure may be considered in the formula revision. References 1
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