- Email: [email protected]
On the Excitation Mechanism of Solar Five-minute Oscillations
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
ChineseAstronomy Astronomy and and Astrophysics Astrophysics 33 Chinese 33(2009) (2009)233–240 233–240
On the Excitation Mechanism of Solar Five-minute Oscillations† 1
XIONG Da-run1
DENG Li-cai2
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008 2 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012 Abstract 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 lowfrequency f- 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 p1modes) 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 ∼ 3700μHz (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 5minute 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. Key words: convection—Sun: oscillations †
Supported by National Natural Science Foundation and Ministry of Science and Techonlogy of China Received 2009–06–08 [email protected]
0275-1062/09/$-see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2009.07.011
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1. INTRODUCTION The solar 5-minute oscillation is an important discovery in the last century (Leighton, Noyer & Simon 1962). It enables us to be able to probe its internal structure by observing oscillations at solar surface. Helioseismology has achieved great progress. However, its excitation mechanism is not yet understood, and it is still a disputed problem. The Sun is located outside the Cepheid instability strip. So most people believe that the Sun is pulsationally stable due to damping of convection, and the oscillations of the Sun and solar-type stars are excited by the so-called turbulent stochastic mechanism (Goldreich & Keeley 1977a,b; Goldreich & Kumar 1988; Kumar, Franklin & Goldreich 1988; Kumar & Goldreich 1989; Goldreich, Murray &Kumar 1994; Samadi, Nordlund, Stein, Goupil & Roxburgh 2003). As shown in a series of our previous works, convection is not a pure damping effect for stellar oscillations as generally regarded, otherwise it is not possible to explain the large amplitude long-period variables and small amplitude red variables in the low-temperature region outside the instability strip. The goal of the present paper is to explore the excitation mechanism of the solar five-minute oscillations. In Section 2 the results of non-adiabatic oscillations of the solar non-radial modes are given, the excitation mechanism of solar five-minute oscillations is discussed in Section 3, and conclusions are given in the last section.
2. THE NUMERICAL RESULTS The Sun has a very extended convective envelope. Convection yields transport and change of energy and momentum in stellar interior, therefore it will influence the structure and pulsation stability of stars. Convection is the main exciting and damping mechanism for the Sun and all the red and yellow stars having extended convective envelopes. Their pulsation stability depends sensitively on the treatment of convection. Following the hydrodynamic equations and turbulence theory, we developed a non-local and time-dependent theory of convection (Xiong 1989; Xiong, Cheng & Deng 1997). It has more solid hydrodynamic base and is able to describe the dynamic behaviors of turbulent convection more exactly than the phenomenological mixing-length theory does. The reliability of our convection theory has been supported by observations. Our theoretical predictions agree with observations very well (Xiong, Cheng & Deng 1998; Xiong, Deng & Cheng 1998; Xiong & Deng 2001a; Xiong & Deng 2007). Using the same theory we calculated the radial non-adiabatic oscillations of the Sun. The results show that all the intermediate-order modes (10 < n < 25) are unstable and all the low-order (n < 11) and high-order (n > 24) modes are stable (Xiong, Cheng & Deng 2000). In the present paper we return to study the non-radial oscillations of the Sun by using the same non-local and time-dependent convection theory and the same convective parameters. The equilibrium model used in the calculations of oscillations is an envelope model of non-local convection of the Sun, however the nuclear energy has been taken into account. The depth on convection zone is 0.715R, which is consistent with the requirement of helioseismology. The working equations for calculation of the equilibrium model can be found in our previous papers (Xiong & Deng 2001b; Deng, Xiong & Chan 2006). The equations of linear non-adiabatic oscillations are in 14 orders. 6 of them are similar to the equations of traditional non-radial non-adiabatic oscillations, however the turbulent pressure in the
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conservation equation of momentum and turbulent thermal convection in the conservation equation of energy are included. They are derived by linearizing the average conservation equations of mass, momentum and energy, the equation of radiation transfer and the Poisson equation of gravity potential. The other 8 equations are derived directly by linearizing the dynamic equations of auto- and cross-correlations of turbulent velocity and temperature fluctuations. These equations are coupled with each other (the dynamic equations of average motion contain correlations of turbulent velocity and temperature fluctuations, while the dynamic equations of correlations include also the quantities of average motion), and they form a complete set of dynamic equations of radiative fluid. For the details of derivation procedure and simplifying hypothesis, please refer to our previous works (Xiong 1989; Xiong, Chen & Deng 1997). Normally, interference through convective energy transfer is referred as the thermodynamic coupling between convection and oscillations, while that through turbulent Reynold’s stress is called dynamic coupling between convection and oscillations. A modified version of the MHD equation of state (Hummer & Mihalas 1988; Mihalas, D¨ appen & Hummer 1988; D¨ appen, Mihalas, Hummer & Mihalas 1988) has been applied. For opacities, we used an analytic expression to approach the OPAL table (Rogers & Iglesias 1992) in intermediate- and high-temperature regions and the low-temperature opacity table (Alexander & Ferguson 1994) in the low-temperature region. The analytic approach turns out to be good within a few percent level, except at some extremely low temperature and high density. The advantage is obvious: smooth transition from the intermediate- and high-temperature OPAL table to the low-temperature Alexander table. We calculated at the same time the linear non-adiabatic oscillations of low- and intermediate-degrees (l = 1 − 25) g4–p39 modes for the Sun both with and without the coupling between convection and oscillations. When the coupling between convection and oscillations is not taken into account, all f- and low-frequency p-modes with frequency ν <∼ 3100μHz (or P >∼ 5.4 min) are pulsationally unstable, and all the high-order p-modes with frequency ν >∼ 3100μHz (or P <∼ 5.4 min) are pulsationally stable. The most linearly unstable mode is at ν ∼ 2600μHz (or P ∼ 6.5 min). Fig. 1a illustrates the variations of the pulsation amplitude growth per period η = −2πωi /ωr (ω = iωi + ωr is the complex angular frequency) with frequency ν = ωr /2π, where the circles, triangles, inverse-triangles and squares are respectively the oscillation modes of l =2, 5, 10 & 20, the large open symbols represent the unstable modes, while the small solid symbols are stable ones. When the coupling between convection and oscillations is taken into account, all the f- and low-order p-modes with ν <∼ 1000μHz (or P >∼ 16 min), and high-order p-modes with ν >∼ 5500μHz (or P <∼ 3 min) are pulsationally stable, while the p-modes with ∼ 1000μHz < ν <∼ 5500μHz are all unstable. The most unstable mode (η reaches maximum) is located at ν ∼ 3700μHz (or P ∼ 4.5 min), which agrees roughly with the observed peak of power spectrum of solar 5-minute oscillations (Libbrecht & Woodard 1991; Libbrecht 1988; Libbrecht & Zirin 1986). Fig. 1b shows the amplitude growth rate per period η as a function of frequency ν, the usage of symbol is the same as in Fig. 1a.
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Fig. 1 The amplitude growth rate per period η = −2πωi /ωr versus frequency for l = 2 (circles), 5 (triangles), 10 (inverse-triangles) and 20 (squares) oscillation modes. The large open and small solid symbols are respectively the unstable and stable modes. a) The coupling between convection and oscillations is not taken into account, b) the coupling between convection and oscillations is considered.
3. MECHANISMS OF EXCITATION AND DAMPING It can be seen from Fig. 1a that all the f- and low-order p-modes are pulsationally unstable, while the coupling between convection and oscillations is neglected. They are excited by the so-called radiative modulation excitation at the top of convective zone (Xiong, Cehng & Deng 1998). We have showed that all the low-order modes are pulsationally unstable for all the red and yellow stars on the right side of the blue edge of the Chepheid instability when the coupling between convection and oscillations is ignored. Therefore, the red edge of the instability strip his never been defined without such coupling (Xiong, Cheng & Deng 1998; Xiong, & Deng 2007). Of course, it is not true, because convection has important effects on the stability for these low-temperature stars with extended convective envelopes. Figs. 2a-d show the accumulated work W (W = WPg + WPt + Wbuoy , in solid lines), its gas pressure WPg (long-dashed lines) and turbulent pressure WPt (dotted lines) components as functions of depth for P5-, P10-, P30- and P39-modes with l = 2, when the coupling between convection and oscillations is taken into account. The component of buoyancy force WPbuoy is too small, so it is not shown in the plots. The accumulated work is defined by integration from stellar center to surface, therefore the value of the integrated work W (M0 ) at the surface is exactly the amplitude growth rate per period η = −2πωi /ωr . Our calculations show that W (M0 ) and η agree with each other within 1%. This is a straightforward and effective justification for numerical calculations. When the coupling between convection and oscillations is neglected, the p5-mode of l = 2 is unstable, while it becomes stable (W (M0 ) < 0 in Fig. 2a) when the coupling is considered. The p30-mode shows also different property. It is stable when the coupling between convection and oscillations is neglected, however it becomes unstable when the coupling is considered (W > 0 in Fig. 2c). Compared with Figs. 2a & b, the curves in Figs. 2c & d have relatively simple shapes. The violent variations of accumulated work curves in Figs. 2a & b happen in the ionization regions of hydrogen and helium, where CP ,
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Γ1 , Γ2 , Γ3 and χ all change violently. This is why the accumulated work shows complicated behavior there. Toward higher order modes, the amplitudes of oscillations decrease quickly towards interior of the Sun. Oscillations tend to occur in surface layers, therefore the curves of accumulated work are shifted to outer layers, so they have simple shapes (Figs. 2c & d).
Fig. 2 The accumulated work W (solid lines) and its gas pressure (dashed lines) and turbulent pressure (dotted lines) components versus depth for P5-(a), P10-(b), P30-(c) and P39(d)-modes of l = 2. The dashed-long dashed lines are the logarithm of the fractional radiation flux. The three horizontal bars indicate respectively the locations of ionization regions of hydrogen and helium.
It can be seen from Figs. 2a-c that, generally, turbulent pressure is always an excitation. This is due to the fact that during the process of stellar oscillations turbulent pressure Pt = ρx2 normally lags behind the variations of density due to the inertia of convective motion. In a Pt − V diagram it forms a positive Carrow circle, i.e. the turbulent kinetic energy converts into the pulsational kinetic energy, therefore it is an excitation. The gas pressure component of accumulated work, WPg , is more complicated. In the deep interior of the solar convection zone and far away from its boundaries, the radiative flux is much smaller than the convective flux, therefore WPg is mostly contributed by convective flux (the thermodynamic coupling between convection and oscillations). As we can see from Figs. 2a & b, in the deep interior of convection zone, the coupling between convection and oscillations is a damping mechanism. It is due to the fact that the variations of
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convective flux and density of gas, in general, are nearly synchronous. This is why the f- and low-order p-modes become stable once the coupling between convection and oscillations is considered. Towards higher-order modes, oscillations tend to happen in the surface layers, the damping resulted from the thermodynamic coupling at the deep convective zone weakens and the excitation resulted from radiative and convective modulation excitation at the top of the convection zone and turbulent pressure (dynamic coupling between convection and oscillations) strengthen, so they become unstable. Such a property of WPg is very clearly demonstrated in Figs. 2a-c. For p39-mode (Fig. 2d) the abrupt drop off of the accumulated work W near the surface is obviously caused by the damping in the outer atmosphere of the Sun. Following the above analysis, it can be known that the coupling between convection and oscillations plays a key role for the stabilization of low-frequency f- and p-modes and the excitation of intermediate-frequency p-modes. The Cepheid instability strip crosses the H-R diagram from upper-right corner to lowerleft corner. Along this strip, there are both populations I and II Cepheid, RR Lyrae, δ Scuti stars and pulsating dwarfs. They are all excited by radiative κ-mechanism. The Sun is located outside the instability strip. Most people believe that the solar oscillations are damped by convection, and the oscillations of the Sun and solar-type stars are excited by the so-called turbulent stochastic mechanism (Goldreich & Keeley 1977a,b; Goldreich & Kumar 1988; Kumar, Franklin & Goldreich 1988; Kumar & Goldreich 1989; Goldreich, Murray & Kumar 1994; Samadi, Nordlund, Stein, Goupil & Roxburgh 2003). Using our non-local and time-dependent statistical theory of convection, we not only explained successfully the red edge of the Chepheid instability strip (Xiong, Cheng & Deng 1998; Xiong & Deng 2001a), but also predicted theoretically the existence of the Mira instability strip in the low temperature region outside the Cepheid instability strip. These red variables are excited by the coupling between convection and oscillations. We also pointed out that the luminous red giants and red supergiants are pulsating at the fundamental mode and the first overtone, whose growth rates of pulsating amplitude are also large. Towards lower luminosity, the fundamental mode and the first overtone become stable, instability moves to high-order modes and the amplitude growth rates also decrease (Xiong, Deng & Cheng 1998; Xiong & Deng 2007). Our theoretical predictions agree with observations of pulsating red- and yellow-variables very well (Eggen 1973a,b, 1975a,b 1977; Percy 1997; Wo´zniak, Williams, Vetrand & Gupta 2004; Wood 2000; Soszy´ nsky, Udalsky, Kubiak et al. 2004,2005). Furthermore, both theories and observations show such an obvious tendency: for giants and supergiants, oscillations tend to be radial, usually single or a few low-order overtones, and to have large amplitudes, such as Cepheids, RR Lyrae and Mira variables. For subgiants and dwarfs towards the main sequence, non-radial oscillations become more popular, and usually have multiple modes at the same time with small amplitudes, such as β Cephei, δ Scuti, rapidly pulsating Ap and slowly pulsating B stars (SPB), and pulsating white dwarf stars. The present study shows that, when considering the coupling between convection and oscillations, the solar 5-minute oscillations can be self-exciting, which follows the above tendency. Therefore, we have reason to believe that the solar 5-minute oscillations are a natural extension of the sequence starting from Miras to small amplitude red and yellow variables as well as from giant-type to dwarftype variables. There is no wide gap between them: they are all excited by the coupling between convection and oscillations. This is true at least for low- and intermediate-degree
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modes. The limited line width of p-modes of the Sun (Libbrecht 1988) means their limited lifetimes , so most people believe that the p-modes of the Sun are damping due to convection. It was the most important argument of turbulent stochastic excitation. If the intermediatefrequency p-modes of the Sun are self-exciting, such as advocated by us, there must be a damping mechanism other than the convective coupling. We think that the refraction, scattering and additional phase shifts of standing acoustic waves caused by turbulent media seem to be reasonable explanations. If one were using the observed line width to estimate the damping, the “regular” coupling between convection and oscillations alone would not be enough to support the oscillations of the high-frequency p-modes of the Sun. Therefore, there must be some other excitation mechanism for these high-frequency modes. We think that the turbulent stochastic excitation should be the most possible candidate. 4. CONCLUSIONS In the present paper we calculated the linear non-adiabatic oscillations of low- and intermediate-degree modes for the Sun and studied their excitation mechanism. The main conclusions can be summaried as follows: 1. The solar oscillations with periods of 3-16 minutes are self-exciting, this is true at least for low- and intermediate-degree p-modes. 2. 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-frequency modes, the coupling between convection and oscillations dominates; while for high-frequency modes, stochastic excitation dominates. 3. Refraction, scattering and additional phase shifts for acoustic waves caused by turbulent media are very likely the major damping mechanisms for p-modes. When the coupling between convection and oscillations is taken into accont, all the f- and low-order p-modes with ν <∼ 1000μHz (or P >∼ 16 min) are pulsationally stable, however the low-degree (l < 6) p1-modes are unstable. Our careful analysis shows that the instability for these low-degree p1-modes is excited by the work of buoyancy force. The velocity amplitude of the p1-mode of l = 1 is estimated to be about 1-2 mm/second by comparing our theoretical amplitude growth rates with the observed amplitudes of the solar five-minute oscillations (Libbrecht 1988). It is still possible to be determined, although this is difficult. The amplitude decreases quickly with increase of l. The p1-mode for l > 2 will be hard to determine. This fact leads to a simple and useful method for the determination of excitation mechanism of solar five-minute oscillations. If only low-degree p1-modes are determined and all the low-degree f- and p2-modes adjacent are not determined, then our excitation mechanism of regular convection coupling for low-frequency modes is correct; otherwise if the low-degree p1-modes and all the adjacent f- and p2-modes are determined, then the solar five-minute oscillations will be excited by the turbulent stochastic excitation.
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ELSEVIER
ChineseAstronomy Astronomy and and Astrophysics Astrophysics 33 Chinese 33(2009) (2009)233–240 233–240
On the Excitation Mechanism of Solar Five-minute Oscillations† 1
XIONG Da-run1
DENG Li-cai2
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008 2 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012 Abstract 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 lowfrequency f- 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 p1modes) 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 ∼ 3700μHz (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 5minute 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. Key words: convection—Sun: oscillations †
Supported by National Natural Science Foundation and Ministry of Science and Techonlogy of China Received 2009–06–08 [email protected]
0275-1062/09/$-see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2009.07.011
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1. INTRODUCTION The solar 5-minute oscillation is an important discovery in the last century (Leighton, Noyer & Simon 1962). It enables us to be able to probe its internal structure by observing oscillations at solar surface. Helioseismology has achieved great progress. However, its excitation mechanism is not yet understood, and it is still a disputed problem. The Sun is located outside the Cepheid instability strip. So most people believe that the Sun is pulsationally stable due to damping of convection, and the oscillations of the Sun and solar-type stars are excited by the so-called turbulent stochastic mechanism (Goldreich & Keeley 1977a,b; Goldreich & Kumar 1988; Kumar, Franklin & Goldreich 1988; Kumar & Goldreich 1989; Goldreich, Murray &Kumar 1994; Samadi, Nordlund, Stein, Goupil & Roxburgh 2003). As shown in a series of our previous works, convection is not a pure damping effect for stellar oscillations as generally regarded, otherwise it is not possible to explain the large amplitude long-period variables and small amplitude red variables in the low-temperature region outside the instability strip. The goal of the present paper is to explore the excitation mechanism of the solar five-minute oscillations. In Section 2 the results of non-adiabatic oscillations of the solar non-radial modes are given, the excitation mechanism of solar five-minute oscillations is discussed in Section 3, and conclusions are given in the last section.
2. THE NUMERICAL RESULTS The Sun has a very extended convective envelope. Convection yields transport and change of energy and momentum in stellar interior, therefore it will influence the structure and pulsation stability of stars. Convection is the main exciting and damping mechanism for the Sun and all the red and yellow stars having extended convective envelopes. Their pulsation stability depends sensitively on the treatment of convection. Following the hydrodynamic equations and turbulence theory, we developed a non-local and time-dependent theory of convection (Xiong 1989; Xiong, Cheng & Deng 1997). It has more solid hydrodynamic base and is able to describe the dynamic behaviors of turbulent convection more exactly than the phenomenological mixing-length theory does. The reliability of our convection theory has been supported by observations. Our theoretical predictions agree with observations very well (Xiong, Cheng & Deng 1998; Xiong, Deng & Cheng 1998; Xiong & Deng 2001a; Xiong & Deng 2007). Using the same theory we calculated the radial non-adiabatic oscillations of the Sun. The results show that all the intermediate-order modes (10 < n < 25) are unstable and all the low-order (n < 11) and high-order (n > 24) modes are stable (Xiong, Cheng & Deng 2000). In the present paper we return to study the non-radial oscillations of the Sun by using the same non-local and time-dependent convection theory and the same convective parameters. The equilibrium model used in the calculations of oscillations is an envelope model of non-local convection of the Sun, however the nuclear energy has been taken into account. The depth on convection zone is 0.715R, which is consistent with the requirement of helioseismology. The working equations for calculation of the equilibrium model can be found in our previous papers (Xiong & Deng 2001b; Deng, Xiong & Chan 2006). The equations of linear non-adiabatic oscillations are in 14 orders. 6 of them are similar to the equations of traditional non-radial non-adiabatic oscillations, however the turbulent pressure in the
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conservation equation of momentum and turbulent thermal convection in the conservation equation of energy are included. They are derived by linearizing the average conservation equations of mass, momentum and energy, the equation of radiation transfer and the Poisson equation of gravity potential. The other 8 equations are derived directly by linearizing the dynamic equations of auto- and cross-correlations of turbulent velocity and temperature fluctuations. These equations are coupled with each other (the dynamic equations of average motion contain correlations of turbulent velocity and temperature fluctuations, while the dynamic equations of correlations include also the quantities of average motion), and they form a complete set of dynamic equations of radiative fluid. For the details of derivation procedure and simplifying hypothesis, please refer to our previous works (Xiong 1989; Xiong, Chen & Deng 1997). Normally, interference through convective energy transfer is referred as the thermodynamic coupling between convection and oscillations, while that through turbulent Reynold’s stress is called dynamic coupling between convection and oscillations. A modified version of the MHD equation of state (Hummer & Mihalas 1988; Mihalas, D¨ appen & Hummer 1988; D¨ appen, Mihalas, Hummer & Mihalas 1988) has been applied. For opacities, we used an analytic expression to approach the OPAL table (Rogers & Iglesias 1992) in intermediate- and high-temperature regions and the low-temperature opacity table (Alexander & Ferguson 1994) in the low-temperature region. The analytic approach turns out to be good within a few percent level, except at some extremely low temperature and high density. The advantage is obvious: smooth transition from the intermediate- and high-temperature OPAL table to the low-temperature Alexander table. We calculated at the same time the linear non-adiabatic oscillations of low- and intermediate-degrees (l = 1 − 25) g4–p39 modes for the Sun both with and without the coupling between convection and oscillations. When the coupling between convection and oscillations is not taken into account, all f- and low-frequency p-modes with frequency ν <∼ 3100μHz (or P >∼ 5.4 min) are pulsationally unstable, and all the high-order p-modes with frequency ν >∼ 3100μHz (or P <∼ 5.4 min) are pulsationally stable. The most linearly unstable mode is at ν ∼ 2600μHz (or P ∼ 6.5 min). Fig. 1a illustrates the variations of the pulsation amplitude growth per period η = −2πωi /ωr (ω = iωi + ωr is the complex angular frequency) with frequency ν = ωr /2π, where the circles, triangles, inverse-triangles and squares are respectively the oscillation modes of l =2, 5, 10 & 20, the large open symbols represent the unstable modes, while the small solid symbols are stable ones. When the coupling between convection and oscillations is taken into account, all the f- and low-order p-modes with ν <∼ 1000μHz (or P >∼ 16 min), and high-order p-modes with ν >∼ 5500μHz (or P <∼ 3 min) are pulsationally stable, while the p-modes with ∼ 1000μHz < ν <∼ 5500μHz are all unstable. The most unstable mode (η reaches maximum) is located at ν ∼ 3700μHz (or P ∼ 4.5 min), which agrees roughly with the observed peak of power spectrum of solar 5-minute oscillations (Libbrecht & Woodard 1991; Libbrecht 1988; Libbrecht & Zirin 1986). Fig. 1b shows the amplitude growth rate per period η as a function of frequency ν, the usage of symbol is the same as in Fig. 1a.
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Fig. 1 The amplitude growth rate per period η = −2πωi /ωr versus frequency for l = 2 (circles), 5 (triangles), 10 (inverse-triangles) and 20 (squares) oscillation modes. The large open and small solid symbols are respectively the unstable and stable modes. a) The coupling between convection and oscillations is not taken into account, b) the coupling between convection and oscillations is considered.
3. MECHANISMS OF EXCITATION AND DAMPING It can be seen from Fig. 1a that all the f- and low-order p-modes are pulsationally unstable, while the coupling between convection and oscillations is neglected. They are excited by the so-called radiative modulation excitation at the top of convective zone (Xiong, Cehng & Deng 1998). We have showed that all the low-order modes are pulsationally unstable for all the red and yellow stars on the right side of the blue edge of the Chepheid instability when the coupling between convection and oscillations is ignored. Therefore, the red edge of the instability strip his never been defined without such coupling (Xiong, Cheng & Deng 1998; Xiong, & Deng 2007). Of course, it is not true, because convection has important effects on the stability for these low-temperature stars with extended convective envelopes. Figs. 2a-d show the accumulated work W (W = WPg + WPt + Wbuoy , in solid lines), its gas pressure WPg (long-dashed lines) and turbulent pressure WPt (dotted lines) components as functions of depth for P5-, P10-, P30- and P39-modes with l = 2, when the coupling between convection and oscillations is taken into account. The component of buoyancy force WPbuoy is too small, so it is not shown in the plots. The accumulated work is defined by integration from stellar center to surface, therefore the value of the integrated work W (M0 ) at the surface is exactly the amplitude growth rate per period η = −2πωi /ωr . Our calculations show that W (M0 ) and η agree with each other within 1%. This is a straightforward and effective justification for numerical calculations. When the coupling between convection and oscillations is neglected, the p5-mode of l = 2 is unstable, while it becomes stable (W (M0 ) < 0 in Fig. 2a) when the coupling is considered. The p30-mode shows also different property. It is stable when the coupling between convection and oscillations is neglected, however it becomes unstable when the coupling is considered (W > 0 in Fig. 2c). Compared with Figs. 2a & b, the curves in Figs. 2c & d have relatively simple shapes. The violent variations of accumulated work curves in Figs. 2a & b happen in the ionization regions of hydrogen and helium, where CP ,
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Γ1 , Γ2 , Γ3 and χ all change violently. This is why the accumulated work shows complicated behavior there. Toward higher order modes, the amplitudes of oscillations decrease quickly towards interior of the Sun. Oscillations tend to occur in surface layers, therefore the curves of accumulated work are shifted to outer layers, so they have simple shapes (Figs. 2c & d).
Fig. 2 The accumulated work W (solid lines) and its gas pressure (dashed lines) and turbulent pressure (dotted lines) components versus depth for P5-(a), P10-(b), P30-(c) and P39(d)-modes of l = 2. The dashed-long dashed lines are the logarithm of the fractional radiation flux. The three horizontal bars indicate respectively the locations of ionization regions of hydrogen and helium.
It can be seen from Figs. 2a-c that, generally, turbulent pressure is always an excitation. This is due to the fact that during the process of stellar oscillations turbulent pressure Pt = ρx2 normally lags behind the variations of density due to the inertia of convective motion. In a Pt − V diagram it forms a positive Carrow circle, i.e. the turbulent kinetic energy converts into the pulsational kinetic energy, therefore it is an excitation. The gas pressure component of accumulated work, WPg , is more complicated. In the deep interior of the solar convection zone and far away from its boundaries, the radiative flux is much smaller than the convective flux, therefore WPg is mostly contributed by convective flux (the thermodynamic coupling between convection and oscillations). As we can see from Figs. 2a & b, in the deep interior of convection zone, the coupling between convection and oscillations is a damping mechanism. It is due to the fact that the variations of
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convective flux and density of gas, in general, are nearly synchronous. This is why the f- and low-order p-modes become stable once the coupling between convection and oscillations is considered. Towards higher-order modes, oscillations tend to happen in the surface layers, the damping resulted from the thermodynamic coupling at the deep convective zone weakens and the excitation resulted from radiative and convective modulation excitation at the top of the convection zone and turbulent pressure (dynamic coupling between convection and oscillations) strengthen, so they become unstable. Such a property of WPg is very clearly demonstrated in Figs. 2a-c. For p39-mode (Fig. 2d) the abrupt drop off of the accumulated work W near the surface is obviously caused by the damping in the outer atmosphere of the Sun. Following the above analysis, it can be known that the coupling between convection and oscillations plays a key role for the stabilization of low-frequency f- and p-modes and the excitation of intermediate-frequency p-modes. The Cepheid instability strip crosses the H-R diagram from upper-right corner to lowerleft corner. Along this strip, there are both populations I and II Cepheid, RR Lyrae, δ Scuti stars and pulsating dwarfs. They are all excited by radiative κ-mechanism. The Sun is located outside the instability strip. Most people believe that the solar oscillations are damped by convection, and the oscillations of the Sun and solar-type stars are excited by the so-called turbulent stochastic mechanism (Goldreich & Keeley 1977a,b; Goldreich & Kumar 1988; Kumar, Franklin & Goldreich 1988; Kumar & Goldreich 1989; Goldreich, Murray & Kumar 1994; Samadi, Nordlund, Stein, Goupil & Roxburgh 2003). Using our non-local and time-dependent statistical theory of convection, we not only explained successfully the red edge of the Chepheid instability strip (Xiong, Cheng & Deng 1998; Xiong & Deng 2001a), but also predicted theoretically the existence of the Mira instability strip in the low temperature region outside the Cepheid instability strip. These red variables are excited by the coupling between convection and oscillations. We also pointed out that the luminous red giants and red supergiants are pulsating at the fundamental mode and the first overtone, whose growth rates of pulsating amplitude are also large. Towards lower luminosity, the fundamental mode and the first overtone become stable, instability moves to high-order modes and the amplitude growth rates also decrease (Xiong, Deng & Cheng 1998; Xiong & Deng 2007). Our theoretical predictions agree with observations of pulsating red- and yellow-variables very well (Eggen 1973a,b, 1975a,b 1977; Percy 1997; Wo´zniak, Williams, Vetrand & Gupta 2004; Wood 2000; Soszy´ nsky, Udalsky, Kubiak et al. 2004,2005). Furthermore, both theories and observations show such an obvious tendency: for giants and supergiants, oscillations tend to be radial, usually single or a few low-order overtones, and to have large amplitudes, such as Cepheids, RR Lyrae and Mira variables. For subgiants and dwarfs towards the main sequence, non-radial oscillations become more popular, and usually have multiple modes at the same time with small amplitudes, such as β Cephei, δ Scuti, rapidly pulsating Ap and slowly pulsating B stars (SPB), and pulsating white dwarf stars. The present study shows that, when considering the coupling between convection and oscillations, the solar 5-minute oscillations can be self-exciting, which follows the above tendency. Therefore, we have reason to believe that the solar 5-minute oscillations are a natural extension of the sequence starting from Miras to small amplitude red and yellow variables as well as from giant-type to dwarftype variables. There is no wide gap between them: they are all excited by the coupling between convection and oscillations. This is true at least for low- and intermediate-degree
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modes. The limited line width of p-modes of the Sun (Libbrecht 1988) means their limited lifetimes , so most people believe that the p-modes of the Sun are damping due to convection. It was the most important argument of turbulent stochastic excitation. If the intermediatefrequency p-modes of the Sun are self-exciting, such as advocated by us, there must be a damping mechanism other than the convective coupling. We think that the refraction, scattering and additional phase shifts of standing acoustic waves caused by turbulent media seem to be reasonable explanations. If one were using the observed line width to estimate the damping, the “regular” coupling between convection and oscillations alone would not be enough to support the oscillations of the high-frequency p-modes of the Sun. Therefore, there must be some other excitation mechanism for these high-frequency modes. We think that the turbulent stochastic excitation should be the most possible candidate. 4. CONCLUSIONS In the present paper we calculated the linear non-adiabatic oscillations of low- and intermediate-degree modes for the Sun and studied their excitation mechanism. The main conclusions can be summaried as follows: 1. The solar oscillations with periods of 3-16 minutes are self-exciting, this is true at least for low- and intermediate-degree p-modes. 2. 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-frequency modes, the coupling between convection and oscillations dominates; while for high-frequency modes, stochastic excitation dominates. 3. Refraction, scattering and additional phase shifts for acoustic waves caused by turbulent media are very likely the major damping mechanisms for p-modes. When the coupling between convection and oscillations is taken into accont, all the f- and low-order p-modes with ν <∼ 1000μHz (or P >∼ 16 min) are pulsationally stable, however the low-degree (l < 6) p1-modes are unstable. Our careful analysis shows that the instability for these low-degree p1-modes is excited by the work of buoyancy force. The velocity amplitude of the p1-mode of l = 1 is estimated to be about 1-2 mm/second by comparing our theoretical amplitude growth rates with the observed amplitudes of the solar five-minute oscillations (Libbrecht 1988). It is still possible to be determined, although this is difficult. The amplitude decreases quickly with increase of l. The p1-mode for l > 2 will be hard to determine. This fact leads to a simple and useful method for the determination of excitation mechanism of solar five-minute oscillations. If only low-degree p1-modes are determined and all the low-degree f- and p2-modes adjacent are not determined, then our excitation mechanism of regular convection coupling for low-frequency modes is correct; otherwise if the low-degree p1-modes and all the adjacent f- and p2-modes are determined, then the solar five-minute oscillations will be excited by the turbulent stochastic excitation.
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