Vol. 44 no. 2

SCIENCE IN CHINA (SeriesA)

February 2001

Non-homogeneous solar models with metal enriched envelopes YANG Jiayan ( ~ ~ - ~ ) ,

LI Yan ( ~

,~- ) & XU Huayin (-/~-~ ~- )

Yunnan Observatory, NAOCAS, Kunming 650011, China Received March 20, 2000 Abstract Solar models with enhancement of heavy elements in the convective envelopes are investigated using the updated input physics. Unlike previous Iow-Z models that adopt quite low central metal abundance to considerably reduce neutrino fluxes, we investigate the effects of moderate enrichment of heavy elements in the solar convection zone on the solar structure and p-mode oscillations. It is found that the metal endched models have less massive convection zones with deeper bottom boundaries, and their temperature profiles are systematically lower while the sound speed profiles are higher in the interior and lower in the envelope than that of the standard model. The contamination of heavy elements at different evolution phases is investigated, which results in little influence on the properties of the solar age models. The surface helium abundance is reduced considerably, and is able to approach the seismically determined value when the enhancement of heavy elements in the convection zone is carefully adjusted. The p-mode frequency patterns of our metal enriched models are systematically 10 pHz lower than those of the standard model, and are in better agreement with the results of observations. Keywords:

solar structure, solar evolution, solar oscillations.

The huge volume of the Sun serves as a sound resonance cavity, and restricts sound waves propagating within it to form standing wave oscillations. Great efforts are made to observe these oscillation modes, and their frequencies have been measured more and more precisely. Recently, these high quality observation data are conversely used to probe the structure of the Sun. The position of the base of the convection zone, the envelope helium abundance, and the sound speed as a function of the radius are derived directly from the observed oscillation frequencies to a high precision. These impose stringent restrictions on the solar modeling, and allow definitive tests and refinements of the input physics. An important assumption involved in the so-called standard solar models is that, except for the central nuclear transmutation, the chemical composition is homogeneous in the solar interior and keeps constant throughout the solar evolution. The standard solar models have a serious problem, i . e . the predicted solar neutrino fluxes are two or three times larger than what have been detected by different experiments Ill . Low Z models are proposed at the beginning to reduce the predicted solar neutrino flux E23. Contrary to the standard models, low Z models take into account the possibility of chemical stratification of the Sun. A pre-main sequence star is chemically homogeneous, since it experiences fully convective mixing during the Hayashi phase evolution. However, the star may capture some interplanetary matter afterwards. Joss E3] suggested that stellar surfaces may be continuously enhanced in heavy elements by in-falling comets. Levy and Ruzmaikina E43 investigated the effects of dust-gas separation during the star formation process. Dust grains that aggregate by turbulence to large sizes will settle from the upper layers toward the center of the

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molecular cloud and merge into the proto-planetary nebula disk, and finally fall onto the star by means of disk accretion. The Sun has a convection zone just below its photosphere, and the falling matter will be mixed into the whole convection zone. As the metal abundance of the interplanetary dust is much higher than that of the solar material, the solar convection zone will be enhanced with heavy elements. Low-Z models can give rather low neutrino fluxes, but usually result in shallow convection zones and very low initial helium abundance. Moreover, their calculated p-mode oscillation frequencies are in worse agreement with the observations than the standard homogeneous solar models. Therefore, low Z models are thought to be unrealistic in recent years. However, lots of evidence confirms that the solar envelope has inevitably been contaminated by interplanetary materials, even if not as much as demanded by previous low Z models. In this paper, we investigate the moderate enhancement of the envelope metallicity, using the updated input physics, and focus our attention on the influences of the structure and p-mode oscillations of the Sun rather than the solar neutrino problem. 1

Physics assumptions and c o m p u t a t i o n m e t h o d s

Solar evolution models are calculated using an evolution code originally written by Kippenhahn et al. [53 and updated by Stix tr] and Stix & Skaley ETl . Some modifications are made to incorporate the recent considerations of the input physics. Energy transfer by convection is treated according to the standard mixing-length theory, and the boundaries of the convection zones are de. 9 E8] . Nuclear reaction rates are taken from Caughlan and termined by the Schwarzschild criterion Fowler' s tables Eg] . Modifications to the equation of state are described by Stix and Skaley[7] , according to the Debye-Htickel approximation. OPAL opacities g91hz series El~ are used in the high temperature region. In the outer envelope of the Sun, low-temperature opacities from Alexander and Ferguson are used to include the contributions from molecules Ell] . These two sets of opacity data are connected around temperatures of 8000K. At that fitting point they coincide very well with each other. The element diffusion, which is known to considerably improve the agreement between calculated and observed p-mode frequencies, is not considered in the present work, in order to distinguish the effects of the envelope metal enrichment. It will be incooperated in the next step of our investigations. Our solar models consist of more than 23000 mass zones and are evolved from the zero-age main sequence for 210 time steps to reach the present solar age. Previous low-Z models usually adopted very low central metal abundance so as to reduce the calculated neutrino emission fluxes. Observations show, however, that the Sun has the normal Population I abundance [12'13] Solar neutrino experiments also put strong restrictions on the metal abundance distribution, i . e . the metal abundance in the solar interior do not deviate too much from its surface observation value Ct4J . Correspondingly, we choose the initial metallieity to be Zo = 0 . 0 1 , which is much higher than the choices of some other investigations[15'16] . In the early history of the solar system, the Sun is thought to be surrounded by a proto-planetary disk of dense gas and dust mixture. The original Sun has high luminosity. Matter can escape because of radiant pressure. At the same time, planets arround the Sun formed gradually and interplanetary space became vacuum, as we see today. If some interplanetary matter falls onto the Sun, we prefer to believe that it happens during the early phase of the solar evolution. When the Sun is still surrounded by an opaque disk, the dust would fall onto the Sun more quickly than the gas does because of its bigger density. As a result, the heavy element fraction of solar envelope is different ,

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from the original fraction. Interplanetary mass accretion can be very complex. According to the present observation, the processes of mass accretion and mass loss may be existent at the same time. We assume, for simplicity, that the Sun acquires an amount of matter made up of pure heavy elements at a certain moment in its early evolution, resulting in the enhancement of its envelope metallicity to the observed value of Z = O. 02. This is also a distinction from the previous approaches of gradual enrichment of the envelope metallicity. The fallen interplanetary matter immediately spread out by convective motion to the whole convection zone, and the Sun then consists of two parts, the envelope enhanced with heavy elements and the core of the original metal abundance, with the bottom of the convection zone as the boundary. Suppose f x M| to be the initial mass of the model and k x f x M| the mass of the convection zone just before the metal contamination. After the acquirement of the interplanetary matter, which is assumed to be composed of pure heavy elements, the mass of the model increases to 1M| = 1.99 x 1033 g, while the metal abundance in the convection zone increases from Z0 to Z . After a few steps of simple manipulations, f is found to be 1-Z f = 1- Z + k(ZZo)

(1)

Properties of evolution models We have calculated three different series of evolution models. Model A is a standard model, which does not consider the effects of metal enrichment. Models B and C are metal enriched models. The metal enrichment takes place for model B at the zero-age main sequence and for model C at the age of 1 Gyrs old for comparison. We want to see if the process of accretion will affect the present-day solar structure seriously. As we will find later, the influence is so little that the simple approximation we used would not lead to serious problems. The initial helium abundance, Y0, and the mixing-length to pressure scale height ratio, c~, are iteratively adjusted to give the solar luminosity (3.844 x 10 33 ergs s - 1 ) and solar radius (6.96 x 10 x~ cm) at the present solar age (4.566 Gym). In the meantime, the mass fraction of the convection zone just before the interplanetary contamination, k, is also found. Table 1 gives the basic parameters of the three models, where Y0 and Z0 are the initial helium and metal abundances. For solar age models, Yc and Zc are the central helium and metal Table 1

Properties of the standard and metal enriched models

Parameters

Model A

Model B

Model C

Y0, Zo

0.264,0.02

0.203,0.01

0.202,0.01

a

1.934

1.933

1. 926

0.0251

0.0236

k 1- f

0. 000261

0. 000245

0.563,0.01 14.79

0.563,0.01 14.79

Y~, Z~ TJlO6K

0.646,0.02 15.62

pig" era- 3 M~/M| Rbc/R|

149.7

14l .4

141.5

0.975 0.714

0.978 0.711

0. 978 0.712

T~/106K

2.237

2.174

2.168

~cl/SNUs ~r

7.99 140.32

3.17 104.87

3.17 104.88

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abundance, Tc and pc are the central temperature and density, Mbc is the mass interior to the base of the convection z o n e , R b c and Tbc a r e the radius and temperature at the base of the convection zone. It can be found from table 1 that 1 - f , which is the mass in unit of M| falling onto the Sun, is about O. 00025 M| just as much as the mass of Saturn but consisting of purely heavy elements. The value (1029g) is a little greater than that of some authors'. For instance, Joss [3] gave his estimated value 1026g. But Joss's estimate was based on the rate at which comets are presently being accreted by the Sun. In the early history of solar system, there were dense interplanetary matter around the Sun. The rate of accretion at that time would be higher than the present. So the mass accreted may approach our result. If we take the metal abundance of the interplanetary dust to be the solar vlaue of 0.02 as the minimum limit, the mass required to provide the captured heavy elements is 0.0125M| which is comparable to the mass of the solar protoplanetary nebula of 0.02M| Els] . Therefore, we cannot rule out the possibility of the enrichment of the solar envelope metallicity by interplanetary contamination. It is very interesting to notice that the bases of the convection zones for the metal enriched models B and C are a little deeper than that of the standard model A, and are in very good agreement with recent helioseismic determinationsE~9] . This is contrary to the results of previous investigations, in which metal enriched models result in shallow convection zones E16'2~ . The temperatures at the bases of the convection zones for our models B and C are, however, a bit lower than that of the standard model A. This can be understood by checking the mass exterior to the base of the convection zone. By table 1, since model A has more mass in the convection zone with a shorter depth to extend into the solar interior than models B and C, the convection envelope of model A is denser, and therefore more opaque than those of models B and C, which gives a higher temperature at its bottom. Table 1 shows that models B and C give quite low neutrino fluxes for either C1 or Ga detectors and initial helium abundance, similar to previous low-Z models E1'16]o This is mainly due to the fact that the central helium abundance of models B and C, which has strong influence on the flux of high energy neutrinos, is considerably lower than that of the standard model A. Further decrease in the central heavy element abundance can still reduce the neutrino fluxes, along with even smaller value of the initial helium abundance, and therefore, is not a reasonable measure -0.05 | to solve the solar neutrino anomaly. Our models leave rooms for recent declarations of possible eto neutrino masses and oscillations. -0.1 Evolution tracks for the cmputed three models are shown in fig. 1. Dots and crosses along the two lines crossing the figure from the upper 3.762 3.360 3.358 3.756 3.754 3.752 left to the lower right part represent respectively log Teer the evolution track of models A and B, while stars Fig. 1. Evolution tracks of the standard and metal enriched models. Dots track represents the standard model A, and and crosses that form a zigzag pattern represent crosses track the metal enriched model B. The track with respectively the evolution track of model C before stars at the beginning part plus crosses behind is for the metal and after the capture of interplanetary matter. It enriched model C. can be noticed that the track of model B is close

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to that of model A, with only a little higher luminosity for a given effective temperature. On the other hand, model C has quite high effective temperature before the capture of interplanetary matter. After the metal enrichment in its convection envelope, the model adjusts its structure and turns to evolve toward the low effective temperature direction. The time consumed by this adjustment phase is about 0 . 6 Gyrs, and after that the evolution track is in good coincidence with that of model B. This feature is quite different from the standard models', which evolve along a track of increasing the effective temperature and luminosity. If the interplanetary contamination takes place for several times on the Sun, and each results in considerable enrichment of the envelope metallicity and later adjustment of the internal structure, the evolution track may show corresponding number of similar zigzag like patterns. However, it is surprising to notice in table 1 that the properties of models B and C at the solar age are almost identical, indicating that, if the Sun has been contaminated at the early phase, when and how the contamination happened have effect on its early evolution track but have little effect on the present solar properties. As shown in fig. 2, the masses exterior to • 1031 the bases of the convection zones for models B 6.0 and C are always smaller than that of model A 5.5 during their evolution. For model C, the mass beyond the base of its convection zone decreases ~ 5.0 considerably at the moment the envelope metal ~ 4.5 enhancement happens, and increases later to approach the corresponding value of model B. On 4.o J the other hand, the temperature at the base of the convection zone for model C responds to the eno i ;, 5 xlo Age/a velope metal enhancement in a different way: it jumps up at the moment of the metal enrichment Fig. 2. Masses included beyond the bases of the convection and increases continuously to approach the corre- zones of models A, B, and C during the evolution9 Dots represent the standard model A, crosses the metal enriched model sponding value of model B. B, and stars the metal enriched model C. Figs. 3 and 4 show for the solar age models the differences in the pressure and temperature profile of models B and C with respect to that of model A. It can be seen that the temperatures over the whole interior of models B and C are about .. .....

...

~ .....

...

-.......

r

<

~

G

-0.05

-0.2 i

0

2X'10 i~

4

X i

l0 i~

Temperature differences between the standard and metal enriched models. Solid line is for model B to A, and dotted line for model C to A.

i

I

2X 10 l~ 4X 101~ r/cm

I

6X 101~

9

r / c m

Fig. 3.

I

0

6XI10 I~

Fig. 4.

Pressure differences between the standard and metal enriched models. Solid line is for model B to A, and dotted line for model C to A.

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4%---9 % cooler than model A, while the pressures of models B and C are about 0 % - - 4 % higher in the nuclear burning core and about 0 % - - 1 6 % lower in the middle radiative and outermost convective zones than that of model A. It is also of interest to notice that the differences in both pressure and temperature of models B and C with respect to model A are almost constant in the convection zones.

3

Frequencies of the p-mode oscillations

The three solar age models are inputted into an updated linear adiabatic pulsation code of Li [22] to calculate the p-mode oscillation frequencies. We calculated the frequencies for the spherical harmonic degree l = 0 - 10, 15, 20. Frequencies of modes with high values of 1 are seriously influenced by the non-adiabatic effects and are not considered in the present paper. We have used the gray atmosphere in our p-mode oscillation calculations. In fig. 5 we compare the calculated frequencies with the observation values obtained by Lib0 . ,a,..s "~, . . 9 ~ . ,-.x;_x." brecht et al. [23]~ It is found that the p-mode frequencies of models B and C are in better agreement ~ -2o with the observations than those of model A. This is an important result, and completely contradicts the conclusions of previous low-Z models. As our metal enriched models should not be necessary to -40 i result in worse p-mode frequencies, a major arguI t I i 1000 2000 3000 4000 5000 ment against the possibility of non-homogeneous soObserved frequency/IxHz lar models maybe no longer exists. Fig. 5. Comparisons of calculated solar p-mode frequenWe notice that the tendency of the frequency cies for the standard and metal enriched models with respect differences between the calculation and observation to the observations. Solid lines are for the standard model A, dashed lines and dotted lines are for models B and C, is similar for all three models. The p-mode frerespectively. The spherical degrees of the modes quencies of models B and C are in good agreement / = 0 - - 1 0 , 15, 20. with each other, while those of the standard model A are systematically about 10 ~tHz higher than those of models B and C. This can be understood with the help of fig. 6, the profiles of sound speed differences of models B and C with respect to model A, as follows. It can be noticed that the sound speed increases by 0 % - - 2 % in the nuclear fusion cores and decreases by 0 % - - 2 % in the , e 0.02 outer adjacent radiative zones for models B and .C ~ compared with model A, and is almost the same m ~ the convection zones of all three models. It is known that the sound speed is much larger in the solar core than in the solar envelope, and a sound wave will therefore spend more time to travel across -0.02 the solar envelope than to travel across the solar 2• ~ 4• 6• r/cm core. As a result, the time shortened due to relatively high sound speed to travel across the cores of Fig. 6. Differences in the sound speed between the stanmodels B and C compared with the case of model A dard and metal enriched models. Solid line is for models B to A, and dotted line for model C to A. is less than the time delayed by relatively slow

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sound speed to travel across the outer radiative zones. The net effect is the increase in the time needed for the p-mode oscillation waves to propagate across the whole solar interior and therefore the decrease in their eigenfrequencies.

4

Discussions and conclusions

Low-Z models are rejected so far due to three major disagreements with the observations: the depths of their convection zones are shallower than the seismically determined value, their pmode frequencies are in worse agreement with the observational, and their initial helium abundance adjusted by matching the model's radius and luminosity to the observed values at the present solar age is too small. However, much evidence supports the hypothesis that the Sun might be enriched with heavy elements, and the metal abundance in its convection zone could be higher than in the nuclear fusion core. We investigate this possibility again using the up-to-date input physics and some improvements. We do not adopt a very low central metal abundance, which was often used to considerably reduce the predicted solar neutrino flux. On the contrary, we calculate models with the central metal abundance just a half of the surface value. We are amazed to find that our metal enriched models result in adequate depths of the convection zones which are in very good agreement with the seismic results. The agreement between the p-mode frequencies of our metal enriched models and the observations are also improved, and even better than the standard homogeneous model. The improvement on the initial helium abundance is not as distinct as the above two aspects. After being calibrated to match the correct solar luminosity and radius at the solar age, low-Z models result in a very low initial helium abundance, for instance, 0 . 2 in our and many other authors' models, which is even smaller than the primordial (Big Bang) helium abundance. We have calculated other models with different central metal abundances, and found that the initial helium abundance is related to the central metal abundance. When the central metal abundance is taken to be 0.004, the initial helium abundance is found to be 0. 143. This is an interesting resuit. As indicated by seismic studies E~'25J , the solar envelope helium abundance is around 0.24, which is less than the standard model's value of about 0 . 2 7 . Element diffusion and settling are therefore introduced to reduce the envelope helium abundance during the evolution, and these models give better agreement between the observed and calculated p-mode frequencies Els] . Now we have another approach to remedying the discrepancy between the calculated and observed solar envelope helium abundance. If we choose the central metal abundance to be lower than the standard value of 0.02 but not as small as what we adopt in the present paper ( 0 . 0 1 ) , we may expect to get a larger initial helium abundance, for the reasons discussed above. Therefore, we may tune the central metal abundance to match the initial helium abundance to the seismic determined value. It should be noticed that, unlike the helium diffusion which brings the helium originally in the solar envelope down to the solar core and leaves a helium abundance profile in the present solar interior, our approach will reduce the helium abundance over the whole solar interior from the beginning of the solar evolution. Investigations based on the solar neutrino experiments made by Fukugita & Hata [~53 set an important restriction to the metal abundance distribution over the solar interior, i . e . the ratio of the metal abundance in the solar core to that in the solar envelope is in the range of 0 . 4 - - 1 . 4 , with the peak of the probability at 0 . 8 . This clearly shows that the central metal abundance can

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be, or even more probably, smaller than the surface metal abundance of the Sun. From the above, we conclude that solar models with enhancement of heavy elements exterior to the base of their convection zones are still of vitality, and could be a solution of problems which the present standard solar models cannot solve. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 19625306 and 19833040) and National Climbing Project "Multi-wavelength Observations and Studies of Violent Activities of Astronomical Objects". Fruitful discussions with R. Q. Huang, G. Q. Luo, and Z. W. Han are highly appreciated.

References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

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