Hiromoto Shibahashi (shibahashi@astron.s.u-tokyo.ac.jp)I discuss the meanings of the new helioseismic results for the solar interior and atmosphere. The eigenfrequencies of p-modes have provided us the sound speed profile as well as the density profile in the Sun. I discuss how to determine the profiles of other physical quantities. The solar oscillations may provide us with a diagnostic tool not only for the solar interior but also for the solar atmosphere. I discuss prospects of the helioseismic investigation of the chromosphere of the Sun.
INTRODUCTION
Needless to say, the Sun is irreplaceable for the terrestrial environment and life on the earth. Understanding the physics of the Sun is closely related to our daily life, hence the Sun has been intensively investigated from the early stage of modern astronomy and astrophysics. There still remain, however, many issues in solar physics. Let me list up some of them:
Studying these issues is worth doing, not only for the understanding of the Sun, but also for establishing the physics of stars which are the fundamental constituents of the universe. It should be stressed that most of these issues are related to the interior of the Sun. Even in the case of the solar surface phenomena such as the sunspots, they originate in a deeper region. Hence it is essential to diagnose the interior of the Sun. Previously, this was impossible to do, but now the situation has been drastically altered by the development of helioseismology. We now have new eyes to see the invisible interior of the Sun (Deubner, Christensen-Dalsgaard, and Kurtz 1998). In this review, I will discuss what the recent helioseismic results mean for the solar interior.
SOUND SPEED PROFILE DETERMINED FROM HELIOSEISMOLOGY
The recent observations of solar oscillations provide us with accurate frequencies of a large number of solar eigenmodes. The relative errors in some of the measured frequencies are as small as 10-6, and they are even smaller than the relative errors in the measurement of the solar radius and of the solar mass. By the inversion of these frequencies, we can determine functional forms of the acoustic quantities; the sound speed profile c(r, θ) and the density profile ρ (r). We now know the sound speed profile within the errors of a few tenths of a percent, and the density profile with the order of 1% errors. Also from these frequencies, the helium abundance and the depth of the convection zone have been accurately determined. From frequency m-splitting, the internal rotation profile has been determined as a function of the distance from the center and of the latitude. The errors in these inverted quantities are caused by the statistical error in the frequencies, the statistical error in the measurement of the solar radius, and by the systematic error in the inversion methods. The most accurately inverted quantity is the sound speed profile. Frequency m-splitting caused by deformation of the solar structure from spherical symmetry is so small that the sound speed is well described as a function of r. Figure 1 shows the squared sound speed profile obtained from helioseismology. As clearly seen in this figure, the sound speed increases slightly from the center and then drops with r. This fact implies that the core consists of more helium than the outer part, and this is regarded as observational evidence for occurrence of nuclear reactions in the core. The apparent kink at r/R‾0.7 implies that the temperature gradient switches from the radiative gradient to the adiabatic gradient at this depth, and hence this depth is regarded as the bottom of the convection zone. We now know the depth of the the convection zone with the precision to a few parts in a thousand of the solar radius.
Fig.1: The inverted sound speed profile determined from the data obtained by LOWL (Tomczyk et al. 1995) and those obtained at the South Pole (Jefferies et al. 1995).
Figure 2 is the relative difference between the inverted squared sound speed profile and the latest solar model (Christensen-Dalsgaard et al. 1996). As seen in this figure, the difference is as small as 0.5% at most. It should be stressed, however, that the difference is larger than the error level. Hence, we should not say that the latest evolutionary model is consistent with helioseismological results. One of the apparent conspicuous differences is a hump at r/R‾0.65, just beneath the bottom of the convection zone. Though errors in the opacity at the relevant temperature, density and chemical composition are not ruled out as a cause for this hump, a mixing process related with convective overshooting or rotational shear is likely to be responsible (Spiegel and Zahn 1992). The inverted squared sound speed has a steeper gradient near r/R‾0.3 than the model. Though the cause for this has not yet been clarified, it should be noted that this layer is where 3He is most accumulated.
Fig.2: Relative difference δc2/c2$ between the square of the sound speed c in the Sun (inverted from various observational data sets) and that in a model computed by Christensen-Dalsgaard et al. (1996). After Takata and Shibahashi (1998a).
One of the recent topics of helioseismology is the apparent line asymmetry of the peaks in the power spectrum. It is not known whether taking account of this line asymmetry affects the inverted profiles. The line asymmetry is found to be dependent mainly on the frequency, and only very weakly on the degree of the mode. Hence it must affect the envelope structure near the surface, but not significantly the deep interior structure.
BUILDING A SEISMIC SOLAR MODEL AND REPLACING THE EVOLUTIONARY STANDARD SOLAR MODEL WITH IT
One of the scientific goals of helioseismology is to discriminate between the possible solutions of the solar neutrino problem: a defect in the modeling of the Sun, or one in the theory of neutrino physics. Knowing the profiles of the acoustic quantities is not enough for this purpose. We need to know not only the density and sound speed profiles but also the temperature and chemical composition profiles to discuss the solar neutrino problem. Nevertheless, it should be stressed here that the sound speed profile has been very accurately determined from helioseismology, and the difference between the profile determined in this way and the evolutionary model is much larger than the observational error. This means that we can also determine the profiles of the thermal quantities in the Sun by using the highly accurately determined profiles for the acoustic quantities and adopting the same knowledge of the microphysics (equation of state, nuclear reaction rates, and opacity) as used in building evolutionary solar models. Benefits of building such a solar model are as follows:
We solve the equations of stellar structure with the imposition of information from helioseismology. That is, we assume thermal balance, radiative heat transport below the base of the convection zone, and microphysics including the relevant nuclear reaction rates, opacity, and equation of state, as well as mass conservation and hydrostatic equilibrium. In addition to the usual equations of stellar structure, we adopt helioseismic constraints on the model: the sound speed profile, the density profile and the depth of the convection zone (Takata and Shibahashi 1998a, 1998b, Shibahashi et al. 1998). Note that we do not follow the evolution of the Sun, but obtain a model which describes the present structure of the Sun directly. We do not have to assume the chemical composition profiles in advance. They are obtained as part of the solutions of the fourth-order differential equations of stellar structure. The outer boundary conditions are set at the base of the convection zone. This means that we do not need to care about the convective heat transport, which has theoretical uncertainties. The location of the base of the convection zone is accurately known from the inverted sound speed profile. The boundary conditions at the center are trivial: Mr=0 and Lr=0, while the outer boundary conditions at the base of the convection zone are ∇rad = ∇ad (neutral stability against convection) and Lr = Lsun. Moreover, chemical homogeneity in the convection zone requires Z/X = (Z/X)photosphere at the base of the convection zone. This last condition is used to fix the chemical composition profiles. The neutrino fluxes theoretically calculated from this seismic solar model are still larger than the actually detected fluxes (Takata and Shibahashi 1998a, 1998b). This fact is unchanged even if we take account of the uncertainties in various microphysics relevant to the modeling of the Sun. This implies that a defect in the theory of the neutrino physics is more likely to be responsible for the deficiency of the solar neutrino capture rates. Among the proposed hypotheses for neutrino physics, the most promising one is the MSW effect, by which some fraction of the electron neutrinos generated in the solar core turn in to muon neutrinos by interaction with the electrons in the Sun (Mikheyev and Smirnov 1985, Wolfenstein 1978). The detected electron neutrinos are hence less than the theoretically expected amount. The physical cause for this neutrino oscillation comes from the nonzero mass of neutrinos. The transition probability from the electron neutrinos to the muon neutrinos is parameterized in terms of the mixing angle between the mass eigenstates and the flavor eigenstates of neutrinos and the mass difference between the two kinds of neutrinos. Unfortunately, these two quantities cannot be theoretically predicted, and they are to be determined by some experiments. Comparing the theoretically expected solar neutrino fluxes and the actual detected fluxes is a new tool for the determination of these parameters, and it contributes to particle physics as well as to astrophysics. In doing so, we should now use the seismic solar model, rather than the evolutionary solar model which involves experimentally less supported assumptions.
INTERNAL ROTATION DETERMINED FROM HELIOSEISMOLOGY
Figure 3 shows a recent seismically determined rotation profile in the solar interior (Schou et al. 1998). The internal rotation is determined from m-splitting of the eigenfrequencies. Since the modes which penetrate into the very deep interior are limited to those of low degree, and their splitting is smaller (being proportional to 2l+1 with the decrease of the degree l, it is hard to determine the internal rotation rate in the very deep interior. The same is true for the polar regions, since the modes which are sensitive to the polar regions are limited to low azimuthal order modes (m‾0). Hence, in figure 3, only the rotation profile of the accurately determinable region is shown. Conspicuous features are as follows:
This thin transition layer is called the ``tachocline,'' in which Eckman circulation currents are expected (Spiegel and Zahn 1992). If the Eckman circulation currents really exist, the tachocline is likely to exchange material with the convection zone, leading to the hump in δc/c beneath the convection zone seen in figure 1. The Eckman flow may transport Li and Be from the convection zone into a deeper layer that is hot enough to destroy these elements by nuclear reactions, thus it may be responsible for the Li depletion problem. If the small scale turbulence in the thin layer is induced by shear instability, it will help to maintain the rigid rotation in the deeper zone.
Fig.3: Rotational angular velocity Ω(r, θ) in the Sun, determined from helioseismology. Adapted from Schou et al. (1998) by T. Sekii.
NEW PROBLEMS RAISED BY HELIOSEISMIC INVESTIGATION
As seen in the previous sections, helioseismology has enabled us to see the invisible interior of the Sun. We learned phenomenologically the internal structure of the present-day Sun. This does not necessarily mean that we understood the physics of the Sun. Rather, various new issues have been raised by helioseismic investigation. For example, the rotation profile clarified by helioseismology is somehow different from that expected naively. It seemed natural that the core would become to spin faster while the convective envelope would lose angular momentum and spin down. The rotation profile in the convective envelope had been expected to be cylindrical, as expected from the Taylor-Proudman theorem. We are now faced with new problems: Why does the convection zone rotate as it does? One possible cause is that the turbulent motion might be anisotropic due the effect of the Coriolis force (Gough 1997). Further theoretical investigation is desired. How does the gas flow in a three-dimensional way in the convection zone? How is the rotation in the radiative core attained? What is the mechanism for extracting angular momentum from the radiative core? How is the angular momentum transported from the core to the envelope? The angular momentum can be transported either by actual material mixing, or by some wave motion without material mixing. The latter possibility has been proposed by Zahn et al. (1997) and Kumar and Quataert (1997; see also Talon and Zahn 1998). To study these new issues raised by the recent helioseismic results, theoretical consideration is obviously required, and such newly inspired thought should in turn then be tested by helioseismic observations.
TIME-DISTANCE HELIOSEISMOLOGY
In the case of geoseismology, contrary to helioseismology, it is hard to resolve the oscillation of the surface of the Earth into high-resolution two-dimensional eigenoscillation patterns, since we, the observers, are on the observed object itself. Rather, geoseismology uses the relation between the travel times of the wave and the distances from the source to the observing stations, which is obtained by local observations. This technique was ambitiously first applied to the Sun by Duvall et al. (1993), and it worked quite successfully. This approach opened a new possibility for helioseismology. The travel time between two different points on the solar surface is influenced by localized structures such as temperature inhomogeneity and the local velocity field. Hence this technique is very useful to get information about localized structure. Kosovichev (1996) and Duvall et al. (1997) succeeded in visualizing the three-dimensional convection flow pattern occurring in the actual Sun. Their results for the surface matches well with the surface flow pattern obtained by other non-helioseismic methods. Seeing the roots of sunspots in the deeper layers is one of the scientific goals of this method. We expect that we will see the magnetic flux tubes located near the base of the convection zone as the origin of sunspots. It will become possible to predict sunspots before they appear by local helioseismology in the near future.
SOUNDING THE SOLAR ATMOSPHERE
Helioseismology is useful to probe not only the solar interior but also the solar atmosphere. This possibility was opened by the discovery of a smooth extension of the ridges on the (l,ν)-diagram into the frequency range higher than the acoustic cut-off frequency νac ‾5.3mHz. Finding such a mode structure had been unexpected, hence it surprised us at the first glance. It is now interpreted as an apparent interference pattern induced by correlation between the waves which reach the surface directly from the source and those emitted earlier which reach the surface after refraction (Kumar and Lu 1991). Analysis of pseudo p-mode frequencies provides us with information about the location of the acoustic sources. It is thought that the solar oscillations are excited by Lighthill's (1952) mechanism, which generates sound waves from turbulence (Goldreich and Keeley 1977). The acoustic power induced by this mechanism is very sensitive to the Mach number of the turbulent motion, hence the source layer should correspond to the most highly turbulent layer. The sound waves propagate from the source upward as well as downward. For ν> νac, most of energy of the upward emitted wave is conveyed to the corona. If no reflection occurs, only one ridge appears on the time-distance diagram. However, Jefferies et al. (1997) found by careful analysis that two weak ridges appear on the time-distance diagram, in addition to the main ridge that had been thought to be unique. This is evidence that a small fraction of the wave energy is partially reflected somewhere near the photosphere. These additional ridges correspond to the first and the second bounces, respectively, near the surface (see figure 4). Furthermore, Jefferies et al. (1997) found that each ridge consists of a main component and a satellite component, which are separated from each other by about 6-11 minutes on the time-distance diagram. This implies that there is one more partial refection layer above the photosphere, probably at the corona-chromosphere transition layer. The time difference should reflect the thickness of the chromosphere, and the amplitude should reflect the structure of the reflecting layers. Careful analyses will provide us a new tool to diagnose the solar atmosphere (cf. Jefferies 1998).
Fig.4: Schematic to describe the reflection of high-frequency acoustic waves in the Sun's atmosphere (not to scale). For simplicity the oscillation source is assumed to lie at the photosphere-chromosphere boundary. After Jefferies et al. (1997).
The recent data taken by the GOLF (Global Oscillations at Low Frequencies) instrument aboard the SOHO (The SOlar Heliospheric Observatory) satellite showed that the high frequency peaks also appear even in data taken in integrated sun light, and that the separation between adjacent peaks is about 70 μHz. This separation is similar to that in the case of ν < νac(Garcia et al. 1998). In the latter case, this is produced by the fact even-degree eigenmodes (l=0, 2, 4) and odd-degree eigenmodes (l=1, 3) appear alternatively with an equal spacing in the power spectrum, while the separation between the adjacent radial-order modes is ‾140 μHz. In the case of ν > νac, however, the oscillations are no longer eigenmodes, and the peaks in the power spectrum are due to the apparent interference pattern. Hence the apparent spacing is expected to be the inverse of the round-trip travel time from the source layer to the inner turning layer and back, which is about 140 μHz. The fact that the actual apparent spacing is half of 140 μHz implies that the waves emitted from the source are reflected at the surface on the other side of the Sun and that the reflected wave and the wave emitted upward from the source are producing the observed interference pattern. Since the waves with long horizontal wavenumbers and ν > νac pass closer to the center of the Sun than any eigenoscillations of p-modes, such oscillations detected by GOLF must involve valuable information near the center unavailable from the true p-modes.
Acknowedgement: I am grateful to M. Takata and T. Sekii for kindly producing figures 1 and 4, respectively, upon my request.
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