2. SOLAR ROTATION


 

    Solar rotation was first accurately measured in the last century (in 1863) by R.C. Carrington (q.v.) who used the position of prominent spots as marker points to determine a synodic period of about 27 days. Beginning from the first year of observation the solar rotations are indentified by "Carrington numbers". The solar surface, however, exhibits differential rotation, as well as a coherent pattern of activity related to magnetic fields, and globally coherent oscillation modes. All three phenomena can be employed to shed light on the structure and dynamics of the Sun. Particularly helioseismology, the study of solar oscillation, made it possible to measure the depth of the solar convection zone, the internal rotation profile, the sound speed throughout the Sun, and the solar helium abundance, (Deubner and Gough 1984; Hill and Kroll 1992). Employing a standard model for the internal structure of the Sun, it has been shown with linear adiabatic perturbation theory that small-amplitude oscillations of the solar body about its equilibrium state can be classified into three types: (i) pressure modes (p-modes), where the pressure is the dominant restoring force; (ii) gravity-modes (g-modes), where gravity or buoyancy is the dominant restoring force; and a class of surface or interface modes (f-modes), which are nearly compressionless surface waves. The existence of all three modes has been confirmed by solar observations. The solar rotation rate through a large part of the solar interior has been estimated, utilizing for the most part observations of the p-mode frequency splittings. Each mode is characterized by an eigenfunction with frequency eigenvalue vnlm, where n, l, and m are integer "quantum" numbers; n counts the number of radial nodes in the wavefunction, and l and m describe the nodes in colatitude and longitude, respectively. Rotation breaks the spherical symmetry of the Sun. Because of that the p-mode frequencies are not completely degenerate in m, and the frequencies vnlm in an nl-multiplet are said to be split in analogy to the Zeeman splitting of degenerate atomic energy levels. Because of observational limits it is not yet possible to observe values of splittings for individual m, to be used for inversion. However, results of observations are available in terms of efficients aj (j <= 5) of least-squares fits of the splittings

v_sub_(nlm) - v_sub_(nl0) = L * {Sum for j of [a_sub_j (n,l) [P_sub_j]^(l)} (m / L) . (16)

where Pj(L) is a polynomial of degree j and L = (l[l + 1])1/2. The coefficients aj(n,l) of odd j reveal the information about the internal rotation of the Sun (Figure 1-5). The analysis of observational data reveal that the latitude-dependent solar rotation profile as observed at the solar surface extends down through the convective envelope. In the radiative zone the rotation seems to have a solid-body profile. (Schou, Christensen-Dalsgaard, and Thompson 1992; Hill, Oglesby, and Gu 1992). Todate there exists no obvious theoretical explanation for this helioseismologically inferred solar rotation profile.

 


 

Figure 1-4. Solar interior profile, as inferred from an inversion of p-mode splitting data, displayed at these latitudes: theta0 = 0o (polar), theta0 = 45o, and theta0 = 90o (equatorial). Dashed lines indicate 1sigma error bars based on the observer's estimates of the of the uncertainties in the measured aj coefficients (Courtesy J. Christensen-Dalsgaard; Schou et al. 1992).

 


 

    Measurements of the individual frequencies of normal modes of the oscillating Sun may reveal the internal rotation profile. The ultimate goal of helioseismology is, however, to use all available pulsation data, including growth rates, phases, different modes - and not just observed frequencies - to search the internal structure and evolution of the Sun. Those data will definitely contribute to improve the inadequate treatment of convective transport of energy in the envelope of the Sun by the mixing length theory as well as to solve the solar neutrino problem for the gravitationally stabilized solar fusion reactor. Eigenmodes of pulsations of different degree carry information of physical conditions in quite different parts of the Sun. High-degree modes (Figures 1-5 and 1-6) are restricted to solar sub-surface layers, where solar activity phenomena have their origin. Contrary to this, low-degree modes (Figure 1-7) propagate all the way through the solar body to the regions where the solar neutrino flux is generated. Figures 1-5, 1-6, and 1-7 are equatorial cross sections from a model of the vibrating Sun (Weiss and Schneider 1991).

 


 

Figure 1-5. Solar p-modes, equatorial cross section l = 40, m = 40, v = 3.175 mHz (Courtesy W.W. Weiss; Weiss and Schneider 1991).

 


 

Figure 1-6. Solar p-modes, equatorial cross section l = 40, m = 0, v = 3.175 mHz (Courtesy W.W. Weiss; Weiss and Schneider 1991).

 


 

Figure 1-7. Solar p-modes, equatorial cross section l = 2, m = 2, v = 3.147 mHz (Courtesy W.W. Weiss; Weiss and Schneider 1991).

 


 

    According to observation and theory of stellar evolution, young stars rotate rapidly. If the central part of the Sun still rotates rapidly, this should lead to a small oblateness in the Sun's disk, about 1 part in 105. The extreme observational values reported for the solar oblateness lie between 5.0 ± 0.7 x 10-5 (Dicke and Goldenberg 1967) and 9.6 ± 6.5 x 10-6 (Hill and Stebbins 1975), with a proposal that this quantity varies with the solar cycle (Dicke et al. 1987). The oblateness of the Sun is still a hotly debated issue in observational and theoretical solar physics.