1. SOLAR STRUCTURE Figure 1-1. Schematic view of the structure of the Sun and modes of outward flow of energy.

Two forces keep the Sun in hydrostatic equilibrium in its current stage of evolution: the gravitational force directed inward and the total pressure force directed outward. The equation of hydrostatic equilibrium is , (1)

where P is the pressure, r the radial distance from the center, Mr the mass within a sphere of radius r, the matter density, and G the gravitational constant. This equation is consistent with radius changes, but requires the kinetic energy involved in expansion or contraction of the solar body to be small compared to the gravitational potential of the Sun. For an order of magnitude estimate, equation (1) can be written , (2)

where R is the solar radius, M the solar mass, the mean matter density of the solar gas sphere, Pcis the central and Po the surface pressure respectively, where the latter can be neglected.

The equation of mass conservation , (3)

constrains the integral of the density over the volume to be equal to the mass and leads to the estimation of the mean matter density for the Sun , (4)

where the symbol means "varies as". In the general case,  M/R3, the constant of proportionality depends on the radial mass distribution and the radial distance R (Schwarzschild 1958; Haubold and Mathai 1987, 1992). Using equation (4) and equation (2), the central pressure of the Sun can be estimated to be . (5)

In the general case,  GM2/R4, the constant of proportionality is determined by the radial distribution of mass in the Sun, and the particular radial distance R at which P is measured (Schwarzschild 1958; Haubold and Mathai 1987, 1992).

The interior of the Sun is entirely gaseous and the great majority of atoms are stripped of their electrons. The solar gas behaves under these physical conditions nearly like a perfect gas, governed by the "equation of state" , (6)

where mp is the mass of the proton, k is Boltzmann's constant, and is the mean molecular weight. This equation of state relates the pressure, temperature, density and chemical composition, and is related to other thermodynamic quantities. Then the central temperature of the Sun can be estimated from the perfect gas law in equation (6), that is . (7)

This formula determines the temperature at the centre of the Sun according to its mass, radius and mean molecular weight of the solar matter. In the general case, T  M/R, the constant of proportionality depends on the mass distribution and the radial distance R (Schwarzschild 1958; Haubold and Mathai 1987, 1992).

When X, Y, and Z are the mass fractions of hydrogen, helium, and heavy elements, respectively, then it holds by definition X + Y + Z = 1. The mean molecular weight in equation (6) can be calculated when the degree of ionization of each chemical element of solar matter has been specified. For solar gas composed of fully ionized hydrogen, there are two particles for every proton and it is = 1/2. For a gas composed of fully ionized helium it is = 4/3. For all elements heavier than helium, usually referred to by astronomers as metals, it holds that their atomic weights are twice their charge and accordingly = 2. Thus the mean atomic weight for fully ionized gas is . (8)

The solar matter is at present approximately 75% hydrogen, 23% helium and 2% metals by mass fraction. Throughout the solar interior,  is approximately 0.59, except at the surface, where hydrogen and helium are not fully ionized, and in the core, where the chemical composition is altering due to nuclear burning (compare Table 1-2) (Kavanagh 1972; Bahcall 1989).

 Table 1-2 Internal structure of the Sun (R = 6.96 105km) Internal region extension in terms of solar radius chemical composition core 0.20 R center only: He: 0.63, H: 0.35, metals: 0.02(almost actively ionized matter) radiative zone 0.50 R He: 0.23, H: 0.75, metals: 0.02(highly ionized) convective zone 0.30 R same (less ionized) photosphere 0.002 R same (less ionized) solar surface 1.000 R chromosphere 0.02 same (less ionized) corona 5 same (highly ionized)

An equation of continuity must also be satisfied by the radiation , (9)

where dE/dt is the rate of energy production per unit thickness of the shell of radius r. The equation of energy conservation is , (10)

where Lr denotes the total net energy flux through a spherical shell of radius r and is the net release of energy per gram per second by thermonuclear reactions occurring in the gravitationally stabilized solar fusion reactor. It is assumed in equation (9) that the energy produced by nuclear reactions equals the photon luminosity of the Sun, thus neglecting gravitational contraction and subtracting energy loss through neutrino emission. The mean energy generation rate for the Sun can be inferred from equation (10), that is . (11)

Finally the thermonuclear energy produced in the solar core is transported by radiation through the solar body to the surface. The force due to the gradient of the radiation pressure equals the momentum absorbed from the radiation streaming through the gas , (12)

where P=aT4/3 is the radiation pressure, is the opacity of the solar matter, 1/  is the mean free path of photons, and c is the velocity of light. The coefficient of the radiation density, a, is related to the Stefan-Boltzmann constant since = ac/4. Equation (12) is the energy transport equation taking into account the fact that energy transport in the deep interior of the Sun is exclusively managed by radiation. From equation (11) follows the temperature gradient driving the radiation flux, that is , (13)

allowing an estimate of the solar luminosity , (14)

taking into account equations (4) and (7). The luminosity is independent of the radius; it depends on the opacity and increases with mass. Equation (14) is an important result of the theory of the internal structure of solar-type stars, called theoretical mass-luminosity relationship. The fundamental result as given by equation (14) is that the luminosity of the star is simply determined by its mass, since this rule is based on the fact that the transfer of energy from the stellar interior towards the surface is managed by radiation. The stellar energy sources must somehow adapt to the stellar opacity. The luminosity of a solar-type star is determined largely by photon opacity and not by the energy source.

Gamma-ray photons produced in thermonuclear reactions in the core of the Sun are being scattered, absorbed or re-emitted by free electrons, ions, and atoms on their way to the surface of the Sun. The opacity in equation (12) is the measure of the solar material's efficiency at inhibiting the passage of the photons through the solar interior. The actual value of the opacity depends on various processes which may operate simultaneously: bound-bound transitions, bound-free transitions, free-free transitions, and scattering of photons by free electrons, ions and atoms. Scattering of photons by free electrons is the most important process for the solar core. Approaching the solar surface, bound-free transitions take over to determine the opacity of solar matter. The structure of the Sun depends in a sensitive way on the opacity, for if changes, the Sun must readjust all its parameters to allow the energy generated in the core to stream to the surface, not being blocked at any point in the solar interior.

Boundary conditions for the system of nonlinear differential equations (equations (1), (3), (10), (12)) have to be specified to arrive at specific solutions: At the solar centre it is r = 0, Mr = 0, Lr = 0, and at the assumed solar surface (this is actually the photosphere) it holds Mr = M , and for an age of t = 4.5 109 years, r = R , L = . Mass, radius, surface temperature, surface chemical composition, and luminosity of the Sun are known by observation. Using the conservation laws and known properties of gases (equation of state, opacity, energy generation rates), the internal structure of the Sun can be calculated in matching the observed properties at the solar surface. However, because the equations of solar structure form a system of first-order nonlinear simultaneous differential equations, they have to be integrated numerically to obtain a very detailed picture of the run of physical variables throughout the Sun. Order of magnitude estimations provided in equations (4), (5), (7), (11), and (14) can be considered only to be a first approach to the problem (Mathai and Haubold 1988). Figure 1-2 shows the numerical results of a standard solar model based on the system of differential equations as described above (Sears 1964; Sackmann, Boothrayd, and Fouler 1990; Guenther et al. 1992). Figure 1-2. A standard solar model of the present solar interior: X = 0.708, Y = 0.272, Z = 0.0020, c = 158gcm-3, Tc = 1.57 107K.

 Chemical composition changes with time (compare equation (10)) due to thermonuclear reactions in the solar core that results in a continuously evolving structure, the calculation of which adds another system of differential equations (kinetic equations) to the set of differential equations described above (Schwarzschild 1958; Kourganoff 1973). Figure 1-3. Schematic view of the structure of the Sun and modes of outward flow of energy.