1. SOLAR STRUCTURE


  1.1 Internal Structure

    The structure of the Sun is determinated by the conditions of mass conservation, momentum conservation, energy conservation, and the mode of energy transport. The Sun is an oblate spheroid, like all the major bodies in the solar system, but in a first simplifying approach to describe the solar structure, the effects of rotation and magnetic fields will be neglected here so that the Sun is taken to be spherically symmetrical. Calculating a solar model means the determination of pressure, temperature and chemical composition as a function of mass or radius through the Sun, (Chandrasekhar 1967; Kourganoff 1973).

 


 

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

(dP / dr) = - rho * [(G * M_sub_r) / (r^2)] , (1)

where P is the pressure, r the radial distance from the center, Mr the mass within a sphere of radius r, rho 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

(dP / dr) round [(P_sub_c - P_sub_0) / (R_sub_sun)] round (P_sub_c / R_sub_sun) round {(G * M_sub_sun * rho_quer) / [(R_sub_sun)^2]} , (2)

where Rsun is the solar radius, Msun the solar mass, rho quer 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

(dM_Sub_r / dr) = 4 * Pi * r^2 * rho , (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

(M_sub_sun / R_sub_sun) round [(R_sub_sun)^2 * rho_quer] => (rho_quer)_sub_sun varies as [M_sub_sun / (R_sub_sun)^3] , (4)

where the symbol varies as means "varies as". In the general case, rho varies as 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

P_sub_c varies as {G * ([(M_sub_sun)^2] / [(R_sub_sun)^4])} . (5)

    In the general case, rho varies as 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"

P = (k / [mu * m_sub_p]) * rho * T , (6)

where mp is the mass of the proton, k is Boltzmann's constant, and mu 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

T_sub_c round [(mu * m_sub_p) / k] * [P_sub_c / rho_quer] => T_sub_c varies as [(m_sub_p * G) / k] * mu * (M_sub_sun / R_sun_sun) . (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 varies as muM/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 mu 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 mu = 1/2. For a gas composed of fully ionized helium it is mu = 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 mu = 2. Thus the mean atomic weight for fully ionized gas is

mu = 1 / [2X + (3/4)Y + (1/2)Z] . (8)

The solar matter is at present approximately 75% hydrogen, 23% helium and 2% metals by mass fraction. Throughout the solar interior, musun 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 (Rsun = 6.96 x 105km)

Internal region extension in terms of solar radius chemical composition

core 0.20 Rsun center only: He: 0.63, H: 0.35, metals: 0.02
(almost actively ionized matter)
radiative zone 0.50 Rsun He: 0.23, H: 0.75, metals: 0.02
(highly ionized)
convective zone 0.30 Rsun same (less ionized)
photosphere 0.002 Rsun same (less ionized)
solar surface 1.000 Rsun
chromosphere 0.02 same (less ionized)
corona round 5 same (highly ionized)

 


 

    An equation of continuity must also be satisfied by the radiation

(dE / dt) + (dL_sub_r / dr) = 0 , (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

(dL_sub_r / dr) = 4 * Pi * r^2 * rho * epsilon = - (dE / dt) , (10)

where Lr denotes the total net energy flux through a spherical shell of radius r and Epsilon 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

epsilon_quer round (L_sub_sun / M_sub_sun) . (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

(dP / dr) = - [(kappa * rho) / c] * [L_sub_r / (4 * Pi * r^2)] , (12)

where P=aT4/3 is the radiation pressure, kappa is the opacity of the solar matter, 1/kapparho 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 sigma since sigma = 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

(dT / dr) = - [3 / (4ac)] * [(kappa * rho) / T^3] * [L_sub_r / (4 * Pi * r^2)] , (13)

allowing an estimate of the solar luminosity

(T_sub_c / R_sub_sun) round [1 / ac] * [(kappa * rho_quer) / {(T_sub_c)^3}] * [L_sub_sun / {(R_sub_sun)^2}] => L_sub_sun varies as ac * {[(G * m_sub_p) / k]^4} * [mu^4 / kappa] * (M_sub_sun)^3 , (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 kappa 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 kappa 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 = Msun, and for an age of tsun = 4.5 x 109 years, r = Rsun, L = sun. 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, rhoc = 158gcm-3, Tc = 1.57 x 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.

 


  1.1.1 Core

    The core of the Sun is a gravitationally stabilized fusion reactor. There, energy is produced by conversion of hydrogen into helium. Each hydrogen atom weighs 1.0078 atomic units and each helium atom is made from four hydrogen atoms thus weighing 4.003 atomic units. Accordingly, the difference of 0.0282 atomic units, or 0.7% of the mass m, is converted into energy E according to Einstein's formula E=mc2, where c is the velocity of light. Most atoms in the core of the Sun are entirely stripped of their electrons by the high temperature and opacity is governed by scattering of photons by free electrons, by inverse bremsstrahlung on ionized hydrogen and helium, and by bound-free scattering by elements heavier than helium.

1.1.2 Radiative Zone

    The radiative zone is a region of highly ionized gas. There the energy transport is primarily by photon diffusion and is described in terms of the Rosseland mean opacity (this is a weighted inverse mean of the opacity over all frequencies, which can be used when the optical depth is very large and radiative transport reduces to a diffusion process).

1.1.3 Convective Zone

    In the outer regions, atoms may keep their electrons because of the low temperature and ions and even neutral hydrogen exist. Here many atomic absorption processes occur, mainly bound-free transitions. The high opacity makes it difficult for photon radiation to continue outward and steep temperature gradients are established which lead to convective currents. The outer envelope of the Sun is in convective equilibrium. It is the location where sunspots and other solar activity phenomena are generated. Observationally, the outer solar atmosphere following the convective zone has been divided into three spherically symmetric layers - the photosphere, chromosphere, and corona - lying successively above one another (Zirin 1988).

1.1.4 Photosphere

    The outer limit of the photosphere is the boundary of the visible solar disk as seen in white light. Most of the radiation emitted by the Sun originates in the photosphere, which is only about 500 km thick. This radiation is in equilibrium and the Stefan-Boltzmann law can be applied to calculate the effective temperature of the solar photosphere, which is Te = 5780 K. According to the Stefan-Boltzmann law each square centimeter of the solar surface having the temperature T emits, in all directions, light of sigmaT4 ergs per second. Subsequently, the total emission of the Sun in one second, i.e. the luminosity, equals

L_sub_sun = 4 * Pi * (R_sub_sun)^2 * sigma * (T_sub_e)^4 . (15)

This fundamental relation also determines the radius of the Sun when its luminosity and surface temperature are known. The spectrum of the photosphere consists of absorption lines superimposed on an approximately blackbody continuum.

1.1.5 Chromosphere

    A thin transition region extending 5000 km above the photosphere is called the chromosphere. Considerably hotter than the photosphere, the chromosphere is heated by hydromagnetic waves and compression waves originated by spicules and granules. The temperature of the chromosphere is about 10,000 K and it has an emission spectrum.

1.1.6 Corona

    During a total solar eclipse the outermost atmosphere of the Sun can be seen. Called the solar corona (q.v.), this is a hot gas merging gradually into the transparent interplanetary medium, and flowing outward from it is the solar wind. Current theories indicate that the corona is heated by the dissipation of mechanical energy stemming from the convection zone, or by dissipation of magnetic energy by field-line reconnection. The kinetic temperature of the solar corona is about 2 x 106 K and its gas has a density of about 10-15 gcm-3. Solar x-ray radiation originates in the corona.

1.2 Solar Activity

    The Sun emits radiation in a wide range of the energy spectrum from long radio waves (300m) to x-rays (0.1nm), including high-energy particles (cosmic rays q.v.). Almost 95% of the radiated energy is concentrated in a relatively narrow band between 250 nm and 2500 nm. The total radiation received from the Sun is called the solar constant; it was formerly regarded as a fixed value, 2.00 ± 0.04 calcm-2min-1, alternatively 1.36 ± 0.7 x 106 ergscm-2s-1 (although difficult to measure), but from satellite observations it is now confirmed as a variable (up to about 0.5% (Herman and Goldberg 1978; Sofia 1981; Schatten and Arking 1990). The transient phenomena occurring in the solar atmosphere can be grouped together under the term solar activity: sunspots and faculae occur in the photosphere; flares and plages belong to the chromosphere; and prominences and coronal structures develop in the corona. All solar activity phenomena are connected in this way or another with the 11 and 22-year sunspot cycle.

1.2.1 Granules

    Granules are huge convective cells of hot gases, 400-1000 km in diameter, spread in a cellular pattern over the entire photosphere except at sunspots. Granulation supports the transfer of energy from the convective zone outward into space. Granules behave as short-lived bubbles, lasting only 3 to 10 minutes, that rise and fall at a velocity of about 0.5 kms-1, thereby moving vertically a distance of the order of 200 km.

1.2.2 Spicules

    Spicules look like hairs of gas rising and falling at the upper chromosphere, reaching into the corona. They last as long as 10 minutes, attaining vertical speeds of up to 20 kms-1 getting upward to as high as 15,000 km. Spicules array themselves into chromospheric networks establishing giant supergranulated cells with gases rising in the center and descending at their outer boundaries.

1.2.3 Sunspots

    Sunspots are relatively cool and dark markings on the Sun's photosphere, which exhibit distinct cycles. They are concentrations of strong magnetic fields (2000-3000 Gauss), with diameters less than about 50,000 km and lifetime of a few days to weeks. A sunspot generally develops a very dark central region, called the umbra, which is surrounded by the penumbra. The 11-year sunspot cycle consists of variations in the size, number, and position of the sunspots (Fairbridge 1987a). It is extremely variable in length (actually, 7 to 17 years), the high-activity cycles (to >= 200 spots) are generally short (9-10 years), and the low activity cycles (sometimes <= 50 spots) are long (12-13 years). In the sunspot cycle the number of sunspots usually peaks 2-3 years after the beginning of each cycle and decays gradually, but low activity cycles may have a reversed symmetry. First spots of the cycle appear at higher latitudes, mostly between 20o and 25o, and as the spots increase in size and number they occur closer to the equator. Very few spots are observed outside the latitude range of 5o - 35o. The magnetic polarity of the sunspot groups reverses in each successive cycle so that the complete cycle lasts 22 years, the so-called "Hale cycle". Recent observations have indicated that the magnetic solar cycle is a coherent phenomenon throughout the solar surface. For each pair of 11-year cycles, the one with north or leading magnetic orientation is usually stronger than the south one. Between 1645 and 1715 very few sunspots were seen, a time period called Maunder minimum. This period was associated with a long cold spell in Europe, known as Little Ice Age. Carbon-14 measurements from tree rings and Beryllium-10 measurements from arctic ice-cores confirm the low solar activity level at that time (Sonett 1984; Beer 1987, Fairbridge 1987b). The solar 11-year cycle has been recorded on a regular basis since the beginning of the 18th century classified by Wolf's quantity N of the number of sunspots Ns plus ten times the number Gof sunspot groups: N = W(Ns + 10G), where W is a weighting factor assigned to an individual observer to account for variation in equipment, atmosphere conditions, and observer enthusiasm (Gibson 1973). The quantity N is widely used as an indicator of sunspot activity and is commonly called the Zurich sunspot number. More recently Bracewell (1989) was able to show that the quantity N = ±(Ns + 10G)3/2, with ± denoting the dipole orientation, is nearly a sinusoidal function of time with a period of 22.2 ± 2 years. Using proxy data for ancient sunspot periods (such as auroral frequency), the average of the 11-year cycle is 11.12 years, and thus coupled to the magnetic period.

1.2.4 Prominences

    They are regions of cool (104K), high-density gas embedded in the hot (106K), low-density corona. Prominences can be observed as flamelike tongues of gas that appear above the limb of the Sun when observed in the light of the Halpha line. They occur in regions of horizontal magnetic fields, because these fields support prominences against the solar gravity, and indicate the transition from one magnetic polarity to the opposite.

1.2.5 Flares

    Sunspots are accompanied by large eruptions called solar flares emitting high-energy particles and radiation in a very broad spectrum of energy. A solar flare is actually the result of an intensely hot electromagnetic explosion in the corona and produces vast quantities of x-rays which brighten the chromospheric gases. Typical lifetimes of solar flares are one to two hours and the temperature in flares can reach several million degrees. Flare particles ejected into outer space reach the Earth in a few hours or days and are the cause of disruptions in radio transmission. Aurora and magnetic storms are due to strong solar flare eruptions. The peak of solar flare activity is lagged by the sunspot cycle, usually 1-2 years, but some high-energy eruptions may occur at any time; the mean cycle of flare frequency is 0.417 years.