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Astron. Astrophys. 334, 703-712 (1998)

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2. Formulation of the problem

2.1. General principles

The term "standard solar model" defines a certain set of simplifying assumptions under which the calculation of stellar evolution may more easily be performed. A uniform initial chemical composition is assumed, rotation, magnetic fields and mass loss are neglected, and spherically-symmetric hydrostatic equilibrium is imposed. Mass is chosen as the independent variable rather than radius, since the equations of stellar structure become less nonlinear.

The partial differential equations of stellar evolution are as usual expressed as two sets of ordinary differential equations: one that describes the equilibrium structure at a particular time given the chemical composition, and another that describes the evolution with time of the chemical composition. The solution of the former is discussed in Sect. 2.2, and the latter in Sect. 2.5.

2.2. Spatial integration

The equations describing the equilibrium structure of a solar model are fourth order, with two boundary conditions each at the centre and surface. These were originally solved by the shooting technique, whereby the equations were integrated inwards from the surface and outwards from the centre, the matching being achieved by varying the four free parameters (central pressure and temperature, surface radius and luminosity). This technique has largely been superceded by the Henyey relaxation method.

An N -point grid [FORMULA] ([FORMULA] with [FORMULA]) is imposed, and the differential equations are replaced by a set of finite-difference equations. Originally these finite-difference equations were second order, but MoSEC uses a fourth-order scheme described by Cash & Moore (1980). [FORMULA] further grid points [FORMULA] are interleaved midway between pairs of the original N grid points. Equations are then constructed for each of the dependent variables, pressure (p), temperature (T), luminosity (L) and radius (R); for example,

[EQUATION]

where [FORMULA] denotes the derivative with respect to the independent variable, m. [FORMULA] is given by

[EQUATION]

Similar equations are used for T, L and R. The derivatives on the right-hand sides of Eqs (1) and (2) are known functions of p, T, L and R through the equations of stellar structure.

This large system of equations is solved by a series of Newton-Raphson iterations until convergence has been obtained. This requires the knowledge of the partial derivatives of [FORMULA], [FORMULA], [FORMULA] and [FORMULA] with respect to p, T, L and R. Some of these derivatives may be evaluated analytically, but for others (such as those involving the equation of state or opacity tables) numerical derivatives have to be employed. Each iteration requires the solution of a linear system of equations, through the inversion of a block diagonal matrix. It is important in such a relaxation scheme to have initial conditions sufficiently close to the true solution; two Runge-Kutta integrations, one outward from the centre and one inward from the surface, are used to provide such an initial trial solution.

2.3. Boundary conditions

Boundary conditions are imposed at the centre and at the surface. At the centre,

[EQUATION]

[EQUATION]

where [FORMULA] denotes the energy-generation rate per unit mass, and [FORMULA] the equilibrium energy-generation rate at the centre. At the surface,

[EQUATION]

[EQUATION]

where [FORMULA] is the effective temperature, given by [FORMULA] ([FORMULA] is the Stefan-Boltzmann constant), and [FORMULA] is the optical depth at which interior solution is matched to the atmosphere. The latter should be made sufficiently large that the diffusion approximation (upon which the radiative-transfer equation is based) is valid (Morel et al. 1994). Assuming the HSRA [FORMULA] law, as fitted by Ando & Osaki (1975),

[EQUATION]

[FORMULA] is obtained directly, and [FORMULA] is obtained by integrating for the structure of the atmosphere using a Runge-Kutta scheme with adaptive step size.

2.4. Truncation error

If h is the step size in mass, then the local truncation error in Eq. (1) is [FORMULA]. This gives a global truncation error which scales like [FORMULA], so the finite-difference scheme is fourth order. This is demonstrated in Fig. 1, which plots the logarithm of the relative errors in radius and luminosity against [FORMULA]. The slight departure of the errors in luminosity from following a straight line is probably due to errors arising from the interpolation of opacity and equation of state tables. A value of [FORMULA] is sufficient to constrain the relative truncation errors in radius and luminosity to [FORMULA].

[FIGURE] Fig. 1. Logarithm of the relative errors in radius (solid line) and luminosity (dashed line) plotted against the logarithm of the number of radial grid points, N.

2.5. Nuclear burning

The treatment of the time evolution of chemical composition is crucial to the success of a stellar evolution code. The principal energy-generating cycle of the proton-proton chain (which dominates in the sun) consists of two halves, the conversion of hydrogen into [FORMULA] via deuterium, and the fusion of [FORMULA] to form [FORMULA]. In the centre of the sun, the second of these reactions proceeds very much faster than the first, so that equilibrium of [FORMULA] is quickly established. Away from the centre, equilibrium takes much longer to be reached, so it is essential to follow the evolution of the [FORMULA] abundance correctly in order to calculate the total luminosity accurately. Since the time scales of the two reactions of the principal branch of the proton-proton chain are so different, the equations constitute a stiff system, and should be integrated by an appropriate implicit technique.

Christensen-Dalsgaard (1991a) used a backward Euler method for evolving the [FORMULA] abundance, and a second-order scheme for the evolution of the hydrogen abundance. Subsequent codes have employed more sophisticated integration schemes; for instance, the CESAM code developed by Morel et al. (1990) uses an implicit, second-order, time-integration scheme. MoSEC uses a similar second-order implicit method for determining the time evolution of the abundances of [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. To simplify the reaction network, [FORMULA] is assumed to be always in equilibrium, while only the dominant branches are considered in the CNO cycle.

The reaction rate [FORMULA], corresponding to process p (as defined in Table 1) at the point [FORMULA], is then given by one of the following equations (in units of [FORMULA]):

[EQUATION]

where [FORMULA] is the fractional abundance by mass of the element with atomic number n, and [FORMULA] is the reaction-rate coefficient for process p. The latter are functions of density and temperature alone. The evolution of the elemental abundances is then calculated according to the following set of equations:

[EQUATION]

where [FORMULA] is the atomic mass of the element with atomic number n.


[TABLE]

Table 1. Simplified nuclear reaction network.


2.6. Diffusion and gravitational settling

The relative abundances of chemical elements in the solar interior are modified not only by nuclear burning, but also by the motion of atoms of different species with respect to one another. Gravitational and radiative forces acting on individual atoms drive such motions, while interactions between atoms tend to redistribute momentum in a random way and thereby counteract these forces. The details of element segregation in the sun are determined by the exact nature of the competition between these processes in a multi-component plasma.

Since the abundances of the heavy elements are small, it is a reasonable approximation to treat them as if they were trace elements in a [FORMULA] - [FORMULA] background (Proffitt 1994). The seven species discussed in the context of nuclear reactions ([FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA]) are treated individually in the settling and diffusion calculation, and are assumed to be fully ionized, while all the other heavy elements are grouped together and assumed to diffuse like fully ionized iron. Since the diffusion of heavy elements has a relatively small effect on the properties of the computed model, this represents a good approximation (Bahcall & Pinsonneault 1995).

The simplified transport equations of Michaud & Proffitt (1993) are used to describe the gravitational settling and diffusion of helium and the heavy elements. The diffusion velocities given by these equations are accurate to within about 10% when compared to the more rigorous derivation of Burgers (1969). The equations are solved using an explicit, second-order, time integration scheme, with time steps determined by the overall truncation error requirement.

2.7. Time evolution

An Q -point grid in time, [FORMULA], is constructed such that [FORMULA] and [FORMULA], where [FORMULA] is the age of the sun. In order to evolve the solar model from time [FORMULA] to time [FORMULA], assuming the structure [FORMULA], [FORMULA], [FORMULA] to be known at [FORMULA], the following procedure is adopted:

1) The coefficients [FORMULA] are evaluated using the known values of [FORMULA] and [FORMULA]. The coefficients [FORMULA] are initially assumed to be equal to [FORMULA].

2) Eqs. (15)-(21) are integrated assuming the coefficients [FORMULA] take the values

[EQUATION]

3) The structure is evaluated at time [FORMULA], using the new hydrogen abundance so obtained. This yields a new set of values for [FORMULA].

4) Steps 2 and 3 are repeated until satisfactory convergence has been obtained.

The grid points [FORMULA] are not distributed uniformly in time: in the early stages of evolution they are much more closely spaced than later on. The distribution is given by the functional form

[EQUATION]

where [FORMULA]. The exact function chosen to define the grid in time does not affect the rate of convergence of the solution as Q is increased, but does affect the actual truncation error for a particular value of Q. The form (23) is chosen as it gives a particularly low truncation error. Since the procedure used to update the chemical composition between [FORMULA] and [FORMULA] is second-order, the truncation error in the final model at time [FORMULA] goes like [FORMULA]. In order to exploit this known dependence, a series of increasing trial values of Q is used, and the final result is obtained by Richardson extrapolation. In order to demonstrate this, Fig. 2 shows how the truncation error in radius and luminosity varies with Q ; the logarithm of the relative error in each of these quantities is plotted against [FORMULA] for [FORMULA]. While the relative error is still larger than [FORMULA] for [FORMULA], Richardson extrapolation enables the radius and luminosity to be estimated to an accuracy of better than [FORMULA] using just [FORMULA].

[FIGURE] Fig. 2. Logarithm of the relative errors in radius (solid line) and luminosity (dashed line) plotted against the logarithm of the number of temporal grid points, Q.

2.8. Calibration of mixing length and initial hydrogen abundance

The final values at time [FORMULA] of the luminosity and radius depend on the initial hydrogen abundance and the mixing-length parameter, [FORMULA]. These are modified by a series of Newton-Raphson iterations until the luminosity and radius agree with [FORMULA] and [FORMULA] (the present-day luminosity and radius) to within one part in [FORMULA]. Sufficiently many grid points are required in the temporal and spatial integrations that their relative truncation errors are less than [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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