2
Solar Structure and Evolution

This chapter introduces the equations of stellar structure and evolution and the most important physical processes that determine the evolution of the Sun. We then present the important concept of standard solar model (SSM) and place it in the broader context of evolution of low mass stars, of which the Sun is a typical representative, by discussing the main characteristics of its evolution, from its formation to advanced evolutionary phases. The present-day properties of SSM are then discussed in detail, including theoretical predictions for solar neutrinos and helioseismic probes of the solar interior and the current theoretical uncertainties in solar models. We close the chapter with a generic discussion of nonstandard solar models (non-SSMs), i.e. in which physical processes considered nonstandard have been included in the models.

2.1 Equations of Stellar Structure and Evolution

The fundamental equations that determine the structure and evolution of stars reflect the conservation laws of mass, energy, and momentum and the physical processes that take place in stellar interiors. The resulting equations form a set of partial differential equations and the problem requires both initial and boundary conditions. Here, we derive the full set of equations and boundary conditions that underlie studies of solar models.

2.1.1 Mechanical Structure

The mechanical structure of stars, the Sun among them, is determined by the conservation laws of mass and momentum. In spherical symmetry, the equation of mass conservation can be established in a convenient form by considering the total variation of the mass inside a sphere of radius :

Here, the first term on the right is the mass contained in a shell of thickness d and the second term is the mass flow with radial velocity at the surface of the sphere in the time interval d. In this description, is the independent coordinate. However, in spherical symmetry it is more convenient to use the Lagrangian description in which is the independent coordinate, i.e. to express and other quantities as functions of and . The transformation between the two descriptions is obtained by considering the partial derivative:

Applying this transformation to , one readily obtains the equation of mass conservation in the Lagrangian description:

By comparing this expression with Eq. (2.1), it becomes clear that using the Lagrangian formulation equations can be written more simply.

The conservation of momentum in spherical symmetry is expressed as

where the net acceleration is the result of the gravitational acceleration and the force per unit mass exerted by the pressure gradient.

The secular evolution of stars driven by stable nuclear burning leads to very slow changes in stellar size, leading to a negligible net acceleration. This is especially true for the Sun, a low mass main sequence star that evolves appreciably only over timescales of  years. Estimates of the acceleration based on the evolution of detailed solar models lead to maximum values of the order of a few times at the solar surface. This can be compared with the gravitational acceleration that, in the solar surface, is , i.e. about 28 orders of magnitude larger. Under these conditions, the conservation of momentum reduces to the equation of hydrostatic equilibrium:

The response timescale to departures from hydrostatic equilibrium is very short. This can be estimated by considering Eq. (2.3). If, for example, a very large increase in the pressure gradient were to occur such that the second term on the right-hand side dominates, then the response time would be of the order . Here, is an average value of the sound speed in the solar interior, of the order of . On the other hand, if the pressure gradient were to become negligible, the response time would be that of free fall: , where is the solar mean density. Any large-scale dynamical instability would relax in timescales extremely short compared to the evolutionary timescales of the Sun, justifying again the assumption of hydrostatic equilibrium in solar evolution models. Finally, transforming Eq. (2.4) to the Lagrangian formulation leads to

Equations (2.2) and (2.5) determine the mechanical structure of the Sun.

2.1.2 Energy Conservation and Transport

The next step in deriving the full set of stellar evolution equations is to consider the energetics of stars. The first law of thermodynamics states:

where and are respectively the heat and internal energy per unit mass for an element of specific volume . Here, the first term represents a change in the internal energy and the second term the work, by expansion or contraction, done on the element. Also, if represents a local rate of energy production per unit mass and is the radial component of the energy flux, the rate of change of heat in the volume element is

where the second term is the divergence of the energy flux per unit mass. This can be expressed as

Here is the energy flux across a sphere of radius , i.e. the luminosity. Using the Lagrangian transformation,

defining

and combining with the first law of thermodynamics:

In standard stellar evolution represents nuclear energy sources and neutrino energy losses, i.e. . The second term, , usually referred to as gravothermal energy, accounts for changes in internal energy and work. The final form of the equation of energy conservation is then:

2.1.2.1 Energy Transport by Radiation and Conduction

In standard stellar models, there are three mechanisms for transporting energy that occur because of the temperature gradient that exists between the inner and outer stellar regions: radiation, (electron) conduction, and convection. From a strict point of view, the first two are always present because they directly depend on the existence of a temperature gradient. Convection, on the other hand, occurs in stars when a region becomes dynamically unstable and macroscopic convective motions set in. The actual temperature gradient in the stellar interior will be determined by the combined action of these transport mechanisms.

Radiation transport in stellar interiors can be treated under the so-called diffusion approximation, i.e. by consideration of Fick's law applied to radiation energy. This is justified because the mean free path of photons in stellar interiors is much smaller than the characteristic length scales over which physical conditions change significantly.1 The opaqueness of matter to radiation is conveniently represented in stellar interiors by the opacity, an absorption coefficient expressed in units of cross section per unit mass. The mean free path of photons is then given by

where is an appropriate average, to be discussed below, of the opacity taken over the whole spectrum of photons. Typical values of determined from detailed solar models range between and in the solar interior. The small values of also mean that the variation of temperature that a photon experiences over is very small and local thermodynamic equilibrium (LTE) is very well satisfied in solar interiors.

Fick's law applied to radiation energy density is

with as the speed of light, the radiation density constant, and the energy density of radiation. Expressing now the radiation flux in terms of the luminosity, replacing the energy density gradient by its corresponding temperature gradient, and with Eq. (2.7):

Applying the Lagrangian transformation, the equation of radiative transport is

Conduction can also be treated as a diffusive process, characterized by a conductive opacity . It is accounted for in the transport of energy by replacing the radiative opacity by

In the Sun, conduction is a very inefficient transport mechanism. In an ideal gas under non-degenerate conditions, electrons have an extremely short mean free path in comparison to photons. In the solar center, where the density of free electrons is highest, the conductive opacity is , in comparison to a typical value for the radiative opacity , i.e. . The mean free path of electrons is then shorter by the same factor, rendering conduction a negligible energy transport mechanism in the solar interior.

The equation for radiative and conductive transport is then

It is useful to express the temperature gradient in a dimensionless form:

where the radiative gradient is defined as

so that Eq. (2.9) reads:

Before, it was mentioned that is an appropriate average over the whole spectrum of photon energies. In fact, the relation used above implicitly assumes LTE and that the radiation field is that of a black body, i.e. described by a Planck distribution. Under these conditions, it can be shown that is the Rosseland mean opacity, given by

where is the radiation frequency, the monochromatic opacity, and

is the Planck function for radiation intensity. The Rosseland mean opacity entering the equation of...