Solar Standard Model PDF

Modelling the Sun Introduction The theory of stellar structure is extremely complex, requiring some knowledge of thermodynamics, atomic physics, nuclear physics, and gravitation theory, among other disciplines. At the same time, stellar structure has contributed in a significant way to the developme...

Modelling the Sun Introduction The theory of stellar structure is extremely complex, requiring some knowledge of thermodynamics, atomic physics, nuclear physics, and gravitation theory, among other disciplines. At the same time, stellar structure has contributed in a significant way to the development of these areas. An example is the boost given to nuclear physics by the identification of the nuclear fusion reactions as the main energy source in the stellar interior. More recently, the study of the late evolutionary stages of massive stars has stimulated the development of the gravitation theory of dense objects, such as neutron stars and black holes. The structure of a star can be described by a set of equations containing variables such as the pressure P, density 𝜌, temperature T, luminosity L, etc. However, based on some simple ideas, it is possible to obtain estimates of the main properties of the stellar interior. The most important hypotheses are: a) spherical symmetry, b) absence of rotation, c) absence of magnetic fields, and d) hydrostatic equilibrium. Moreover, it is implicitly assumed that the physical laws derived from laboratory measurements also hold in the whole universe. 2 2 Ref. 01 W. J. Maciel, Introduction to Stellar Structure (Springer, 2016). page 19 Basics of stellar modelling 2 Stellar models are generally calculated under a number of simplifying approximations, of varying justification. In most cases rotation and other effects causing departures from spherical symmetry are neglected and hence the star is regarded as spherically symmetric. Also, with the exception of convection, hydrodynamical instabilities are neglected, while convection is treated in a highly simplified manner. The mass of the star is assumed to be constant, so that no significant mass loss is included. In contrast to these simplifications of the ‘macrophysics’ the microphysics is included with considerable, although certainly inadequate, detail. In recent calculations effects of diffusion and settling are typically included, at least in computations of solar models. The result of these approximations is what is often called a ‘standard solar model’, although still obviously depending on the assumptions made in the details of the calculation.* Even so, such models computed independently, with recent formulations of the microphysics, give rather similar results. In this paper I generally restrict the discussion to standard models, although discussing the effects of some of the generalizations. It might be noted that the present Sun is in fact one case where the standard assumptions may have some validity: at least the Sun rotates sufficiently slowly that direct dynamical effects of rotation are likely to be negligible. On the other hand, rotation was probably faster in the past and the loss and redistribution of angular momentum may well have led to instabilities and hence mixing affecting the present composition profile. 3 Ref. 03 Solar structure and evolution - Jørgen Christensen-Dalsgaard * The notion of ‘standard model’ develops over time; for example, until around 1995 diffusion and settling would not generally be regarded as part of ‘standard’ solar modelling. How Do We Model the Sun? http://large.stanford.edu/courses/2017/ph241/stayner2/ Overview How does a star work? It all starts in the core, where atoms are flying around so quickly that they fuse together when they hit each other. The fusion of those atoms produces energy in the form of fast-moving particles that increase the temperature of whatever they bump into. The result of fusion in the core of the star is a self-sustaining explosion of particles at temperatures up to 15 million Kelvin. The energy from the explosions leaves the core as heat (which spreads toward the outer edges of the sun) and sunlight, which is known scientifically as radiation. This process is described in four main equations that we can use to make a rough model of a star's shape. This treatment of the model draws heavily on Maciel, which derives all the equations in more detail. Fig. 1: Hydrostatic equilibrium in the sun: pressure pushes outward and gravity pulls inward. (Source: J. Stayner) Hydrostatic (Pressure) Equilbrium To get our first equation, we look at what makes fusion happen in the star. The main interaction here is a battle between gravity and pressure. Gravity pulls inward while pressure pushes outward. A star is formed when the force of gravity is so large that it compresses gases to such temperatures and densities that fusion occurs in its core, where the pressure is the highest. This interaction is also what gives a star its spherical shape: gravity pulls everything toward the core, while pressure pushes outward (as in Fig. 1). Because the radius of a star is fairly constant, we assume that the force due to gravity is equal to the force from pressure, obtaining (1) where G is the Newton gravitation constant, M(r) is the total mass contained inside the sphere of radius r, ρ(r) is the mass density at radius r, and P(r) is the pressure at r. Thermal Equilibrium through Radiation Loss Our second equation comes from the observation that the sun radiates light but has a fairly constant temperature. Sunlight has quite a bit of energy in it, and if the temperature is constant, then all the energy in sunlight has to be coming from somewhere. Since fusion is the primary source of energy in a star, we relate the luminosity of the star (L) with the rate at which energy is produced per unit mass by nuclear fusion (ε) by (2) Continuity of Mass The third equation we will use is based on a way of thinking about the mass of the star; we will use it to find the relationship between the density of the star and its size. The idea is that if the star has a certain density, then we can divide the star into shells with very, very thin thicknesses. Each shell will have a mass dM = 4π r2p(r)dr. We can turn that into a differential equation by dividing by dr, obtaining (3) Energy Transport Within the Star Finally, we know that the sun is hotter at its core than it is on the outside; there are three primary ways this can happen, but only two of them (convection and radiation) are significant to the star's structure. Radiation in the context of heat flow happens when a part of the star near the core emits light that is then reabsorbed by another part of the star before it can leave. In convection, particles move from one area of the star to another - this is equivalent to a cold wind bringing cooler temperatures to an area. We describe them using the equation  (4 ) The second expression is used in the convective regions of the star and the radiative expression is used everywhere else. In the Sun, the core is essentially radiative while the outer layers are convective. Here κR is the radiative opacity, c is the speed of light, T(r) is the temperature in Kelvin, a is Stefan's contant (radiative energy density = aT4), and Γ2 is the dimensionless specific heat ratio of the stellar atmosphere (typically 5/3). Solving the Equations We are left with five differential equations, and five obvious variables: M(r), L(r), T(r), P(r), p(r). But, it turns out that ε, Γ2, and κR all depend on the chemical composition of the star as well as some of the other five variables. As a result, we have six variables with five equations. The solution to that problem is to guess one of the variables - usually the mass M - and then calculate the rest based off of that result. The process is iterated until one comes out with a solution that fits one's observations of the star. In general, this is done with computers due to the tedium of iteration, particularly if you are using the two different energy transport equations for different parts of the star. © John Stayner. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author. References B. Olson, "Fusion Regulation in the Sun," Physics 241, Stanford University, Winter 2011. W. J. Maciel, Introduction to Stellar Structure (Springer, 2016). {3} Jørgen Christensen-Dalsgaard: 2021, Solar structure and evolution - Living Reviews in Solar Physics (2021) 18:2.