## Stars

### Stellar Models

In most cases, the density, the temperature, and the chemical composition of a star change appreciably only over very long time intervals. For the Sun, only 1% of the hydrogen is depleted and converted into helium in one billion years . Thus the change induced by nuclear fuel depletion is entirely negligible. A static stellar model is appropriate for the Sun and most of the main sequence stars. Time does not appear in any equation under this circumstance. Whether it is static or dynamic, the stellar structure is governed by five basic equations. In mathematical terms, they are a set of inter-dependent differential equations (see Figure 08-27). A verbal description is given below for simulating the structure of a main sequence star.

#### Figure 08-27 Stellar Model [view large image]

• The equation of hydrostatic equilibrium - it is essentially the balance between the gaseous pressure and gravity as a function of the distance from the center of the star. An additional term to describe the acceleration of a layer (within the star) is required if it undergoes an expansion/contraction phase.
• The equation of mass - this equation calculates the mass within a certain distance from the center of the star.
• The equation of conservation of energy - this equation governs the generation of energy (luminosity) from the nuclear fuel and gravitation as a function of the distance from the center of the star. It is applicable to the core only as indicated in Figure 07-03. In the pre-main-sequence phase, nuclear fuel would not be available; only the gravitational energy is converted to heat as the protostar contracts. In the post-main-sequence phase, the nuclear energy production rate would become more sensitive to the progress of time.
• The equation of radiative transport - it describes temperature variation related to the propagation of the luminosity in the radiative zone of the star as shown in Figure 07-03.
• The equation of convective transport - it describes temperature variation related to the change of pressure in the convective zone of the star as shown in Figure 07-03.

• In addition to these equations which characterize general conditions, there are three explicit relations to describe more specifically the behavior of the interior gases.

• The equation of state - this is a formula to describe the relationship between the pressure, the density, the temperature, and the composition of the gases.
• The equation for the absorption coefficient - this is a formula to describe the opacity of the gases.
• The equation for energy generation - this is a formula to determine the energy production rate by the nuclear fuel and gravity.
It is not possible to obtain an analytic solution to the stellar models. All realistic models are computed by numerical methods. Anyone who is interested in constructing stellar models, can find more information from many text books on stellar structure such as "Structure and Evolution of the Stars" by M. Schwarzschild, Princeton University Press, 1958 (Dover paperback, 1965). This classical text was the first modern monograph on stellar structure, and has since taught a generation of astro-physicists how to apply electronic computers to the computation of stellar models. It is still being recommended in reading list for course in stellar structure.

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