Assumptions made

1. The static configuration of a gaseous sphere held together by self-gravitation must satisfy the so-called condition of hydrostatic equilibrium, a local equation applicable at local points in the star. The essence of the equation of hydrostatic equilibrium is that the variation of the pressure P throughout the star is sufficient to just balance the gravitational field of the star. Furthermore, the pressure is determined by an equation of state applicable to the local conditions in the stellar interior.



The above considerations do not in themselves determine the structure of a star. At each point in a stellar interior any specified increase of pressure that may be required to balance the gravitational force is obtainable from an unlimited number of combinations of density (rho) and temperature (T) values. So one or more additional conditions relating the physical variables in the stellar interior are needed.

An approximation introduced at the beginning of this century is that of eliminating T by assuming that the pressure can be expressed as some power of the density only. This "polytropic" pressure-density relationship is written as P = K*rho^Y, where the exponent Y is a free parameter which enables us to consider model stars (called polytropes) of various types, and K is constant which depends upon the nature itself of the polytrope.




2. The polytrope relation was selected more on mathematical than physical grounds since it cannot be deduced from physical arguments. But certain idealized physical circumstances for a star lead naturally to a polytropic pressure-density relationship. Here follow some classical polytropes that were important in the past:


• Y = 5/3: model of a fully convective star (i.e. a star in which the energy is transported by convection) where radiation pressure is not important; also used for a white dwarf supported by the pressure of a non-relativistic degenerate electron gas.


• Y = 4/3: the standard model of Eddington, used to describe a star in which the gas pressure is a constant fraction of the total pressure throughout the star (this particular polytrope corresponds more closely to stars in radiative equilibrium, i.e. stars for which the energy is transported by radiative transfer rather than by convection); also used to describe a white dwarf with relativistic degeneration.


• Y = 1: an approximation for the distribution of stars in a globular cluster (it represents a self-gravitating isothermal gas, and a globular cluster can be seen as an isothermal gaseous sphere in which the particles are not molecules, but stars).






3. A polytropic model is only space dipendent, as it gives the variation of the pressure (P), the density (rho), and the mass inside r (Mr) as a function of r, the distance to the center of the model. No time dependent processes - such as nuclear reactions, chemical mixing, energy transport, or mass loss by stellar wind - are included. Moreover the model does not contain any information on the temperature throughout the star.



The star is divided into successive concentric thin shells, or layers, from the center to the surface. The relationships to be satisfied by the physical variables of the stellar model are then imposed at each layer. The distance between two layers - or the thickness of a single layer - is given by dr, which will be taken as constant throughout the star.

The thickness dr of a layer should be small compared to the radius of the star. For a central density of 10^10, 10^11, or 10^12 g/cm³ our suggested values for dr are 1.6, 0.8, or 0.4 km, respectively. Adopting them the pressure vanishes after about 800-900 layers, meaning that we have reached the surface of the star. The values of r and Mr at that point are therefore the total radius and the total mass of the star.