2⋅ **Basic** **Assumptions**, **Theorems**, **and** **Polytropes**(2.2.11)Now we define β * to be the value of β which makes Equation (2.2.11) anequality, **and** then we obtain the st**and**ard result thatSince (1-β)/β 4 is a monotone increasing function of (1-β),(2.2.12)(2.2.13)Equation (2.2.11) can be solved directly for M in terms of β * **and** thus it places limitson the ratio of radiation pressure to total pressure for stars of a given mass.Ch**and**rasekhar 1 (p.75) provides the brief table of values shown in Table 2.1.As we shall see later, m is typically of the order of unity (for example µ is ½for pure hydrogen **and** 2 for pure iron). It is clear from Table 2.1, that by the time thatradiation pressure accounts for half of the total pressure, we are dealing with a verymassive star indeed. However, it is equally clear that the effects of radiation pressuremust be included, **and** they can be expected to have a significant effect on thestructure of massive stars.39

1. Stellar Interiors2.3 Homology TransformationsThe term homology has a wide usage, but in general it means "proportional to" **and** isdenoted by the symbol ~ . One set is said to be homologous to another if the two canbe put into a one-to-one correspondence. If every element of one set, say z i , can beidentified with every element of another set, say z' i , then z i ~ z' i **and** the two sets arehomologous. Thus a homology transformation is a mapping which relates theelements of one set to those of another. In astronomy, the term homology has beenused almost exclusively to relate one stellar structure to another in a special way.One can characterize the structure of a star by means of the five variablesP(r), T(r), M(r), µ(r), **and** ρ(r) which are all dependent on the position coordinate r. Inour development so far, we have produced three constraints on these variables, theideal-gas law, hydrostatic equilibrium, **and** the definition of M(r). Thus specifyingthe transformation of any two of the five dependent variables **and** of the independentvariable r specifies the remaining three. If the transformations can be written assimple proportionalities, then the two stars are said to be homologous to each other.For example, ifthen(2.3.1)(2.3.2)where ξ, ζ, η, **and** χ st**and** for any of the remaining structure variables. However,because of the constraints mentioned above, C 4 , C 5 , **and** C 6 are not linearlyindependent but are specified in terms of the remaining C's. Consider the definitionof M(r) **and** a homology transformation from r → r'. Then(2.3.3)In a similar manner, we can employ the equation of hydrostatic equilibriumto find the homology transformation for the pressure P, since40(2.3.4)