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Chapter 2 Basic Assumptions, Theorems, and Polytropes

Chapter 2 Basic Assumptions, Theorems, and Polytropes

2⋅

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 standard 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.Chandrasekhar 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 stand 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)

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