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Chapter 2 Basic Assumptions, Theorems, and Polytropes

Chapter 2 Basic Assumptions, Theorems, and Polytropes

1. Stellar

1. Stellar InteriorsReplacing the potential gradient from equation (1.2.28), we have(2.1.6)This is the equation of hydrostatic equilibrium for spherical stars. Because of itsgenerality and the fact that virtually no assumptions are required to obtain it, we canuse its integral to place fairly narrow limits on the conditions that must prevail insidea star.In equation (2.1.2), we introduced a new variable M(r). Note that itsinvocation is equivalent to invoking a conservation law. The conservation of massbasically requires that the total mass interior to r be accounted for by summing overthe density interior to r. Thus,or its differential form(2.1.7)(2.1.8)2.2 Integral Theorems from Hydrostatic EquilibriumaLimits on State VariablesFollowing Chandrasekhar, 1 we wish to define a quantity I σ,ν (r) whichis effectively the σth moment of the mass distribution further weighted by r −ν .Specifically(2.2.1)There are quite a variety of physical quantities which can be related to I σ,ν . Forexample,is just the absolute value of the total gravitational energy of the star.(2.2.2)We can use this integral quantity to place limits on physical quantities ofinterest if we replace ρ by as defined by equation (2.1.2). Since36

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