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Neutron stars are well modeled by polytropes with index between n = 0.5 and n = 1. A polytrope with index n = 1.5 is a good model for fully convective star cores [5] [6] (like those of red giants), brown dwarfs, giant gaseous planets (like Jupiter). With this index, the polytropic exponent is 5/3, which is the heat capacity ratio (γ) for ...
A polytropic process is a thermodynamic process that obeys the relation: = where p is the pressure , V is volume , n is the polytropic index , and C is a constant. The polytropic process equation describes expansion and compression processes which include heat transfer.
The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating spheres of plasma such as stars, but typically it is a rather limiting assumption.
The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). For a linear triatomic molecule such as CO 2 , there are only 5 degrees of freedom (3 translations and 2 rotations), assuming vibrational modes are not ...
The formula for the exponential results from reducing the powers of G in the series expansion and identifying the respective series coefficients of G 2 and G with −cos(θ) and sin(θ) respectively. The second expression here for e Gθ is the same as the expression for R ( θ ) in the article containing the derivation of the generator , R ( θ ...
A highly detailed study on the Modified Tait equation of state with the physical interpretation of the two empirical parameters and is given in chapter 4 of reference. [2] Expressions as a function of entropy for the two empirical parameters n {\displaystyle n} and B {\displaystyle B} are given for water, helium-3 and helium-4.
reduced temperature minus 1, T − T c / T c f: specific free energy: C: specific heat; −T ∂ 2 f / ∂T 2 J: source field (e.g. P − P c / P c where P is the pressure and P c the critical pressure for the liquid-gas critical point, reduced chemical potential, the magnetic field H for the Curie point) χ
The method of exponent pairs gives a class of estimates for functions with a particular smoothness property. Fix parameters N , R , T , s ,δ. We consider functions f defined on an interval [ N ,2 N ] which are R times continuously differentiable , satisfying
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