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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.
Characteristic velocity or , or C-star is a measure of the combustion performance of a rocket engine independent of nozzle performance, and is used to compare different propellants and propulsion systems. c* should not be confused with c, which is the effective exhaust velocity related to the specific impulse by: =. Specific impulse and ...
The largest Lyapunov exponent is as follows [2] = (()). Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable .
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 matrix exponential satisfies the following properties. [2] We begin with the properties that are immediate consequences of the definition as a power series: e 0 = I; exp(X T) = (exp X) T, where X T denotes the transpose of X. exp(X ∗) = (exp X) ∗, where X ∗ denotes the conjugate transpose of X. If Y is invertible then e YXY −1 = Ye ...
The compressed exponential function (with β > 1) has less practical importance, with the notable exceptions of β = 2, which gives the normal distribution, and of compressed exponential relaxation in the dynamics of amorphous solids. [1] In mathematics, the stretched exponential is also known as the complementary cumulative Weibull distribution.
In other words, the configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B is in state 1 and particle A is in state 2. This assumption leads to the proper (Boltzmann) statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the Gibbs ...