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A polytrope with index n = 3 is a good model for the cores of white dwarfs of higher masses, according to the equation of state of relativistic degenerate matter. [7] A polytrope with index n = 3 is usually also used to model main-sequence stars like the Sun, at least in the radiation zone, corresponding to the Eddington standard model of ...
Some specific values of n correspond to particular cases: = for an isobaric process, = + for an isochoric process. In addition, when the ideal gas law applies: = for an isothermal process,
The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian profile, and is associated with many processes, including resonance energy distribution, impact and natural spectral line broadening and quadratic stark line broadening.
This yields a polytrope of index 3, which has a total mass, M limit, depending only on K 2. [9] For a fully relativistic treatment, the equation of state used interpolates between the equations P = K 1 ρ 5/3 for small ρ and P = K 2 ρ 4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at M limit.
For a polytrope solution with n=3 (as in the Eddington stellar model for radiative zone), P is proportional to T 4 and the left-hand side is constant and equals 1/4, smaller than the ideal monatomic gas approximation for the right-hand side giving / = /. This explains the stability of the radiative zone against convection.
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For example, in the Sun the convection at the base of the convection zone, near the core, is adiabatic but that near the surface is not. The mixing length theory contains two free parameters which must be set to make the model fit observations, so it is a phenomenological theory rather than a rigorous mathematical formulation.