Non-extensive statistics and the stellar polytrope index

In astrophysics the polytropic law with index n is commonly used as a means of imposing a simple and ordered physical structure on a gaseous (or smoothed discrete) system. In many instances it would be preferable to be able to introduce a polytropic density variation analytically into the basic theory rather than numerically at the computational phase. It is perhaps unfortunate that the three well known classical analytical E type solutions of the Lane-Emden equation for n = 0, 1 and 5 all have some constraining physical features; specifically, the polytrope n = 0 has uniform density and hence arbitrary radius, when n = 1 the mass and radius are independent of each other and the solution cannot be transformed homologically, and because the first zero ξ1 = ∞ for n = 5 the corresponding polytropic model has infinite extent and central condensation. In contrast, and unlike most stars, the two finite radius models have central condensations which ~ 1.

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A Finite Radius Solution for the Polytrope Index 5

Publications of the Astronomical Society of Australia ◽

10.1017/s1323358000018683 ◽

1980 ◽

Vol 4 (1) ◽

pp. 37-39 ◽

Cited By ~ 2

Author(s):

J. O. Murphy

Keyword(s):

Large Scale ◽

Developmental Stages ◽

Stellar Structure ◽

Structure Theory ◽

Finite Radius ◽

Density Distributions ◽

Wide Range ◽

Stellar Pulsations ◽

Polytrope Index ◽

Modern Astronomy

Polytropic models have played a significant part in the historical development of stellar structure theory and in other related branches of theoretical astronomy such as stellar pulsations and rotation. In modern astronomy polytropic solutions, because of their wide range of density distributions and our ability to introduce them analytically if necessary, now have an acknowledged role in large scale numerical experiments especially during the developmental stages.

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A variable polytrope index applied to planet and material models

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stv1397 ◽

2015 ◽

Vol 452 (2) ◽

pp. 1375-1393 ◽

Cited By ~ 3

Author(s):

S. P. Weppner ◽

J. P. McKelvey ◽

K. D. Thielen ◽

A. K. Zielinski

Keyword(s):

Material Models ◽

Polytrope Index

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MATHEMATICAL MODEL FOR THE STUDY AND DESIGN OF A ROTARY-VANE GAS REFRIGERATION MACHINE

Refrigeration Engineering and Technology ◽

10.15673/ret.v52i3.116 ◽

2016 ◽

Vol 52 (3) ◽

Author(s):

V. V. Trandafilov ◽

M. G. Khmelniuk ◽

O. Y. Yakovleva

Keyword(s):

Mathematical Model ◽

Heat Exchanger ◽

Coefficient Of Performance ◽

Operating Cycle ◽

Regenerative Heat ◽

Regenerative Heat Exchanger ◽

Proposed Model ◽

Compression And Expansion ◽

Geometric Variables ◽

Polytrope Index

This paper presents a mathematical model of calculating the main parameters the operating cycle, rotary-vane gas refrigerating machine that affect installation, machine control and working processes occurring in it at the specified criteria. A procedure and a graphical method for the rotary-vane gas refrigerating machine (RVGRM) are proposed. A parametric study of the main geometric variables and temperature variables on the thermal behavior of the system is analyzed. The model considers polytrope index for the compression and expansion in the chamber. Graphs of the pressure and temperature in the chamber of the angle of rotation of the output shaft are received. The possibility of inclusion in the cycle regenerative heat exchanger is appreciated. The change of the coefficient of performance machine after turning the cycle regenerative heat exchanger is analyzed. It is shown that the installation of a regenerator RVGRM cycle results in increased COP more than 30%. The simulation results show that the proposed model can be used to design and optimize gas refrigerator Stirling.

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Mass Distributions of Galaxies by Globular Cluster Systems

Symposium - International Astronomical Union ◽

10.1017/s0074180900043436 ◽

1988 ◽

Vol 126 ◽

pp. 639-640

Author(s):

Kazutomo Takayanagi

Keyword(s):

Globular Cluster ◽

Surface Density ◽

Spiral Galaxies ◽

Cluster Systems ◽

Mass Distributions ◽

Density Distributions ◽

Polytrope Index

Density curves of polytrope index 5 were fitted to the surface density distributions of 26 globular cluster systems. Though we have shown only two of them in Fig. 1, the remaining systems resemble the above cases. No differences exist between elliptical and spiral galaxies.

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The flow helicity in quasi-orderedcellular convection

Proceedings of the International Astronomical Union ◽

10.1017/s1743921313002755 ◽

2012 ◽

Vol 8 (S294) ◽

pp. 359-360

Author(s):

A. V. Getling

Keyword(s):

Finite Difference ◽

Rotational Velocity ◽

Vertical Axis ◽

Horizontal Layer ◽

Thermal Perturbation ◽

Convective Flows ◽

Maximum Value ◽

The Mean ◽

Prandtl Numbers ◽

Polytrope Index

AbstractThe helicity of cellular convective flows in a horizontal layer of compressible fluid (gas) heated from below and rotating about a vertical axis is studied using finite-difference numerical simulations. The medium is assumed to be polytropically stratified. An initial thermal perturbation is introduced so as to produce a system of Bénard-type hexagonal convection cells. The flow gradually becomes less ordered, and the mean helicity grows initially and decreases sharply after the substantial chaotisation of the flow. Given the Rayleigh and Prandtl numbers, the maximum value reached by the mean helicity increases with the decrease of the polytrope index and has a maximum at a certain rotational velocity of the layer.

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Nonlinear theta-pinch equilibria

Journal of Plasma Physics ◽

10.1017/s0022377800010813 ◽

1981 ◽

Vol 26 (3) ◽

pp. 419-430 ◽

Cited By ~ 1

Author(s):

B. Abraham-Shrauner

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Particle Density ◽

Nonlinear Differential Equations ◽

Radial Distance ◽

Analytic Solutions ◽

Line Density ◽

Euler’S Method ◽

Theta Pinch ◽

Polytrope Index

Analytic solutions for cylindrical, rigid-drift equilibria were searched for by checking the invariance of the second-order, nonlinear differential equations of the line density and particle density under one-parameter Lie groups. No invariance was found for the cylindrical screw-pinch for the polytrope index γ ╪ 2 of the pressure equation of state. For the Z-pinch, the differential equation for the line density for arbitrary γ is invariant under one group and reduces by Euler's method to an intractable first-order, nonlinear differential equation. In the case of a θ-pinch, the nonlinear differential equations for the particle density and line density are invariant at least under the translation group. The particle density is found as an implicit function of the radial distance r, involving incomplete beta functions of the γth power of the particle density.

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