scholarly journals Structure of a Sequence of Two Zone Polytropic Stellar Models with Indices 0 and 1

1983 ◽  
Vol 36 (3) ◽  
pp. 453 ◽  
Author(s):  
JO Murphy
Keyword(s):  
Radial Displacement ◽  
Radial Distance ◽  
Density Variation ◽  
Uniform Density ◽  
Constant Density ◽  
Surface Phenomena ◽  
Composite Function ◽  
Emden Equation ◽  
Stellar Models ◽  
Type Composite

The structure and physical properties of a sequence of two zone polytropic stellar models, based on E-type composite analytical solutions of the Lane-Emden equation for indices n = 0 and 1, have been determined. The resulting models are characterized by an inner zone of constant density, corresponding to the n = 0 component, and a relatively small envelope with a steep polytropic density gradient and index n = 1. At the surface of these near uniform density composite configurations the physically significant condition p = 0 is always satisfied, a condition which is not fulfilled by the complete poly trope of index n = O. In each model the surface corresponds to the first zero of the associated composite function, which in turn is given by one of the higher order zeros of the E solution of the Lane-Emden equation for n = 1. The radial pulsational properties of these low central condensation type models have also been investigated with the relative amplitudes of radial displacement and density variation given as a function of radial distance for the first six modes of some selected models. Overall, these results establish that the radial displacement amplitudes are only significant in the n = 1 outer layer and accordingly the pulsations can be classified as essentially surface phenomena

A Sequence of E-Type Composite Analytical Solutions of the Lane-Emden Equation

1982 ◽  
Vol 4 (4) ◽  
pp. 376-378 ◽  
Author(s):  
J.O. Murphy
Keyword(s):  
Density Distribution ◽  
Analytical Solutions ◽  
Surface Density ◽  
Radial Distance ◽  
Stellar Model ◽  
The Other ◽  
Uniform Density ◽  
Density Condition ◽  
Emden Equation ◽  
Type Composite

The polytropic stellar model with index n = 0 has a uniform density distribution throughout, and consequently its physical radius is essentially arbitrary because the surface density condition, ϱ = 0, is never satisfied. This surface anomaly, which is not associated with the other polytopic models for 0 < n ≤ 5, could be a constraining factor in certain astrophysical applications involving the n = 0 polytrope. For example, in some circumstances it may be appropriate to utilize the simple physical formulation of the model but on the other hand inappropriate to disregard any zero boundary requirements for the surface density. A sequence of new E-type (as defined below) composite analytical solutions to the Lane-Emden equation, based on the indices 0 and 1, has been developed which eliminates this physical indetermination. The associated polytropic models can be classified as essentially uniform density models. Specifically, they have a large central uniform density n = 0 zone matched, in a physically consistent way, to a small outer n = 1 zone which has a steep density gradient giving ϱ = 0, along with T = 0 and P = 0, at the radial distance corresponding to the first zero of the composite solution.

Physical Characteristics of a Polytrope Index 5 with Finite Radius

1981 ◽  
Vol 4 (2) ◽  
pp. 205-208
Author(s):  
J. O. Murphy
Keyword(s):  
Discrete System ◽  
Physical Structure ◽  
Density Variation ◽  
Uniform Density ◽  
Finite Radius ◽  
Polytropic Model ◽  
Emden Equation ◽  
Infinite Extent ◽  
Physical Features ◽  
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.

scholarly journals Decayless Kink Oscillations Excited by Random Driving: Motion in Transitional Layer

Solar Physics ◽  
2021 ◽  
Vol 296 (8) ◽  
Author(s):  
M. S. Ruderman ◽  
N. S. Petrukhin ◽  
E. Pelinovsky
Keyword(s):  
Coronal Loop ◽  
Radial Displacement ◽  
Oscillation Amplitude ◽  
Radial Distance ◽  
Magnetic Pressure ◽  
Pressure Perturbation ◽  
Transitional Layer ◽  
Thin Boundary Layer ◽  
Plasma Displacement ◽  
Thin Tube

AbstractIn this article we study the plasma motion in the transitional layer of a coronal loop randomly driven at one of its footpoints in the thin-tube and thin-boundary-layer (TTTB) approximation. We introduce the average of the square of a random function with respect to time. This average can be considered as the square of the oscillation amplitude of this quantity. Then we calculate the oscillation amplitudes of the radial and azimuthal plasma displacement as well as the perturbation of the magnetic pressure. We find that the amplitudes of the plasma radial displacement and the magnetic-pressure perturbation do not change across the transitional layer. The amplitude of the plasma radial displacement is of the same order as the driver amplitude. The amplitude of the magnetic-pressure perturbation is of the order of the driver amplitude times the ratio of the loop radius to the loop length squared. The amplitude of the plasma azimuthal displacement is of the order of the driver amplitude times $\text{Re}^{1/6}$ Re 1 / 6 , where Re is the Reynolds number. It has a peak at the position in the transitional layer where the local Alfvén frequency coincides with the fundamental frequency of the loop kink oscillation. The ratio of the amplitude near this position and far from it is of the order of $\ell$ ℓ , where $\ell$ ℓ is the ratio of thickness of the transitional layer to the loop radius. We calculate the dependence of the plasma azimuthal displacement on the radial distance in the transitional layer in a particular case where the density profile in this layer is linear.

scholarly journals Collapse, Equilibrium, and Fragmentation of Rotating, Adiabatic Clouds

1980 ◽  
Vol 58 ◽  
pp. 255-259
Author(s):  
A. P. Boss
Keyword(s):  
Thermal Energy ◽  
Binary Systems ◽  
Initial Conditions ◽  
Density Variation ◽  
Critical Values ◽  
Numerical Calculations ◽  
Uniform Density ◽  
Dynamic Fragmentation

AbstractNumerical calculations of the collapse of adiabatic clouds from uniform density and rotation initial conditions show that when restricted to axisymmetry, the clouds form either near-equilibrium spheroids or rings. Rings form in the collapse of low thermal energy clouds and have β = T/|w|≳ 0.43. When the axisymmetric constraint is removed and an initial m=2 density variation is introduced, clouds either collapse to form near-equilibrium ellipsoids or else fragment into binary systems through a bar phase. Ellipsoids form in the collapse of high thermal energy clouds and have β ≲ 0.27. The results are consistent with the critical values of β for instabilities in Maclaurin spheroids, and suggest that protostellar clouds may undergo a dynamic fragmentation in the nonisothermal collapse regime.

scholarly journals Outflows from inflows: the nature of Bondi-like accretion

2019 ◽  
Vol 491 (1) ◽  
pp. L76-L80 ◽  
Author(s):  
Tim Waters ◽  
Aycin Aykutalp ◽  
Daniel Proga ◽  
Jarrett Johnson ◽  
Hui Li ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Black Hole ◽  
Angular Momentum ◽  
Rotational Velocity ◽  
Cosmic Time ◽  
Meridional Flow ◽  
Outer Radius ◽  
Uniform Density ◽  
Constant Density ◽  
Starting Point

ABSTRACT The classic Bondi solution remains a common starting point both for studying black hole growth across cosmic time in cosmological simulations and for smaller scale simulations of active galactic nuclei (AGN) feedback. In nature, however, there will be inhomogeneous distributions of rotational velocity and density along the outer radius (Ro) marking the sphere of influence of a black hole. While there have been many studies of how the Bondi solution changes with a prescribed angular momentum boundary condition, they have all assumed a constant density at Ro. In this Letter, we show that a non-uniform density at Ro causes a meridional flow and due to conservation of angular momentum, the Bondi solution qualitatively changes into an inflow–outflow solution. Using physical arguments, we analytically identify the critical logarithmic density gradient |$\partial \ln \rho/\partial \theta$| above which this change of the solution occurs. For realistic Ro, this critical gradient is less than 0.01 and tends to 0 as Ro → ∞. We show using numerical simulations that, unlike for solutions with an imposed rotational velocity, the accretion rate for solutions under an inhomogeneous density boundary condition remains constant at nearly the Bondi rate $\dot{M}_\mathrm{ B}$, while the outflow rate can greatly exceed $\dot{M}_\mathrm{ B}$.

Transient Stresses and Displacement Around a Wellbore Due to Fluid Flow in Transversely Isotropic, Porous Media: II. Finite Reservoirs

1968 ◽  
Vol 8 (01) ◽  
pp. 79-86 ◽  
Author(s):  
M.S. Seth ◽  
K.E. Gray
Keyword(s):  
Porous Media ◽  
Radial Displacement ◽  
Outer Boundary ◽  
Stress Gradient ◽  
Fluid Pressure ◽  
Radial Distance ◽  
Transversely Isotropic ◽  
Vertical Stress ◽  
State Flow ◽  
Constant Rate

Abstract In Part 1 of this work,1 equations of elasticity were formulated for transversely isotropic, axisymmetric, homogeneous, porous media exhibiting pore fluid pressure. Equations of elasticity and the thermal analogy method were used to determine transient horizontal, tangential, and vertical stresses and radial displacement in a semi-infinite cylindrical region when either a constant rate of pressure or a constant rate of flow is maintained at the wellbore. In this paper, the approach presented earlier is extended to finite reservoirs for the cases ofsteady-state flow,constant pressures at the well bore and outer boundary andconstant pressure at the wellbore and no flow at the outer boundary. Results of this work show that radial and tangential stress gradients are high near the wellbore but diminish rapidly away from the well; the vertical stress gradient behaves in the same way but is less severe. Radial stresses are compressive or neutral, whereas tangential and vertical stresses may be tensile, neutral or compressive, depending upon the boundary conditions, the physical properties of the system and the radial distance involved (vertical stresses are always compressive in an unbounded system1). For constant boundary pressures, both radial and tangential stresses increase with time whereas they both decrease for a closed outer boundary and constant pressure at the wellbore. The vertical stress decreases with time for both systems. For steady-state systems, radial displacement may be positive or negative, depending upon the dimensions of the system, the pressure differential and the porosity. Radial displacement may be positive or negative for a closed outer boundary but is positive for constant pressures at both boundaries. INTRODUCTION The importance, utility and complexity of a realistic appraisal of the stress state at and local to a wellbore were indicated in Part 1. In this paper the analytical approach presented earlier is extended to finite, cylindrical reservoir geometry for the cases ofsteady-state flow,constant pressures at wellbore and outer boundary andconstant pressure at the wellbore and no flow at the outer boundary. Other than the outer boundary of the reservoir being finite, the physical system and assumptions pertinent thereto are the same as before. The reader may wish to review the mathematical development through Eq. 49 of Part 1 before proceeding here.

Equilibrium theory of the nuclear symmetry energy of infinite nuclear matter

1969 ◽  
Vol 47 (20) ◽  
pp. 2171-2209 ◽  
Author(s):  
Richard A. Weiss ◽  
A. G. W. Cameron
Keyword(s):  
Kinetic Energy ◽  
Nuclear Matter ◽  
Symmetry Energy ◽  
Density Variation ◽  
Average Kinetic Energy ◽  
Constant Density ◽  
Nuclear Symmetry Energy ◽  
Neutron Excess ◽  
Order Coefficient ◽  
Excess Parameter

A set of generalized nuclear matter curves is calculated as a function of density and ξ = 1−(2Z/A), which maps out the energy versus density plane for 0 ≤ ξ ≤ 1 and determines the nuclear matter equilibrium curve (NMEC) as the locus of their saturation points. The NMEC immediately determines the equilibrium energy and density as a function of the neutron excess, and thereby automatically gives the nuclear symmetry energy. The component parts of the equilibrium energy are also determined, and we find that the average kinetic energy per nucleon is a decreasing function of the neutron excess parameter, so that the contribution of the kinetic energy to the second order coefficient, β2∞, is negative. By noting that the density variation along the NMEC is determined by kFE = k∞(1−F2ξ2 + F4ξ4−… ) f−1 with k∞ = 1.4 f−1, F2 ~ 0.45, and F4 ~ 0.07, we find a general connection between the equilibrium and nonequilibrium symmetry energy coefficients, i.e. β0∞ = β0NE(k∞), β2∞ = β2NE(k∞), β4∞ = β4NE(k∞)[Formula: see text], etc., where K0(2) is the standard nuclear compressibility. We find a large negative value for the fourth order coefficient, β4∞ ~ −25 MeV, and a large positive value for the sixth order coefficient, β6∞ ~ 15 MeV, while the corresponding nonequilibrium values of these two coefficients are small and positive. Nuclear matter systems with neutron excess are found to be more bound than is predicted by constant density calculations, and we find that a negative isospin compression energy term is required to be added to the previous constant density calculations.

Composite Analytical Solutions of the Lane-Emden Equation with Polytropic Indicies n = 1 and n = 5

1983 ◽  
Vol 5 (2) ◽  
pp. 175-179 ◽  
Author(s):  
J. O. Murphy
Keyword(s):  
Analytical Solutions ◽  
Total Pressure ◽  
Radial Distance ◽  
Emden Equation ◽  
Symmetrical Distribution ◽  
Physical Variables ◽  
Gravitational Equilibrium

For a spherically symmetrical distribution of gaseous matter in gravitational equilibrium the total pressure, density and other physical variables are all functions of the radial distance measured from the centre.

On the chaotic behaviour induced by surface phenomena in some general relativistic stellar models

1990 ◽  
Vol 146 (3) ◽  
pp. 105-109
Author(s):  
A. Di Prisco ◽  
L. Herrera ◽  
J. Carot
Keyword(s):  
Chaotic Behaviour ◽  
Surface Phenomena ◽  
General Relativistic ◽  
Stellar Models

Physical Structure of a Sequence of Two Zone Polytropic Stellar Models

1985 ◽  
Vol 6 (2) ◽  
pp. 219-222 ◽  
Author(s):  
J. O. Murphy ◽  
R. Fiedler
Keyword(s):  
Analytical Solutions ◽  
Outer Zone ◽  
Physical Structure ◽  
Emden Equation ◽  
Stabilizing Effect ◽  
Stellar Models

AbstractExpressions for the physical structure of a sequence of two zone polytropic stellar models, based on composite analytical solutions of the Lane-Emden equation with indices n = 1 and n = 5, hâve been determined. The coefficients of vibrational stability for radial oscillations of this sequence of models have also been calculated and it is found that increasing the extent of the n = 5 outer zone has a stabilizing effect.

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