Stability of polytropes
- Physics Department, University of California, Los Angeles, California 90095-1547 (United States)
This paper is an investigation of the stability of some ideal stars. It is intended as a study in general relativity, with emphasis on the coupling to matter, aimed at a better understanding of strong gravitational fields and 'black holes'. This contrasts with the usual attitude in astrophysics, where Einstein's equations are invoked as a refinement of classical thermodynamics and Newtonian gravity. Our work is based on action principles for systems of metric and matter fields, well-defined relativistic field models that we hope may represent plausible types of matter. The thermodynamic content must be extracted from the theory itself. When the flow of matter is irrotational, and described by a scalar density, we are led to differential equations that differ little from those of Tolman, but they admit a conserved current, and stronger boundary conditions that affect the matching of the interior solution to an external metric and imply a relation of mass and radius. We propose a complete revision of the treatment of boundary conditions. An ideal star in our terminology has spherical symmetry and an isentropic equation of state, p=a{rho}{sup {gamma}}, a and {gamma} piecewise constant. In our first work it was assumed that the density vanished beyond a finite distance from the origin and that the metric is to be matched at the boundary to an exterior Schwartzchild metric. But it is difficult to decide what the boundary conditions should be and we are consequently skeptical of the concept of a fixed boundary. We investigate the double polytrope, characterized by a polytropic index n{<=}3, in the bulk of the star and a value larger than five in an outer atmosphere that extends to infinity. It has no fixed boundary but a region of critical density where the polytropic index changes from a value that is appropriate for the bulk of the star to a value that provides a crude model for the atmosphere. The boundary conditions are now natural and unambiguous. The existence of a relation between mass and radius is confirmed, as well as an upper limit on the mass. The principal conclusion is that all the static configurations are stable. There is a solution that fits the Sun. The masses of white dwarfs respect the Chandrasekhar limit. The application to neutron stars has surprising aspects.
- OSTI ID:
- 21210196
- Journal Information:
- Physical Review. D, Particles Fields, Vol. 77, Issue 10; Other Information: DOI: 10.1103/PhysRevD.77.104019; (c) 2008 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 0556-2821
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
GENERAL PHYSICS
ASTROPHYSICS
BLACK HOLES
BOUNDARY CONDITIONS
DIFFERENTIAL EQUATIONS
EINSTEIN FIELD EQUATIONS
EQUATIONS OF STATE
GENERAL RELATIVITY THEORY
GRAVITATION
GRAVITATIONAL FIELDS
ISENTROPIC PROCESSES
MASS
MATHEMATICAL SOLUTIONS
NEUTRON STARS
RELATIVISTIC RANGE
SPHERICAL CONFIGURATION
SUN
SYMMETRY
THERMODYNAMICS
WHITE DWARF STARS