Structure of polytropic stars in General Relativity

• Published in: Astrophysics and space science Vol. 364; no. 7; pp. 1 - 9
• Main Authors: Humi, Mayer, Roumas, John
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.07.2019; Springer Nature B.V
• Subjects: Astrobiology; Astronomy; Astrophysics; Astrophysics and Astroparticles; Coefficients; Cosmology; Equations of state; Gas density; Gravitation; Interstellar clouds; Interstellar matter; Irreversible processes; Mass distribution; Observations and Techniques; Original Article; Physics; Physics and Astronomy; Relativity; Scientific papers; Space Exploration and Astronautics; Space Sciences (including Extraterrestrial Physics; Stars; Stellar models; Stellar structure; Theory of relativity; Polytropic; General Relativity; Star interiors
• ISSN: 0004-640X; 1572-946X
• DOI: 10.1007/s10509-019-3608-y

The inner structure of a star or primordial interstellar cloud is a topic of major importance in classical and relativistic astrophysics. The impact that General Relativity has on this structure has been the subject of many research papers. In this paper we consider within the context of General Relativity a prototype model for stellar structure in which the pressure and density, but not temperature and density, are related polytropically. To justify this assumption, we note that stars undergo thermodynamically irreversible processes, including the loss of heat to their surroundings. Because of these processes, the temperature may not be controlled by local pressure and gas density. The usual polytropic equation of state relating pressure p and density ρ may now be replaced the generalized equation p = A ( r ) ρ α ( r ) , where the isentropy coefficient A ( r ) and isentropy index α ( r ) are functions of radius r . Solutions for the interior stellar structure are then derived within the framework of Einstein’s equations for General Relativity. A single equation for the cumulative mass distribution of the star is obtained and the Tolman-Oppenheimer-Volkoff equation is used to derive formulae for the isentropic index and coefficient. We present analytic and numerical solutions for the generalized polytropic structure of self-gravitating stars and examine their stability. We also prove that if the isentropic index and isentropic coefficient are known, the corresponding distribution of mass within the star is uniquely determined.