Tolman–Oppenheimer–Volkoff limit

Tolman–Oppenheimer–Volkoff limit is the maximum stable mass of a neutron star, a theoretical boundary derived from general relativity and the equation of state for degenerate matter. It represents the point where gravitational collapse into a black hole becomes inevitable, marking a critical threshold in stellar evolution. The concept was first calculated in a seminal 1939 paper by Richard C. Tolman, J. Robert Oppenheimer, and George Volkoff, building upon earlier work by Subrahmanyan Chandrasekhar on white dwarf stars.

Definition and significance

The Tolman–Oppenheimer–Volkoff limit defines the upper mass bound for a neutron star supported against gravitational collapse by neutron degeneracy pressure. It is a cornerstone of relativistic astrophysics, distinguishing the fate of massive stellar remnants between stable compact stars and singularities. This limit is crucial for understanding the endpoints of stellar evolution for progenitors between the Chandrasekhar limit and approximately 20-30 solar masses. Observations of systems like the Hulse–Taylor binary and GW170817 provide empirical constraints on this theoretical value, informing models of gamma-ray bursts and kilonovae.

Derivation and theoretical basis

The derivation stems from solving the Tolman–Oppenheimer–Volkoff equation, a modification of the hydrostatic equilibrium condition within the framework of general relativity. This equation integrates a prescribed equation of state for ultra-dense matter, a domain probing physics beyond the Standard Model. Key assumptions include spherical symmetry, described by the Schwarzschild metric, and a static, perfect fluid. The calculation requires numerical integration from the star's center to its surface, where pressure vanishes, with the total gravitational mass serving as the output. The inherent uncertainty in the equation of state near nuclear density makes the limit dependent on theoretical models of the strong interaction.

Numerical value and astrophysical implications

The precise numerical value is not a universal constant but ranges approximately from 1.5 to 3.0 solar masses, contingent on the uncertain equation of state of neutron-rich matter. Modern constraints from observatories like LIGO and NICER suggest a likely value near 2.2 solar masses. A measured mass exceeding this limit, such as the PSR J0740+6620 or the PSR J0348+0432, would imply either a stiffer equation of state or the presence of exotic matter like quark–gluon plasma. The limit directly influences predictions for supernova remnants, the mass distribution of black holes, and the gravitational-wave signals from mergers observed by the Virgo interferometer.

Relation to other mass limits

This limit exists in a hierarchy of mass thresholds governing stellar remnants. It is superseded by the Chandrasekhar limit of about 1.4 solar masses for electron-degenerate matter in white dwarfs. For masses above the Tolman–Oppenheimer–Volkoff limit, the Buchdahl's theorem and the concept of the Photon sphere become relevant en route to black hole formation. The theoretical Quark star and Boson star would have their own distinct mass limits, while the ultimate absence of a pressure-supported limit defines the Kerr metric for rotating black holes. These interconnected concepts are central to the Landau–Thorne–Żytkow object hypothesis.

Historical context and development

The limit emerged from the rapid development of relativistic astrophysics following Albert Einstein's formulation of general relativity. Lev Landau first speculated on neutron cores in 1932, shortly after the discovery of the neutron by James Chadwick. The pivotal work by Richard C. Tolman, J. Robert Oppenheimer, and George Volkoff in 1939, published in Physical Review, was directly influenced by Subrahmanyan Chandrasekhar's earlier treatment of white dwarfs. Subsequent refinements came from theorists like John Archibald Wheeler and Yakov Zel'dovich, with observational impetus provided by the discovery of PSR B1919+21 by Jocelyn Bell Burnell and the Crab Nebula pulsar.

Category:Astrophysics Category:Stellar astronomy Category:General relativity