The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. These stars resist gravitational collapse primarily through electron degeneracy pressure, compared to main sequence stars, which resist collapse through thermal pressure. The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction.Sean Carroll, Ph.D., Caltech, 2007, The Teaching Company, Dark Matter, Dark Energy: The Dark Side of the Universe, Guidebook Part 2 page 44, Accessed Oct. 7, 2013, "...Chandrasekhar limit: The maximum mass of a white dwarf star, about 1.4 times the mass of the Sun. Above this mass, the gravitational pull becomes too much, and the star collapses to a neutron star or black hole ..." The value of the Chandrasekhar limit depends upon the ratio of the number of electrons to nucleons (neutrons plus protons) in the star. For small stars this ratio is around 1/2 and the limit is about . The limit was named after Subrahmanyan Chandrasekhar who won the 1983 Nobel prize together with William Alfred Fowler for work on stellar models.
Physics
Normal stars fuse gravitationally compressed hydrogen into helium, generating vast amounts of heat. As the hydrogen is consumed, the stars' core compresses further allowing the helium and heavier nuclei to fuse ultimately resulting in stable iron nuclei, a process called stellar evolution. The next step depends upon the mass of the star. Stars below the Chandrasekhar limit become stable white dwarf stars, remaining that way throughout the rest of the history of the universe (assuming the absence of external forces). Stars above the limit can become neutron stars or black holes.
The Chandrasekhar limit is a consequence of competition between gravity and electron degeneracy pressure. Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels. Compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons increases on compression, so pressure must be exerted on the electron gas to compress it, producing electron degeneracy pressure. With sufficient compression, electrons are forced into nuclei in the process of electron capture, relieving the pressure.
In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form , where is the pressure, is the mass density, and is a constant. Solving the hydrostatic equation leads to a model white dwarf that is a polytrope of index – and therefore has radius inversely proportional to the cube root of its mass, and volume inversely proportional to its mass.
As the mass of a model white dwarf increases, the typical energies to which degeneracy pressure forces the electrons are no longer negligible relative to their rest masses. The speeds of the electrons approach the speed of light, and special relativity must be taken into account. In the strongly relativistic limit, the equation of state takes the form . This yields a polytrope of index 3, which has a total mass, , depending only on .
For a fully relativistic treatment, the equation of state used interpolates between the equations for small and for large . When this is done, the model radius still decreases with mass, but becomes zero at . This is the Chandrasekhar limit. The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively. has been set equal to 2. Radius is measured in standard solar radiiStandards for Astronomical Catalogues, Version 2.0 , section 3.2.2, web page, accessed 12-I-2007. or kilometers, and mass in standard solar masses.
Calculated values for the limit vary depending on the nuclear composition of the mass. ChandrasekharOn Stars, Their Evolution and Their Stability , Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983. gives the following expression, based on the equation of state for an ideal Fermi gas:
where: is the reduced Planck constant is the speed of light is the gravitational constant is the average molecular weight per electron, which depends upon the chemical composition of the star is the mass of the hydrogen atom is a constant connected with the solution to the Lane–Emden equation As is the Planck mass, the limit is of the order of
The limiting mass can be obtained formally from the Chandrasekhar's white dwarf equation by taking the limit of large central density.
A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature. Lieb and Yau have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.
History In 1926, the British physicist Ralph H. Fowler observed that the relationship between the density, energy, and temperature of white dwarfs could be explained by viewing them as a gas of nonrelativistic, non-interacting electrons and nuclei that obey Fermi–Dirac statistics. This Fermi gas model was then used by the British physicist Edmund Clifton Stoner in 1929 to calculate the relationship among the mass, radius, and density of white dwarfs, assuming they were homogeneous spheres. Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately . In 1930, Stoner derived the internal energy–density equation of state for a Fermi gas, and was then able to treat the mass–radius relationship in a fully relativistic manner, giving a limiting mass of approximately (for ). Stoner went on to derive the pressure–density equation of state, which he published in 1932. These equations of state were also previously published by the Soviet physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter.. Frenkel's work, however, was ignored by the astronomical and astrophysical community.
A series of papers published between 1931 and 1935 had its beginning on a trip from India to England in 1930, where the Indian physicist Subrahmanyan Chandrasekhar worked on the calculation of the statistics of a degenerate Fermi gas.Chandrasekhar's biographical memoir at the National Academy of Sciences , web page, accessed 12-01-2007. In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state, and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above. Chandrasekhar reviews this work in his Nobel Prize lecture.
The existence of a related limit, based on the conceptual breakthrough of combining relativity with Fermi degeneracy, was first established in separate papers published by Wilhelm Anderson and E. C. Stoner for a uniform density star in 1929. Eric G. Blackman wrote that the roles of Stoner and Anderson in the discovery of mass limits were overlooked when Freeman Dyson wrote a biography of Chandrasekhar.Eric G. Blackman, "Giants of physics found white-dwarf mass limits", Nature 440, 148 (2006) Michael Nauenberg claims that Stoner established the mass limit first.Michael Nauenberg, "Edmund C. Stoner and the Discovery of the Maximum Mass of White Dwarfs", Journal for the History of Astronomy, Vol. 39, p. 297–312, (2008) The priority dispute has also been discussed at length by Virginia Trimble who writes that: "Chandrasekhar famously, perhaps even notoriously did his critical calculation on board ship in 1930, and ... was not aware of either Stoner's or Anderson's work at the time. His work was therefore independent, but, more to the point, he adopted Eddington's polytropes for his models which could, therefore, be in hydrostatic equilibrium, which constant density stars cannot, and real ones must be."Virginia Trimble, "Chandrasekhar and the history of astronomy", Fluid Flows to Black Holes, pp. 49–50 (2011) This value was also computed in 1932 by the Soviet physicist Lev Landau,On the Theory of Stars, in Collected Papers of L. D. Landau, ed. and with an introduction by D. ter Haar, New York: Gordon and Breach, 1965; originally published in Phys. Z. Sowjet. 1 (1932), 285. who, however, did not apply it to white dwarfs and concluded that quantum laws might be invalid for stars heavier than 1.5 solar mass.
Chandrasekhar–Eddington dispute
In 1935, Chandrasekhar presented his work on the limit at a scientific conference. It was immediately opposed by the established British astrophysicist Arthur Eddington. Eddington was aware that the existence of black holes was theoretically possible, and also realized that the existence of the limit made their formation possible. However, he was unwilling to accept that this could happen. After the talk by Chandrasekhar on the limit, Eddington remarked:
Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law universally applicable, even for large . Although Niels Bohr, Fowler, Wolfgang Pauli, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar.Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes, Arthur I. Miller, Boston, New York: Houghton Mifflin, 2005, ; reviewed at The Guardian: The battle of black holes . Through the rest of his life, Eddington held to his position in his writings,Relativity Theory of Protons and Electrons, Sir Arthur Eddington, Cambridge: Cambridge University Press, 1936, chapter 13. including his work on his fundamental theory.Fundamental Theory, Sir A. S. Eddington, Cambridge: Cambridge University Press, 1946, §43–45. The drama associated with this disagreement is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar. In Miller's view:
However, Chandrasekhar chose to move on, leaving the study of stellar structure to focus on stellar dynamics. In 1983 in recognition for his work, Chandrasekhar shared a Nobel prize "for his theoretical studies of the physical processes of importance to the structure and evolution of the stars" with William Alfred Fowler.
Applications The core of a star is kept from collapsing by the heat generated by the fusion of nuclei of lighter elements into heavier ones. At various stages of stellar evolution, the nuclei required for this process are exhausted, and the core collapses, causing it to become denser and hotter. A critical situation arises when iron accumulates in the core, since iron nuclei are incapable of generating further energy through fusion. If the core becomes sufficiently dense, electron degeneracy pressure will play a significant part in stabilizing it against gravitational collapse.