Gamma-ray burst progenitors

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Eta Carinae, in the constellation of Carina, one of the nearer candidates for a hypernova EtaCarinae.jpg
Eta Carinae, in the constellation of Carina, one of the nearer candidates for a hypernova

Gamma-ray burst progenitors are the types of celestial objects that can emit gamma-ray bursts (GRBs). GRBs show an extraordinary degree of diversity. They can last anywhere from a fraction of a second to many minutes. Bursts could have a single profile or oscillate wildly up and down in intensity, and their spectra are highly variable unlike other objects in space. The near complete lack of observational constraint led to a profusion of theories, including evaporating black holes, magnetic flares on white dwarfs, accretion of matter onto neutron stars, antimatter accretion, supernovae, hypernovae, and rapid extraction of rotational energy from supermassive black holes, among others. [1] [2]

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There are at least two different types of progenitors (sources) of GRBs: one responsible for the long-duration, soft-spectrum bursts and one (or possibly more) responsible for short-duration, hard-spectrum bursts. The progenitors of long GRBs are believed to be massive, low-metallicity stars exploding due to the collapse of their cores. The progenitors of short GRBs are thought to arise from mergers of compact binary systems like neutron stars, which was confirmed by the GW170817 observation of a neutron star merger and a kilonova.

Long GRBs: massive stars

Collapsar model

As of 2007, there is almost universal agreement in the astrophysics community that the long-duration bursts are associated with the deaths of massive stars in a specific kind of supernova-like event commonly referred to as a collapsar or hypernova. [2] [3] Very massive stars are able to fuse material in their centers all the way to iron, at which point a star cannot continue to generate energy by fusion and collapses, in this case, immediately forming a black hole. Matter from the star around the core rains down towards the center and (for rapidly rotating stars) swirls into a high-density accretion disk. The infall of this material into the black hole drives a pair of jets out along the rotational axis, where the matter density is much lower than in the accretion disk, towards the poles of the star at velocities approaching the speed of light, creating a relativistic shock wave [4] at the front. If the star is not surrounded by a thick, diffuse hydrogen envelope, the jets' material can pummel all the way to the stellar surface. The leading shock actually accelerates as the density of the stellar matter it travels through decreases, and by the time it reaches the surface of the star it may be traveling with a Lorentz factor of 100 or higher (that is, a velocity of 0.9999 times the speed of light). Once it reaches the surface, the shock wave breaks out into space, with much of its energy released in the form of gamma-rays.

Three very special conditions are required for a star to evolve all the way to a gamma-ray burst under this theory: the star must be very massive (probably at least 40 Solar masses on the main sequence) to form a central black hole in the first place, the star must be rapidly rotating to develop an accretion torus capable of launching jets, and the star must have low metallicity in order to strip off its hydrogen envelope so the jets can reach the surface. As a result, gamma-ray bursts are far rarer than ordinary core-collapse supernovae, which only require that the star be massive enough to fuse all the way to iron.

Evidence for the collapsar view

This consensus is based largely on two lines of evidence. First, long gamma-ray bursts are found without exception in systems with abundant recent star formation, such as in irregular galaxies and in the arms of spiral galaxies. [5] This is strong evidence of a link to massive stars, which evolve and die within a few hundred million years and are never found in regions where star formation has long ceased. This does not necessarily prove the collapsar model (other models also predict an association with star formation) but does provide significant support.

Second, there are now several observed cases where a supernova has immediately followed a gamma-ray burst. While most GRBs occur too far away for current instruments to have any chance of detecting the relatively faint emission from a supernova at that distance, for lower-redshift systems there are several well-documented cases where a GRB was followed within a few days by the appearance of a supernova. These supernovae that have been successfully classified are type Ib/c, a rare class of supernova caused by core collapse. Type Ib and Ic supernovae lack hydrogen absorption lines, consistent with the theoretical prediction of stars that have lost their hydrogen envelope. The GRBs with the most obvious supernova signatures include GRB 060218 (SN 2006aj), [6] GRB 030329 (SN 2003dh), [7] and GRB 980425 (SN 1998bw), [8] and a handful of more distant GRBs show supernova "bumps" in their afterglow light curves at late times.

Possible challenges to this theory emerged recently, with the discovery [9] [10] of two nearby long gamma-ray bursts that lacked the signature of any type of supernova: both GRB060614 and GRB 060505 defied predictions that a supernova would emerge despite intense scrutiny from ground-based telescopes. Both events were, however, associated with actively star-forming stellar populations. One possible explanation is that during the core collapse of a very massive star a black hole can form, which then 'swallows' the entire star before the supernova blast can reach the surface.[ citation needed ]

Short GRBs: degenerate binary systems

Short gamma-ray bursts appear to be an exception. Until 2007, only a handful of these events have been localized to a definite galactic host. However, those that have been localized appear to show significant differences from the long-burst population. While at least one short burst has been found in the star-forming central region of a galaxy, several others have been associated with the outer regions and even the outer halo of large elliptical galaxies in which star formation has nearly ceased. All the hosts identified so far have also been at low redshift. [11] Furthermore, despite the relatively nearby distances and detailed follow-up study for these events, no supernova has been associated with any short GRB.

Neutron star and neutron star/black hole mergers

While the astrophysical community has yet to settle on a single, universally favored model for the progenitors of short GRBs, the generally preferred model is the merger of two compact objects as a result of gravitational inspiral: two neutron stars, [12] [13] or a neutron star and a black hole. [14] While thought to be rare in the Universe, a small number of cases of close neutron star - neutron star binaries are known in our Galaxy, and neutron star - black hole binaries are believed to exist as well. According to Einstein's theory of general relativity, systems of this nature will slowly lose energy due to gravitational radiation and the two degenerate objects will spiral closer and closer together, until in the last few moments, tidal forces rip the neutron star (or stars) apart and an immense amount of energy is liberated before the matter plunges into a single black hole. The whole process is believed to occur extremely quickly and be completely over within a few seconds, accounting for the short nature of these bursts. Unlike long-duration bursts, there is no conventional star to explode and therefore no supernova.

This model has been well-supported so far by the distribution of short GRB host galaxies, which have been observed in old galaxies with no star formation (for example, GRB050509B, the first short burst to be localized to a probable host) as well as in galaxies with star formation still occurring (such as GRB050709, the second), as even younger-looking galaxies can have significant populations of old stars. However, the picture is clouded somewhat by the observation of X-ray flaring [15] in short GRBs out to very late times (up to many days), long after the merger should have been completed, and the failure to find nearby hosts of any sort for some short GRBs.

Magnetar giant flares

One final possible model that may describe a small subset of short GRBs are the so-called magnetar giant flares (also called megaflares or hyperflares). Early high-energy satellites discovered a small population of objects in the Galactic plane that frequently produced repeated bursts of soft gamma-rays and hard X-rays. Because these sources repeat and because the explosions have very soft (generally thermal) high-energy spectra, they were quickly realized to be a separate class of object from normal gamma-ray bursts and excluded from subsequent GRB studies. However, on rare occasions these objects, now believed to be extremely magnetized neutron stars and sometimes termed magnetars, are capable of producing extremely luminous outbursts. The most powerful such event observed to date, the giant flare of 27 December 2004, originated from the magnetar SGR 1806-20 and was bright enough to saturate the detectors of every gamma-ray satellite in orbit and significantly disrupted Earth's ionosphere. [16] While still significantly less luminous than "normal" gamma-ray bursts (short or long), such an event would be detectable to current spacecraft from galaxies as far as the Virgo cluster and, at this distance, would be difficult to distinguish from other types of short gamma-ray burst on the basis of the light curve alone. To date, three gamma-ray bursts have been associated with SGR flares in galaxies beyond the Milky Way: GRB 790305b in the Large Magellanic Cloud, GRB 051103 from M81 and GRB 070201 from M31. [17]

Diversity in the origin of long GRBs

HETE II and Swift observations reveal that long gamma-ray bursts come with and without supernovae, and with and without pronounced X-ray afterglows. It gives a clue to a diversity in the origin of long GRBs, possibly in- and outside of star-forming regions, with otherwise a common inner engine. The timescale of tens of seconds of long GRBs hereby appears to be intrinsic to their inner engine, for example, associated with a viscous or a dissipative process.

The most powerful stellar mass transient sources are the above-mentioned progenitors (collapsars and mergers of compact objects), all producing rotating black holes surrounded by debris in the form of an accretion disk or torus. A rotating black hole carries spin-energy in angular momentum [18] as does a spinning top:

where and denote the moment of inertia and the angular velocity of the black hole in the trigonometric expression [19] for the specific angular momentum of a Kerr black hole of mass . With no small parameter present, it has been well-recognized that the spin energy of a Kerr black hole can reach a substantial fraction (29%) of its total mass-energy , thus holding promise to power the most remarkable transient sources in the sky. Of particular interest are mechanisms for producing non-thermal radiation by the gravitational field of rotating black holes, in the process of spin-down against their surroundings in aforementioned scenarios.

By Mach's principle, spacetime is dragged along with mass-energy, with the distant stars on cosmological scales or with a black hole in close proximity. Thus, matter tends to spin-up around rotating black holes, for the same reason that pulsars spin down by shedding angular momentum in radiation to infinity. A major amount of spin-energy of rapidly spinning black holes can hereby be released in a process of viscous spin-down against an inner disk or torus—into various emission channels.

Spin-down of rapidly spinning stellar mass black holes in their lowest energy state takes tens of seconds against an inner disk, representing the remnant debris of the merger of two neutron stars, the break-up of a neutron star around a companion black hole or formed in core-collapse of a massive star. Forced turbulence in the inner disk stimulates the creation of magnetic fields and multipole mass-moments, thereby opening radiation channels in radio, neutrinos and, mostly, in gravitational waves with distinctive chirps shown in the diagram [20] with the creation of astronomical amounts of Bekenstein-Hawking entropy. [21] [22] [23]

Diagram of van Putten (2009) showing the gravitational radiation produced in binary coalescence of neutron stars with another neutron star or black hole and, post-coalescence or following core-collapse of a massive star, the expected radiation by high-density turbulent matter around stellar mass Kerr black holes. As the ISCO (ellipse) relaxes to that around a slowly rotating, nearly Schwarzschild black hole, the late-time frequency of gravitational radiation provides accurate metrology of the black hole mass. Chirp diagram vanputten.jpg
Diagram of van Putten (2009) showing the gravitational radiation produced in binary coalescence of neutron stars with another neutron star or black hole and, post-coalescence or following core-collapse of a massive star, the expected radiation by high-density turbulent matter around stellar mass Kerr black holes. As the ISCO (ellipse) relaxes to that around a slowly rotating, nearly Schwarzschild black hole, the late-time frequency of gravitational radiation provides accurate metrology of the black hole mass.

Transparency of matter to gravitational waves offers a new probe to the inner-most workings of supernovae and GRBs. The gravitational-wave observatories LIGO and Virgo are designed to probe stellar mass transients in a frequency range of tens to about fifteen hundred Hz. The above-mentioned gravitational-wave emissions fall well within the LIGO-Virgo bandwidth of sensitivity; for long GRBs powered by "naked inner engines" produced in the binary merger of a neutron star with another neutron star or companion black hole, the above-mentioned magnetic disk winds dissipate into long-duration radio-bursts, that may be observed by the novel Low Frequency Array (LOFAR).

See also

Related Research Articles

Neutron star Collapsed core of a massive star

A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes, and some hypothetical objects, neutron stars are the smallest and densest currently known class of stellar objects. Neutron stars have a radius on the order of 10 kilometres (6.2 mi) and a mass of about 1.4 solar masses. They result from the supernova explosion of a massive star, combined with gravitational collapse, that compresses the core past white dwarf star density to that of atomic nuclei.

Supernova Star exploding at the end of its stellar evolution

A supernova is a powerful and luminous stellar explosion. This transient astronomical event occurs during the last evolutionary stages of a massive star or when a white dwarf is triggered into runaway nuclear fusion. The original object, called the progenitor, either collapses to a neutron star or black hole, or is completely destroyed. The peak optical luminosity of a supernova can be comparable to that of an entire galaxy before fading over several weeks or months.

Gamma-ray burst Flashes of gamma rays from distant galaxies

In gamma-ray astronomy, gamma-ray bursts (GRBs) are immensely energetic explosions that have been observed in distant galaxies. They are the brightest and most energetic electromagnetic events known to occur in the universe. Bursts can last from ten milliseconds to several hours. After an initial flash of gamma rays, a longer-lived "afterglow" is usually emitted at longer wavelengths.

Magnetar Type of neutron star

A magnetar is a type of neutron star believed to have an extremely powerful magnetic field (∼109 to 1011 T, ∼1013 to 1015 G). The magnetic-field decay powers the emission of high-energy electromagnetic radiation, particularly X-rays and gamma rays. The theory regarding these objects was proposed in 1992 by Robert Duncan and Christopher Thompson. The theory was subsequently developed by Bohdan Paczyński and by its proposers. This theory explained a burst of gamma rays from the Large Magellanic Cloud that had been detected on March 5, 1979, and other less bright bursts from within our galaxy. During the following decade, the magnetar hypothesis became widely accepted as a likely explanation for soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs). In 2020, a fast radio burst (FRB) was detected from a magnetar.

Superluminous supernova Supernova at least ten times more luminous than a standard supernova

A super-luminous supernova is a type of stellar explosion with a luminosity 10 or more times higher than that of standard supernovae. Like supernovae, SLSNe seem to be produced by several mechanisms, which is readily revealed by their light-curves and spectra. There are multiple models for what conditions may produce an SLSN, including core collapse in particularly massive stars, millisecond magnetars, interaction with circumstellar material, or pair-instability supernovae.

Fermi Gamma-ray Space Telescope Space telescope for gamma-ray astronomy launched in 2008

The Fermi Gamma-ray Space Telescope (FGST), formerly called the Gamma-ray Large Area Space Telescope (GLAST), is a space observatory being used to perform gamma-ray astronomy observations from low Earth orbit. Its main instrument is the Large Area Telescope (LAT), with which astronomers mostly intend to perform an all-sky survey studying astrophysical and cosmological phenomena such as active galactic nuclei, pulsars, other high-energy sources and dark matter. Another instrument aboard Fermi, the Gamma-ray Burst Monitor, is being used to study gamma-ray bursts and solar flares.

In astroparticle physics, an ultra-high-energy cosmic ray (UHECR) is a cosmic ray with an energy greater than 1 EeV (1018 electronvolts, approximately 0.16 joules), far beyond both the rest mass and energies typical of other cosmic ray particles.

A stellar black hole is a black hole formed by the gravitational collapse of a star. They have masses ranging from about 5 to several tens of solar masses. The process is observed as a hypernova explosion or as a gamma ray burst. These black holes are also referred to as collapsars.

An astrophysical jet is an astronomical phenomenon where outflows of ionised matter are emitted as an extended beam along the axis of rotation. When this greatly accelerated matter in the beam approaches the speed of light, astrophysical jets become relativistic jets as they show effects from special relativity.

Type Ia supernova Type of supernova in binary systems

A type Ia supernova is a type of supernova that occurs in binary systems in which one of the stars is a white dwarf. The other star can be anything from a giant star to an even smaller white dwarf.

Type II supernova Explosion of a star 8 to 45 times the mass of the Sun

A Type II supernova results from the rapid collapse and violent explosion of a massive star. A star must have at least 8 times, but no more than 40 to 50 times, the mass of the Sun (M) to undergo this type of explosion. Type II supernovae are distinguished from other types of supernovae by the presence of hydrogen in their spectra. They are usually observed in the spiral arms of galaxies and in H II regions, but not in elliptical galaxies; those are generally composed of older low-mass stars, with few of the young highly massive stars necessary to cause a supernova.

Pair-instability supernova Type of high-energy supernova in very large stars

A pair-instability supernova is a type of supernova predicted to occur when pair production, the production of free electrons and positrons in the collision between atomic nuclei and energetic gamma rays, temporarily reduces the internal radiation pressure supporting a supermassive star's core against gravitational collapse. This pressure drop leads to a partial collapse, which in turn causes greatly accelerated burning in a runaway thermonuclear explosion, resulting in the star being blown completely apart without leaving a stellar remnant behind.

WR 104 Triple star system

WR 104 is a triple star system located about 2,580 parsecs (8,400 ly) from Earth. The primary star is a Wolf–Rayet star, abbreviated as WR, with a B0.5 main sequence star in close orbit and another more distant fainter companion.

Neutron star merger Type of stellar collision

A neutron star merger is a type of stellar collision. It occurs in a fashion similar to the rare brand of type Ia supernovae resulting from merging white dwarfs.

Kilonova Supernova formed from a neutron star merger

A kilonova is a transient astronomical event that occurs in a compact binary system when two neutron stars or a neutron star and a black hole merge into each other. Kilonovae are thought to emit short gamma-ray bursts and strong electromagnetic radiation due to the radioactive decay of heavy r-process nuclei that are produced and ejected fairly isotropically during the merger process.

Multi-messenger astronomy is astronomy based on the coordinated observation and interpretation of disparate "messenger" signals. Interplanetary probes can visit objects within the Solar System, but beyond that, information must rely on "extrasolar messengers". The four extrasolar messengers are electromagnetic radiation, gravitational waves, neutrinos, and cosmic rays. They are created by different astrophysical processes, and thus reveal different information about their sources.

Hypernova Supernova that ejects a large mass at unusually high velocity

A hypernova is a very energetic supernova thought to result from an extreme core-collapse scenario. In this case, a massive star collapses to form a rotating black hole emitting twin energetic jets and surrounded by an accretion disk. It is a type of stellar explosion that ejects material with an unusually high kinetic energy, an order of magnitude higher than most supernovae, with a luminosity at least 10 times greater. They usually appear similar to a type Ic supernova, but with unusually broad spectral lines indicating an extremely high expansion velocity. Hypernovae are one of the mechanisms for producing long gamma ray bursts (GRBs), which range from 2 seconds to over a minute in duration. They have also been referred to as superluminous supernovae, though that classification also includes other types of extremely luminous stellar explosions that have different origins.

GW170817

GW 170817 was a gravitational wave (GW) signal observed by the LIGO and Virgo detectors on 17 August 2017, originating from the shell elliptical galaxy NGC 4993. The GW was produced by the last minutes of two neutron stars spiralling closer to each other and finally merging, and is the first GW observation which has been confirmed by non-gravitational means. Unlike the five previous GW detections, which were of merging black holes not expected to produce a detectable electromagnetic signal, the aftermath of this merger was also seen by 70 observatories on 7 continents and in space, across the electromagnetic spectrum, marking a significant breakthrough for multi-messenger astronomy. The discovery and subsequent observations of GW 170817 were given the Breakthrough of the Year award for 2017 by the journal Science.

NGC 4993 Lenticular galaxy in the constellation of Hydra

NGC 4993 is a lenticular galaxy located about 140 million light-years away in the constellation Hydra. It was discovered on 26 March 1789 by William Herschel and is a member of the NGC 4993 Group.

Gamma-ray burst precursor Event that occurs before gamma-ray bursts

A gamma-ray precursor is a short X-ray outburst event that comes before the main outburst of the gamma-ray burst progenitor. There is no consensus on the mechanism for this event, although several theories have been suggested.

References

  1. Ruderman, M. (1975). "Theories of gamma-ray bursts". Texas Symposium on Relativistic Astrophysics . 262 (1 Seventh Texas): 164–180. Bibcode:1975NYASA.262..164R. doi:10.1111/j.1749-6632.1975.tb31430.x. S2CID   83006091.
  2. 1 2 "Gamma-ray burst supports hypernova hypothesis". cerncourier.com. September 4, 2003. Retrieved 2007-10-14.
  3. MacFadyen, A. I.; Woosley, S. E.; Heger, A. (2001). "Supernovae, Jets, and Collapsars". Astrophysical Journal. 550 (1): 410–425. arXiv: astro-ph/9910034 . Bibcode:2001ApJ...550..410M. doi:10.1086/319698. S2CID   1673646.
  4. Blandford, R.D. & McKee, C. F. (1976). "Fluid Dynamics of relativistic blast waves". Physics of Fluids. 19 (8): 1130–1138. Bibcode:1976PhFl...19.1130B. doi:10.1063/1.861619.
  5. Bloom, J.S.; Kulkarni, S. R. & Djorgovski, S. G. (2002). "The Observed Offset Distribution of Gamma-Ray Bursts from Their Host Galaxies: A Robust Clue to the Nature of the Progenitors". Astronomical Journal. 123 (3): 1111–1148. arXiv: astro-ph/0010176 . Bibcode:2002AJ....123.1111B. doi:10.1086/338893. S2CID   6939747.
  6. Sollerman, J.; et al. (2006). "Supernova 2006aj and the associated X-Ray Flash 060218". Astronomy and Astrophysics. 454 (2): 503–509. arXiv: astro-ph/0603495 . Bibcode:2006A&A...454..503S. doi:10.1051/0004-6361:20065226. S2CID   16069010.
  7. Mazzali, P.; et al. (2003). "The Type Ic Hypernova SN 2003dh/GRB 030329". Astrophysical Journal. 599 (2): L95–L98. arXiv: astro-ph/0309555 . Bibcode:2003ApJ...599L..95M. doi:10.1086/381259. S2CID   558757.
  8. Kulkarni, S.R.; et al. (1998). "Radio emission from the unusual supernova 1998bw and its association with the gamma-ray burst of 25 April 1998". Nature. 395 (6703): 663–669. Bibcode:1998Natur.395..663K. doi:10.1038/27139. S2CID   4429303.
  9. Fynbo; et al. (2006). "A new type of massive stellar death: no supernovae from two nearby long gamma-ray bursts". Nature. 444 (7122): 1047–9. arXiv: astro-ph/0608313 . Bibcode:2006Natur.444.1047F. doi:10.1038/nature05375. PMID   17183316. S2CID   4367010.
  10. "New type of cosmic explosion found". astronomy.com. December 20, 2006. Retrieved 2007-09-15.
  11. Prochaska; et al. (2006). "The Galaxy Hosts and Large-Scale Environments of Short-Hard Gamma-Ray Bursts". Astrophysical Journal. 641 (2): 989–994. arXiv: astro-ph/0510022 . Bibcode:2006ApJ...642..989P. doi:10.1086/501160. S2CID   54915144.
  12. Blinnikov, S.; et al. (1984). "Exploding Neutron Stars in Close Binaries". Soviet Astronomy Letters. 10: 177. arXiv: 1808.05287 . Bibcode:1984SvAL...10..177B.
  13. Eichler, David; Livio, Mario; Piran, Tsvi; Schramm, David N. (1989). "Nucleosynthesis, neutrino bursts and gamma-rays from coalescing neutron stars". Nature. 340 (6229): 126. Bibcode:1989Natur.340..126E. doi:10.1038/340126a0. S2CID   4357406.
  14. Lattimer, J. M. & Schramm, D. N. (1976). "The tidal disruption of neutron stars by black holes in close binaries". Astrophysical Journal. 210: 549. Bibcode:1976ApJ...210..549L. doi:10.1086/154860. hdl: 2152/35059 .
  15. Burrows, D. N.; et al. (2005). "Bright X-ray Flares in Gamma-Ray Burst Afterglows". Science. 309 (5742): 1833–1835. arXiv: astro-ph/0506130 . Bibcode:2005Sci...309.1833B. doi:10.1126/science.1116168. PMID   16109845. S2CID   19757528.
  16. Hurley et al., 2005. Nature v.434 p.1098, "An exceptionally bright flare from SGR 1806-20 and the origins of short-duration gamma-ray bursts"
  17. Frederiks 2008
  18. Kerr, R.P. (1963). "Gravitational field of a spinning mass: as an example of algebraically special metrics". Phys. Rev. Lett. 11 (5): 237. Bibcode:1963PhRvL..11..237K. doi:10.1103/PhysRevLett.11.237.
  19. van Putten, M.H.P.M., 1999, Science, 284, 115
  20. Maurice H.P.M. van Putten (2009). "On the origin of long gamma-ray bursts". MNRAS Letters . 396 (1): L81–L84. Bibcode:2009MNRAS.396L..81V. doi: 10.1111/j.1745-3933.2009.00666.x .
  21. Bekenstein, J.D. (1973). "Black holes and entropy". Physical Review D . 7 (8): 2333. Bibcode:1973PhRvD...7.2333B. doi:10.1103/PhysRevD.7.2333.
  22. Hawking, S.W. (1973). "Black holes and entropy". Nature. 248 (5443): 30. Bibcode:1974Natur.248...30H. doi:10.1038/248030a0. S2CID   4290107.
  23. Strominger, A.; Vafa, C. (1996). "Microscopic Origin of the Bekenstein-Hawking Entropy". Phys. Lett. B. 379 (5443): 99–104. arXiv: hep-th/9601029 . Bibcode:1996PhLB..379...99S. doi:10.1016/0370-2693(96)00345-0. S2CID   1041890.