Tolman–Oppenheimer–Volkoff limit

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The Tolman–Oppenheimer–Volkoff limit (or TOV limit) is an upper bound to the mass of cold, non-rotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. Stars more massive than the TOV limit collapse into a black hole. The original calculation in 1939, which neglected complications such as nuclear forces between neutrons, placed this limit at approximately 0.7 solar masses (M☉). Later, more refined analyses have resulted in larger values.

Contents

Theoretical work in 1996 placed the limit at approximately 1.5 to 3.0 M☉, [1] corresponding to an original stellar mass of 15 to 20 M☉; additional work in the same year gave a more precise range of 2.2 to 2.9 M☉. [2]

Data from GW170817, the first gravitational wave observation attributed to merging neutron stars (thought to have collapsed into a black hole [3] within a few seconds after merging [4] ) placed the limit in the range of 2.01 to 2.17 M☉. [5]

In the case of a rigidly spinning neutron star, meaning that different levels in the interior of the star all rotate at the same rate, the mass limit is thought to increase by up to 18–20%. [4] [5]

History

The idea that there should be an absolute upper limit for the mass of a cold (as distinct from thermal pressure supported) self-gravitating body dates back to the 1932 work of Lev Landau, based on the Pauli exclusion principle. Pauli's principle shows that the fermionic particles in sufficiently compressed matter would be forced into energy states so high that their rest mass contribution would become negligible when compared with the relativistic kinetic contribution (RKC). RKC is determined just by the relevant quantum wavelength λ, which would be of the order of the mean interparticle separation. In terms of Planck units, with the reduced Planck constant ħ, the speed of light c, and the gravitational constant G all set equal to one, there will be a corresponding pressure given roughly by

.

At the upper mass limit, that pressure will equal the pressure needed to resist gravity. The pressure to resist gravity for a body of mass M will be given according to the virial theorem roughly by

,

where ρ is the density. This will be given by ρ = ⁠m/λ3⁠, where m is the relevant mass per particle. It can be seen that the wavelength cancels out so that one obtains an approximate mass limit formula of the very simple form

.

In this relationship, m can be taken to be given roughly by the proton mass. This even applies in the white dwarf case (that of the Chandrasekhar limit) for which the fermionic particles providing the pressure are electrons. This is because the mass density is provided by the nuclei in which the neutrons are at most about as numerous as the protons. Likewise the protons, for charge neutrality, must be exactly as numerous as the electrons outside.

In the case of neutron stars this limit was first worked out by J. Robert Oppenheimer and George Volkoff in 1939, using the work of Richard Chace Tolman. Oppenheimer and Volkoff assumed that the neutrons in a neutron star formed a degenerate cold Fermi gas. They thereby obtained a limiting mass of approximately 0.7  solar masses, [6] [7] which was less than the Chandrasekhar limit for white dwarfs.

Oppenheimer and Volkoff's paper notes that "the effect of repulsive forces, i.e., of raising the pressure for a given density above the value given by the Fermi equation of state ... could tend to prevent the collapse." [7] And indeed, the most massive neutron star detected so far, PSR J0952–0607, is estimated to be much heavier than Oppenheimer and Volkoff's TOV limit at 2.35±0.17M☉. [8] [9] More realistic models neutron stars including baryon strong force repulsion predict a neutron star mass limit of 2.2 to 2.9 M☉. [10] [11] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known.

Applications

In a star less massive than the limit, the gravitational compression is balanced by short-range repulsive neutron–neutron interactions mediated by the strong force and also by the quantum degeneracy pressure of neutrons, preventing collapse. [12] : 74  If its mass is above the limit, the star will collapse to some denser form. It could form a black hole, or change composition and be supported in some other way (for example, by quark degeneracy pressure if it becomes a quark star). Because the properties of hypothetical, more exotic forms of degenerate matter are even more poorly known than those of neutron-degenerate matter, most astrophysicists assume, in the absence of evidence to the contrary, that a neutron star above the limit collapses directly into a black hole.

A black hole formed by the collapse of an individual star must have mass exceeding the Tolman–Oppenheimer–Volkoff limit. Theory predicts that because of mass loss during stellar evolution, a black hole formed from an isolated star of solar metallicity can have a mass of no more than approximately 10 solar masses. [13] :Fig. 16 Observationally, because of their large mass, relative faintness, and X-ray spectra, a number of massive objects in X-ray binaries are thought to be stellar black holes. These black hole candidates are estimated to have masses between 3 and 20 solar masses. [14] [15] LIGO has detected black hole mergers involving black holes in the 7.5–50 solar mass range; it is possible – although unlikely – that these black holes were themselves the result of previous mergers.

Oppenheimer and Volkoff discounted the influence of heat, stating in reference to work by Landau (1932), 'even [at] 107 degrees... the pressure is determined essentially by the density only and not by the temperature' [7] – yet it has been estimated [16] that temperatures can reach up to approximately >109 K during formation of a neutron star, mergers and binary accretion. Another source of heat and therefore collapse-resisting pressure in neutron stars is 'viscous friction in the presence of differential rotation.' [16]

Oppenheimer and Volkoff's calculation of the mass limit of neutron stars also neglected to consider the rotation of neutron stars, however we now know that neutron stars are capable of spinning at much faster rates than were known in Oppenheimer and Volkoff's time. The fastest-spinning neutron star known is PSR J1748-2446ad, rotating at a rate of 716 times per second [17] [18] or 43,000 revolutions per minute, giving a linear (tangential) speed at the surface on the order of 0.24c (i.e., nearly a quarter the speed of light). Star rotation interferes with convective heat loss during supernova collapse, so rotating stars are more likely to collapse directly to form a black hole [19] : 1044 

List of least massive black holes

NameMass
(M☉)		Distance
(ly)		Companion class Mass determination methodNotesRefs.
V723 Monocerotis 3.04±0.061,500K0IIISpectroscopic radial velocity measurements of companion.The companion star was identified in 2022 to actually be a warm subgiant [20] [20] [21]
2MASS J05215658+43592203.3+2.8
−0.7		10,000K-type (?) giantSpectroscopic radial velocity measurements of noninteracting companion.In Milky Way outskirts. Recent Hubble ultraviolet data has suggested the companion to this object is actually a warm subgiant, and ruling out smaller compact companions such as a white dwarf. [22] [22] [23] [24] [25]
GW190425's remnant3.4+0.3
−0.1		518,600,000N/AGravitational wave data of neutron star merger from LIGO and Virgo interferometers.97% chance of prompt collapse into a black hole immediately after merger. Alternative study suggests collapse 2.5 hours later. [23] [26] [27] [28]
NGC 3201-14.36±0.4115,600(see Notes)Spectroscopic radial velocity measurements of noninteracting companion.In globular cluster NGC 3201. Companion is 0.8M☉ main sequence turn-off. [23] [29]
GRO J1719-24/
GRS 1716−249		≥4.98,500K0-5 V Near-infrared photometry of companion and Eddington flux. LMXB system. [23] [30]
4U 1543-475.0+2.5
−2.3		30,000 ± 3,500A2 (V)?Spectroscopic radial velocity measurements of companion.SXT system. [23] [31]
XTE J1650-500 ≥5.18,500 ± 2,300K4V Orbital resonance modeling from QPOs Transient binary X-ray source [32] [33]
GRO J1655-40 5.31±0.07<5,500F6IVPrecision X-ray timing observations from RossiXTE.LMXB system. [34] [35]
GX 339-4 5.9±3.626,000N/A [23]

List of objects in mass gap

This list contains objects that may be neutron stars, black holes, quark stars, or other exotic objects. This list is distinct from the list of least massive black holes due to the undetermined nature of these objects, largely because of indeterminate mass, or other poor observation data.

See also

Notes

    Related Research Articles

    <span class="mw-page-title-main">Black hole</span> Object that has a no-return boundary

    A black hole is a region of spacetime wherein gravity is so strong that no matter or electromagnetic energy can escape it. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of no escape is called the event horizon. A black hole has a great effect on the fate and circumstances of an object crossing it, but it has no locally detectable features according to general relativity. In many ways, a black hole acts like an ideal black body, as it reflects no light. Quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is of the order of billionths of a kelvin for stellar black holes, making it essentially impossible to observe directly.

    The Chandrasekhar limit is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M☉ (2.765×1030 kg). The limit was named after Subrahmanyan Chandrasekhar.

    <span class="mw-page-title-main">Globular cluster</span> Spherical collection of stars

    A globular cluster is a spheroidal conglomeration of stars that is bound together by gravity, with a higher concentration of stars towards its center. It can contain anywhere from tens of thousands to many millions of member stars, all orbiting in a stable, compact formation. Globular clusters are similar in form to dwarf spheroidal galaxies, and though globular clusters were long held to be the more luminous of the two, discoveries of outliers had made the distinction between the two less clear by the early 21st century. Their name is derived from Latin globulus. Globular clusters are occasionally known simply as "globulars".

    <span class="mw-page-title-main">Neutron star</span> Collapsed core of a massive star

    A neutron star is the collapsed core of a massive supergiant star. It results 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. Except for black holes, neutron stars are the smallest and densest known class of stellar objects. They have a radius on the order of 10 kilometers (6 mi) and a mass of about 1.4 M☉. Stars that collapse into neutron stars have a total mass of between 10 and 25 solar masses (M☉), or possibly more for those that are especially rich in elements heavier than hydrogen and helium.

    <span class="mw-page-title-main">Supernova</span> Explosion of a star at its end of life

    A supernova is a powerful and luminous explosion of a star. A supernova 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 to form a diffuse nebula. The peak optical luminosity of a supernova can be comparable to that of an entire galaxy before fading over several weeks or months.

    <span class="mw-page-title-main">Stellar evolution</span> Changes to stars over their lifespans

    Stellar evolution is the process by which a star changes over the course of its lifetime and how it can lead to the creation of a new star. Depending on the mass of the star, its lifetime can range from a few million years for the most massive to trillions of years for the least massive, which is considerably longer than the current age of the universe. The table shows the lifetimes of stars as a function of their masses. All stars are formed from collapsing clouds of gas and dust, often called nebulae or molecular clouds. Over the course of millions of years, these protostars settle down into a state of equilibrium, becoming what is known as a main-sequence star.

    <span class="mw-page-title-main">White dwarf</span> Type of stellar remnant composed mostly of electron-degenerate matter

    A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to Earth's. A white dwarf's low luminosity comes from the emission of residual thermal energy; no fusion takes place in a white dwarf. The nearest known white dwarf is Sirius B, at 8.6 light years, the smaller component of the Sirius binary star. There are currently thought to be eight white dwarfs among the hundred star systems nearest the Sun. The unusual faintness of white dwarfs was first recognized in 1910. The name white dwarf was coined by Willem Jacob Luyten in 1922.

    A Thorne–Żytkow object, also known as a hybrid star, is a conjectured type of star wherein a red giant or red supergiant contains a neutron star at its core, formed from the collision of the giant with the neutron star. Such objects were hypothesized by Kip Thorne and Anna Żytkow in 1977. In 2014, it was discovered that the star HV 2112, located in the Small Magellanic Cloud (SMC), was a strong candidate, though this view has since been refuted. Another possible candidate is the star HV 11417, also located in the SMC.

    In astronomy, the term compact object refers collectively to white dwarfs, neutron stars, and black holes. It could also include exotic stars if such hypothetical, dense bodies are confirmed to exist. All compact objects have a high mass relative to their radius, giving them a very high density, compared to ordinary atomic matter.

    <span class="mw-page-title-main">Supermassive black hole</span> Largest type of black hole

    A supermassive black hole is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions, of times the mass of the Sun (M☉). Black holes are a class of astronomical objects that have undergone gravitational collapse, leaving behind spheroidal regions of space from which nothing can escape, including light. Observational evidence indicates that almost every large galaxy has a supermassive black hole at its center. For example, the Milky Way galaxy has a supermassive black hole at its center, corresponding to the radio source Sagittarius A*. Accretion of interstellar gas onto supermassive black holes is the process responsible for powering active galactic nuclei (AGNs) and quasars.

    <span class="mw-page-title-main">Intermediate-mass black hole</span> Class of black holes with a mass range of 100 to 100000 solar masses

    An intermediate-mass black hole (IMBH) is a class of black hole with mass in the range of tens to tens thousand (102–105) solar masses: significantly higher than stellar black holes but lower than the tens thousand to hundreds trillion (105–1015) solar mass supermassive black holes. Several IMBH candidate objects have been discovered in the Milky Way galaxy and others nearby, based on indirect gas cloud velocity and accretion disk spectra observations of various evidentiary strength.

    <span class="mw-page-title-main">Stellar black hole</span> Black hole formed by a collapsed star

    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. They are the remnants of supernova explosions, which may be observed as a type of gamma ray burst. These black holes are also referred to as collapsars.

    <span class="mw-page-title-main">R136a1</span> Wolf–Rayet star with one of the highest mass and luminosity of any known star

    R136a1 is one of the most massive and luminous stars known, at nearly 200 M☉ and nearly 4.7 million L☉, and is also one of the hottest, at around 46,000 K. It is a Wolf–Rayet star at the center of R136, the central concentration of stars of the large NGC 2070 open cluster in the Tarantula Nebula in the Large Magellanic Cloud. The cluster can be seen in the far southern celestial hemisphere with binoculars or a small telescope, at magnitude 7.25. R136a1 itself is 100 times fainter than the cluster and can only be resolved using speckle interferometry.

    Stellar mass is a phrase that is used by astronomers to describe the mass of a star. It is usually enumerated in terms of the Sun's mass as a proportion of a solar mass (M☉). Hence, the bright star Sirius has around 2.02 M☉. A star's mass will vary over its lifetime as mass is lost with the stellar wind or ejected via pulsational behavior, or if additional mass is accreted, such as from a companion star.

    <span class="mw-page-title-main">Hypernova</span> Supernova that ejects a large mass at unusually high velocity

    A hypernova is a very energetic supernova which is believed to result from an extreme core collapse scenario. In this case, a massive star collapses to form a rotating black hole emitting twin astrophysical 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. Hypernovae release such intense gamma rays that they often 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.

    A shell collapsar is a hypothetical compact astrophysical object, which might constitute an alternative explanation for observations of astronomical black hole candidates. It is a collapsed star that resembles a black hole, but is formed without a point-like central singularity and without an event horizon. The model of the shell collapsar was first proposed by Trevor W. Marshall and allows the formation of neutron stars beyond the Tolman–Oppenheimer–Volkoff limit of 0.7 M☉.

    <span class="mw-page-title-main">Direct collapse black hole</span> High-mass black hole seeds

    Direct collapse black holes (DCBHs) are high-mass black hole seeds that form from the direct collapse of a large amount of material. They putatively formed within the redshift range z=15–30, when the Universe was about 100–250 million years old. Unlike seeds formed from the first population of stars (also known as Population III stars), direct collapse black hole seeds are formed by a direct, general relativistic instability. They are very massive, with a typical mass at formation of ~105 M☉. This category of black hole seeds was originally proposed theoretically to alleviate the challenge in building supermassive black holes already at redshift z~7, as numerous observations to date have confirmed.

    <span class="mw-page-title-main">UY Volantis</span> Low mass X-ray binary in the constellation Volans

    UY Volantis, also known as EXO 0748-676, is a low mass X-ray binary system located in the constellation Volans. With an apparent magnitude of 16.9, it requires a powerful telescope to see. With a radial velocity of 20 km/s, it is drifting away from the Solar System, and is currently located 26,000 light years away.

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