# In-between neutron stars and black holes

Current knowledge and theories,...Suggest a maximum mass for neutron stars about 2-3 solar masses, and is generally assumed that black holes are ANY compact object above that or more generally 5 solar masses. Question: can new physics introduce new populations of intermediate objects between 2-3 solar masses and 5 solar masses? Extreme Compact Objects are sometimes mentioned but...Could also enlarge the assumed minimal mass of black holes (observed indirectly)?

There could certainly be something strange (literally) that might happen that could produce stable stars up to about 3 solar masses (the highest observed and precise neutron star masses are at 2 solar masses), but probably not much higher.

The thing is, even if you postulate some crazy material with the hardest possible equation of state, General Relativity ensures that the pressure at the core of the star contributes to the curvature of space; and the increasing pressure required to support a more massive star actually results in its collapse. The exact mass of this limit depends on the rotation of the star, but I don't think can be far above 3 solar masses.

At present there appears to be a notable gap between the most massive neutron stars and lowest mass black holes. Either compact remnants don't form (or neutron stars can't accrete enough) in this mass range or something stops us finding them (perhaps they can't form in binaries).

Current knowledge and theories, ... Suggest a maximum mass for neutron stars [of] about 2-3 solar masses, and [it] is generally assumed that black holes are ANY compact object above that, or more generally 5 solar masses.

Question: Can new physics introduce new populations of intermediate objects between 2-3 solar masses and 5 solar masses?

Extreme Compact Objects are sometimes mentioned but ... Could also enlarge the assumed minimal mass of black holes (observed indirectly)?

The range of object sizes involved in your question are from almost 3.2 M$_\odot\!$ and 5 M$_\odot$. Any objects in that mass range would not be explained by the cosmological equation of state. I will quote sources to support the upper and lower limits, then I'll discuss the theory behind what happens to matter within that range of masses.

If you accept those numbers without proof you can save a lot of reading by skipping to three quarters of the way through this answer.

Quoting from Wikipedia: For stellar black holes the minimum size is 5 M$_\odot$, any smaller black holes are a hypothetical type of black hole that formed soon after the Big Bang referred to as a primordial black hole. Since primordial black holes did not form from stellar gravitational collapse, their masses can be far below stellar mass (c. $2×10^{30}$ kg). Hawking calculated that primordial black holes could weigh as little as 10$^{−8}$ kg, about the weight of a human ovum.

"A stellar black hole (or stellar-mass black hole) is a black hole formed by the gravitational collapse of a massive 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.

...

The gravitational collapse of a star is a natural process that can produce a black hole. It is inevitable at the end of the life of a star, when all stellar energy sources are exhausted. If the mass of the collapsing part of the star is below the Tolman–Oppenheimer–Volkoff (TOV) limit for neutron-degenerate matter, the end product is a compact star — either a white dwarf (for masses below the Chandrasekhar limit) or a neutron star or a (hypothetical) quark star. If the collapsing star has a mass exceeding the TOV limit, the crush will continue until zero volume is achieved and a black hole is formed around that point in space.

The maximum mass that a neutron star can possess (without becoming a black hole) is not fully understood. In 1939, it was estimated at 0.7 solar masses, called the TOV limit. In 1996, a different estimate put this upper mass in a range from 1.5 to 3 solar masses.

In the theory of general relativity, a black hole could exist of any mass. The lower the mass, the higher the density of matter has to be in order to form a black hole. (See, for example, the discussion in Schwarzschild radius, the radius of a black hole.) There are no known processes that can produce black holes with mass less than a few times the mass of the Sun. If black holes that small exist, they are most likely primordial black holes. Until 2016, the largest known stellar black hole was 15.65±1.45 solar masses. In September 2015, a black hole of 62±4 solar masses was discovered in gravitational waves as it formed in a merger event of two smaller black holes. As of April 2008, XTE J1650-500 was reported by NASA and others to be the smallest-mass black hole currently known to science, with a mass 3.8 solar masses and a diameter of only 24 kilometers (15 miles). However, this claim was subsequently retracted. The more likely mass is 5–10 solar masses.

There is observational evidence for two other types of black holes, which are much more massive than stellar black holes. They are intermediate-mass black holes (in the centre of globular clusters) and supermassive black holes in the centre of the Milky Way and other galaxies.".

That sets the upper limit at 5 M$_\odot$.

Now to address how the lower limit was derived.

"2.3. Maximum Mass

The mass and the composition of neutron stars (NSs) are intricately related. One of the most important empirical clues that would lead to constraints on a wide range of physical processes is the maximum mass of NSs. For instance, secure constraints on the maximum mass provide insight into the range of viable equations of state (EOS) for matter at supranuclear densities.

A first order theoretical upper limit can be obtained by numerically integrating the Oppenheimer-Volkoff equations (also called the Tolman–Oppenheimer–Volkoff equation, TOV) for a low-density EOS at the lowest energy state of the nuclei (Baym et al. 1971). This yields an extreme upper bound to the maximum mass of a NS at M$_{max}$ ∼ 3.2 M$_\odot\!$ (Rhoades & Ruffini 1974). Any compact star to stably support masses beyond this limit requires stronger short-range repulsive nuclear forces that stiffens the EOSs beyond the causal limit. For cases in which causality is not a requisite (v→∞) an upper limit still exist in general relativity ≈ 5.2 M$_\odot\!$ that considers uniform density spheres (Shapiro & Teukolsky 1983). However, for these cases the extremely stiff EOSs that require the sound speed to be super-luminal (or FTL) (dP/dρ ≥ $c^2$) are considered non physical. [See: Exotic Matter].

Differentially rotating NSs that can support significantly more mass than uniform rotators can be temporarily produced by binary mergers (Baumgarte et al. 2000). While differential rotation provides excess radial stability against collapse, even for modest magnetic fields, magnetic braking and viscous forces will inevitably bring differentially rotating objects into uniform rotation (Shapiro 2000). Therefore, radio pulsars can be treated as uniform rotators when calculating the maximum NS mass.

While general relativity along with the causal limit put a strict upper limit on the maximum NS mass at ∼ 3.2 M$_\odot\!$, the lower bound is mostly determined by the still unknown EOS of matter at these densities and therefore is not well constrained. There are modern EOSs with detailed inclusions of nuclear processes such as kaon condensation and nucleon-nucleon scattering which affect the stiffness. These EOSs give a range of 1.5–2.2 M⊙ as the lower bound for the maximum NS mass (Thorsson et al. 1994; Kalogera & Baym 1996). Although these lower bounds for a maximum NS mass are implied for a variation of more realistic EOSs, it is still unclear whether any of these values are favored. Therefore,

$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad M_{max} \, ∼ \, 1.5–3.2 \; \text{M}_\odot \qquad\qquad\qquad\qquad\qquad (5)$$

can be considered a secure range for the maximum NS mass value.

• The paper "Neutron Stars and Black Holes in X-Ray Binaries" (13 Feb 1998), by Jan van Paradijs is a little out of date (for exact sizes and more precise equations of the range of mass in question) but it contains a few informative diagrams that assist one to understand that mass can only accumulate in particular ranges of mass.

On page 12 is this diagram:

"The current lack of knowledge of the EoS and the corresponding uncertainties in the predicted NS masses are illustrated in Fig. 1.

Fig. 1. Left panel: range of equations of state of dense matter (pressure $P$ versus mass density $ρ$), as predicted by various models and consistent with the existence of massive neutron stars. The dotted lines labeled CL and FFG correspond to the causal limit and the free Fermi gas equations of state, respectively (see Sec. 4). Right panel: corresponding range of allowed masses $M$ for nonrotating neutron stars as a function of the central baryon number density $n_c$. The horizontal lines correspond to the precisely measured masses of three pulsars (see Sec. 6).".

...

"Fig. 2. Fractional mass $M_{in}/M$ contained in the inner region of a static spherical NS of mass $M$ and radius $R$, at density $ρ > ρ_⋆$, for two different cases: $ρ_⋆ = 3 × 10^{14} \, g \, cm^{−3}$ (left) and $ρ_⋆ = 5 × 10^{14} \, g \, cm^{−3}$ (right). The shaded areas reflect the uncertainties in the EoS $^{28, 30}$ at $ρ < ρ_⋆$. Only the ranges of $M$ and $R$ allowed by the compactness constraint$^{16, 77}$ $r_g/R ≤ 6/8$ are shown. See the text for details.".

...

5. Effect of rotation on the maximum mass

Rotation increases the maximum mass of NSs because the centrifugal force acts against gravity. We will consider two different cases: (i) rigidly rotating NSs, and (ii) differentially rotating NSs.

...

Setting M = 2M$_\odot\!$ and R = 10 km and using Eq. (17) we find that rotation increases the maximum mass by ∼ 3% only for PSR J1748−2446, whose frequency f = Ω/(2π) = 716 Hz is the highest measured.

...

On page 26 is this useful diagram showing matter's aversion to forming certain sized masses. This text starts on page 25:

"... A few years later, McClintock & Remillard (1986) measured the mass function of the transient source A 0620–00 (which also had a very soft X-ray spectrum during its outburst in 1975) after it had returned to quiescence, to be 3.18±0.16 M$_\odot$.

This immediately (see below) showed that the compact star in this system is too massive to be a neutron star, and gave some confidence in the idea that X-ray spectra may be an efficient way to select BHXBs.

In spite of the fact that some X-ray spectral characteristics of black holes, and rapid variability are also seen in some neutron stars, their combined presence, in particular in X-ray transients, has remained strikingly effective in singling out black holes.

As implied in the above discussion, the main argument that the compact object in a particular X-ray binary is a black hole, is that neutron star masses cannot exceed a certain maximum value. This assumption rests on very general considerations, e.g., that sound cannot travel faster than light, on the basis of which Nauenberg & Chapline (1973) and Rhoades & Ruffini (1974) concluded that any neutron star, independent of the equation of state (EOS) of high-density matter, must have a mass $\small{\lesssim}$ 3 M$_\odot$. Rotation of the neutron star (ignored in the above analyses) does not increase the mass limit by more than 20% (Shapiro & Teukolsky 1983). Detailed modelling of neutron stars, for a wide range of equations of state, leads (see Fig. 10) to upper mass limits between ∼ 1.5 M$_\odot\!$ (very soft EOS) and ∼ 2 M$_\odot\!$ (very stiff EOS) (see, e.g., Arnett & Bowers 1977; Datta 1988; Cheng et al. 1993; Cook et al. 1994; Engvik et al. 1996; see also the contribution of N. Glendenning to this Volume).

The fact that compact objects with dynamical mass estimates exceeding ∼ 3 M$_\odot\!$ cannot be neutron stars, is not equivalent to their being black holes, as defined by the particular space-time structure described by Schwarzschild and Kerr metrics, which are characterized, in particular, by the absence of a hard surface. This has led to the extensive use of the term “black-hole candidate” for these objects. Of course, detection of X-ray pulsations or X-ray bursts immediately disqualifies a compact star as a black hole, but positive evidence for the absence of a hard surface has been very hard to obtain. This should not come as a surprise, since a nominal (M = 1.4 M$_\odot$, R = 10 km) neutron star is just 2.5 times larger than its Schwarzschild radius, and one may expect the accretion flow to be very similar to that of a black hole of comparable mass. The energy release at the neutron star surface, which is absent for a black hole, might lead to observable differences in spectra and variability, but unless the origin of the spectra and variability of X-ray binaries is much better understood than it is nowadays, the conclusion that a black hole has been found on the basis of such phenomena must be considered weak at best.".

[The authors refer again to Figure 10 much later, on page 41.]

"4. Mass determinations of Compact Stars in X-ray binaries

4.1. NEUTRON STAR MASSES AND EQUATION OF STATE

Apart from their crucial role in distinguishing black holes from neutron stars, the importance of measuring the masses of compact stars in X-ray binaries is that they may provide constraints on the properties of the high-density matter in the interior of neutron stars.

These properties are described by an equation of state (EOS), which together with the Oppenheimer-Volkov equations allows one to calculate models of the interior structure of neutron stars (see, e.g., Shapiro & Teukolsky 1983). Since neutron stars can be considered to be zero-temperature objects these models form a one-parameter sequence in which mass, M, and radius, R, depend only on the central density. For a given equation of state one thus has a unique mass-radius relation. Extensive calculations of neutron star models have been made by Arnett & Bowers (1977) and Datta (1988); for a detailed discussion I refer to the contribution of N. Glendenning to this Volume.

Equations of state can be conveniently distinguished by the compressibility of the neutron star matter; for very “stiff” and very “soft” EOS one finds that neutron stars have radii of ∼ 15 km, and ∼ 8 km, respectively (see Fig. 10). Also, the maximum possible neutron star mass depends on the EOS; it is ∼ 1.5 M$_\odot\!$ for very soft EOS, and up to ∼ 2.5 M$_\odot\!$ for the stiffest EOS.

As will be discussed in more detail below, most neutron star masses are consistent with a value close to 1.4 M$_\odot$. From Fig. 10 it appears that at this value masses do not allow one to draw conclusions about the stiffness of the EOS of neutron star matter. For that, one would need observed masses in excess of 1.6 M$_\odot$, which would exclude the softest EOS (note that stiff equations of state are not excluded by low neutron star masses). Similarly, measurements of the gravitational redshift, $z$, at the neutron star surface alone are not a sensitive EOS discriminant, since both stiff and soft equations of state allow $M/R$ ratios up to ∼ 0.2 M$_\odot km^{-1}$ (see Fig. 10), corresponding to redshifts up to ∼ 0.6.

Very accurate neutron star masses have been determined from a variety of general-relativistic effects on the radio pulse arrival times of double neutron star systems. These results will be briefly summarized in Sect. 4.2.1. Neutron star masses have been determined for six HMXB pulsars from pulse arrival time measurements, in combination with radial-velocity observations of their massive companions (see Sect. 4.3). Masses have also been estimated for the low-mass binary radio pulsar PSR J1012+5307, whose companion is a white dwarf, and for the neutron stars in the LMXBs Cyg X-2 (a Z source), Cen X-4 (an SXT) and 4U 1626–67 (an X-ray pulsar). These results are described in Sections 4.2.1, 4.3.3, and 4.3.4, respectively.

In addition to direct measurements of mass and radius, a variety of other ways to obtain observational constraints on the EOS of neutron stars have been proposed.". ...

$$\tiny{\begin{array}{c} \hline & BHF & BHF & DBHF & VCS & pQCD & RMF & RMF & RMF/NJL & RMF/MBM \\ & (N) & (NH) & (N) & (N) & (NQ) & (N) & (NH) & (NQ) & (NQ) \\ \hline Mmax/M_\odot & 2.0-2.5 & 1.3-1.6 & 2.0-2.5 & 2.0-2.2 & 2.0 & 2.1-2.8 & 2.0-2.3 & 2.0-2.2 & 2.0-2.5 \\ \hline \end{array}}$$

"Table 1. Maximum neutron-star mass as predicted by different theories of dense matter. The core is assumed to contain nucleons (N), nucleons and hyperons (NH), nucleons and quarks (NQ).

Microscopic calculations: Brueckner Hartree-Fock (BHF),$^{35, 50–52}$ Dirac Brueckner Hartree-Fock (DBHF),$^{31, 36}$ variational chain summation method (VCS),$^{40}$ perturbative quantum chromodynamics (pQCD).$^{64}$

Effective models: Relativistic Mean Field (RMF),$^{57, 60, 70}$ Nambu-Jona-Lasinio (NJL),$^{59, 65, 71}$ Modified Bag Model (MBM).$^{72, 73}$ If the largest maximum mass M$_{max 2}$ for a given class of models exceeds 2.0M$_\odot$, and the smallest maximum mass M$_{max 1}$ is lower than 2.0M$_\odot\!$ we present the narrower range of masses 2M$_\odot\!$ − M$_{max 2}$ consistent with observations. If, however, M$_{max 2}$ < 2.0M$_\odot$, then the range of M$_{max}$ shown is M$_{max 1}$ − M$_{max 2}$; such a class of models is ruled out by observations.

For further explanations see the text.".

References:

31. F. Sammarruca, Int. J. Mod. Phys. E 19(2010) 1259.
35. Z. H. Li and H.-J. Schulze, Phys. Rev. C 78 (2008) 028801.
36. C. Fuchs, J. Phys. G: Nucl. Part. Phys. 35 (2008) 014049.
50. I. Vida˜na, D. Logoteta, C. Providˆencia, A. Polls, I. Bombaci, Europhys. Lett. 94 (2011) 11002.
51. G. F. Burgio, H.-J. Schulze, A. Li, Phys. Rev. C 83 (2011) 025804.
52. H.-J. Schulze and T. Rijken, Phys. Rev. C 84 (2011) 035801.
59. L. Bonanno, A. Sedrakian, Astron. Astrophys. 539 (2012) A16.
60. G. Colucci, A. Sedrakian, Phys. Rev. C 87 (2013) 055806.
65. J. L. Zdunik and P. Haensel, Astron. Astrophys. 551 (2013) A61.
69. X. Y. Lai and R. X. Xu, MNRAS 398 (2009) L31.
70. H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, Astrophys. J. Suppl. 197 (2011) 20.
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72. S. Weissenborn, I. Sagert, G. Pagliara, M. Hempel, J. Schaeffner Bielich, Astrophys. J. Lett. 740 (2011) L14.
73. F. Ozel, D. Psaltis, S. Ransom, P. Demorest, M. Alford, ¨ Astrophys. J. Letters 724 (2010) L199.


That sets the lower limit at around 3.2 M$_\odot\!$ or less. Practically, rather than theoretically, it's less than three in observed objects.

Now, how can we get mass within that range to occur. Hint: add or subtract.

"Binary star systems are very important in astrophysics because calculations of their orbits allow the masses of their component stars to be directly determined, which in turn allows other stellar parameters, such as radius and density, to be indirectly estimated. This also determines an empirical mass-luminosity relationship (MLR) from which the masses of single stars can be estimated.

...

Configuration of the system

Another classification is based on the distance between the stars, relative to their sizes:[33]

Detached binaries are binary stars where each component is within its Roche lobe, i.e. the area where the gravitational pull of the star itself is larger than that of the other component. The stars have no major effect on each other, and essentially evolve separately. Most binaries belong to this class.

Semidetached binary stars are binary stars where one of the components fills the binary star's Roche lobe and the other does not. Gas from the surface of the Roche-lobe-filling component (donor) is transferred to the other, accreting star. The mass transfer dominates the evolution of the system. In many cases, the inflowing gas forms an accretion disc around the accretor.

A contact binary is a type of binary star in which both components of the binary fill their Roche lobes. The uppermost part of the stellar atmospheres forms a common envelope that surrounds both stars. As the friction of the envelope brakes the orbital motion, the stars may eventually merge. W Ursae Majoris is an example.

Cataclysmic variables and X-ray binaries

When a binary system contains a compact object such as a white dwarf, neutron star or black hole, gas from the other (donor) star can accrete onto the compact object. This releases gravitational potential energy, causing the gas to become hotter and emit radiation. Cataclysmic variable stars, where the compact object is a white dwarf, are examples of such systems. In X-ray binaries, the compact object can be either a neutron star or a black hole. These binaries are classified as low-mass or high-mass according to the mass of the donor star. High-mass X-ray binaries contain a young, early-type, high-mass donor star which transfers mass by its stellar wind, while low-mass X-ray binaries are semidetached binaries in which gas from a late-type donor star or a white dwarf overflows the Roche lobe and falls towards the neutron star or black hole. ...

...

Formation

While it is not impossible that some binaries might be created through gravitational capture between two single stars, given the very low likelihood of such an event (three objects being actually required, as conservation of energy rules out a single gravitating body capturing another) and the high number of binaries currently in existence, this cannot be the primary formation process. The observation of binaries consisting of stars not yet on the main sequence supports the theory that binaries develop during star formation. Fragmentation of the molecular cloud during the formation of protostars is an acceptable explanation for the formation of a binary or multiple star system.

The outcome of the three-body problem, in which the three stars are of comparable mass, is that eventually one of the three stars will be ejected from the system and, assuming no significant further perturbations, the remaining two will form a stable binary system.

Mass transfer and accretion

As a main-sequence star increases in size during its evolution, it may at some point exceed its Roche lobe, meaning that some of its matter ventures into a region where the gravitational pull of its companion star is larger than its own. The result is that matter will transfer from one star to another through a process known as Roche lobe overflow (RLOF), either being absorbed by direct impact or through an accretion disc. The mathematical point through which this transfer happens is called the first Lagrangian point. It is not uncommon that the accretion disc is the brightest (and thus sometimes the only visible) element of a binary star.

If a star grows outside of its Roche lobe too fast for all abundant matter to be transferred to the other component, it is also possible that matter will leave the system through other Lagrange points or as stellar wind, thus being effectively lost to both components. Since the evolution of a star is determined by its mass, the process influences the evolution of both companions, and creates stages that cannot be attained by single stars.

Studies of the eclipsing ternary Algol led to the Algol paradox in the theory of stellar evolution: although components of a binary star form at the same time, and massive stars evolve much faster than the less massive ones, it was observed that the more massive component Algol A is still in the main sequence, while the less massive Algol B is a subgiant at a later evolutionary stage. The paradox can be solved by mass transfer: when the more massive star became a subgiant, it filled its Roche lobe, and most of the mass was transferred to the other star, which is still in the main sequence. In some binaries similar to Algol, a gas flow can actually be seen.

Runaways and novae

It is also possible for widely separated binaries to lose gravitational contact with each other during their lifetime, as a result of external perturbations. The components will then move on to evolve as single stars. A close encounter between two binary systems can also result in the gravitational disruption of both systems, with some of the stars being ejected at high velocities, leading to runaway stars.

If a white dwarf has a close companion star that overflows its Roche lobe, the white dwarf will steadily accrete gases from the star's outer atmosphere. These are compacted on the white dwarf's surface by its intense gravity, compressed and heated to very high temperatures as additional material is drawn in. The white dwarf consists of degenerate matter and so is largely unresponsive to heat, while the accreted hydrogen is not. Hydrogen fusion can occur in a stable manner on the surface through the CNO cycle, causing the enormous amount of energy liberated by this process to blow the remaining gases away from the white dwarf's surface. The result is an extremely bright outburst of light, known as a nova.

In extreme cases this event can cause the white dwarf to exceed the Chandrasekhar limit and trigger a supernova that destroys the entire star, another possible cause for runaways. An example of such an event is the supernova SN 1572, which was observed by Tycho Brahe. The Hubble Space Telescope recently took a picture of the remnants of this event.

Now that we have that range of mass within a small area, what do we end up with?

We don't end up with a single object with a mass between ~ \3.2 M$_\odot\!$ and 5 M$_\odot\!$ (except, possibly, during the Big Bang where $v→c$) since some mass is converted and emitted as x-rays, some is expelled as an orbiting Keplerian velocity field accretion disk, and some may be transferred back to the other stars in a hierarchical system.

The amount of compression is limited by the Pauli exclusion principle. An excellent physics website with somewhat simple explanations is Hyperphysics.phy-astr.gsu.edu.

The other limit (to this answer) is our understanding of Quark degeneracy:

"At densities greater than those supported by neutron degeneracy, quark matter is expected to occur. Several variations of this hypothesis have been proposed that represent quark-degenerate states. Strange matter is a degenerate gas of quarks that is often assumed to contain strange quarks in addition to the usual up and down quarks. Color superconductor materials are degenerate gases of quarks in which quarks pair up in a manner similar to Cooper pairing in electrical superconductors. The equations of state for the various proposed forms of quark-degenerate matter vary widely, and are usually also poorly defined, due to the difficulty of modeling strong force interactions.

Quark-degenerate matter may occur in the cores of neutron stars, depending on the equations of state of neutron-degenerate matter. It may also occur in hypothetical quark stars, formed by the collapse of objects above the Tolman–Oppenheimer–Volkoff mass limit for neutron-degenerate objects. Whether quark-degenerate matter forms at all in these situations depends on the equations of state of both neutron-degenerate matter and quark-degenerate matter, both of which are poorly known. Quark stars are considered to be an intermediate category among neutron stars and black holes. Few scientists claim that quark stars and black holes are one and the same. Not enough data exist to support any hypothesis but neutron stars with awkward spectrums have been used in arguments.".

See also: "Mass Transfer and Disc Formation in AGB Binary Systems" (13 Apr 2017), by Chen, Frank, Blackman, Nordhaus, and Carroll-Nellenback.