Kurzgesagt – In a Nutshell
Sources – Black Hole Star
Thanks to our experts
Dr. Warrick Ball
University of Birmingham
Dr. Matt Caplan
Illinois State University
– Black hole stars may have been the largest stars to ever exist. They burned brighter than galaxies and were larger than any star today or that could ever exist in the future. But besides their scale, what makes them special and weird is that deep inside, they were occupied by a cosmic parasite, an endlessly hungry black hole. How is that even possible?
Black hole stars, also known as “quasi-stars”, were proposed a few years ago as a new kind of astronomical object possible in the early universe, consisting on a huge “envelope” of almost pure hydrogen hosting a black hole in its center.
#Begelman, Mitchell C. et al. (2006): “Formation of supermassive black holes by direct collapse in pre-galactic haloes”, Monthly Notices of the Royal Astronomical Society, Volume 370, Issue 1.
https://academic.oup.com/mnras/article/370/1/289/1026607
Quote: “We describe a mechanism by which supermassive black holes (SMBHs) can form directly in the nuclei of protogalaxies, without the need for ‘seed’ black holes left over from early star formation. Self-gravitating gas in dark matter haloes can lose angular momentum rapidly via runaway, global dynamical instabilities, the so-called ‘bars within bars’ mechanism. This leads to the rapid build-up of a dense, self-gravitating core supported by gas pressure – surrounded by a radiation pressure-dominated envelope – which gradually contracts and is compressed further by subsequent infall. We show that these conditions lead to such high temperatures in the central region that the gas cools catastrophically by thermal neutrino emission, leading to the formation and rapid growth of a central black hole.”
Concrete values for their masses, sizes and luminosities will be given below.
– Black hole stars take the weirdness of black holes and go beyond to break everything we know about how stars form and grow. They were only possible during a short window of time in the early Universe, but if they existed, they would solve one of the largest mysteries of cosmology.
The main motivation to consider the properties of such objects is that they could explain the origin of the supermassive black holes that today occupy the centers of most large galaxies:
#Begelman, Mitchell C. et al. (2006): “Quasi-stars: accreting black holes inside massive envelopes”, Monthly Notices of the Royal Astronomical Society, Volume 387, Issue 4
https://academic.oup.com/mnras/article/387/4/1649/1091917
Quote: “We study the structure and evolution of ‘quasi-stars’, accreting black holes embedded within massive hydrostatic gaseous envelopes. These configurations may model the early growth of supermassive black hole seeds.”
The formation of quasi-stars (and the seeds of the supermassive black holes they may have produced) is supposed to have been possible in the very early universe, roughly between 100 million and 500 million years after the Big Bang:
#Begelman, Mitchell C. et al. (2006): “Formation of supermassive black holes by direct collapse in pre-galactic haloes”, Monthly Notices of the Royal Astronomical Society, Volume 370, Issue 1.
https://academic.oup.com/mnras/article/370/1/289/1026607
Quote: “We have presented a scenario for the accumulation of gas in the centres of dark matter haloes with Tvir ≳ 104-K, the initial collapse of the gas to form seed black holes, and the subsequent early growth of SMBHs. This mechanism can lead naturally to the super-Eddington growth of black holes up to masses ∼106-M⊙, as early as redshifts 10–20. Given additional growth to ∼109-M⊙ at close to the Eddington rate, the model can account for the population of quasars observed at z ∼ 6 (Fan et al. 2004)”
Where “redshift” (z) is a measure often used by astronomers to indicate the age of the universe. A redshift of z = 0 corresponds to the present-day universe, while the Big Bang has formally an infinite redshift. Typical values for the formation of the first stars (and therefore quasi-stars, as will be explained below) are supposed to be between z = 20 and z = 10. In order of magnitude, this corresponds to an age of the universe between 100 and 500 million years after the Big Bang (the exact figures depend on the values of the cosmological parameters used, which have some uncertainties).
#Wright, Edward L. (2006): “Ned Wright’s Javascript Cosmology Calculator”, UCLA Division of Astronomy and Astrophysics (retrieved 2022)
– Black Hole Stars were excessive any way you look at them. The most massive stars today may have about 300 solar masses – a black hole star had up to 10 million solar masses of nearly pure hydrogen.
One of the most massive stars known to date is R136a1, in the Large Magellanic Cloud. Its mass is uncertain, but some surveys have estimated up to 315 solar masses:
#Crowther, Paul A. et al. (2016): “The R136 star cluster dissected with Hubble Space Telescope/STIS. I. Far-ultraviolet spectroscopic census and the origin of He II λ1640 in young star clusters”, Monthly Notices of the Royal Astronomical Society, Volume 458, Issue 1.
The specific mass and radius of a quasi-star is currently not known in full detail and depends on the theoretical model used to study them. However, they had to be orders of magnitude more massive and bigger than today’s largest stars. The value of 107 solar masses has been cited as “representative” in the literature:
#Czerny, Bozena (2012): “Quasi-star Jets as Unidentified Gamma-Ray Sources”, The Astrophysical Journal Letters, Volume 755, Number 1.
https://iopscience.iop.org/article/10.1088/2041-8205/755/1/L15
As quoted above, the luminosity of such a protostar would be in the order of 1045 erg/s and its radius would be in the order of 1017 cm. In order of magnitude, this implies a luminosity about 1012 times larger than that of the Sun, and a radius about 106 times bigger:
#NASA (2022): “Sun Fact Sheet” (retrieved 2022)
https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
In the above table J stands for “Joule”, which equals 107 ergs, and a volume of 1.4·1018 km3 corresponds to a radius of about 700,000 km. In order of magnitude, a quasi-star of 107 solar masses would therefore be as luminous as 1012 sunlike stars. The Milky Way is considered a large galaxy and is estimated to have about 1011 stars:
#ESA: “How many stars are there in the Universe?” (retrieved 2022)
Quote: “Astronomers estimate there are about 100 thousand million stars in the Milky Way alone.”
Regarding the composition of quasi-stars, one has to take into account that the only chemical elements created at the Big Bang were hydrogen, a bit of helium and tiny traces of lithium and beryllium. All other elements were first synthesized hundreds of millions of years later by the nuclear reactions taking place at the cores of the first stars. Therefore, the first stars of the universe were very different from today’s, in the sense that they originated from “pristine gas” extremely rich in hydrogen and not yet “contaminated” with heavier chemical elements. This first generation of stars is known as “Population III” stars.
#Swinburne University of Technology: “Population III”, COSMOS - The SAO Encyclopedia of Astronomy (retrieved 2022)
https://astronomy.swin.edu.au/cosmos/p/Population+III
Quote: “Stars observed in galaxies were originally divided into two populations by Walter Baade in the 1940s. Although a more refined means of classifying stellar populations has since been established (according to whether they are found in the thin disk, thick disk, halo or bulge of the galaxy), astronomers have continued to coarsely classify stars as either Population I (Pop I, metal-rich) or Population II (Pop II, metal-poor). However, even the most metal-poor Pop II stars have metallicities (commonly denoted [Z/H]) far above that of the gas left over from the Big Bang.
For this reason, astronomers have introduced a third class of star. Population III (Pop III) stars are composed entirely of primordial gas – hydrogen, helium and very small amounts of lithium and beryllium. This means that the gas from which Pop III stars formed had not been ‘recycled’ (incorporated into, and then expelled) from previous generations of stars, but was pristine material left over from the Big Bang. As such, these stars would have a [Z/H] ~ -10 and would constitute the very first generation of stars formed within a galaxy. These Pop III stars would then produce the metals observed in Pop II stars and initiate the gradual increase in metallicity across subsequent generations of stars.”
Quasi-stars were also born from the same clouds of primordial gas, i.e. they were “Population III objects” too.
#Begelman, Mitchell C. et al. (2006): “Quasi-stars: accreting black holes inside massive envelopes”, Monthly Notices of the Royal Astronomical Society, Volume 387, Issue 4
https://academic.oup.com/mnras/article/387/4/1649/1091917
Quote: “Motivated by this, we develop in this paper a model proposed by Begelman, Volonteri & Rees (2006b hereafter BVR), in which supermassive black holes form, not from Pop III stars themselves, but rather from the evolution of a new class of Pop III objects that might form in metal-free haloes too massive to yield individual stars.”
As we will see later, it was this lack of heavier chemical elements what would allow Population III objects to become much larger than present-day stars.
– Let us take a moment to look at what this means visually. The Sun. Wezen. LL Pegasi. The largest star.
Like many of the figures used here, the following sizes are subject to considerable uncertainties and should be understood as nothing more than reasonable estimates.
Wezen, also known as Delta Canis Majoris, is a star in the constellation of Canis Major. Its radius has been estimated in about 200 solar radii:
#Davis, J. et al. (2013): “The Emergent Flux and Effective Temperature of δ Canis Majoris”, Publications of the Astronomical Society of Australia, Volume 24, Issue 3
LL Pegasi, also known as AFGL 3068 or CRL 3068, is in the constellation of Pegasus and its radius has been estimated in about 900 solar radii:
#De Beck, E. et al. (2010): “Probing the mass-loss history of AGB and red supergiant stars from CO rotational line profiles II – CO line survey of evolved stars: derivation of mass-loss rate formulae”, Astronomy & Astrophysics, vol. 523, A18.
https://www.aanda.org/articles/aa/full_html/2010/15/aa13771-09/aa13771-09.html
Finally, one of the largest stars known, popularly known as “the largest star”, is Stephenson 2-18, or St2-18, a red supergiant in the constellation of Scutum. It’s radius is uncertain, but it can be estimated from its surface temperature and its total luminosity (total energy radiated per unit time):
#Fok, Thomas K. T. (2012): “Maser Observations of Westerlund 1 and Comprehensive Considerations on Maser Properties of Red Supergiants Associated with Massive Clusters”, The Astrophysical Journal, Volume 760, Number 1.
https://iopscience.iop.org/article/10.1088/0004-637X/760/1/65
From these numbers one sees that St2-18 is significantly colder than the Sun but radiates 105.64 = 436,515.8 times more energy per unit time, so it has to be considerably bigger.
Stars obey the Stefan-Boltzmann law: if their surface temperature is T, their flux F (energy emitted per unit time and unit area) is very well approximated by the flux of a black body at the same temperature. This is given by the well-known formula:
F = σT 4
where σ is the Stefan-Boltzmann constant. For the Sun, this formula means that:
L☉/(4πR☉2) = σT☉4
(where L☉, R☉ and T☉ denote the solar luminosity, the solar radius and the solar temperature). Applying the same formula to St2-18 and solving for R2-18 in terms of the solar parameters, one finds:
R2-18 = (T☉ / T2-18) 2 (L2-18 / L☉)½ R☉ ~ 2,150 R☉
where the temperature of the Sun is given by T☉ = 5,772 K
#NASA (2022): “Sun Fact Sheet” (retrieved 2022)
https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
So the radius of Stephenson 2-18 could well be about 2,000 solar radii.
– And finally the black hole star. Its scale is beyond words: over 800,000 times wider than our Sun, 380 times larger than the largest star we know today.
As explained above, a quasi-star of 107 solar masses would typically be about a million times larger than the Sun. The precise value will depend on the model, however. Other studies have provided provides slightly different values for the early stages of the quasi-star:
#Schleicher, Dominik R. G. (2013): “Massive black hole factories: Supermassive and quasi-star formation in primordial halos”, Astronomy & Astrophysics, Vol. 558, A59
https://www.aanda.org/articles/aa/abs/2013/10/aa21949-13/aa21949-13.html
For a mass of 107 solar masses, the above formula yields a radius of around 800,000 solar radii.
– And far below its surface is a black hole, growing rapidly as it devours billions upon billions of tons of matter per second.
Once again, the accretion rates of the central black hole will vary depending on the model, and will also depend on the mass of the black hole. Some values are given here:
#Ball, Warrick H. (2011): “The structure and evolution of quasi-stars”, Monthly Notices of the Royal Astronomical Society, Volume 414, Issue 3
Where MBH represents the mass of the central black hole in solar masses and the same symbol with a dot indicates its accretion rate in ten-thousandths of solar masses per year. Other references consider much higher values. An accretion rate of 3·10–4 solar masses per year would translate into some 2·1016 tonnes per second, or twenty million billion tonnes per second.
– Normally, stars are born from gigantic clouds, collections of thousands to millions of solar masses of mostly hydrogen. In these clouds, matter starts to accumulate around the densest spots inside. As these spots get denser, their gravitational pull intensifies and they grow faster.
Stars are born of molecular clouds composed of mostly hydrogen and with masses of up to several millions of solar masses:
#NASA (2022): “Stars” (retrieved 2022)
https://science.nasa.gov/astrophysics/focus-areas/how-do-stars-form-and-evolve
Quote: “Stars are born within the clouds of dust and scattered throughout most galaxies. A familiar example of such a dust cloud is the Orion Nebula. Turbulence deep within these clouds gives rise to knots with sufficient mass that the gas and dust can begin to collapse under its own gravitational attraction. As the cloud collapses, the material at the center begins to heat up. Known as a protostar, it is this hot core at the heart of the collapsing cloud that will one day become a star.”
#Swinburne University: “Molecular Cloud”, COSMOS – The SAO Encyclopedia of Astronomy (retrieved 2022)
https://astronomy.swin.edu.au/cosmos/m/Molecular+Cloud
Quote: “Dust and gas primarily in the form of hydrogen molecules are the main constituents of the coldest, densest clouds in the interstellar medium.”
#Dobbs, Clare L. et al. (2013): “Formation of Molecular Clouds and Global Conditions for Star Formation”, Protostars and Planets VI, University of Arizona Press.
https://arxiv.org/abs/1312.3223
Quote: “Molecular cloud masses range from ∼ 102 M⊙ for small clouds at high Galactic latitudes (e.g., Magnani et al. 1985) and in the outer disk of the Milky Way (e.g., Brand and Wouterloot 1995; Heyer et al. 2001) up to giant ∼ 107 M⊙ clouds in the central molecular zone of the Galaxy (Oka et al. 2001)”
These clouds do not collapse into a single point, but usually fragment into smaller structures
#NASA (2016): “Discovering the Origin of Stars Through 3D Simulation” (retrieved 2022)
https://www.nas.nasa.gov/publications/articles/feature_origin_of_stars_Klein.html
Quote: “The simulations show the entire evolution of these clusters—starting with a giant molecular cloud that collapses due to gravitational forces, to the formation of multiple turbulent clumps of interstellar gas inside the cloud, which in turn collapse into stellar clusters and cores that ultimately form individual stars.”
– Eventually, they generate so much heat and pressure that they ignite fusion reactions, and a new star is born. But this puts a limit on their size: Nuclear fusion releases enough radiation energy that the surrounding gas cloud is blown away. The new baby star can’t gather more mass.
The radiation pressure from a nascent star blows away the surrounding gas, a process that limits how big a star can get.
#Rosen, Anna L et al. (2020): “The Role of Outflows, Radiation Pressure, and Magnetic Fields in Massive Star Formation”, The Astronomical Journal, Volume 160, Number 2
https://iopscience.iop.org/article/10.3847/1538-3881/ab9abf
Quote: “Stellar feedback in the form of radiation pressure and magnetically driven collimated outflows may limit the maximum mass that a star can achieve and affect the star formation efficiency of massive prestellar cores.”
– From now on the star is living on the edge between two forces: Gravity pulling in, trying to squash the star, and radiation created by fusion, pushing outwards, trying to blow the star apart. After millions to billions of years, the core runs out of fuel and the balance breaks, destroying the star. But black hole stars were very, very different.
A star is a self-regulating system:
#George Djorgovski (2004): “How Stars Work”, California Institute of Technology (retrieved 2020)
– A few hundred million years after the Big Bang, when the universe was much smaller, all the matter in existence was much more concentrated. The universe was much denser and hotter. Dark matter was a dominant player, forming giant structures called dark matter halos. These dark matter halos were so massive that they pulled in and concentrated unthinkably gigantic amounts of hydrogen gas, becoming the birthplaces of the first stars and galaxies.
Dark matter is considered to be the key element driving the formation of structures in the early universe, leading to the birth of the first stars and galaxies:
#ESA: “History of cosmic structure formation” (retrieved 2022)
https://www.esa.int/Science_Exploration/Space_Science/Planck/History_of_cosmic_structure_formation
Quote: “Between the release of the cosmic microwave background and the formation of the first stars and galaxies (t = 380,000 years to t = a few hundred million years) [...] Since dark matter particles had already created a network of dense and empty structure, ordinary matter particles could feel the gravitational attraction from the densest concentrations of dark matter and fall toward them. [...] A few hundred million years after the Big Bang, the distribution of matter in the Universe had produced very dense knots at the intersections of the sheets and filaments that make up the cosmic web. In these knots, the density of ordinary matter was so high that the formation of stars and galaxies became possible.”
Dark matter haloes had a whole range of masses and sizes, but a typical length scale would be a few thousands of light-years:
#Wise, John H. et al. (2008): “Resolving the Formation of Protogalaxies. II. Central Gravitational Collapse”, The Astrophysical Journal, Volume 682, Number 2
(Third and fourth columns not shown here.)
Quote: “At the 10 kpc scale, the filamentary large-scale structure is shown, and the protogalactic halo exists at the intersection of these filaments.”
The two simulations above (A and B) show the distribution of the protogalactic gas at a redshift z ~16, which corresponds to an age of the universe of about 250 million years:
#Wright, Edward L. (2006): “Ned Wright’s Javascript Cosmology Calculator”, UCLA Division of Astronomy and Astrophysics (retrieved 2022)
As explained in the quote above, the dark matter haloes are located at the intersections of the gas filaments. The figures show that the typical length scale of those intersections is in the order of the kiloparsec. This amounts to several light-years (1 parsec = 3.26 light-years).
– Epic clouds of hydrogen formed, some as massive as 100 million Suns, more than the mass of small galaxies.
The mass of the dark matter haloes in the early universe that could have hosted the formation of quasi-stars has been estimated in between 107 and 109 solar masses:
#Begelman, Mitchell C. et al. (2006): “Formation of supermassive black holes by direct collapse in pre-galactic haloes”, Monthly Notices of the Royal Astronomical Society, Volume 370, Issue 1.
https://academic.oup.com/mnras/article/370/1/289/1026607
Quote: “A Tvir ∼ 104-K halo has a mass between 107-M⊙ and 109-M⊙ at redshift 6 < z < 20. The black hole forming in such a host could grow at the super-Eddington rate given by equation (13) until it reaches ∼106-M⊙, at which point its mass would approach that of the quasi-star.”
Given the fact that about 1/5 of all the mass in the universe (visible + dark matter) is made of ordinary matter, a dark matter pregalactic halo of 108-109 solar masses would easily have assembled an amount of hydrogen of the order of 108 solar masses.
Some (present-day) dwarf galaxies are clearly smaller. Two examples are the Ursa Major II Dwarf Galaxy (UMa II dSph) and the Sagittarius Dwarf Irregular Galaxy (SagDIG), two satellite galaxies of the Milky Way whose respective masses have been estimated in around 5 million and 38 million solar masses:
#Simon, Joshua D. et al. (2007): “The Kinematics of the Ultra-faint Milky Way Satellites: Solving the Missing Satellite Problem”, The Astrophysical Journal, Volume 670, Number 1
#Karachentsev, I. et al. (1999): “The Sagittarius dwarf irregular galaxy (SagDIG): distance and star formation history”, Astronomy and Astrophysics, Vol. 352.
https://arxiv.org/abs/astro-ph/9910136
Quote: “For the H I mass and the total (virial) mass of SagDIG Lo et al. (1993) derived MHI = 7.4 · 106 M⊙ and Mvt = 3.8 · 107 M⊙ respectively.”
– In this unique environment, that will never exist again, the enormous gravitational pull of the dark matter halos drew gas into its center and created extremely massive stars.
As explained above, the first stars of the universe were “Population III objects”, i.e. were made almost exclusively of hydrogen and helium and lacked any other chemical elements, since these had still to be synthesized for the first time in the cores of the stars. The chemical properties of this pristine gas made the first stars to be much bigger than today’s.
#Larson, Richard B. et al. (2001): “The First Stars in the Universe”, Scientific American
http://user.astro.columbia.edu/~gbryan/C2900/gbryan_SciAm01.pdf
Quote: “The first star-forming clumps were much warmer than the molecular gas clouds in which most stars currently form. Dust grains and molecules containing heavy elements cool the present-day clouds much more efficiently to temperatures of only about 10 kelvins. The minimum mass that a clump of gas must have to collapse under its gravity is called the Jeans mass, which is proportional to the square of the gas temperature and inversely proportional to the square root of the gas pressure. The first star-forming systems would have had pressures similar to those of present-day molecular clouds. But because the temperatures of the first collapsing gas clumps were almost 30 times higher than those of molecular clouds, their Jeans mass would have been almost 1,000 times larger.”
However, quasi-stars were different from regular primordial stars. They were so much bigger that, eventually, a black hole formed at the center.
– As we said before, when a star is born it blows away the gas cloud that created it – but these titanic gas clouds in the early universe were so large and massive that even after their birth, more and more gas piled on the newborn star, making it grow to unbelievable proportions. The young star is forced to grow and grow and grow, getting more and more massive, until in some cases, it reaches up to ten million times the mass of our sun. Crushed by gravity, its core gets hotter and hotter, desperately pushing outward, trying to blow itself apart – but to no avail. There is too much mass and too much pressure. The balance is impossible to uphold. Like a supernova on fast forward, the core gets crushed into a black hole.
The gravitational collapse of the core of a quasi-star into a black hole is one of the defining characteristics of quasi-stars:
#Ball, Warrick H. (2011): “The structure and evolution of quasi-stars”, Monthly Notices of the Royal Astronomical Society, Volume 414, Issue 3
https://academic.oup.com/mnras/article/414/3/2751/1046448
Quote: “If the rate of mass infall is much higher then the envelope of the star does not reach thermal equilibrium during the lifetime of the star (Begelman 2010). In this case, the core collapses after hydrogen burning is complete. The structure that remains after core collapse is a stellar-mass BH embedded within a giant-like envelope, or a quasi-star (Begelman et al. 2006). The attractive feature of a quasi-star is that the accretion rate on to the BH is limited by the Eddington rate of the entire object, which is initially much larger than that of the BH alone. The excess energy is carried away by convection.”
#Begelman, Mitchell C. et al. (2006): “Quasi-stars: accreting black holes inside massive envelopes”, Monthly Notices of the Royal Astronomical Society, Volume 387, Issue 4
https://academic.oup.com/mnras/article/387/4/1649/1091917
Quote: “First, gas in metal-free haloes with a virial temperature above T≃ 104K flows towards the centre of the potential as a result of gravitational instabilities, forming a massive, pressure-supported central object. Nuclear reactions may start, but the very high infall rate continues to compress and heat the core, precluding formation of an ordinary star. Eventually, when the core temperature attains T ~ 5 × 108 K, neutrino losses result in a catastrophic collapse of the core to a black hole.”
#Schleicher, Dominik R. G. (2013): “Massive black hole factories: Supermassive and quasi-star formation in primordial halos”, Astronomy & Astrophysics, Vol. 558, A59
https://www.aanda.org/articles/aa/abs/2013/10/aa21949-13/aa21949-13.html
Quote: “While a supermassive star denotes a conventional star with very high masses of 103−106 M⊙ (Shapiro & Teukolsky 1986), a quasi-star refers to an object with similar mass, but where the central core has collapsed into a low-mass black hole (Begelman et al. 2006; Begelman 2010).”
– Normally that would be the end – today’s stars go supernova, a black hole forms and things calm down.
In the present universe, when massive stars reach the end of their lives their core also collapses into a neutron star or a black hole, a process accompanied by a violent supernova explosion:
#Encyclopaedia Britannica: “Supernova” (retrieved 2022)
https://www.britannica.com/science/supernova
Quote: “The supernova detonation occurs when material falls in from the outer layers of the star and then rebounds off the core, which has stopped collapsing and suddenly presents a hard surface to the infalling gases. The shock wave generated by this collision propagates outward and blows off the star’s outer gaseous layers. The amount of material blasted outward depends on the star’s original mass.
If the core mass exceeds three solar masses, the core collapse is too great to produce a neutron star; the imploding star is compressed into an even smaller and denser body—namely, a black hole. Infalling material disappears into the black hole, the gravitational field of which is so intense that not even light can escape. The entire star is not taken in by the black hole, since much of the falling envelope of the star either rebounds from the temporary formation of a spinning neutron core or misses passing through the very centre of the core and is spun off instead.”
– But in this case, the star survives its own death. A tremendous explosion rocks the star from the inside, but it is not enough – the star is so large and massive that not even a supernova can destroy it – but now it has a black hole for a heart.
As mentioned, the core-collapse process usually comes with a supernova explosion. However, it is thought that very massive stars (those with masses larger than 250 or 260 solar masses) don’t go supernova – they are so massive that the would-be explosion cannot make it through the shells of the star:
#Ho, Anna Y. Q. (2020): “Strange Supernovae Upend Expectations”, Scientific American
https://www.scientificamerican.com/article/strange-supernovae-upend-expectations/
The mass of a quasi-star is several orders of magnitude higher, so there is no chance that a supernova-like event happening at the core would trigger an explosion that blows the star.
– It is tiny, a few tens of kilometers, in the center of a thing the size of the solar system.
The collapse of the core of a quasi-star is expected to give rise to a “stellar-mass” black hole of just a few solar masses or several tens of solar masses at most:
#Begelman, Mitchell C. et al. (2006): “Formation of supermassive black holes by direct collapse in pre-galactic haloes”, Monthly Notices of the Royal Astronomical Society, Volume 370, Issue 1.
https://academic.oup.com/mnras/article/370/1/289/1026607
Quote: “Details of the collapse depend on the angular momentum in the core as well as the precise core mass at the time of collapse. If angular momentum is initially unimportant [...] Continued infall similarly circumvents neutron degeneracy, with the result that a black hole of a several solar masses forms in a few times the core free-fall time. [...] therefore the prompt black hole mass is unlikely to exceed several tens of solar masses.”
And such black holes have sizes in the order of a few tens of kilometers:
#NASA-Chandra (2019): “A Black Hole Primer with Chandra” (retrieved 2022)
https://chandra.si.edu/blackhole/
Quote: “The radius of the event horizon (proportional to the mass) is very small, only 30 kilometers for a non-spinning black hole with the mass of 10 Suns.”
As explained above the size of a quasi-star is uncertain and the exact figure can vary depending on the theoretical model used. At any rate, is in the order of hundreds or thousands of astronomical units:
#Begelman, Mitchell C. et al. (2006): “Quasi-stars: accreting black holes inside massive envelopes”, Monthly Notices of the Royal Astronomical Society, Volume 387, Issue 4
https://academic.oup.com/mnras/article/387/4/1649/1091917
Quote: “Thus, quasi-stars should have radii of the order of 102–103 au and temperatures of a few thousand degrees.”
There is no unambiguous number for “the size of the solar-system”. As in other videos, here we have chosen to define it at the location of the heliopause, the border marking the region of influence of the solar wind. This border is located at some 123 astronomical units from the Sun:
#Encyclopaedia Britannica: “Heliopause” (retrieved 2022)
https://www.britannica.com/science/heliopause
Quote: “heliopause, boundary of the heliosphere, the spherical region around the Sun that is filled with solar magnetic fields and the outward-moving solar wind consisting of protons and electrons. [...] The heliopause is about 123 astronomical units (AU; 18 billion km [11 billion miles]) from the Sun.”
(This is much further away than the orbit of Neptune, which is about 30 AU from the Sun, and beyond the main region of the Kuiper belt.)
– Stars are born from ever faster spinning and collapsing gas, and so they also spin. When a black hole is born from the core of a star, it keeps its angular momentum. This means that matter that gets drawn in doesn’t just fall in a straight line, but instead begins orbiting the black hole, in smaller and smaller circles going faster and faster. The result is an accretion disk where gas orbits at nearly the speed of light.
The matter around a black hole forms a disk of inspiralling hot gas:
#NASA-Chandra (2019): “A Black Hole Primer with Chandra” (retrieved 2022)
https://chandra.si.edu/blackhole/
Quote: “accretion disk – A disk of gas and dust that can accumulate around a center of gravitational attraction, such as a normal star, a white dwarf, neutron star, or black hole. As the gas spirals in due to friction, it becomes hot and emits radiation.”
Closest to the black hole, the gas moves at nearly the speed of light:
#NASA’s Goddard Space Flight Center (2019): “Black Hole Accretion Disk Visualization” (retrieved 2022)
https://svs.gsfc.nasa.gov/13326
Quote: “Nearest the black hole, the gas orbits at close to the speed of light, while the outer portions spin a bit more slowly. This difference stretches and shears the bright knots, producing light and dark lanes in the disk.”
– Only a small amount of gas actually falls in at any given moment. Basically, black holes put a lot of food on the table and only nibble at it. But the matter trapped in the accretion disk doesn’t have a good time: Friction and collisions between particles heat it up to temperatures of millions of degrees. Actively feeding black holes have accretion disks that are incredibly hot and powerful. This heat from the disk further restricts how much a black hole can devour, just like the core of stars, the superhot material creates radiation that blows away most of the food within its reach. So even if a black hole had access to as much food as it desired, it can only grow slowly.
Friction in the accretion disc causes the gas to heat up to temperatures of millions of degrees:
#NASA-Chandra (2019): “A Black Hole Primer with Chandra” (retrieved 2022)
https://chandra.si.edu/blackhole/
Quote: “Searching for black holes is a tricky business. One way to locate black holes is to search for the X-radiation from a disk of hot gas swirling toward a black hole. Friction between particles in the disk heats them to many millions of degrees, and they produce X-rays. Such disks have been found in binary star systems composed of a normal star in a close orbit around a stellar-mass black hole and, on a much larger scale, around the supermassive black holes in the centers of galaxies.”
And all these phenomena make the accretion rate (the speed at which the black hole absorbs matter) to be very small when compared to the mass of the black hole itself:
#Daly, Ruth A. (2020): “Black hole mass accretion rates and efficiency factors for over 750 AGN and multiple GBH”, Monthly Notices of the Royal Astronomical Society, Volume 500, Issue 1.
This table shows accretion rates for several supermassive black holes. The logarithm of the accretion rate (in solar masses per year) is given in column 5, and the logarithm of the mass of the black hole (in solar masses) is given in column 10. We see that the highest accretion rate in this list is that of 3C 268.4, a black hole with a mass of 6 billion solar masses:
Mass = 109.80 M⊙ = 6.3·109 M⊙
However, its accretion rate is just a few solar masses per year:
Acc. rate = 101.10 M⊙ / year = 12.6 M⊙ / year
So, at that rate, the black hole would need 5 million years to get a mass increase of 1%.
The same applies to stellar-mass black holes:
Where we see that V404 Cyg, a black hole of 10 solar masses located in the Milky Way in the constellation of Cygnus, accretes mass at a rate of about
Acc rate = 10–8.5 M⊙ / year = 3·10–9 M⊙ / year
At that rate, the black hole would need more than 30 million years to experience a mass increase of 1%
– A black hole embedded inside a black hole star is different. The enormous pressure surrounding it pushes down matter directly into the black hole, overcoming all restrictions on how fast it can consume.
One of the main characteristics of quasi-stars is that the central black hole can accrete mass at a much higher rate due to the presence of the massive hydrogen envelope surrounding the black hole:
#Begelman, Mitchell C. et al. (2006): “Formation of supermassive black holes by direct collapse in pre-galactic haloes”, Monthly Notices of the Royal Astronomical Society, Volume 370, Issue 1.
https://academic.oup.com/mnras/article/370/1/289/1026607
Quote: “We estimate the initial mass and growth rate of the black hole for typical conditions in metal-free haloes with Tvir ∼ 104-K, which are the most likely to be susceptible to runaway infall. The initial black hole should have a mass of ≲ 20 M⊙, but in principle could grow at a super-Eddington rate until it reaches ∼104–106 M⊙.”
#Ball, Warrick H. (2011): “The structure and evolution of quasi-stars”, Monthly Notices of the Royal Astronomical Society, Volume 414, Issue 3
https://academic.oup.com/mnras/article/414/3/2751/1046448
Quote: “The attractive feature of a quasi-star is that the accretion rate on to the BH is limited by the Eddington rate of the entire object, which is initially much larger than that of the BH alone.”
–This process is so violent and releases so much energy that the accretion disk becomes hotter and releases more radiation pressure than any star core ever could – enough to push back against the weight of 10 million Suns. An impossibly dangerous balance has been created – millions of solar masses pushing in, the angry radiation of a force fed black hole pushing out. For the next few million years, the black hole star is consumed from within. The black hole grows to thousands of solar masses and the bigger it gets, the faster it eats, which heats up the star even more and causes it to expand.
These high accretion rates imply a very energetic accretion disc that pushes back against the envelope of the quasi-star, causing it to inflate. The weight of the envelope will hold the equilibrium for some time; but, as the black hole grows, the feedback of its accretion disk will eventually blow the quasi-star away:
#Rossi, Elena Maria (2017): “Supermassive black hole seeds: updates on the ‘quasi-star model’ ”, Journal of Physics: Conference Series, Volume 840.
https://iopscience.iop.org/article/10.1088/1742-6596/840/1/012027
Quote: “As the core of a MS collapses into a black hole, the surrounding envelope follows the fall. However, if in the innermost region the gas has enough angular momentum to circularise outside the last stable circular orbit of the black hole and form an accretion disc, accretion energy may be injected back into the envelope stopping its collapse. The envelope would inflate and expand until equilibrium is found, which corresponds to a surface luminosity roughly equal to the Eddington luminosity for the total mass of the envelope. This is a highly super Eddington luminosity for the black hole, by a factor equal to the envelope to black hole mass ratio. This is possible because energy is transported through most of the envelope via convection, which is not subject to the Eddington limit [14]. The embryon black hole thus grows at a highly super-Eddington pace, until it has consumed enough envelope mass that the equilibrium for a quasi-star object is not longer possible. Here the feedback from the black hole is likely to blow away the remaining envelope, leaving a bear black hole, that will eventually accrete at a more subdued pace from the pre-galactic disc”.
Before the equilibrium breaks down, the growth of the inner black hole is expected to last a few million years:
#Ball, Warrick H. (2011): “The structure and evolution of quasi-stars”, Monthly Notices of the Royal Astronomical Society, Volume 414, Issue 3
https://academic.oup.com/mnras/article/414/3/2751/1046448
Quote: “In light of these results, it appears that quasi-stars produce BHs that are on the order of at least 0.1 of the mass of the quasi-star and around 0.5 if all the material within the inner radius is accreted. For conservative parameters, this growth occurs within a few million years after the BH initially forms. Realistic variations in the parameters (e.g. larger initial mass, lower radiative efficiency) lead to shorter lifetimes.”
At the end of the process, the mass of the black hole will depend on the mass of the envelope, but typical values would be between several thousands and even a million of solar masses:
#Begelman, Mitchell C. et al. (2006): “Formation of supermassive black holes by direct collapse in pre-galactic haloes”, Monthly Notices of the Royal Astronomical Society, Volume 370, Issue 1.
https://academic.oup.com/mnras/article/370/1/289/1026607
Quote: “We estimate the initial mass and growth rate of the black hole for typical conditions in metal-free haloes with Tvir ∼ 104-K, which are the most likely to be susceptible to runaway infall. The initial black hole should have a mass of ≲ 20 M⊙, but in principle could grow at a super-Eddington rate until it reaches ∼104–106 M⊙.”
– In its final phase, the black hole star has become over 30 times wider than our solar system – truly, the largest star to ever exist in the universe.
As explained above, a quasi-star of 10 million solar masses could be about 800,000 times wider than the Sun. The Sun has a radius of almost 700,000 km:
#NASA (2021): “Our Sun”, Solar System Exploration – Our Galactic Neighborhood (retrieved 2022)
https://solarsystem.nasa.gov/solar-system/sun/in-depth/
Quote: “Our Sun is a medium-sized star with a radius of about 435,000 miles (700,000 kilometers).”
This means that our quasi-star would have a radius of about 8·105 × 7·105 = 5.6·1011 kilometers, or about 3,700 astronomical units (AU). As also mentioned above, for the radius of the solar system we’ve chosen the location of the heliopause, at 123 AU from the Sun. This means that our quasi-star would be 3700/123 ~ 30 times wider than the solar system.
This makes it much bigger than TON 618, one of the largest supermassive black holes known to date:
#NASA: “Black Holes”, Nasa Science – Universe Exploration (retrieved 2022)
https://universe.nasa.gov/black-holes/basics/
Quote: “The most massive black hole observed, Ton 618, tips the scales at 66 billion times the Sun’s mass.”
The radius of a black hole is proportional to its mass. If we neglect the effects of rotation, the relation is very well approximated by the equation:
RBH (km) = 3 MBH / M⊙
where MBH is the mass of the black hole, M⊙ is the mass of the Sun, and the result gives the radius of the black hole in kilometers:
NASA Goddard Space Flight Center: “Estimating the Size and Mass of a Black Hole”, Mathematics Problems about Black Holes
https://spacemath.gsfc.nasa.gov/blackh/7Page81.pdf
Quote: “The radius of a black hole is related to its mass by the simple formula R = 3 M ,where M is the mass of the black hole in units of the sun's mass, and R is the radius of the Event Horizon in kilometers.”
This means that for TON 618 we would get a radius of 66·109 × 3 ~ 2·1011 kilometers, or about 1300 AU, which is about 10 times larger than the solar system. Our quasi-star is 3 times bigger than that.
–The intense magnetic fields at its core spew out jets of plasma from the black hole’s poles, which pierce through the star and shoot out into space, turning it into a cosmic beacon. It must have been one of the most awe inducing sights to ever exist in the universe. But this also marks the end. It becomes too stretched and the accretion disk within too powerful: the parasite destroys its host, blowing it apart. A black hole with the mass of 100,000 Suns rips its way out to hunt for new prey, while leaving behind nothing but a star carcass.
Black holes with active accretion disks can generate powerful jets of high-energy particles emerging from the black hole’s poles:
#NASA-Chandra (2019): “A Black Hole Primer with Chandra” (retrieved 2022)
https://chandra.si.edu/blackhole/
Quote: “Is all matter in the disk around a black hole doomed to fall into the black hole? No, sometimes gas will escape as a hot wind that is blown away from the disk at high speeds. Even more dramatic are the high-energy jets that X-ray and radio observations show exploding away from the vicinity of some supermassive black holes. These jets can move at nearly the speed of light in tight beams and can travel hundreds of thousands of light years.”
NASA (2013): “Black Holes: Monsters in Space”, NuSTAR Mission (retrieved 2022)
https://www.nasa.gov/mission_pages/nustar/multimedia/pia16695.html
Quote: “Also shown is an outflowing jet of energetic particles, believed to be powered by the black hole's spin. The regions near black holes contain compact sources of high energy X-ray radiation thought, in some scenarios, to originate from the base of these jets. This high energy X-radiation lights up the disk, which reflects it, making the disk a source of X-rays. The reflected light enables astronomers to see how fast matter is swirling in the inner region of the disk, and ultimately to measure the black hole's spin rate.”
The same process is expected to have happened at the end of the life of a quasi-star:
#Czerny, Bozena (2012): “Quasi-star Jets as Unidentified Gamma-Ray Sources”, The Astrophysical Journal Letters, Volume 755, Number 1.
https://iopscience.iop.org/article/10.1088/2041-8205/755/1/L15
Quote: “If the quasi-star material possesses some angular momentum, a jet may form, as in gamma-ray bursts or active galactic nuclei."
– Black holes born from regular supernovas can be a few tens of solar masses at most. And because of the process we explained before, they grow slowly after that. If black holes merge together, they can form slightly larger black holes of over a hundred solar masses. It should take billions and billions of years to make black holes with hundreds of thousands or even millions of solar masses.
As explained above, the accretion rate of stellar-mass black holes is very slow, and has been estimated in a few billionths of solar masses per year for a few Milky Way black holes. Similar rates have been deduced for the black holes formed by the first stars of the universe:
#Alvarez, Marcelo A. et al. (2009): “Accretion onto the first stellar-mass black holes”, The Astrophysical Journal, Volume 701, Number 2.
https://iopscience.iop.org/article/10.1088/0004-637X/701/2/L133/meta
Quote: “The first stars, forming at redshifts z > 15 in minihalos with M ∼ 105–6 M☉ may leave behind remnant black holes, which could conceivably have been the "seeds" for the supermassive black holes observed at z ≲ 7. We study remnant black hole growth through accretion, including for the first time the radiation emitted due to accretion, with adaptive mesh refinement cosmological radiation–hydrodynamical simulations. The effects of photoionization and heating dramatically affect the large-scale inflow, resulting in negligible mass growth.”
In other words, stellar-mass black holes just grow too slowly to become supermassive in a short time. Even a “black hole seed” with a mass of the order of 100 solar masses would grow too slowly:
#Ball, Warrick H. (2011): “The structure and evolution of quasi-stars”, Monthly Notices of the Royal Astronomical Society, Volume 414, Issue 3
https://academic.oup.com/mnras/article/414/3/2751/1046448
Quote: “How long would it take such a seed BH to grow to 109 M⊙? If the BH were able to accrete constantly at its Eddington-limited rate, then a 100 M⊙ BH would take about 7 × 108 yr to grow into a 109 M⊙ SMBH for a typical radiative efficiency of the accretion (ε = 0.1). This is barely the age of the Universe at z = 6 and already brings the scenario into doubt.”
The age of the universe at z = 6 is about one billion years:
#Wright, Edward L. (2006): “Ned Wright’s Javascript Cosmology Calculator”, UCLA Division of Astronomy and Astrophysics (retrieved 2022)
This means that, under normal conditions, a big stellar-mass black hole would need billions of years to reach the mass of a typical supermassive black hole.
– And yet, we know that some super massive black holes already had 800 million solar masses only 690 million years after the Big Bang.
#Bañados, Eduardo et al. (2017): “An 800-million-solar-mass black hole in a significantly neutral Universe at a redshift of 7.5”, Nature, vol. 553
https://www.nature.com/articles/nature25180
Quote: “This quasar has a bolometric luminosity of 4 × 1013 times the luminosity of the Sun and a black-hole mass of 8 × 108 solar masses. The existence of this supermassive black hole when the Universe was only 690 million years old—just five per cent of its current age—reinforces models of early black-hole growth that allow black holes with initial masses of more than about 104 solar masses2,3 or episodic hyper-Eddington accretion”
– Black Hole Stars are a sort of black hole cheat code. If they formed very early in our Universe and the black holes that emerged from them were thousands of solar masses, then they could be the seeds for supermassive black holes. These seeds could take root in the center of the earliest galaxies, merging with others and drawing in enough matter to grow quickly and reliably.
Black Hole Stars aren’t the only possible explanation for supermassive black holes appearing so early. Among other hypotheses, there’s also the potential for the direct collapse of large clouds into black holes:
#Latif, M. A. (2022): “Turbulent cold flows gave birth to the first quasars”, Nature, vol. 607
https://www.nature.com/articles/s41586-022-04813-y
Quote: “How quasars powered by supermassive black holes formed less than a billion years after the Big Bang is still one of the outstanding problems in astrophysics, 20 years after their discovery1,2,3,4. Cosmological simulations suggest that rare cold flows converging on primordial haloes in low-shear environments could have created these quasars if they were 104–105 solar masses at birth, but could not resolve their formation5,6,7,8. Semi-analytical studies of the progenitor halo of a primordial quasar found that it favours the formation of such seeds, but could not verify if one actually appeared9. Here we show that a halo at the rare convergence of strong, cold accretion flows creates massive black holes seeds without the need for ultraviolet backgrounds, supersonic streaming motions or even atomic cooling. Cold flows drive violent, supersonic turbulence in the halo, which prevents star formation until it reaches a mass that triggers sudden, catastrophic baryon collapse that forms 31,000 and 40,000 solar-mass stars. This simple, robust process ensures that haloes capable of forming quasars by a redshift of z > 6 produce massive seeds.”
– Very soon, we may be able to verify their past existence. The James Webb Space Telescope is turning its sensors to explore the farthest reaches of the Universe, looking back in time, back to the early universe that we could not see before.
The James Webb Space Telescope will be able to see astrophysical processes located much further (and therefore much older) than never before. Its “lookback time” is expected to reach the first few hundred million years after the Big Bang, and it will hunt signs of the first stars and black holes of the universe:
#NASA (2018): “NASA’s James Webb Space Telescope Could Potentially Detect the First Stars and Black Holes” (retrieved 2018)
Quote: The first stars in the universe blazed to life about 200 to 400 million years after the big bang. Observing those very first individual stars across such vast distances of space normally would be a feat beyond any space science telescope. However, new theoretical work suggests that under the right circumstances, and with a little luck, NASA’s upcoming James Webb Space Telescope will be able to capture light from single stars within that first generation of stars. [...] In addition to the first stars, Windhorst and his colleagues investigated the possibility of seeing accretion disks surrounding the first black holes. Such a black hole, formed by the cataclysmic death of a massive star, could shine brightly if it pulled gas from a companion star.
– So, with luck, we might be able to witness glimpses of these tragic titans in the brief moment between their formation and destruction. Until then, let us do the visual journey again, just for fun. Stars are big – Black hole stars are bigger.
To get a sense of proportion we can compare the size of a quasi-star with that of other known objects. Here we give their radii rounded off in units of the solar radius. Sources for all values can be found below.
OBJECT RADIUS (R⊙)
Earth 0.01
Neptune 0.035
Jupiter 0.1
Sun 1
Sirius A 1.7
Beta Centauri Aa/Ab 9
Sagittarius A* (BH) 17
R136a1 40
Gacrux 120
BL Lacertae (BH) 130
Pistol Star 420
Rho Cassiopeiae 740
Stephenson 2-18 2,000
Cygnus A (BH) 11,000
Messier 87 (BH) 28,000
OJ 287 (BH) 77,000
TON 618 (BH) 280,000
BH Star 800,000
The radius of the Sun is about 700,000 km:
#NASA (2021): “Our Sun”, Solar System Exploration – Our Galactic Neighborhood (retrieved 2022)
https://solarsystem.nasa.gov/solar-system/sun/in-depth/
Quote: “Our Sun is a medium-sized star with a radius of about 435,000 miles (700,000 kilometers).”
Solar System objects:
#NASA (2021): “Planetary Fact-Sheet” (retrieved 2022)
Sirius A (the brightest star in the night sky):
#Kervella, P. et al. (2003): “The interferometric diameter and internal structure of Sirius A”, Astronomy & Astrophysics, Volume 408, Number 2
https://www.aanda.org/articles/aa/abs/2003/35/aa3846/aa3846.html
Quote: “We obtain a uniform disk angular diameter of θUD = 5.936±0.016 mas in the K band and a limb darkened value of θLD = 6.039 ± 0.019 mas. In combination with the Hipparcos parallax of 379.22±1.58 mas, this translates into a linear diameter of 1.711±0.013 D⊙”
Beta Centauri Aa/Ab (twin stars):
#Kaler, Jim (2000): “Hadar (Beta Centauri)”, University of Illinois (retrieved 2022)
http://stars.astro.illinois.edu/sow/hadar.html
Quote: “A spin velocity of at least 140 kilometers per second coupled with a radius of 9 solar gives a rotation period for at least one of the stars of under 3 days.”
Sagittarius A* (supermassive black hole of the Milky Way)
The mass of this black hole is estimated at about 4 million solar masses:
#Event Horizon Telescope (2022): “Astronomers Reveal First Image of the Black Hole at the Heart of Our Galaxy” (retrieved 2022)
https://eventhorizontelescope.org/blog/astronomers-reveal-first-image-black-hole-heart-our-galaxy
Quote: “The new view captures light bent by the powerful gravity of the black hole, which is four million times more massive than our Sun.”
And by using the relation between the mass and the radius of a black hole explained above:
RBH (km) = 3 MBH / M⊙
for a mass of 4·106 M⊙ we get a radius of 12 million km, or about 17 solar radii.
R136a1 (one of the most massive stars known to date)
#Bestenlehner, Joachim M. (2020): “The R136 star cluster dissected with Hubble Space Telescope/STIS – II. Physical properties of the most massive stars in R136”, Monthly Notices of the Royal Astronomical Society, Volume 499, Issue 2
Gacrux (also known as Gamma Cru, a red giant and one of the brightest stars in the night sky):
#Rau, Gioia (2018): “HST/GHRS Observations of Cool, Low-gravity Stars. VI. Mass-loss Rates and Wind Parameters for M Giants”, The Astrophysical Journal, Volume 869, Number 1
https://iopscience.iop.org/article/10.3847/1538-4357/aaf0a0/meta
BL Lacertae (a supermassive black hole in a distant galaxy):
#Titarchuk, Lev et al. (2017): “BL Lacertae: X-ray spectral evolution and a black-hole mass estimate”, Astronomy and Astrophysics, vol. 602, A113
https://www.aanda.org/articles/aa/abs/2017/06/aa30280-16/aa30280-16.html
Quote: “This Γ − M˙ correlation allows us to estimate the black-hole (BH) mass in BL Lac to be MBH ∼ 3 × 107 M⊙ for a distance of 300 Mpc.”
As above, by using RBH (km) = 3 MBH / M⊙ for a mass of 3·107 M⊙ we get a radius of about 130 solar radii.
Pistol star (a blue hypergiant):
This work measured the temperature and luminosity of the star:
#Lau, R. M. (2014): “Nature versus Nurture: Luminous Blue Variable Nebulae in and near Massive Stellar Clusters at the Galactic Center”, The Astrophysical Journal, Volume 785, Number 2
https://iopscience.iop.org/article/10.1088/0004-637X/785/2/120
And, as explained above, from these parameters we can estimate the size of the star by using the Stefan-Boltzmann Law (the amount of energy per unit time and unit surface emitted by a hot body at a given temperature). By applying the Stefan-Boltzmann Law to the Sun and to the Pistol (PP) star, we get the simple relation:
RPP = (T☉ / TPP) 2 (LPP / L☉)½ R☉ ~ 420 R☉
where for the effective temperature of the Sun we’ve taken T☉ = 5,772 K.
#NASA (2022): “Sun Fact Sheet” (retrieved 2022)
https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
Rho Cassiopeiae (a yellow hypergiant):
#van Genderen, A. M. (2018): “Pulsations, eruptions, and evolution of four yellow hypergiants”, Astronomy & Astrophysics, vol. 631, A48
https://www.aanda.org/articles/aa/full_html/2019/11/aa34358-18/aa34358-18.html
This table quotes different temperature estimates calculated with different methods, and also different moments of the stellar cycle (Rho Cassiopeiae is a variable star). For definiteness, we’ll choose the plateau of the middle temperature estimate. This gives a stellar radius of about 740 solar radii
Stephenson 2-18 (one of the largest stars known to date, probably the largest one):
This was calculated above, with the result of about 2,000 solar radii.
Cygnus A (a supermassive black hole in a distant galaxy)
#Tadhunter, C. et al. (2003): “Spectroscopy of the near-nuclear regions of Cygnus A: estimating the mass of the supermassive black hole”, Monthly Notices of the Royal Astronomical Society, Volume 342, Issue 3
https://academic.oup.com/mnras/article/342/3/861/965048
Quote: “Assuming that the ionized gas is circularly rotating in a thin disc and that the large linewidths are due to activity-induced turbulence, the circular velocities measured from both the Keck and HST data lead to an estimate of the mass of the supermassive black hole of 2.5 ± 0.7 × 109 M⊙.”
By using the relation RBH (km) = 3 MBH / M⊙ for a mass of 2.5·109 M⊙ we get a radius of about 11,000 solar radii.
Messier 87 (also known as M87, a supermassive black hole in a distant galaxy)
#Event Horizon Telescope (2019): “Astronomers Capture First Image of a Black Hole” (retrieved 2022)
By using the relation RBH (km) = 3 MBH / M⊙ for a mass of 6.5·109 M⊙ we get a radius of about 28,000 solar radii.
OJ 287 (a supermassive black hole binary in a distant galaxy, whose primary member is one of the largest black holes known to date)
#Laine, Seppo et al. (2020): “Spitzer Observations of the Predicted Eddington Flare from Blazar OJ 287”, The Astrophysical Journal Letters, Volume 894, Number 1
https://iopscience.iop.org/article/10.3847/2041-8213/ab79a4/meta
Quote: “The present BBH model, extracted from the accurate timing of 10 flares between 1913 and 2015 (D18), is specified by the following parameters: primary with mass 1.835 × 1010 M⊙ and Kerr parameter a = 0.38, and a 1.5 × 108 M⊙ secondary in an eccentric (e ∼ 0.65) orbit with a redshifted orbital period of 12 yr.”
By using the relation RBH (km) = 3 MBH / M⊙ for a mass of 1.8·1010 M⊙ we get a radius of about 77,000 solar radii.
TON 618 (one of the largest supermassive black holes known to date, probably the largest one)
This was calculated above, with the result of 2·1011 kilometers, or about 280,000 solar radii.