Tully-Fisher vs. Kepler
The Omega Centauri star cluster (Fig. 1) with a mass of 4 million solar masses has a radius of about 90 light years. On July 10, 2024, a sensational article was published in the journal Nature (https://www.nature.com/articles/s41586-024-07511-z) about the discovery in this cluster of a black hole with a mass 8200 times greater than the mass of the Sun. The main witnesses of the existence of this invisible object were fast stars, which without the gravitational field of the massive hole should have left the cluster very quickly – within a thousand years.
This discovery comes amidst dramatic changes in the cosmological paradigm. Globular clusters are fascinating objects whose role has been greatly underestimated. They appear to be responsible for the mysterious Tully-Fisher relation. What is its mystery?
According to Kepler's law, which even schoolchildren know, the square of the speed V of circular rotation of bodies around a massive object is proportional to the mass M of this object:
Artificial satellites, planets and asteroids obey this law, and it is logical to assume that stars at the edges of spiral galaxies should also obey this law (Fig. 2).
But in 1977, observers R. Tully and J. Fisher discovered the M∝V pattern.4which relates the mass of the disk galaxy M and the rotation velocity V at its edge. The dependence M∝V4 seems to contradict formula (1), from which it follows that M∝V2. Note that at the edge of our galaxy the gravitational acceleration is extremely small ~10-8 cm/sec2therefore, factors may be at work there that do not manifest themselves under conditions of stronger gravitational fields (on the Earth’s surface, gravitational acceleration is 11 orders of magnitude greater).
The sharp contradiction between the Tully-Fisher relation and the classical Kepler law is simply shocking. The shock of the Israeli scientist Mordechai Milgrom was so strong that he abandoned the Newton-Einstein theory of gravity and proposed a modified theory of gravitation (MOND), where he introduced a phenomenological gravitational potential at the edge of the galaxy, ensuring strict fulfillment of the Tully-Fisher law. MOND features a new constant – the value of gravitational acceleration 1.2*10-8 cm/sec2Generally speaking, this is a classic case of the theory ad hoc – that is, a theory that does not follow from any fundamental laws, but is introduced specifically to explain some specific observational phenomenon.
Let us clarify that the Tully-Fisher mystery lies separately from the problem of dark matter – after all, the latter was introduced to explain the constant linear rotation speed of galaxies and does not impose any restrictions on the magnitude of this speed at the edge of galaxies.
The Tully-Fisher law is not the only mysterious phenomenon in the complex relationships of galactic masses and velocities. In 1976, S. Faber and R. Jackson discovered a similar function M ∝ σ4 for elliptical galaxies (where σ is the one-dimensional velocity dispersion). It is striking that this mysterious fourth power of velocity appears in such different systems as flat spiral galaxies, where stars move in nearly circular orbits, and massive elliptical galaxies, which rotate slowly and whose stars move in highly elongated and inclined trajectories.
Then something completely incredible happened: in 2000, Ferrarese & Merritt and Gebhardt et al. established the so-called M-σ or “M-sigma” relationship between supermassive black holes (SMBH) and the velocity dispersion of stars in the galactic bulge (see Fig. 2): M0∝σ4. The mass of the central SMBH M0 is usually a small fraction (~0.1%) of the bulge mass. However, the “M-sigma” ratio indicates a close relationship between such a small relative mass of the SMBH and such a global parameter as the stellar velocity dispersion in the bulge. (Note that the orbits of stars in the galactic bulge have a large eccentricity, comparable to the eccentricity of orbits in elliptical galaxies.)
The amazing “M-sigma” relation turned out to be in a series of similar Tully-Fisher and Faber-Jackson dependencies. Obviously, some single mechanism is at work in all three phenomena, but how were the velocities of circular rotation at the edge of spiral galaxies related to the velocities of stars on eccentric orbits in bulges and elliptical galaxies? And how did a bulge component a thousand times less massive than the bulge itself manage to influence the distribution of velocities in this bulge? For many decades, there were no answers to these pressing questions.
In recent years, the standard theory of the hierarchical and gradual formation of galaxies, developed by inflationists, has been questioned by observers (O.K. Silchenko, Origin and Evolution of Galaxies, Vek2, Fryazino, 2017). An alternative theory has emerged, which suggests that central supermassive black holes were formed not after the formation of galaxies, but before them – and it was the supermassive holes that became the center of condensation of galactic matter (A.M. Cherepashchuk, UFN, 184, 387, 2014). This concept of the importance of black holes in the dynamics of the Universe is logically complemented by the convincing hypothesis that dark matter consists of numerous black holes of stellar masses.
We will show that in this scenario (massive black holes are primary, galaxies are secondary) there is every possibility for solving the Tully-Fisher puzzle and other similar relations.
When the age of the expanding Universe reached 380 thousand years, the proton-electron plasma cooled to 3000 Kelvin and turned into neutral hydrogen. The isotropy of the relic radiation that escaped at that moment from the transparent medium of neutral hydrogen shows that the Universe at that moment was very homogeneous. According to new views, in this medium there was a certain number of supermassive holes, as well as a large number of black holes of stellar mass in the role of dark matter.
In a homogeneous medium of hydrogen and dark matter, a Jeans gravitational instability with a wavelength of:
The mass of the forming Jeans cloud can be estimated as follows (Zel'dovich and Novikov, 1975):
The gravitational acceleration at the boundary of such a Jeans cloud with a diameter of 50 light years (i.e. equal to the region of increasing density at half the wavelength of oscillations) can be estimated as aj~GMj/r2 ~10-8 cm/sec2. Here, in the brain of every scientist (or just homo sapiens) a red alarm goes off: why, in the Universe, whose parameters vary by tens of orders of magnitude, does the magnitude of gravitational acceleration at the boundary of Jeans clouds (a hundred light years in size) and at the edge of galaxies (a hundred thousand light years in size) coincide? The difference in size is a thousand times!
When the question is correctly posed, the answer is not difficult to find. Let us imagine the formation of a galaxy around a supermassive black hole, which captures the inadvertently approaching Jeans clouds, tears them apart and builds a galactic disk (or ellipsoid) around itself. This disk grows and grows until it reaches such a size that the gas (or stars) at its edge begin to be attracted to the center of the galaxy so weakly that the acceleration of attraction reaches a value of 10-8 cm/sec2. And yet numerous Jeans clouds continue to float around the young galaxy, gradually turning into globular clusters of black holes and stars. These globular clusters disturb at a level of ~10-8 cm/sec2 stars at the edge of the galaxy, driving them inward or tearing them away completely. Thus, gravitational disturbances from Jeans clouds limit R – the radius of a galaxy with mass M by the condition:
In celestial mechanics, this region is called the “gravitational sphere.” Now let's do a simple mathematical exercise (you can do it yourself!): we obtain from equation (4) an expression for the radius R∝M1/2 and substitute it into formula (1). By squaring the resulting expression, we obtain the Tully-Fisher law in analytical form (N. Gorkavyi, Galaxies, 10: 73, https://doi.org/10.3390/galaxies10030073, 2022):
Relation (5) is amazingly simple and logical in explaining the famous mysterious Tully-Fisher relationship between the mass of galaxies and their peripheral rotation velocity.
Our reasoning refers to the early Universe. Will constraint (4) work in the modern era? Of course, because the Jeans clouds are still here, they are stable clusters of black holes and stars. These clusters were gradually captured in the halos of galaxies, where they move along eccentric and highly inclined orbits (note that the Omega Centauri cluster moves along a retrograde orbit – that is, it rotates around the center of the galaxy in the direction opposite to the rotation of the galactic disk). The gravitational parameters of these ancient clouds have changed little – and they still limit the sizes of galaxies by condition (4). Globular clusters, like piranhas, bite off stars from the edges of galaxies whose gravitational bond with the main galaxy has weakened.
Similar logic works for elliptical galaxies: no matter what orbit a star moves along, when it reaches the edge of a galaxy, it experiences disturbances from globular clusters and obeys condition (4). The only difference is that to calculate the speed of motion of bodies in elliptical galaxies, one must use not the condition of circular motion (1), but the virial theorem for one-dimensional velocity dispersion, which gives an analytical expression for the Faber-Jackson law:
Let us consider the origin of the “M-sigma” relation. From condition (4) it follows that the area of the galactic disk is proportional to its mass: πR2∝M. The accretion growth of a galaxy (and its bulge) directly depends on the area of its disk, and, consequently, on its mass:
Equation (7) leads to an exponential growth law for the galaxy M=M0exp(γt), where γ is the growth increment. Note that the area of the initial accretion disk was directly dependent on the size (or mass) of the black hole that served as the seed for the formation of the galaxy. Consequently, under the mass M0 we can understand the mass of the initial black hole (maybe with some coefficient). Given equation (6), we get the justification of the “M-sigma” relationship: the mass connection SMBH M0 and one-dimensional dispersion of chaotic velocities σ in the bulge:
Thus, the difficult to explain “M-sigma” dependence was easily obtained for the model of the Universe with a population of relic supermassive black holes. The “M-sigma” relation can be considered a direct consequence and confirmation of the accretion theory of galaxy formation around supermassive black holes (Fig. 4).
The size distribution of black holes includes not only stellar-mass black holes (<100 solar masses) and supermassive holes (>100,000 solar masses), but also some intermediate-mass black holes. When the vast number of Jeans clouds formed in the early Universe, they were mostly made up of stellar-mass black holes. But some clusters were lucky enough to contain intermediate-mass black holes. Such a hole alone has a greater ability to capture gas than all the other small holes in a globular cluster. It is this gas, captured by the intermediate-mass black hole, that collapses into stars, turning the dark Jeans cloud into a shining globular cluster (Fig. 4).
The hypothesis that all globular clusters must have an intermediate-mass black hole in their center was first confirmed by the Mayall II star cluster near the Andromeda Galaxy, which has a mass of 10 million solar masses and a central black hole of twenty thousand solar masses. The Omega Centauri cluster, with a central hole of 8,200 solar masses, is located in our Galaxy and serves as another important confirmation of the hypothesis of massive holes inside globular clusters of stars.
To summarize the above: globular clusters are the most ancient collective structures of the Universe, having arisen before galaxies. Interacting with supermassive holes, clusters actively participate in the growth of spiral and elliptical galaxies. Gravitational disturbances from globular clusters limit the peripheral velocity and size of galaxies, leading to Tully-Fisher type dependencies. Globular clusters are formed by stellar-mass black holes and are practically invisible, constituting dark matter – the bulk of the mass of the Universe (https://elementy.ru//novosti_nauki/434235/Gipoteza_o_tsiklicheskoy_Vselennoy_poluchila_nablyudatelnuyu_podderzhku). If the cluster contains an intermediate-mass black hole, the gas captured by this hole turns the dark cluster into a globular star cluster. Masses of gas captured by supermassive black holes form larger stellar structures – galaxies.
The process of the emergence of the observed distribution of black holes, as well as the reasons why spiral disk galaxies form near some black holes, and elliptical galaxies near others, are described in the author’s book “The Pulsating Universe” (Piter, 2024, 2nd edition).
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