# Formation of Supermassive Black Holes

August 21, 2017

## Abstract

Supermassive Black Hole (SMBH) formation is still a topic of much discussion due to the great difficulty that arises when attempting to directly study these SMBHs. There are many theories that exist as far as formations, and there are a couple common themes in these theories; Supermassive stars play a key role in the formation of these SMBH and SMBH can stem from direct collapse. For black holes to become supermassive they must surpass a specific mass and once they become massive enough they also end up achieving a pre-determined accretion rate. Quasi-stars, similar to supermassive stars in composition, are thought to produce SMBHs under specific circumstances. These quasi-stars have structures that begin similar to pop III stars but as the stars evolve they grow into something entirely different, the core of the star collapses into a black hole as the surrounding gas still holds as a star. It is also thought that after a long enough period of time, a black hole has the potential to accrete matter until it reaches the SMBH status.

## Introduction

Supermassive Black Holes have been a largely debated topic since their discovery. The debate stems from the difficulty of studying them. Black holes bend the light around them, thereby making it impossible to look directly at them. To study them the area and light around them needs to be carefully studied. A SMBH is typically characterized by a black hole of a mass ~$$10^{5}M_{\odot}$$. These black holes have an accretion rate $$\geq.1M_{\odot}yr^{-1}$$ (Fiacconi 2016). Studying these SMBH is very important in learning about the early universe as a SMBH has a very strong gravitational force. The strong gravitational force plays a crucial role in the development of galaxies as they are at the center of a vast majority of galaxies. The debate over the creation of these SMBH’s seems to come down to a few different theories. The first of which is that they are the direct result of a Supermassive Star. It is thought that when stars with a large enough mass collapse they have a chance of collapsing directly into a SMBH. Another theory is that of Direct Collapse. Direct Collapse is a method that would have been more prevalent in the early universe. The process occurs in pregalactic halos when matter begins to lose angular momentum and leads to a quick buildup of a dense self-gravitating core which gradually condenses into a black hole and then a supermassive black hole. Fairly recently, observations of redshifts $$z\geq 6$$ have proven the existence of quasars with masses $$\geq 10^{9}M_{\odot}$$. The existence of these quasars presents an issue on the formation of these SMBHs. Through the use of models it is not thought that SMBH’s would be able to grow at a rate fast enough to reach such a mass at $$z\sim 6$$ (Fiacconi 2016). A purposed solution to this issue is the existence of Quasi-stars. These stars have a mass $$\sim 10^{5-6}M_{\odot}$$ with a core mass of $$\sim 10^{4-5}M_{\odot}$$. The core of a quasi-star is a black hole, which is constantly fed by the surrounding star. The black hole at the core forms after the core has exhausted the nuclear reactions and then collapses. The newly formed black hole is then surrounded by the mass of the star’s original envelope which continues to contract, but on a longer timescale. At this point it is purely speculation on how such a star would maintain itself without collapsing into the black hole at the core or bursting outward from the accretion luminosity. However, a central accretion disc must form around the sphere of influence of the black hole in order to send the accretion energy outward. Black holes that exist within these quasi-stars have the ability to grow at the Eddington rate of the hosting quasi-star, rather than that of the black hole itself, until the black hole reaches a mass  $$10^{3}M_{\odot}$$.

## Direct Collapse

When self-gravitating gas in dark matter halos quickly loses angular momentum and collapses in on itself there is potential to form a black hole, and eventually a SMBH. Not every dark matter halo has the potential to eventually contain a SMBH has certain conditions, other than the amount of gas in the halo, must be met. The gas in these halos has a virial temperature($$T_{vir}$$) $$\geq 10^{4}K$$ which plays an important role in the dynamical stability of the gas (Begelman 2006). Such gas would likely exist in larger halos as opposed to smaller halos due to the likeliness of the gas being more dense at the center. In halos where the $$T_{vir}$$ does not reach the appropriate range it is unlikely that direct collapse could occur since tidal forces would prevent a large amount of gas from collapsing and would cause fragmentation before a black hole could form. Fragmentation creates many problems for forming black holes, and massive stars. When forming supermassive stars and SMBH it is important that gas does not fragment as very early stages of stars can be seen and the gas begins to clump around several areas as opposed to just a single central point. For such a collapse to be possible, observations need to be made a $$z\geq 6$$ where gas clouds would not be polluted by metals and therefore metal line cooling of the collapsing gas would not happen.

It is also of importance that since these gas clouds do not contain metals they primarily contain hydrogen and helium allowing for more massive objects to form. Rotational energy is crucial in a system before the collapse, if the rotational energy becomes too large then the system would become unstable. An equation to calculate the stability looks like

$$(\dfrac{1}{2}f\dfrac{T}{W})^{1/2}<0.34\\$$

which was produced by Christoudoulou et al.(Christodoulou 1995). Where $$T$$ is the rotational kinetic energy, $$W$$ is the gravitational potential energy, and $$f$$ is a geometric dependent parameter and for discs $$f=1$$. A system that is unstable is expected to result in runaway infall so long as the gas collapsing remains cooler than the local $$T_{vir}$$. Such large-scale gravitational instability becomes more difficult as the radius of the halo decreases.

As the black hole begins to form the timescale is limited by the free fall time at the initial radius $$r_{0}$$, i.e. $$t_{0}\sim\dfrac{r_{0}}{v_{0}}$$ where $$v_{0}$$ is the virial speed. As the observation is shifted inward to smaller radii the timescale becomes noticeably longer. As the radius becomes smaller and smaller the free fall time becomes much shorter but the matter that is flowing into these smaller radii occurs on a longer timescale which results in a mass flux through the region. Begelman(2006) finds that the free fall time, $$t_{ff}$$ is correlated to the radius and the mass enclosed within that radius as:

$$t_{ff}\sim(\dfrac{GM(r)}{r^{3}})^{-1/2}\\$$

which can then be translated to the mass flow rate through a region

$$\dot{M}_{0}\sim\dfrac{M(r)}{t_{ff}(r)}\propto(\dfrac{M(r)}{r})^{3/2}\\$$

and since $$\dot{M}_{0}$$ is constant, then $$M(r)\propto r$$ and $$\rho\propto r^{-2}$$. These remain true as long as the gas does not begin to fragment. Should the gas begin to fragment it may begin star formation, although this is unlikely to produce an issue as the potential energy of the gas is more or less constant for $$r<r_{0}$$ and the $$T_{vir}$$ does not differ much from that of the halo it exists in rendering fragmentation inefficient.

As disc accretion occurs towards the center of the mass there is a constant outflow of energy that has been liberated from the mass flowing inward. As the radiation is released it also generates a luminosity emitted from a radius $$r$$ as

$$L(r,t)\sim\alpha^{2}\dfrac{v_{0}^{5}}{G}[\dfrac{r}{r_{1}(t)}]^{-1}\\$$

where $$\alpha$$ is the viscosity parameter (Begelman 2006). Out to a certain radius the radiation is trapped and results in a build-up of pressure which then supports a cloud of gas around it which has been named a quasi-star.

## Quasi-Stars

As the mass of the quasi-star exceeds that of $$3-4M_{\odot}$$ the pressure becomes radiation dominated. At this point the interior quasi-star is hot enough for hydrogen ionization and makes way for the assumption that opacity will be dominated by electron scattering. The radius of the newly formed quasi-star is found to be time independent and is expressed by Begelman(2008)

$$r_{*}\sim\dfrac{\alpha\kappa V_{0}^{3}}{4\pi Gc}=1.6\times 10^{13}\alpha v_{10}^{3}cm\\$$

where $$v_{10}$$ is $$\dfrac{1}{10}$$ of the virial speed $$v_{0}=(\dfrac{GM_{0}}{r_{0}})^{1/2}$$. When the mass of the quasi-star is in the range of $$10^{2-3}\alpha v_{10}^{3}$$ and core temperature is $$10^{6-7}$$K nuclear burning begins. Once burning begins in the core, the core appears similar to that of population III stars, also known as the first stars. The first stars differ from other stars as they were primarily composed of hydrogen and helium, and it is because of this composition that the first stars were much more massive than those that form today in clouds of gas that contain metals and have hydrogen mass fractions that are much larger than observed in modern times. While the core appears similar to that of pop III stars, it is the extreme infall rate that ultimately decides the quasi-star evolution. The constant inward flow and compression of matter prevents the outward radiation of energy from the core and therefore prevents the core from becoming degenerate or collapsing. However, once the core temperature $$T_{c}\sim 5\times 10^{8}K$$ and the mass $$M_{*}\sim 3600\alpha v_{10}^{3}$$ the core collapses into a black hole (Begelman 2006).

In regards to the quasi-star angular momentum is very important as it limits the amount of mass that falls into the growing black hole at the core. If angular momentum played no role in the accretion of the black hole, the entire quasi-star would collapse into the black hole in a single free fall time. Due to this importance, some of the binding energy from the inward falling mass is released near the black hole and is transported outward through torque. The energy is radiated outward from the core and therefore acting on the mass flowing inward slowing it down. The black hole continues to grow at a rate, nearly Eddington due to feedback radiation, until it reaches a mass that is only a couple factors smaller than that of the quasi-star at which point the black hole’s accretion begins to slow to a sub-Eddington rate and becomes limited solely by the inflow of gas into the quasi-star. The black hole has not yet reached the classification of SMBH as the growth becomes sub-Eddington in the range of $$\sim 10^{4}M_{\odot}$$. While the growth would slow down, it would still have ample time to grow to a SMBH classification as these quasi-stars form exclusively in halos that are un-polluted by metals and therefore form in the early stages of the universe.

## Supermassive Stars

In addition to quasi-stars, stars with masses $$M_{*}\geq 10^{5}$$ also hold the potential to collapse into black holes. It is feasible that supermassive stars can collapse into black holes paving the way for SMBH by a redshift of $$z=6$$ due to the nature of supermassive stars living short lives. These lives are typically on the order of $$10^{6}yr$$ which leaves plenty of time for massive black holes left over to accrete mass from the surrounding halo to reach the classification of supermassive. For such a star to form the self-gravitating gas must fall inward at a very large rate $$\geq.5(\dfrac{M_{*}}{10^{6}M_{\odot}})M_{\odot}yr^{-1}$$ (Begelman 2009). This leaves room for problems in the formation as fragmentation and forming stars can occur before the gas reaches the center. Such a star will only be able to form in a dense gas cloud with $$T_{vir}\geq 10^{4}K$$ and the gas must remain at temperatures close to $$T_{vir}$$.

Supermassive stars and quasi-stars are closely related in formation. The final and most important point that decides the evolution of the star is that of the mass accreted as the protostar evolves. Once the protostar reaches a critical mass of $$3.6\times 10^{8}\dot{m}M_{\odot}$$, it is $$\dot{m}$$ the decides which it will evolve into. ˙m is the rate at which the mass of the star changes, which is characterized by the halo the star formed in and how fast the gas is flowing in towards the protostar. for a protostar to transition into a supermassive main-sequence star ˙m must be $$\leq.14$$ whereas if $$\dot{m}>>.14$$ the star’s core will collapse into a black hole and transition into a quasi-star (Schleicher 2013).

Supermassive stars are generally fully convective and therefore have the ability to burn the majority of hydrogen during the main sequence. For supermassive stars that are not fully convective problems arise that have the capability of shortening the life of the star. There is the matter of the envelope of the supermassive star being convectively stable, but if the star is not fully convective then the envelope will not mix with the core and in turn restricts the flow of the fuel to the core. In addition, to maintain equilibrium in the star the luminosity required is greater than that of the Eddington limit of the star. This luminosity calculated by Begelman(2009) is found to be

$$L_{nuc}\approx(1+\dfrac{R_{tr}}{R_{*}})L_{E}(M_{*})\approx\dfrac{8}{3}\dfrac{4\pi G\dot{M}c}{\kappa}t\\$$

where $$R_{tr}$$ is the trapping radius and $$L_{E}$$ is the Eddington Luminosity. Therefore a supermassive star, to remain stable, must become fully convective. To support such a massive star the source of the energy must be the CNO cycle (HOYLE 1963). Using the CNO cycle the supermassive star will reach equilibrium rather quickly after producing the required CNO elements through the $$triple-\alpha-process$$. After the supermassive star burns through the hydrogen the core contracts and therefore heats up which leads to nuetrino losses and then a collapse. In order for the collapse to occur the angular momentum must be taken into account. If the angular velocity is below a threshold defined by:

$$(\dfrac{\Omega}{\Omega_{K}})^{2}<\dfrac{GM(R)}{c^{2}R}\\$$

where $$\Omega$$ is the angular velocity and $$\Omega_{K}$$ is the Keplerian angular velocity $$\Omega_{K}=(\dfrac{GM}{R^{3}})^{1/2}$$, then the star will collapse into a black hole without any angular momentum transportation (Begelman 2009). On the other hand if the angular momentum is too great transport will be required for the star to collapse into a black hole. To transport angular momentum outward, the star must inevitably also transport energy outward (Blandford 1999). One way in which the angular momentum can be transported is through an astrophysical jet. A typical timescale on which the star collapses on a black hole is expressed in Begelman(2009)

$$t_{BH}=\dfrac{M_{BH}}{\dot{M}_{BH}}=450\epsilon_{-1}m_{BH}^{-1}m_{*,6}^{3/4}T_{c,8}^{-5/2}yr\\$$

Where the normalization factor used to account for black hole efficiency is $$\epsilon_{-1}$$, $$m_{*,6}$$ is the mass of the star $$\dfrac{m_{*}}{10^{6}M_{\odot}}$$, and $$T_{c,8}$$ is the core temperature $$\dfrac{T_{c,8}}{10^{8}}$$. Through the use of simulations it has been found that the black hole will engulf $$\sim\dfrac{2}{3}$$ of the mass of the star and it is expected that the black hole will continue to accrete the stars mass up until it reaches about $$\dfrac{9}{10}$$ of the total mass (Shibata 2002). Since these supermassive stars form very early in the universe and live such short lives, the remaining black holes have masses $$M_{BH}\geq 10^{4}$$.

## Discussion and Conclusion

While the formation of SMBH is not yet fully understood there are several cases with compelling arguments and data that are plausible. In the early universe when clouds of gas were still unpolluted by metals and elements other than hydrogen and helium supermassive stars were able to form more easily than in the current universe. The first stars, pop III, have been thought to have been much more massive than that of stars observed today on the scale of $$M_{*}>10^{6}M_{\odot}$$. Due to the nature of massive stars these supermassive stars lived short lives as the fusion reactions occur at a greater rate than that of a less massive star. Theories have been produced about the origination of SMBH and fingers point to the existence of the theoretical quasi-star. The quasi-star is similar to a supermassive star in mass and outer structure, but the core of such a star is a black hole. The star continually feeds that black hole at it’s core until the $$M_{BH}$$ grows comparable to that of the quasi-star. At which point the star falls apart leaving a black hole on the order of $$M_{BH}\sim 10^{4}M_{\odot}$$. Supermassive stars produce similar black holes that have been simulated to engulf up to ninety percent of the stars original mass. Leaving black holes with masses on the order of $$M_{BH}\sim 10^{5}$$.

Both black holes leftover from supermassive stars and quasi-stars would appear $$\sim 10^{6}yr$$ after the star formation. With redshift surveys producing images of bright quasars at redshifts of $$z\geq 6$$, it implies that SMBHs existed $$\leq 10^{9}yr$$ after the Big Bang (Ball 2011). The leftover black holes would continue to accrete mass from the clouds they appear in and have the potential to reach such masses. While there are conditions that need to be accounted for, mainly the existence of gas clouds large enough to produce such a SMBH, these methods seem much more likely than that of a black hole being produced by a star of mass $$M_{*}\sim 100M_{\odot}$$. While still possible a black hole of mass $$M_{BH}\sim 100M_{\odot}$$ would take approximately $$7\times 10^{8}$$ yr to grow to $$10^{9}M_{\odot}$$ while accreting at Eddington Limit which is about the age of the universe at $$z$$ = 6. This is even less plausible when taking into account the chances that a black hole could have enough surrounding mass to accrete at it’s Eddington Limit for such a period of time. While SMBH formation is still mostly speculative there are arguments with a plethora of data supporting them. It is important for SMBH to be continually studied as they hold the key to learning how galaxies, such as the Milky Way and many others, form as SMBH lie at the center of most galaxies.