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| Fig. 1: Artist conception of a black hole accreting from a companion star. The gas being stripped off the stellar companion forms an accretion disk around the black hole. (Source: Wikimedia Commons) |
Although this may be almost too far in the future to even consider, eventually the energy needs of humanity may vastly exceed the capabilities that we can extract from things such as fossil fuels. In this situation we will need to turn to alternative energy sources. Solar energy is an obvious candidate because it will allow us to harness the energy naturally freed from thermonuclear reactions in the sun. Alternatively, fusion itself when it is realized will be lucrative and should prove able to provide a substantial source of energy. However there still might come a point where fusion itself will not be able to be conducted on a large enough scale to fully satisfy our energy needs. Notably, even though the efficiency of fusion vastly exceeds that of fossil fuels by roughly a factor of ten thousand (due to the much larger energy stored in nuclear bonds rather than chemical bonds), the total energy liberated in turning hydrogen into iron is roughly 8 MeV/nucleon. [1] Given that a proton or neutron has a total energy of roughly 1 GeV, this amounts to a total efficiency of 0.8 percent of the rest mass energy. In other words, for a lot of fuel, this provides a relatively small amount of energy.
Although fusion is often considered to be the best form of energy production, it is surprisingly not the most efficient means that exists in the universe. Several alternative methods of energy production exist in theory that are able to produce a larger yield of energy for a smaller input of fuel. In the text that follows I will describe one of these methods. Interestingly, it involves a black hole. Thus these methods are attainable only in science fiction at the moment, and are unlikely to be realized in the lifetime of anyone that is reading.
When a black hole siphons gas off of a star (usually one that is orbiting a black hole, as illustrated in Fig. 1), but it can also be fed by a wind from a massive star), the gas has angular momentum that keeps it from being immediately fed into the black hole. This gas forms into an accretion disk that orbits the black hole. Viscous dissipation in the accretion disk heats up the gas to millions of Kelvin, producing substantial amounts of radiation (mostly X-rays). This radiation escapes the gravitational pull of the black hole and thus the energy is liberated from its gravitational bonds.
Where does this energy come from? It comes from gravitational binding energy. In general, binding energy is the energy that is necessary to break a system apart. This means that a system with a higher binding energy requires more energy to be put in for the bonds holding it together to be broken. Binding energy is not unique to particle orbits around a black hole, and in fact can be found in nuclear interactions, as well as electromagnetic interactions. In order to increase the binding energy of a system, you have to extract energy from the system. This seems slightly counterintuitive, since we are more used to speaking about energy as something that is stored and able to be extracted. Binding energy is the opposite. It is increased by extracting energy, and in order to lower it energy must be injected into the system.
In an accretion disk, gas becomes more strongly bound to the black hole as it moves inward. Thus its gravitational binding energy increases. This necessitates that energy was released from the gas, which can in principle be extracted by a future civilization. For now, lets ignore how the civilization is able to do this, but instead focus on how much energy they would have available for extraction.
In an accretion disk, a number of processes work to release energy. If we think of the gas as a perfect fluid with no friction or viscosity, then particles would radiate away energy in the form of light (thermal emission typically), but they could not remove their angular momentum. Thus eventually all particles would exist in stable, circular orbits (since circular orbits are the lowest energy orbit for a given angular momentum). By including the effects of viscosity, we provide a means by which the gas is able to transport angular momentum outward. However, it is generally thought that throughout the majority of their time in the accretion disk, particles travel in quasi-circular orbits because they are able to remove energy much faster than angular momentum. [2]
A typical black hole in our galaxy has a radius of a few kilometers, while the radius of the typical accretion disk varies, but is thought to be a few hundred times larger. [3] Near to the event horizon of the black hole, the effects of general relativity become important, but they are very small at the outer radius of the accretion disk. The binding energy of a particle orbiting in a circular orbit at a distance r away from the center of a black hole with mass M in general relativity is:
where rg=GM/c2 is the gravitational radius. [2] As we can see, binding energy approaches 0 very far away from the black hole. This makes sense, because we wouldn't expect a particle infinitely far away from a black hole to be bound to it. As the particle moves inward and r decreases, the binding energy increases, until it peaks at 6rg. At this radius, the binding energy is 6 percent of the rest mass energy of the particle. This means that in the process of falling from the outermost edge of the disk (where the binding energy is very low) to 6rg an amount of energy equivalent to nearly 6 percent of the rest mass energy of the particle must be released. A different way of putting this is that the efficiency of an accretion "reaction" is 6 percent.
An obvious question to address here is "why does the particle stop at 6rg rather than continue to slowly spiral inward? The answer to this is that orbits with radii less than 6rg become unstable, meaning that once a particle reaches that point, any perturbation on the orbit causes the particle to rapidly decay to the surface of the black hole. This happens so rapidly that it likely does not have time to cause any additional changes to the energy budget. Because of this, 6 percent is the typical efficiency quoted for an accretion disk around a non-rotating black hole. [2] This efficiency is different for a rotating black hole depending on how rapidly the black hole is spinning, but it can be as high as 42 percent. [2]
How does this compare to other means of energy generation? To illustrate the power of accretion relative to other processes, lets consider what happened if we were to use a tonne of fuel in a chemical process, fusion process, and accretion process. A typical chemical reaction (such as burning a tonne of fuel) process produces one TOE of energy. A fusion process with the same amount of material can, at best, produce roughly 107 TOE of energy. This is substantially higher than chemical reactions. However, by accreting onto a rotating black hole, this same tonne will produce 109 TOE of energy. Because the energy released in accretion is so large compared to everything else, the energy produced by other processes (chemical reactions and nuclear reactions) can safely be ignored when calculating the efficiency of accretion.
Although accretion has a high theoretical efficiency, there are a number of reasons why this picture may be complicated, particularly when it comes to extracting energy using an accretion disk. The simplest problem comes from the fact that the energy being extracted all comes in the form of emitted light. However, not all the light will make it away from the black hole. Some of it is re-absorbed by the disk or falls onto the black hole (meaning that energy will be unable to ever be collected or used). It is difficult to say exactly how much radiation does this for the following reason. When a particle produces a photon, that photon proceeds in a random direction. Because the particle is traveling at a substantial fraction of the speed of light, because of light bending effects of the black hole, and due to gravitational redshift, the paths and energies of observed photons produced in the disk require sophisticated ray tracing software to compute, and it is difficult (though not impossible given the correct tools) to say exactly how much energy escapes compared to how much goes into the black hole. This is not strictly speaking an uncertain quantity, but an exact determination is beyond the scope of this paper, since extracting this energy in a meaningful way is impractical.
In fact, many of the other complications occur due to the impracticality of producing a device to harness the energy that is being produced. First, is obviously that the nearest known black hole is roughly 1 kiloparsec away (A0620-00), so it would take a long time to get to one even if traveling at near the speed of light. [3] Second is that the radiation is produced more or less isotropically, and so in order to reach the peak theoretical efficiency of 6-42 percent one would need to surround the entire black hole binary (since the companion star provides the fuel). This would make the material requirements for such a structure quite large.
There are a few other potential pitfalls that arise due to aspects of accretion that have been observed in black hole binary systems. Mainly, black holes are observed to go through many different phases of accretion and it is not yet well understood how these might impact the efficiency of radiation production. In particular, a radiatively inefficient accretion flow may be able to be sustained, meaning that matter accretes onto the black hole without producing a substantial amount of harnessable energy. In fact, it has been suggested that black holes accrete using an advection-dominated accretion flow (a type of radiatively inefficient accretion flow) during periods of low accretion rate - meaning that these systems may lose additional efficiency due to the fact that they are not always running in their most efficient state. [3] There is also evidence for winds and jets from these systems. both of which could wreck havoc on the tools designed to collect the radiated light from the disk. [3]
To quickly summarize, accretion of material onto a black hole liberates six percent of the rest mass energy of that material into radiation. This vastly exceeds the peak efficiency of any conventional fuel and is a factor of up to 52 times more efficient than nuclear fusion. However what accretion offers in efficiency, it certainly makes up for in impracticality. It is likely that the energetic costs of producing any sort of device to harness the energy of an accretion disk will exceed the energy returns from that device. Therefore although accretion is an interesting concept, it is better suited for science fiction at the moment.
© Warren Morningstar. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] P. Schneider, Extragalactic Astronomy and Cosmology (Springer, 2006), 186.
[2] M. J. Rees, "Black Hole Models for Active Galactic Nuclei," Annu. Rev. Astron. Astr. 22, 471 (1984).
[3] R. A. Remillard, J. E. McClintock, "X-ray Properties of Black Hole Binaries," Ann. Rev. Astron. Astr. 44, 49 (2006).
