Most recent, theoretical research on Active Galactic Nuclei, AGN, has been based upon impressive simulations (e.g., [
11]). These have taken a variety of forms—fluid, MHD and kinetic and incorporated radiative transfer, plasma physics—as well as general relativity. However, an AGN is much more than the region captured by the EHT image of M87 and it is impossible to achieve the dynamic range needed to span the full range of relevant length scales. One approach to connecting simulations across such a large range of scales is to focus on the physics conservation laws and it now seems time to pay more attention to these.
2.3. Energy
Gas flowing inward through a disk carries with it a negative binding energy . So, in contrast to the angular momentum, most of the energy resides at small radius. In a standard accretion disk, this energy is liberated and radiated away. However, in addition, the torque that was invoked to balance the flow of angular momentum will do work at a rate , where is the Keplerian angular velocity. The divergence of this flow of energy trebles the release of energy and the luminosity. The extra luminosity derives from a reduction of the power radiated at small radius, where most of the binding energy resides, and requires efficient outward energy transport by the internal torque. If the inflow is adiabatic with small radiative loss, then power is needed to give the gas more than the escape energy and, again, this must be transported outward by the torque in the disk. Provided that there is still sufficient gas flowing inward at small radius, it can release enough binding energy to account for the extra energy given to the escaping gas.
However, if there are significant radiative losses from the disk or there is too little mass flow remaining at small radius so that it releases relatively little binding energy, then an additional power source must be invoked beyond the gravitational energy of the accreting gas. (This is reminiscent of the development of solar physics, where nuclear power was invoked to prolong the sun’s life from millions to billions of years.) In this case, the natural power source is the spinning black hole.
Rotational energy can amount to up to ∼0.3 of the black hole mass [
2]. In sources such as M87, it is invoked to power the jets electromagnetically. In the simplest, stationary, axisymmetric models the power is carried exclusively as a Poynting flux along the jet. However, in less idealized models, only a small fraction of this power suffices to overwhelm the release of binding energy by the inflowing gas and it seems quite likely that essentially all of the mass supplied at large radius is driven off between a few gravitational radii and the radius where an infall transitions to an outflow which we call the magnetopause. This is perhaps a million times larger than the black hole. Where and how this happens depends upon a more detailed physics investigation and the history of the mass supply. Indeed the flow is quite likely to be time-dependent. (It is commonly supposed that the flow is self-similar but this could be a poor approximation.) This is a very different flow of energy from that of an Advection-Dominated Accretion Flow [
12] where essentially all of the mass supply crosses the event horizon but with a large internal energy and, consequently, a low radiative efficiency. Instead, we propose that the gas is always able to cool, the disk is thin and its most important function is to trap and concentrate the magnetic field around the black hole so as to extract its spin energy.
2.4. Magnetic Flux
Magnetic field plays a large role in contemporary models of AGN. It takes two basic forms (
Figure 1). Within an accretion disk, it can develop as the nonlinear evolution of the magnetorotational instability. Essentially this involves magnetic turbulence evolving on a dynamical timescale. This produces a tangled magnetic field leading to a non-zero mean value for the Maxwell shear stress tensor component <
> that is responsible for the torque
G. The ratio of the shear stress to the total pressure is conventionally called
, though simulations typically exhibit more interesting behavior than is captured by a universal, constant value. Similar magnetic field is found in simulations of thick ion tori and is responsible for the synchrotron emission that is observed by EHT. Of course, electrical current accompanies the magnetic field and is computable from its curl, but MHD is most commonly transacted through the magnetic field. As with all forms of turbulence, there will be significant energy dissipated on small scales, especially within a corona. In a hot, collisionless plasma, the form of this dissipation is likely to involve the production of wave modes, especially Alfvén waves. It will also involve magnetic reconnection. Both processes are associated with the acceleration of high energy particles, especially relativistic electrons.
By contrast, the magnetic flux which threads the horizon of the spinning black hole is thought to be slowly varying. The black hole spacetime acts like a modest electric conductor with an effective resistance ∼
[
2]. (The associated dissipation occurs, invisibly, behind the event horizon.) This implies that the magnetic flux is not line-tied to the horizon as it is when it threads a spinning neutron star. The field lines end up “moving” with an angular velocity about half that associated with the black hole and adjust themselves so as to balance the transverse electromagnetic stress. If we assume that the electromagnetic field adopts the stationarity and axisymmetry of the underlying Kerr spacetime, then there are conserved flows of electrical current, electromagnetic energy and angular momentum along (equipotential and isorotational) magnetic flux surfaces in a non-rotating frame. Close to the horizon, physical “observers” must rotate and if they hover just outside the event horizon, they will see energy flowing into the black hole.
A useful way to think about the extraction of energy is to note that along a given flux surface rotating with a fixed angular velocity, there will be an inner and an outer light surface. Within the former, a physical observer, on a timelike geodesic must move radially inward; beyond the latter, the motion must be outward. Gravitational energy is not localized within general relativity and, it seems better to consider the electromagnetic power as being extracted from the region between these two surfaces and not from the event horizon. In this way a black hole magnetosphere differs from that of a neutron star. For the neutron star, the outflow is determined by conditions at the surface. For the black hole, the behavior derives causally from actions between the light surfaces; and the event horizon can be seen as an absorber of electromagnetic energy in much the same way as an imaginary surface at infinity.
To make this more quantitative, a spinning black hole immersed in magnetic flux supported by external toroidal current, and held in place eventually by orbiting gas, generates an EMF which for M87 is roughly
V∼
, where E is the SI prefix for
. Given the resistance, an estimate for the associated electrical current per jet is ∼
. This flow of electromagnetic Poynting flux is what powers the jet and is essentially invisible because the amount of plasma needed to carry the electrical current and produce the electrical charge density is tiny and unlikely to have dynamical consequence. The minimum value of the ratio of the electron pressure needed to support the current to the magnetic pressure, conventionally
, is of order the ratio of the Larmor radius to the length scale which, in turn is of order the ratio of the rest mass of an electron to
, which is ∼
[
11]. In reality,
will be many orders of magnitude greater than this, but this still suggests that the plasma may be dynamically unimportant close to the black hole. In this case, the electromagnetic field may be best described by force-free electrodynamics which essentially equates the divergence of the purely electromagnetic stress-energy tensor to zero. In addition, so little plasma is required to carry electrical current that there may be no need to invoke pair production above the black hole. Instead, sufficient disk plasma may be transported across the magnetic field by small-scale magnetic interchange instabilities [
7]. Implicit in the force-free description is this assumption that supply of sufficient charge to support the electromagnetic field does not materially affect the electromagnetic field itself. In particular, gaps with ∼teravolt potential differences (∼
of the total potential difference available in M87) do not change the overall flow of electrical current. Current of either sign can cross the horizon simply by having slightly more protons/positrons or electrons flow inwards.
As we move radially outward from the horizon, towards and through the disk, the magnetic field may be either highly ordered, strongly turbulent or somewhere in between. Most descriptions of the accretion disk presume that its large scale magnetic field is constantly regenerated by local disk dynamos. There is no large-scale polarity. By contrast, we propose that there is a single polarity maintained over the entire disk, for at least a dynamical time at the infall radius. It can be argued that unipolarity is inevitable. Differential rotation is likely to render the magnetic field approximately axisymmetric. If we move radially inward and the sign of the poloidal magnetic field reverses twice then there will be magnetic reconnection that will allow the ring of reversed magnetic flux to be released. Through this mechanism a constant sign of high-latitude polidal field anchored to the disk can be established from the outside in as most of the magnetic flux threading the disk is found at large disk radius. This proposal is not inconsistent with there being an active corona, just as happens with the sun. The detailed distribution of the magnetic flux will reflect the disk physics and the history. Note that only a tiny fraction of the total magnetic flux threading the disk is needed to power the jets.
It is not known how far out in radius the disk extends. In M87, it has been observed out to ∼
m or ∼
where the deprojected orbital speed is ∼
, consistent with the black hole mass [
13]. The behavior of the gas beyond this radius is complex and poorly understood despite impressive X-ray imaging by Chandra. One possible scenario is that the inflow from the surrounding cluster is quasi-spherical until its relatively small angular momentum, causes it to hang up and fall into equatorial plane. This is a natural location for a magnetopause—an interface where the infall meets the wind from the disk in our model. The magnetic state of the wind is hard to guess and we must be guided by observations.
2.5. Current
We should also consider the toroidal component of magnetic field around the black hole, in the ergomagnetosphere and in the ejection disc. This is best characterized using the flow of poloidal current, I. If the poloidal field and the angular velocity are aligned with the direction, then I flows inward from the jet to the horizon at high latitude and, in a steady state, must be part of a stationary circuit. This current must exit the horizon at lower latitude. (This is achieved by having opposite charges fall preferentially in the two zones.) The combined current associated with the two jets then flows preferentially along the equatorial plane. Some of this current may complete as a return current flowing the outer boundary of the jets and be associated with Ohmic dissipation/particle acceleration, radio emission—the observed jet sheath.
However, there will still be net negative current within cylindrical radius supporting a toroidal field which allows the disk wind to carry away angular momentum from the disk. If falls off slower than , then there will be an inward Lorentz force causing the wind to collimate towards the jet axis. Even if , the radial magnetic stress associated with it will be larger at the jet than at ( and being the jet and disk outer cylindrical radii, respectively). In principle, this allows the wind to transmit and amplify stress exerted at the magnetopause onto the jet. Simple pinches are notoriously unstable, but winds are more complicated.