Measuring the Angular Momentum of Supermassive Black Holes (SpringerBriefs in Astronomy)
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We simply compare two background metrics: the Kerr solution against the metric of the alternative theory of gravity of interest. However, there is another issue. Usually we know non-rotating solutions in alternative theories of gravity, and in a small number of cases we know the rotating solution in the slow-rotation approximation. Rotating non-Kerr BH solutions in alternative theories of gravity are difficult to find.
The spin plays a crucial role in the properties of the radiation emitted close to BH candidates and therefore, without the rotating solution, it is not possible to test the Kerr metric, because we are not able to distinguish the effects of the spin from those due to possible deviations from the Kerr geometry. General statements or properties that hold in the case of the Kerr metric may not be true in other backgrounds.
This may lead to new phenomena with specific astrophysical implications and observational signatures. An alternative theory of gravity may potentially have several kinds of BHs, which may be created by gravitational collapse from different initial conditions. In a similar context, it is possible that astrophysical BH candidates are not all of the same type and therefore the possible confirmation that a specific object is a Kerr BH does not necessarily imply that all BH candidates are Kerr BHs.
An event horizon is a null surface in spacetime. In the case of a stationary and axisymmetric spacetime, the procedure can significantly simplify. In a coordinate system adapted to the two Killing isometries stationarity and axisymmetry , and such that f is also compatible with the Killing isometries, Eq.
The surface must be closed and non-singular namely geodesically complete in order to be an event horizon and not just a null surface. The problem is thus reduced to finding the solution of the differential equation. A Killing horizon is a null hyper-surface on which there is a null Killing vector field. In a stationary and axisymmetric spacetime and employing a coordinate system adapted to the two Killing isometries, the Killing horizon is given by the largest root of.
In a generic stationary and axisymmetric spacetime, there are three constants of motion, namely the mass m , the energy E , and the axial component of the angular momentum L z. If the spacetime has a forth constant of motion, it is always possible to choose a coordinate system in which the equations of motion are separable. In the Kerr metric, this is the case for the Boyer-Lindquist coordinates and the calculations of the photon trajectories from the observer to the region around the BH can be reduced to the calculations of some elliptic integrals, which is computationally more efficient.
If the spacetime has no Carter-like constant, it is necessary to solve a system of coupled second order differential equations. In accretion disk models and astrophysical measurements, the ISCO radius plays an important role. In non-Kerr metrics, the picture is more complicated. Moreover, in the Kerr metric the usual picture is that a particle 12 of the accretion disk reaches the ISCO and then quickly plunges onto the BH, without emitting much radiation after leaving the ISCO.
On the contrary, there may be an accumulation of gas between the ISCO and the compact object, and the gas has to lose additional energy and angular momentum before plunging to the central body. There are also spacetimes in which there is no ISCO, namely the orbits are always stable. In the Kerr metric, the ISCO radius is located at the minimum of the energy of equatorial circular orbits At larger radii, the specific energy monotonically increases to approach 1 at infinity.
From the ISCO radius to smaller radii, the specific energy monotonically increases to diverge to infinity at the photon orbit. When the ISCO is marginally vertically unstable, the energy of equatorial circular orbits may be a monotonic function without minimum the energy decreases as the radial coordinate decreases. In this case, there is no marginally bound circular orbit. Since astrophysical observations are often sensitive to the position of the ISCO, it is useful to have an idea of the correlation between the spin and the deformation parameter in the determination of the ISCO radius.
The left panel of Fig. As already mentioned, throughout this article I do not impose that the deformation parameter must be a small quantity. The contour levels of the ISCO can give a simple idea of which spacetimes may look similar in astrophysical observations. The right panel in Fig. Second, it is indeed a better estimator to figure out which spacetimes may look similar in astrophysical observations see the two panels in Fig.
Finally, note that non-Kerr metrics often have some pathological features for some choices of the deformation parameters. Naked singularities, regions with closed time-like curves, etc, are possible. Some caution has to be taken. However, even pathological spacetimes can be used to test the Kerr metric if we assume that the spacetime solution is only valid outside of some interior region. For example, the interior would be different because of matter source terms. In the case of a compact object made of exotic matter, the vacuum solution would hold up to the surface of the object, while at smaller radii the metric would be described by an interior solution.
As seen from Eq. For instance, the spin parameter of Earth is about 10 3 and there is no violation of any principle because the vacuum solution holds up to that of the Earth surface. For instance, Giacomazzo et al. Assuming it is somehow possible to create a similar object, Pani et al. Third, if the measurement of the spin parameter of a BH candidate gave a value larger than 1 assuming the Kerr metric, the measurement would be presumably wrong, but it would be a clear indication of new physics. The metric around the BH candidate should thus be different from that of the Kerr metric.
Since spin measurements strongly depend on the exact background metric, the measurement would be wrong and it is possible that the actual value of the spin parameter is instead smaller than 1. In the case of non-Kerr BHs, the critical bound is typically different, and it may be either larger or smaller than 1, depending on the spacetime geometry.
In order to test the Kerr metric with electromagnetic radiation, we need to study the properties of the radiation emitted by the gas in the accretion disk or by stars orbiting the BH candidate. The spectrum of the BH candidate depends on the motion of the gas in the accretion flow or of orbiting stars and by the propagation of the photons from the point of emission to the distant observer.
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We thus note the following:. With this approach, we test the Kerr metric, just like with the PPN formalism we can test the Schwarzschild solution. We do not directly test the Einstein equations. For instance, we cannot distinguish a Kerr BH of general relativity from a Kerr BH in an alternative metric theory of gravity, because the motion of particles and photons is the same. However, the observation of a non-Kerr BH could rule out the Einstein equations because, within general relativity, astrophysical BHs should be well described by the Kerr solution.
Even if we assume the Kerr metric, analyze the data, and obtain a very good fit, it is not enough to claim that the object is a Kerr BH. This is made clearer in the next sections. There is typically a degeneracy among the parameters of the model, and in particular between the spin and possible deviations from the Kerr background. For this reason we adopt a PPN-like approach.
From Stars to States
The X-ray radiation in the spectrum of BH candidates is thought to be generated in the vicinity of these objects. Some features depend on the strong gravitational field near the BH candidate and therefore, if properly understood, they may be used to test the Kerr BH hypothesis. There is some difference between stellar-mass and supermassive BH candidates, because of the different masses and environments. In the case of stellar-mass BH candidates, a source can change its spectral state on a timescale of weeks or months.
For supermassive BH candidates, the timescales are too long and the source can only be observed in its current spectral state. Note that the spectral state classification is still a work in progress, some spectral states and their physical interpretation are not yet well understood, and different authors may use a different nomenclature. The key-point in any measurement is to have the correct astrophysical model.
Within the corona-disk model with lamppost geometry , the set-up is shown in Fig. The accretion disk is geometrically thin and optically thick. It radiates like a blackbody locally and as a multi-color blackbody when integrated radially. The temperature of the disk depends on the BH mass and the mass accretion rate [see e. Bambi et al. The corona is often approximated as a point source located on the axis of the accretion disk and just above the BH. This arrangement is often referred to as a lamppost geometry.
The lamppost geometry requires a plasma of electrons very close to the BH and such a set-up may be realized in the case of the base of a jet. The direct radiation from the hot corona produces a power-law component in the X-ray spectrum. X-ray techniques to test the Kerr metric are discussed in the next subsections: continuum-fitting method III. Both techniques have been developed to measure the spin parameter of BH candidates under the assumption of the Kerr background, but they can be naturally extended to test the Kerr metric.
Current spin measurements of stellar-mass BH candidates reported in the literature under the assumption of the Kerr background are summarized in Tab. Iron line spin measurements of supermassive BH candidates under the assumption of the Kerr background are reported in Tab. QPOs are seen as peaks in the X-ray power density spectra of BH candidates and they may be used to measure the properties of the metric around these objects. For the time being, we do not know the exact mechanism responsible for these oscillations, and therefore QPOs cannot yet be used to test fundamental physics.
Different models provide different results. However, QPOs are a promising tool for the future, because the value of their central frequency can be measured with high precision. The observation of the X-ray polarization of the thermal spectrum of thin accretion disks is another potential technique to test the Kerr metric. The plane of the disk is assumed to be perpendicular to the BH spin The particles of the gas move on nearly geodesic circular orbits, and the inner edge of the disk is at the ISCO radius. The latter assumption plays a crucial role and it is confirmed by some observations that show that the inner edge of the disk does not change appreciably over several years when the source is in the thermal state.
The most natural interpretation is that the inner edge is associated with some intrinsic property of the geometry of the spacetime, namely the radius of the ISCO, and it is not affected by variable phenomena like the accretion process. This is true only for stellar-mass BH candidates, because the temperature of the disk depends on the BH mass and the mass accretion rate. In the latter case, extinction and dust absorption limit the ability to make accurate measurements.
The continuum-fitting method is thus normally applied to stellar-mass BH candidates For the validity of the method, see e. McClintock et al.
The impact of the model parameters on the shape of the spectrum is shown in Fig. Current spin measurements with the continuum-fitting method are reported in Tab. Some of these objects are not dynamically confirmed BH candidates, so their mass is not estimated from the motion of the companion star. The technique can be naturally extended to non-Kerr backgrounds. The impact of a possible non-vanishing deformation parameter on the thermal spectrum of a thin disk is shown in the bottom right panel in Fig.
Even without a quantitative analysis, it is clear that the effect of the spin and of the deformation parameter is very similar. The radiative efficiency in the Novikov-Thorne model is. If we relax the Kerr BH hypothesis and we allow for a non-vanishing deformation parameter, the same value of the radiative efficiency can be obtained for different combinations of the spin parameter and the deformation parameter.
The result is that there is a degeneracy and it is impossible to measure both the spin and the deformation parameter. In general, we can measure only a combination of them.
We obtained the constraints on the spin parameter — deformation parameter plane within the JP background. Some examples are shown in Figs. This approach can be justified a posteriori because the constraints provided by the dash-dotted green lines and by the dashed blue lines are very similar considering the spin uncertainty. There is a quasi-degeneracy in the theoretical prediction of the spectra and therefore a spin measurement inferred in the Kerr background can be translated into an allowed region on the spin parameter — deformation parameter plane.
In both cases, the Kerr measurement is 0. Note the similarity of the shapes of the allowed regions in Figs. The reason is that, in general, if one considers very large deviations from Kerr, in both directions in the deformation parameter, the ISCO radius increases and therefore the Novikov-Thorne radiative efficiency decreases. The result is that very deformed objects cannot mimic a fast-rotating Kerr BH.
However, this is not a general statement, and some deformations may be extremely large. For the time being, there are no observations of this kind and therefore all the data are consistent with the Kerr metric. The continuum-fitting method is probably the most robust technique among those available today. However, the method also has some weak points. The measurements of M , i , and d from optical observations are sometimes difficult and may be affected by systematic effects.
Assuming the systematics are under control, the thermal spectrum of a thin disk has a very simple shape, and it cannot provide much information on the spacetime geometry around the BH candidate. If we assume the Kerr metric, we can determine the spin parameter. If we have just one possible non-vanishing deformation parameter, we meet a degeneracy and, in general, we cannot constrain the spin and possible deviations from the Kerr solution at the same time. If we have a source that looks like a very fast-rotating Kerr BH, we can constrain some deformation parameters e.
The reason is that the spectrum is simply a multi-color blackbody spectrum without additional features. Different parameters of the model have a quite similar impact on the shape of the spectrum and therefore there is a strong parameter degeneracy. The best that we can do is to combine the continuum-fitting measurements with other observations to break the parameter degeneracy.
The illumination of a cold disk by a hot corona produces a reflection component as well as some spectral lines by fluorescence in the X-ray spectrum of the source. This line is intrinsically narrow in frequency, while the one observed in the X-ray spectrum of BH candidates appears broadened and skewed. The iron line is the strongest feature aside of the continuum, see Fig. This technique relies on fits of the whole reflected spectrum, but the spin measurement or possible tests of the Kerr metric is mainly determined by the iron line.
For this reason the technique is often called the iron line method. The local spectrum I e the power radiated per unit area of emitting surface per unit solid angle per unit frequency is determined by the reflection processes and the geometry of the system. Assuming axisymmetry, I e depends on the photon energy, the emission radius, and possibly on the emission angle the angle of propagation of the photon with respect to the normal to the disk if the emission is not isotropic.
With this choice, we have two free parameters for the emissivity profile, q and r b r e a k. In the case of AGN, the cosmological redshift of the source can instead be important for some objects and it can be an additional parameter of the model.
While the continuum-fitting method requires independent measurements of the mass, the distance, and the viewing angle of the source, the iron line analysis does not require them. Mass and distance have no impact on the line profile, as the physics is essentially independent of the size of the system, while the viewing angle can be inferred during the fitting procedure from the effect of the Doppler blueshift.
As in the continuum-fitting method, the iron line measurement assumes that the accretion disk can be described by the Novikov-Thorne model, and crucially depends on the inner edge of the disk being located at the ISCO radius. These states are those most frequently employed in reflection modeling. At present, the evidence indicates that if truncation is present, it is likely to be mild a factor of a few. In the case of supermassive BH candidates, this technique is more widely used, see the third column in Tab.
In part, this is because we never have a precise estimate of the Eddington scaled accretion luminosity, due to the large uncertainties in the bolometric correction and mass estimates. This works both ways so that it is hard to say whether we are in the thin disk range or not. In part, it is because we cannot choose a different spectral state due to the much longer timescales of supermassive BHs than stellar-mass BHs.
If one were to fit data for a system in which the disk were truncated, and make the usual assumption that the inner-edge was at the ISCO, the fit would then incorrectly underestimate the value of spin relative to the resulting Kerr prediction if it were truncated at the ISCO radius.
The same question is at play for using reflection to test the Kerr metric with data from faint hard-states. In the case of too high accretion luminosities, the inner part of the disk may instead be geometrically thick, and this would lead to overestimate the BH spin or, otherwise, get a wrong constraint on the deformation parameter. Some very high values of the BH spins reported in Tab. In subsequent sections, we shall develop an eigenvalue scheme to identify which critical point will be of what category. It is to be noted in this context that a physical flow can be constructed only through a saddle type critical point.
At this stage the fundamental distinction between a formal multi-critical configuration and a realizable multi- transonic flow requires certain clarification. Such configurations, however, does not necessarily produce multi-transonic flows. Critical point behavior is a formal property of a differential equation of certain class describing certain behaviors of the first order autonomous dynamical systems . Considering the fact that out of three formal critical points, the middle one being the center type and hence not allowing any transonic solutions to be constructed through it, we explain why the realizable multi-transonic solutions form a sub-class of the formal multi-critical configurations.
Subsonic flow starting at the large distance becomes supersonic after crossing the saddle type outer sonic point. Once supersonic, flow cannot access another regular sonic point until it is made subsonic again. Certain additional physical mechanism a discontinuous shock transition in the present work.
A shock-free multi-critical flow for solution is thus a mathematical construct only, for which the in-going accretion solution remains mono-transonic in practice. The physically realizable multi-transonic configuration is obtained by allowing the flow solution constructed through the outer sonic point with the flow solution passing through the inner sonic point through a steady standing discontinuous shock transition.
In what follows, we discuss the criteria of such shock formation in somewhat detail. Shock formation in axisymmetric inviscid accretion. Certain kind of perturbation may produce discontinuities in large scale astrophysical flows. Based on specific boundary conditions to be satisfied across the surface of discontinuity, such surfaces are classified into various categories - the most important in astrophysical fluid dynamics being the shock waves or shocks.
Supersonic astrophysical flows are prone to shock formation phenomena in general and become subsonic after the shock. The angular momentum supported centrifugal potential barrier acts as a repulsive agent against the attractive force of the gravity to break the supersonic incoming flow matter heading toward a black hole for our case. The issue of shock formation in this work will be addressed along two different directions.
The energy preserving Rankine - Hugoniot [62—66] shocks will be considered for the adiabatic flow, where the post shock temperature shoots up compared to the position dependent temperature of the pre-shock flow. The shock thick- ness is considered to be infinitesimally small - in principle we deal with shocks with shocks with zero effective thickness. For isothermal flow we consider temperature preserving shocks. Flow dissipates energy to maintain the invariance of the temperature. The post-shock flow thickness does not get altered due to such shocks.
Hence the accretion flow through the outer sonic point may jump discontinuously through DD1 to the other branch with higher specific entropy being subsonic again and through that branch D1 JI it may become again supersonic. The other branch FBG in the fig. In fig. In such region, however, accretion flow remains mono-transonic. Dashed blue in online version lines for constant height, dotted red in online lines for conical and solid green in online lines for vertical equilibrium model. We find that the shock location non-linearly correlates with black hole spin same result has been obtained for full general relativistic flow in the Kerr metric, see, e.
This is because of the fact that more amount of gravitational potential energy is available to be released for smaller shock location. We quantitatively demonstrate such finding through fig. Clearly, stronger shocks are produced for smaller shock locations. It is also observed that for the same shock location, the shock is strongest for constant height flow and is weakest for flow in vertical equilibrium. Isothermal Flow. The symbols O and I represent mono-transonic accretion constructed through the outer and the inner sonic points, respectively. We have a large overlapping region here for the formation of the isothermal shock.
The variation of the shock location with the black hole spin parameter is shown in fig. The variation of the shock strength with the Kerr parameter has been shown in fig. Stronger shocks are formed closer to the horizon and constant height flow produces the strongest possible shock variation of the density compression ratio as well as the ratio of the post to pre-shock pressure on the black hole spin is shown in fig. Stability of the stationary flow.
Following the method depicted in refs. These quantities are introduced into the time dependent counterparts of continuity equation, eq. Following the same line of argument as mentioned in the ref. The figure in right magnifies the common region of the figure in left. Hence following the argument of refs. Acoustic Surface Gravity : Dependence on the black hole spin parameter. This we have demonstrated in the previous section.
This indicates that the numerical value of the acoustic surface gravity correlates with the strength of the gravitational attraction of the background gravitational field. This is intuitively obvious because at the outer acoustic horizon which forms a large distance away from the black hole , space-time becomes asymptotically flat and the effect of the black hole spin does not really affect the dynamics of the flow, and hence the nature of the sonic geometry embedded within. One can, however, bypass such constraints by studying the dependence of the acoustic surface gravity as a function of the black hole spin for mono-transonic flow constructed through the inner acoustic horizon.
Sen would like to acknowledge the kind hospitality provided by HRI, Allahabad, India, under a visiting student research programme. The visits of S. Liang and K. ApJ, , Frank, A. King, and D. Accretion Power in Astrophysics. Cambridge University Press, Cambridge, Kato, J. Fukue, and S. Black Hole Accretion Disc.
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