Inner accretion disk edges in a Kerr-like spacetime

According to the no-hair theorem, astrophysical black holes are uniquely described by the Kerr metric. In order to test this theorem with observations in either the electromagnetic or gravitational-wave spectra, several Kerr-like spacetimes have been constructed which describe potential deviations from the Kerr spacetime in parametric form. For electromagnetic tests of the no-hair theorem, such metrics allow for the proper modeling of the accretion flows around candidate black holes and the radiation emitted from them. In many of these models, the location of the inner edge of the accretion disk is of special importance. This inner edge is often taken to coincide with the innermost stable circular orbit, which can serve as a direct probe of the spin and the deviation from the Kerr metric. In certain cases, however, an innermost stable circular orbit does not exist, and the inner edge of an accretion disk is instead determined by an instability against small perturbations in the direction vertical to the disk. In this paper, I analyze the properties of accretion disks in the Kerr-like metric proposed by Johannsen and Psaltis [Phys. Rev. D 83, 124015 (2011)], whose inner edges are located at the radii where this vertical instability occurs. I derive expressions of the energy and axial angular momentum of disk particles that move on circular equatorial orbits and calculate the locations of the inner disk edges. As a possible observable of such accretion disks, I simulate profiles of relativistically broadened iron lines and show that they depend significantly on the values of the spin and the deviation parameter.

Figure 1

Different shapes of the (singular) Killing horizon for values of the spin $|a|=0.8M$ and deviation parameter ${ϵ}_3$. Left: ${ϵ}_3=0.85<{ϵ}_3^{\mathrm{bound}}(0.8M)$, where ${ϵ}_3^{\mathrm{bound}}(a)$ denotes the boundary between the regions of the parameter space with different topologies of the Killing horizon. The inner and outer Killing horizons have spherical topology. Center: ${ϵ}_3=0.92799\approx {ϵ}_3^{\mathrm{bound}}(0.8M)$. The inner and outer Killing horizons merge in the equatorial plane. Right: ${ϵ}_3=1.0>{ϵ}_3^{\mathrm{bound}}(0.8M)$. The inner and outer Killing horizons have split into two spherelike surfaces located above and below the equatorial plane, respectively. For all values of the parameter ${ϵ}_3$, the origin is likewise singular.

Figure 2

$P_5(r)$ as a function of the radius $r$ with parameters $a=0.7M$ and ${ϵ}_3=2.0$. Top panel: The function $P_5(r)$ is very steep and has several local minima and maxima, as well as five roots marked by red vertical dashed lines, one single root labeled “${\mathrm{n}}_5$” and two double roots labeled “${\mathrm{n}}_{5\mathrm{a}}$” and “${\mathrm{n}}_{5\mathrm{b}}$,” respectively. Lower panel: Enlarged view of the function $P_5$ around the roots on the $r$ axis.

Figure 3

Energy $E$ (top panel) and axial angular momentum $L_z$ (bottom panel) in units of the rest mass of a particle on a circular equatorial orbit as observed at radial infinity for a spin value $a=0.8M$ and several values of the parameter ${ϵ}_3$. If the Killing horizon has spherical topology [i.e., if ${ϵ}_3<{ϵ}_3^{\mathrm{bound}}(0.8M)\approx 0.92799$], both $E$ and $L_z$ diverge to plus infinity at the horizon and are imaginary inside. Otherwise, they change sign near the outermost projected location of the Killing horizon onto the equatorial plane and diverge to minus infinity at the origin.

Figure 4

Energy $E_{\mathrm{local}}$ in units of the rest mass of a particle on a circular equatorial orbit as observed by a local observer for values of the spin $a=0.8M$ and the parameter ${ϵ}_3=0.92799$. The energy is positive at all radii $r>0$. In particular, at least at the radii shown in this figure, the energy is greater than or equal to the rest mass of the particle, where equality holds only at a radius $r\approx 1.31M$ (see inset).

Figure 5

Radial (top) and vertical (bottom) epicyclic frequencies ${\nu }_r\equiv {\Omega }_r/2\pi$ and ${\nu }_{\theta }\equiv {\Omega }_{\theta }/2\pi$ for a $10M_{\odot }$ black hole for $a=0.8M$ and several values of the parameter ${ϵ}_3$. If an ISCO exists, the radial epicyclic frequency vanishes at the ISCO radius, and circular equatorial orbits are radially unstable inside of this radius. If an ISCO does not exist, all circular orbits are radially stable but vertically unstable roughly in the interval $0.1M\lesssim r\lesssim 1.7M$.

Figure 6

Contours of (blue curves, labeled $2M–10M$) constant ISCO radius and (green curves, labeled $1.5M–2.5M$) constant radius, at which the vertical epicyclic frequency vanishes, if no ISCO exists. The boundary between these regions (red curve) is determined by the function ${ϵ}_3^{\mathrm{bound}}(a)$, $a>0$, given by Eq. (6).

Figure 7

Top: Regions where an ISCO exists (labeled “radial instability”) and where all circular equatorial orbits are radially stable but vertically unstable (labeled “vertical instability”). The boundary between these regions (blue curve) is determined by the function ${ϵ}_3^{\mathrm{bound}}(a)$, $a>0$, given by Eq. (6). If an ISCO exists, circular equatorial motion is also vertically unstable, but this vertical instability only occurs inside of the ISCO except for values of the spin and the parameter ${ϵ}_3$, which lie in a very narrow (red shaded) region along the boundary in the range $0.57M\lesssim a\le M$. Bottom: Thickness $\Delta {ϵ}_3$ of the red shaded region in the top panel.

Figure 8

Profiles of relativistically broadened iron lines in units of the rest frame energy $E_0$ for values of the spin $a=0.9M$, emissivity index $\alpha =2$, outer disk radius $r_{\mathrm{out}}=15M$, disk inclination $i=30°$, and several values of the deviation parameter. In the case of a Kerr black hole (${ϵ}_3=0$), the inner edge is located at the ISCO. For values of the parameter ${ϵ}_3=1.0$ and ${ϵ}_3=5.0$, the inner edge is located at the radius, at which the vertical epicyclic frequency vanishes. Changing the parameter ${ϵ}_3$ primarily affects the flux ratio of the two peaks of a given profile as well as the low-energy spectrum corresponding to the location of the inner disk edge.

Figure 9

Profiles of relativistically broadened iron lines in units of the rest frame energy $E_0$ for values of the spin $a=0.9M$, emissivity index $\alpha =2$, outer disk radius $r_{\mathrm{out}}=15M$, disk inclination $i=30°$, and two values of the deviation parameter. In both cases, the value of the deviation parameter is chosen so that the inner edge of the accretion disk as determined by either the ISCO (${ϵ}_3=-0.0775$) or the vertical instability (${ϵ}_3=5.0$) is located at the same radius ($r\approx 2.423M$). The flux ratio of the peaks is much larger in the former case than in the latter case, and the red tail extends to much lower energies.

Figure 10

Iron line profiles in units of the rest frame energy $E_0$ with values of the spin and the deviation parameter chosen so that the inner disk edges coincide. The line profiles are very similar.