Photon emission from inside the innermost stable circular orbit

We consider a situation where a light source orbiting the innermost stable circular orbit (ISCO) of the Kerr black hole is gently falling from the marginally stable orbit due to an infinitesimal perturbation. Assuming that the light source emits photons isotropically, we show that the last radius at which more than 50% of emitted photons can escape to infinity is approximately halfway between the ISCO radius and the event horizon radius. To evaluate them, we determine emitter orbits from the vicinity of the ISCO, which are uniquely specified for each black hole spin, and identify the conditions for a photon to escape from any point on the equatorial plane of the Kerr spacetime to infinity by specifying regions in the two-dimensional photon impact parameter space completely. We further show that the proper motion of the emitter affects the photon escape probability and blueshifts the energy of emitted photons.

Figure 1

Typical shape of allowed region for the radial photon motion and the parameter range of $b$ for escape photons. Gray curves denote $b_i$ ($i=1$, 2) as a function of $r$ for fixed $q$, and gray shaded regions denote forbidden regions, while other regions denote allowed regions. Vertical dashed lines are the radial coordinate value of the emission point $r=r_{*}$. The functions $b_i$ in (a)–(c) have extrema $b_i^{\mathrm{s}}$ at $r=r_i$, respectively, and are typical shape for $0\le q<27$, where (a) $r_{\mathrm{H}}<r_{*}<r_1$, (b) $r_1\le r_{*}<r_2$, and (c) $r_{*}\ge r_2$. The functions $b_i$ in (d) are typical shape for $q>27$, in which the allowed region is divided into two disconnected ones, and no longer have extrema. Blue and orange shaded regions show the parameter range of $b$ for an escape photon initially with ${\sigma }_r=1$ and ${\sigma }_r=-1$, respectively.

Figure 2

Characteristic radii of particle circular orbits as functions of $a$. Blue solid curve: the ISCO radius given by Eq. (35). Green solid curve: the radius of photon circular prograde orbit given by Eq. (24). Green dashed curve: the radius of photon circular retrograde orbit given by Eq. (25). Gray dotted line is $r=3$, and gray solid curve denotes the horizon radius. The equality $r_{\mathrm{I}}=r_2^c$ holds at $a=0.6382\dots$, and $r_{\mathrm{I}}=3$ holds at $a=0.7818\dots$.

Figure 3

Radial motion of the emitter for each value of $a$. It falls from the vicinity of the ISCO into the horizon. Each curve terminates at the horizon radius.

Figure 4

Top view of the plunge trajectories of an emitter from the vicinity of the ISCO into the horizon. The blue solid curves show the plunge orbits, and the gray dashed circles denote the horizon radii. The left figure shows the case $a=0$, for which $r_{\mathrm{H}}=2$ and $r_{\mathrm{I}}=6$, and the initial condition is chosen as $\epsilon =0.1118$. The right figure shows the case $a=0.9$, for which $r_{\mathrm{H}}\simeq 1.4358$ and $r_{\mathrm{I}}\simeq 2.3208$, and the initial condition is chosen as $\epsilon =0.06379$. The size of each frame coincides with each ISCO radius. The coordinates are defined by $(x,y)=(r\cos \psi ,r\sin \psi )$, where $\psi$ is the angle in the ingoing Kerr-Schild coordinates (see Appendix pp2).

Figure 5

Emission angles $(\alpha ,\beta )$ defined at the rest frame of the emitter.

Figure 6

Photon escape probability $P(r_{*};a)$ for each value of $a$. Solid curves: the escape probabilities of a photon emitted from the emitter plunging from the ISCO into the horizon. Dashed curves and dots are the escape probabilities of a photon emitted from an emitter on circular geodesics and the ISCO, respectively.

Figure 7

Plots of $r_0$ and $r_{\mathrm{z}}$. Orange dots: $r=r_0$. Red dots: $r=r_{\mathrm{z}}$. Blue solid curve: the ISCO radius given by Eq. (35). Green solid curve: the radius of photon circular prograde orbit given by Eq. (24). Gray solid curve: the horizon radius.

Figure 8

Escape probabilities of a photon emitted from an emitter in various states of motion. Solid curves denote $P$, and dotted curves denote $P_{\mathrm{L}}$, as functions of $r_{*}$. Dashed curves: the escape probabilities of a photon emitted from an emitter on circular geodesics.

Figure 9

Maximum blueshift. Solid curves: $z_{\max }$ in the range $r_1^{\mathrm{c}}\le r_{*}<r_{\mathrm{I}}$. Dash-dotted curves: $z_{\max }$ in the range $r_{\mathrm{H}}<r_{*}<r_1^{\mathrm{c}}$. Dashed curves and dots: the maximum blueshift of an escape photon emitted from an emitter orbiting circular geodesics and the ISCO, respectively.