Radiation from an emitter revolving around a magnetized nonrotating black hole

One of the methods of study of black holes in astrophysics is based on broadening of the spectrum of radiation of ionized iron atoms. The line $\mathrm{K}\alpha$ associated with iron emission at 6.4 keV is very narrow. If such an ion is revolving around a black hole, this line is effectively broadened as a result of the Doppler and gravitational redshift effects. The profile of the broadened spectrum contains information about the gravitational field of the black hole. In the presence of a regular magnetic field in the vicinity of a black hole, the characteristics of the motion of charged ions are modified. In particular, their innermost stable circular orbits become closer to the horizon. The purpose of this work is to study how this effect modifies the spectrum broadening of lines emitted by such an ion. Our final goal is to analyze whether the change of the spectrum profiles can give us information about the magnetic field in the black hole vicinity.

Figure 1

Magnetic field $b$ of a magnetized black hole is a function of ISCO radius $\rho$. Labels $+$ and $-$ stand for the anti-Larmor and Larmor orbit branches, respectively.

Figure 2

Specific energy $\mathcal{E}$ of a charged particle at ISCO in a magnetized black hole as a function of ISCO radius $\rho$. Labels $+$ and $-$ stand for the anti-Larmor and Larmor orbit branches, respectively.

Figure 3

Angular velocity $|\mathrm{\Omega }|$ as a function ISCO radius in a magnetized black hole. Labels $+$ and $-$ stand for the anti-Larmor and Larmor orbit branches, respectively.

Figure 4

Velocity $v$ of a charged particle at the ISCO in a magnetized black hole at a function of ISCO radius $\rho$. Labels $+$ and $-$ stand for the anti-Larmor and Larmor orbit branches, respectively.

Figure 5

Motion of a photon in the equatorial plane. The photon emitted at $P_e$ propagates to a distant observer along a trajectory without radial turning points (a direct ray). The ray emitted at $P_e^{\prime }$ is indirect. It at first moves to the black hole and only after it passes through a radial turning point does it propagate to the distant observer.

Figure 6

Angular definitions.

Figure 7

Impact plane.

Figure 8

The boundary $\mathrm{\Gamma }$ between the regions $I$ and $II$ for different values of the observer’s angle ${\theta }_o$: Curve 1: ${\theta }_o=5°$; Curve 2: ${\theta }_o=45°$; Curve 3: ${\theta }_o=85°$. A large circle is a curve $|\mathit{\xi }|={\ell }_{*}=3\sqrt{3}/2$.

Figure 9

Function $C(z)$. It monotonically grows with $z$ from $\pi /2$ at $z=0$ and becomes infinite at $z=2/3$.

Figure 10

The critical inverse radius ${\zeta }_{*}$ as a function the inclination angle ${\theta }_o$.

Figure 11

Images of some orbits corresponding to ${\theta }_o=85°$. The innermost curve is the image of the ${\zeta }_e=5/6$ orbit, it is formed by direct null rays. The next curve is the image of the ${\zeta }_e=2/3$ orbit, it is formed by direct null rays as well. And the outermost curve is the image of the ${\zeta }_e=1/3$ orbit. The lower part of the curve is formed by direct null rays. The upper part of the curve is formed by indirect rays. The circle represents the rim of the black hole shadow.

Figure 12

Diagram illustrating orbit of the emitter. The arrows show the direction of the emitter’s motion ($\mathrm{\Omega }>0$). For the emitter located in the right semicircle, $\phi \in [0,\pi ]$, photons have Doppler redshift and for the emitter located in the left semicircle, $\phi \in (-\pi ,0]$, photons have Doppler blue-shift. The spectral function diverges at $\phi ={\phi }_m$ and $\phi =-{\phi }_m$, where $|{\phi }_m|>\pi /2$. The portion of the orbit corresponding to indirect null rays is defined by the angle $|\phi |>{\phi }_{*}$.

Figure 13

Spectral function for ISCO, $b=0$, at ${\zeta }_e=1/3$. The inclination angle is ${\theta }_o=30°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.136$ and its specific energy is $\mathcal{E}=0.943$. The spectrum has peaks at $w_{-}=0.571$ (${\phi }_m=97°{85}^{\prime }$) and at $w_{+}=0.928$ (at $-{\phi }_m$). The minimal values (0.290 and 0.370) of $n_w$ for the two spectral branches are at $w_0=0.707$. The width parameter is $\mathrm{\Delta }=0.476$ and the asymmetry parameter is $\delta =0.236$. One also has $N_o=0.285$.

Figure 14

Spectral function for ISCO, $b=0$, at ${\zeta }_e=1/3$. The inclination angle is ${\theta }_o=60°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.136$ and its specific energy is $\mathcal{E}=0.943$. The spectrum has peaks at $w_{-}=0.497$ (${\phi }_m=107°9^{\prime }$) and at $w_{+}=1.224$ (at $-{\phi }_m$). The minimal values (0.127 and 0.222) of $n_w$ for two spectral branches are at $w_0=0.707$. The width parameter is $\mathrm{\Delta }=0.845$ and the asymmetry parameter is $\delta =0.423$. One also has $N_o=0.356$.

Figure 15

Spectral function for ISCO, $b=0$, at ${\zeta }_e=1/3$. The inclination angle is ${\theta }_o=85°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.136$ and its specific energy is $\mathcal{E}=0.943$. The spectrum has peaks at $w_{-}=0.472$ (${\phi }_m=118°4^{\prime }$) and at $w_{+}=1.407$ (at $-{\phi }_m$). The minimal values (0.079 and 0.196) of $n_w$ for two spectral branches are at $w_0=0.707$. The width parameter is $\mathrm{\Delta }=0.995$ and the asymmetry parameter is $\delta =0.498$. One also has $N_o=0.397$.

Figure 16

Spectral function for ISCO, $b=2.251$, at ${\zeta }_e=5/6$. The inclination angle is ${\theta }_o=30°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.162$ and its specific energy is $\mathcal{E}=0.465$. The spectrum has peaks at $w_{-}=0.315$ (${\phi }_m=104°7^{\prime }$) and at $w_{+}=0.415$ (at $-{\phi }_m$). The minimal values (0.128 and 0.186) of $n_w$ for two spectral branches are at $w_0=0.358$. The width parameter is $\mathrm{\Delta }=0.274$ and the asymmetry parameter is $\delta =0.137$. One also has $N_o=0.050$.

Figure 17

Spectral function for ISCO, $b=2.251$, at ${\zeta }_e=5/6$. The inclination angle is ${\theta }_o=60°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.162$ and its specific energy is $\mathcal{E}=0.465$. The spectrum has peaks at $w_{-}=0.284$ (${\phi }_m=124°6^{\prime }$) and at $w_{+}=0.486$ (at $-{\phi }_m$). The minimal values (0.055 and 0.117) of $n_w$ for two spectral branches are at $w_0=0.358$. The width parameter is $\mathrm{\Delta }=0.525$ and the asymmetry parameter is $\delta =0.262$. One also has $N_o=0.058$.

Figure 18

Spectral function for ISCO, $b=2.251$, at ${\zeta }_e=5/6$. The inclination angle is ${\theta }_o=85°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.162$ and its specific energy is $\mathcal{E}=0.465$. The spectrum has peaks at $w_{-}=0.264$ (${\phi }_m=158°6^{\prime }$) and at $w_{+}=0.558$ (at $-{\phi }_m$). The minimal values (0.036 and 0.105) of $n_w$ for two spectral branches are at $w_0=0.358$. The width parameter is $\mathrm{\Delta }=0.714$ and the asymmetry parameter is $\delta =0.357$. One also has $N_o=0.085$.

Figure 19

Spectral function for SCO, $b=2.251$, at ${\zeta }_e=1/3$. The inclination angle is ${\theta }_o=30°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.005$ and its specific energy is $\mathcal{E}=0.817$. The spectrum has peaks at $w_{-}=0.809$ (${\phi }_m=97°{85}^{\prime }$) and at $w_{+}=0.823$ (at $-{\phi }_m$). The minimal values (10.46 and 13.35) of $n_w$ for two spectral branches are at $w_0=0.816$. The width parameter is $\mathrm{\Delta }=0.018$ and the asymmetry parameter is $\delta =0.009$. One also has $N_o=0.390$.

Figure 20

Spectral function for SCO, $b=2.251$, at ${\zeta }_e=1/3$. The inclination angle is ${\theta }_o=60°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.005$ and its specific energy is $\mathcal{E}=0.817$. The spectrum has peaks at $w_{-}=0.804$ (${\phi }_m=107°9^{\prime }$) and at $w_{+}=0.829$ (at $-{\phi }_m$). The minimal values (4.578 and 8.018) of $n_w$ for two spectral branches are at $w_0=0.816$. The width parameter is $\mathrm{\Delta }=0.031$ and the asymmetry parameter is $\delta =0.016$. One also has $N_o=0.382$.

Figure 21

Spectral function for SCO, $b=2.251$, at ${\zeta }_e=1/3$. The inclination angle is ${\theta }_o=85°$. The angular velocity of the emitter is $\mathrm{\Omega }=0.005$ and its specific energy is $\mathcal{E}=0.817$. The spectrum has peaks at $w_{-}=0.802$ (${\phi }_m=118°4^{\prime }$) and at $w_{+}=0.832$ (at $-{\phi }_m$). The minimal values (2.836 and 7.051) of $n_w$ for two spectral branches are at $w_0=0.816$. The width parameter is $\mathrm{\Delta }=0.037$ and the asymmetry parameter is $\delta =0.019$. One also has $N_o=0.388$.