Modeling the motion of a bright spot in jets from black holes M87* and SgrA*

Abstract

We study the general relativistic motion of a bright spot in a jet from an accreting black hole. The corresponding lensed images of the moving bright spot are calculated numerically in discrete time intervals along the bright spot trajectory in the Kerr space-time framework. As representative examples, we consider the cases of supermassive black holes SgrA* and M87*. Astrophysical observations of the moving bright spots in the jets from black holes provides the unique possibility for the verification of different gravitation theories in the strong field limit.

A Appendix

We describe here some mathematical details and physical approaches for our numerical calculations of the bright spot motion along the black hole rotation axes.

1.1 A.1 Locally non-rotating frames

The most physically appropriate coordinate frame for Kerr black hole is a so-called Locally Non-Rotating Frames (LNRF) [53, 55]. The orthonormal tetrad

(7)

relates the Boyer–Lindquist coordinates $(t,r,\theta ,\phi )$ with the similar ones for physical observers in the LNRF:

$$\begin{aligned} \mathbf{e }_{(t)}\!=\! & {} e^{-\nu }\left( \frac{\partial }{\partial t} \!+\!\omega \frac{\partial }{\partial \varphi }\right) \!=\! \left( \frac{A}{\Sigma \Delta }\right) ^{1/2}\!\frac{\partial }{\partial t}\!+\! \frac{2Mat}{(A\Sigma \Delta )^{1/2}}\frac{\partial }{\partial \varphi }, \nonumber \\ \mathbf{e }_{(r)}= & {} e^{-\mu _1}\frac{\partial }{\partial r} = \left( \frac{\Delta }{\Sigma }\right) ^{1/2}\frac{\partial }{\partial r}, \nonumber \\ \mathbf{e }_{(\theta )}= & {} e^{-\mu _2}\frac{\partial }{\partial \theta } = \frac{1}{\Sigma ^{1/2}}\frac{\partial }{\partial \theta }, \nonumber \\ \mathbf{e }_{(\varphi )}= & {} e^{-\psi }\frac{\partial }{\partial \varphi } = \left( \frac{\Sigma }{A}\right) ^{1/2}\!\frac{1}{\sin \theta } \frac{\partial }{\partial \varphi }. \end{aligned}$$

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The related basis differential 1-forms for LNRF are

$$\begin{aligned} \mathbf{e }^{(t)}= & {} e^{\nu }\mathbf{d }t= \left( \frac{\Sigma \Delta }{A}\right) ^{1/2}\!\mathbf{d }t, \end{aligned}$$

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$$\begin{aligned} \mathbf{e }^{(r)}= & {} e^{\mu _1}\mathbf{d }t= \left( \frac{\Sigma }{\Delta }\right) ^{1/2}\!\mathbf{d }r, \end{aligned}$$

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$$\begin{aligned} \mathbf{e }^{(\theta )}= & {} e^{\mu _2}\mathbf{d }t= \Sigma ^{1/2}\,\mathbf{d }\theta , \end{aligned}$$

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$$\begin{aligned} \mathbf{e }^{(\varphi )}= & {} -\omega e^{\psi }\mathbf{d }t+e^{\psi }\mathbf{d }\varphi =-\frac{2Mar\sin \theta }{(\Sigma A)^{1/2}}\,\mathbf{d }t+ \left( \frac{A}{\Sigma }\right) ^{1/2}\!\!\sin \theta \,\mathbf{d }\varphi . \end{aligned}$$

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1.2 A.2 Equations of motion for test particles

Brandon Carter [49, 51, 55, 56] derived the remarkable first order differential equations of motion in the Kerr space-time, which meant as for the analytical and also for numeric calculations of the test particle trajectories. These trajectories depend on the integrals of motion: $\mu$—the test particle mass, E—the test particle total energy, L—test particle azimuth angular momentum, and the very specific Carter constant Q, defining the non-equatorial motion of the test particle. The motion of test particles is bounded to an equatorial plane of the metric if $Q = 0$.

$$\begin{aligned} \Sigma \frac{dr}{d\tau }= & {} \pm \sqrt{R(r)}, \end{aligned}$$

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$$\begin{aligned} \Sigma \frac{d\theta }{d\tau }= & {} \pm \sqrt{\Theta (\theta )}, \end{aligned}$$

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$$\begin{aligned} \Sigma \frac{d\phi }{d\tau }= & {} L\sin ^{-2}\theta +a(\Delta ^{-1}P-E), \end{aligned}$$

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$$\begin{aligned} \Sigma \frac{dt}{d\tau }= & {} a(L-aE\sin ^{2}\theta )+(r^2+a^2)\Delta ^{-1}P, \end{aligned}$$

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where $\tau$ is a proper time of the massive ($\mu \ne 0$) test particles or a corresponding affine parametrization for the massless ($\mu =0$) test particles. The radial potential R(r) in these equations, which governs the radial motion of test particles, is

$$\begin{aligned} R(r) = P^2-\Delta [\mu ^2r^2+(L-aE)^2+Q], \end{aligned}$$

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where

$$\begin{aligned} P=E(r^2+a^2)-a L. \end{aligned}$$

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Respectively, the polar potential $\Theta (\theta )$ is

$$\begin{aligned} \Theta (\theta ) = Q-\cos ^2\theta [a^2(\mu ^2-E^2)+L^2\sin ^{-2}\theta ]. \end{aligned}$$

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Note that zeros of these potentials define the radial and polar turning points $dR/d\tau =0$ and $d\Theta /d\tau =0$, correspondingly.

It is useful to define the orbital parameters for the massive test particles, $\gamma =E/\mu$, $\lambda =L/E$ and $q^2=Q/E^2$. Notice that there are also possible particle trajectories with $Q<0$, which do not reach the space infinity and, consequently, a distant observer (formally at $r=\infty$). We will not consider such trajectories in this article.

1.3 A.3 Integral equations for test particle motion

In our numerical calculations, we also use the integral equations for test particle motion (13)–(16):

$$\begin{aligned}&\fint \frac{dr}{\sqrt{R(r)}} =\fint \frac{d\theta }{\sqrt{\Theta (\theta )}}, \end{aligned}$$

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$$\begin{aligned}&\tau =\fint \frac{r^2}{\sqrt{R(r)}}\,dr +\fint \frac{a^2\cos ^2\theta }{\sqrt{\Theta (\theta )}}\,d\theta , \end{aligned}$$

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$$\begin{aligned}&\phi =\fint \frac{aP}{\Delta \sqrt{R(r)}}\,dr +\fint \frac{L-aE\sin ^2\theta }{\sin ^2\theta \sqrt{\Theta (\theta )}}\,d\theta , \end{aligned}$$

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$$\begin{aligned}&t=\fint \frac{(r^2+a^2)P}{\Delta \sqrt{R(r)}}\,dr +\fint \frac{(L-aE\sin ^2\theta )a}{\sqrt{\Theta (\theta )}}\,d\theta . \end{aligned}$$

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The integrals in (20)–(23) are the line (or curve) integrals monotonically growing along the test particle trajectories. For example, the line integrals in (20) add up to the ordinary ones in the absence of both the radial and polar turning points on the particle test particle trajectory:

$$\begin{aligned} \int ^{r_s}_{r_0}\frac{dr}{\sqrt{R(r)}}= \int _{\theta _0}^{\theta _s}\frac{d\theta }{\sqrt{\Theta (\theta )}}. \end{aligned}$$

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Here $r_s$ and $\theta _s$ are the initial test particle (e. g., photon) radial and polar coordinates, while $r_0\gg r_\mathrm{h}$ and $\theta _0$ is the corresponding final (finishing) points on the trajectory (e. g., the photon detection point by a distant telescope). The second example is a case when there is only one turning point in the polar direction, $\theta _\mathrm{min}(\lambda ,q)$ (derived from the equation $\Theta (\theta )=0$). The corresponding line integrals in (20) add up now to the ordinary ones:

$$\begin{aligned} \int _{r_s}^{r_0}\frac{dr}{\sqrt{R(r)}} =\int _{\theta _\mathrm{min}}^{\theta _s}\frac{d\theta }{\sqrt{\Theta (\theta )}} +\int _{\theta _\mathrm{min}}^{\theta _0}\frac{d\theta }{\sqrt{\Theta (\theta )}}. \end{aligned}$$

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The most complicated case, which we consider in this paper, corresponds to the test particle trajectory with the one turning point in the polar direction, $\theta _\mathrm{min}(\lambda ,q)$ (derived from the equation $\Theta (\theta )=0$), and the one turning point in the radial direction, $r_\mathrm{min}(\lambda ,q)$ (derived from the equation $R(r)=0$). The corresponding line integrals in (20) in the our most complicated case add up to the following ordinary ones:

$$\begin{aligned} \int _{r_\mathrm{min}}^{r_s}\!\!\frac{dr}{\sqrt{R(r)}} +\int _{r_\mathrm{min}}^{r_0}\!\!\frac{dr}{\sqrt{R(r)}} =\!\!\int _{\theta _\mathrm{min}}^{\theta _s}\!\!\frac{d\theta }{\sqrt{\Theta (\theta )}} +\!\!\int _{\theta _\mathrm{min}}^{\theta _0}\!\!\frac{d\theta }{\sqrt{\Theta (\theta )}}. \end{aligned}$$

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It is clear that integral equations (20)–(23) for test particle trajectories with more numbers of turning points add up to the ordinary integrals in similar ways.

1.4 A.4 Energy shift of photons emitted by the moving bright spot

We suppose that a bright spherical massive blob of hot plasma with a mass $\mu$ and total energy $E=\mu$ (parabolic motion) is moving away from the extreme Kerr black hole ($a=1$) with ballistic velocity along the black hole rotation axis, starting very close to the event horizon radius, $r_\mathrm{h}=1$. To disregard the tidal effects, we additionally suppose that the radius of bright massive blob, $r_\mathrm{b}$, is negligible in comparison with the event horizon radius $r_\mathrm{b}\ll r_\mathrm{h}$. It is also supposed that the radius of the blob remains constant during the motion.

From equation (19) for the polar potential $\Theta (\theta )$ it follows that all photons reaching a distant observer and starting from the black hole rotation axis (at $\theta _s=0$ and $r\ge r_\mathrm{h}$) must have orbital parameter $\lambda =L/E=0$. So, our task is reduced to finding only one orbital parameter, q(r) for these photons.

Orbital parameters $\lambda =L/E$ and $q=\sqrt{Q}/E$ of the photon trajectory, reaching a distant telescope (placed at the radius $r_0>>r_\mathrm{h}$, at the polar angle $\theta _0$ and at the azimuth angle $\varphi _0$), are directly related with the corresponding impact parameters, viewed at the celestial sphere [43, 44, 70]:

$$\begin{aligned} \alpha =-\frac{\lambda }{\sin \theta _0}, \quad \beta = \pm \sqrt{\Theta (\theta _0)}, \end{aligned}$$

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where $\Theta (\theta )$ defined in equation (19). The impact parameters $\alpha$ and $\beta$ are called, respectively, the horizontal and vertical impact parameters.

We also take into account the gravitational red-shift and Doppler effect of photons emitted by the luminous blob of plasma moving along the jet and detected by a distant observer. The orthonormal Locally Non-Rotating Frames (LNRF) from equations (7)–(12) is suitable for calculations of the corresponding energy shift of these photons and energy flux from the bright blob detected by a distant observer.

A radial velocity component of the luminous blob with a mass $\mu$, moving along the jet with the azimuth angular momentum $L=0$ in the LNRF is

$$\begin{aligned} V\equiv V^{(r)}=\frac{u^\mu e^{(r)}_\mu }{u^\nu e^{(t)}_\nu } =\frac{\sqrt{R(r)}}{(r^2+a^2)\gamma }. \end{aligned}$$

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Here, $\gamma =E/\mu$ is the Lorentz gamma-factor, $u^\mu =dx^\mu /ds$ is the 4-velocity of the blob, defined by differential equations (13)–(16), and R(r) is the radial potential from equation (17) with the parameter $L=0$.

All photons, reaching a distant observer, start from the moving blob with the orbital parameter $\lambda =0$. In this case, the corresponding components of the photon 4-momentum $p^{(\mu )}$ in the LNRF are

$$\begin{aligned} p^{(\varphi )}= & {} g^{\mu \nu }p_\nu e_\mu ^{(\varphi )} =0, \end{aligned}$$

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$$\begin{aligned} p^{(t)}= & {} g^{\mu \nu }p_\nu e_\mu ^{(t)} =\sqrt{\frac{r^2+a^2}{\Delta }}, \end{aligned}$$

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$$\begin{aligned} p^{(r)}= & {} g^{\mu \nu }p_\nu e_\mu ^{(r)} =\sqrt{\frac{r^2+a^2}{\Delta }-\frac{r^2+q^2}{r^2+a^2}}, \end{aligned}$$

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where $\Delta$ defined in equation (5). The condition $p^{(i)}p_{(i)}=0$ determines the fourth component of the photon 4-momentum. The photon energy in the LNRF is $E_\mathrm{LNRF}=p^{(t)}$. Meantime, the corresponding photon energy in the comoving frame of the massive blob moving with a radial velocity V relative to the LNRF is

$$\begin{aligned} {\mathcal {E}}(r,q)=\frac{p^{(t)}-Vp^{(r)}}{\sqrt{1-V^2}}. \end{aligned}$$

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As a result, the photon energy shift (ratio of the photon frequency detected by a distant observer to the frequency of the same photon in the comoving frame of the blob) is $g(r,q)=1/\mathcal{{E}}(r,q)$. This energy takes into account gravitational red-shift and the Doppler effect.

1.5 A.5 Energy flux from the moving bright spot

We incorporate the photon energy shift g(r, q) from equation (32) into the very useful Cunningham–Bardeen formalism [43, 44] for numerical calculation of the energy flux detected by a distant observer from the moving blob of finite size. We also numerically calculate the elliptic deformation of the lensed prime image of the small spherical blob in the strong gravitational field of extreme Kerr black hole (see more details of these calculations in [57,58,59,60, 71,72,73])

The corresponding flux of energy detected by a distant observer from the bright spot in the jet is the double integral of the surface brightness over the angular spread of the bright spot image [44]:

$$\begin{aligned} F_0=\iint \frac{d\alpha d\beta }{r_0^2} \frac{L}{4\pi ^2b^2}g(r,q)^2. \end{aligned}$$

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In this equation $\alpha$ and $\beta$ are respectively the horizontal and vertical impact parameters from (27), $r_0$ is a distance to the bright spot, b is a radius of the bright spot (bright spherical massive blob of hot plasma) and g(r, q) is the detected photon energy shift from equation (32). The double integral (33) in the case of a distant observer at $r_0\gg r_\mathrm{h}$ is well approximated in the deviation of the photon trajectory from the central one in the gravitational field of the black hole:

$$\begin{aligned} F_0=\frac{L}{4\pi ^2b^2r_0^2}g(r,q)^2\iint d\alpha d\beta . \end{aligned}$$

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In the used Cunningham–Bardeen formalism it is developed the effective perturbation method for numerical calculation of the double integral in this equation by taking into account the intersection of photon trajectory with the plane of the bright spot disk perpendicular to the radial direction in the local rest frame the bright spot (see details of this effective perturbation method in [44]).

Results of all our numerical calculations are illustrated in Figs. 1, 2, 3, 4, 5, 6, 7, 8 and 9 and in numerical animation [69].