Imprints of black hole charge on the precessing jet nozzle of M87*

As a prediction of general relativity (GR), the existence of black holes has been supported by the observation of LIGO LIGO . It is widely believed that there exists supermassive black hole in the center of each galaxy. Excitingly, the Event Horizon Telescope (EHT) collaboration released the first-ever image of the black hole at the center of the M87* Akiyama1 , strongly suggesting the presence of a supermassive black hole at the center of M87 and providing a pathway for strong field tests of gravity. Subsequently, the EHT collaboration published the image of SgrA* at the center of our Milky Way Akiyama2 , further advancing the progression of the observation and theory. However, due to limitations in image resolution, there remains significant room for further studies.

One of the most striking features of M87* is the bright jet of energy and matter emanating from its core. Previous studies of the inner region of M87* indicated that the jet near the black hole exhibits a large opening angle WJunor ; KHada ; RCWalker ; RLu . Recently, Cui et al. Cui reported an analysis of 22 years of radio observations, showing that the jet’s position angle varies periodically. They hypothesized that this is due to a misaligned accretion disk around a rotating black hole, leading to Lense-Thirring precession. Through their analysis, they derived a half-opening angle of the precession cone of $1.25^{\circ}\pm 0.18^{\circ}$ and a corresponding period of $11.24\pm 0.47$ years, with a precession angular velocity of $0.56\pm 0.02$ radians per year. This observation strongly indicates that the central black hole in M87* has a tilted accretion disk deviating from the equatorial plane. Subsequently, they examined the imprints of M87’s jet precession on the black hole-accretion disk system, including the disk’s size and the jet’s non-collinear structure Cui2 .

For a tilted accretion disk, the disk plane varies with radius. Within the innermost stable orbit (ISCO), matter particles will rapidly fall into the black hole, and thus it can generally regarded that the accretion disk starts at this orbit radius and then extends towards the distance. The inner disk typically undergoes Bardeen-Petterson alignment BardeenPetterson , which means it usually aligns with the equatorial plane. The outer edge of the inner disk is defined by a characteristic radius known as the warp radius. Beyond the warp radius, the tilt angle of the disk relative to the equatorial plane gradually increases. The tilted accretion disk is found in a wide variety of systems, e.g., protostars, X-ray binaries, and active galactic nuclei (AGN) Papaloizou ; Herrnstein ; Begelman ; Wijers ; Chiang ; Martin ; Lodato ; Casassus . Modeling such complex accretion disks is difficult if one considers its internal dynamical mechanisms. However, if the characteristics of the tilted disk are grasped, one can simulate the motion of the accretion disk with simple and manageable model. For example, the quasi-periodic oscillations observed in certain astrophysical black hole systems can be explored by studying the precession of spherical orbits Zahrani . Another interesting application was first proposed in Ref. Wei that the observed jet precession period can be used to constrain the black hole parameters. In the study, the warp radius and black hole spin parameter are constrained based on the following three assumptions. First, the motion of the disk particles at each radial distance can be accurately described by spherical orbit with a constant radius, deviating from the equatorial plane Wilkins ; Goldstein ; Dymnikova ; Shakura ; ETeo ; PRana ; Kopek . Second, the jet is assumed to originate near the warp radius and be oriented perpendicular to the accretion disk. Finally, the precession axis is considered as the axis of the black hole spin.

The black holes of general relativity can be completely specified by only three parameters: their mass $M$, spin angular momentum $J$, and the electric charge $Q$ according to the “no-hair theorem” Israel1 ; Israel2 ; Carter2 ; Hawking ; Gravitation ; Robinson . The effects caused by the charge are minimal, making it difficult to constrain the charge through observations in weak gravitational fields Sereno ; Ebina . The EHT collaboration, using observations of black hole shadows in strong gravitational fields, constrained multiple parameters, including the charge, but only ruled out certain regions corresponding to specific physical charges Kocherlakota . The precession of M87*’s jet presents another strong gravitational observation following black hole shadow, and we expect to use this observation to constrain multiple parameters. Although very recent study have explored a rotating black hole immersed in a Melvin magnetic field CChen , a simpler and equally meaningful case is the Kerr-Newman black hole. One might argue that the charge of a charged black hole would quickly neutralize in the surrounding plasma, but here we do not consider the neutralization process or the electromagnetic interaction with the astrophysical environment. This assumption is consistent with that made in Refs. Kocherlakota ; Tsukamoto . Our goal is to explore the potential of using black hole jet precession to constrain multiple parameters, with a particular focus on constraining parameters other than the spin, specifically the charge $Q$ in the Kerr-Newman scenario. Besides, some accretion scenario also involve the study of charged rotating black holes Wilson ; Damour ; Ruffini . Thus, here we consider the charged rotating black holes described by the Kerr-Newman solution at the center of M87* Newman . Building on the assumptions proposed in Ref. Wei , we model the tilted accretion disk using spherical orbits around a Kerr-Newman black hole and constrain the black hole parameters through the observed jet precession period.

First, we calculate the energy and angular momentum of spherical orbits around a Kerr-Newman black hole, as well as the radii of the innermost stable spherical orbit (ISSO) and the last spherical orbit (LSO), and focus on the impact of the charge on these quantities. Next, we numerically solve the equations of motion in the $\theta$ and $\phi$ directions to obtain the precession angular velocity of the spherical orbits and investigate its dependence on the black hole parameters. Finally, based on these calculations, we derive the precession period and use the spherical orbits of the Kerr-Newman black hole to model the accretion disk of M87*, constraining the relationships between the spin, charge, and warp radius with the observed jet precession period. We also established a relationship between the maximum warp radius and the charge.

Our paper is organized as follows. In Sec. II, we carry out a detailed study of spherical orbits around a Kerr-Newman black hole, including its special subclasses ISSO and LSO. In Sec. III, we further analyze the precession of spherical orbits. Then we provide some constraints on the black hole parameters using the observed jet precession period in Sec. IV. Finally, we discuss and conclude our results in Sec. V. Here we adopt the metric convention $(-,+,+,+)$ and use geometrical units with $G=c=1$ in addition to recovering dimensionality in Sec. IV.

In this section, we study the properties of the spherical orbits for test particles around a Kerr-Newman black hole, including angular momentum, energy, and stability of the orbits. In addition, the ISSOs and LSOs are also examined. Our main focus is on the effect of a black hole’s charge $Q$ and orbital tilt angle $\zeta$ on spherical orbits in the case of small and large black hole spin.

We start with a brief review of the motion of test particles in the Kerr-Newman spacetime. In the Boyer-Linquist coordinates, the Kerr-Newman black hole reads

where

Here $a,Q,M$ represent the black hole’s spin, charge, and mass. By solving $\Delta=0$, we easily obtain the radii of the black hole horizons

When the black hole exists, $M^{2}-a^{2}-Q^{2}\geq 0$ must be satisfied, otherwise, naked singularities are presented.

For a given geometry, the geodesics of test particles are governed by the Hamilton Jacobi equation,

where $\tau$ is an affine parameter along the geodesics and $S$ is the Jacobi action. When $S$ is separable, it can be written as

where the energy $E$ and the angular momentum $L$ per unit mass of the test particle are constants of motion associating with the Killing fields $\partial_{t}$ and $\partial_{\phi}$, respectively. We mainly study the motion of massive test particles by setting $\delta=1$. Substituting it into Eq. (3), the four equations of motion in the directions $\{t,r,\theta,\phi\}$ can be obtained

with

Here $\mathcal{K}$ is separation constant and called Carter constant corresponding to the Killing-Yano tensor Carter .

In this paper, we focus on spherical orbits as a special class of bound geodesics in Kerr-Newman spacetime with radial coordinate $r=const$. According to Eq. (8), it is apparent that the $\theta$-motion of particles exhibits symmetry about $\theta=\frac{\pi}{2}$. Therefore, for an off-equatorial orbit, the $\theta$-motion will oscillate about the equatorial plane confined within the range $\left(\frac{\pi}{2}-\zeta,\ \frac{\pi}{2}+\zeta\right)$, where $\zeta\in\left(0,\frac{\pi}{2}\right)$ is the tilt angle to the equatorial plane. Because of the particle turning back at two points $\theta=\frac{\pi}{2}\pm\zeta$, we have $\frac{d\theta}{d\tau}=0$, which gives

For a spherical orbit, besides the constants $E$ and $L$, if the tilt angle $\zeta$ is given, the Carter constant $\mathcal{K}$ will also be determined. We then obtain $\theta$- and $\phi$-motions by solving Eqs. (8) and (6).

In the static spherically symmetric spacetime, if the particles are not constrained to the equatorial plane, the spherical orbits of the particles must be a tilted ring. Considering the symmetry, the spherical orbits can be cast in the equatorial plane by reselecting the coordinate axis. Differentially, in the stationary axisymmetric spacetime, the periods of the $\theta$ and $\phi$ directions of the spherical orbits of particles are different due to the dragging effect of black holes in the $\phi$ direction, resulting in Lense-Thirring precession. In this section, we focus on the precession of spherical orbits around a Kerr-Newman spacetime.

Here, we consider the precession of spherical orbits in the vicinity of the black hole as seen by a distant observer, which differs from the local observer given in Ref. Zahrani . In astrophysics, observers and astrophysical events are generally at significant distances from each other. Thus, it is necessary to parameterise the motion of $\theta$ and $\phi$ using the coordinate time $t$. From Eqs. (5), (8), and (6), we obtain

where $E$ and $L$ are solved in Eqs. (11) and (12).

By numerically integrating these two equations, we can obtain the motion of $\theta$ and $\phi$ directions with initial conditions $\theta(0)=\frac{\pi}{2}$ and $\phi(0)=0$. From the evolution curves of $\theta$ and $\phi$ with respect to the coordinate time $t$, we can extract the precession angular velocity of the motion in the $\phi$ direction relative to that in the $\theta$ direction. For specific calculation details, we refer to Ref. Wei .

Now, let us calculate the precession angular velocity $\omega_{t}$ of the spherical orbits. Integrating Eq. (14), we obtain the period of $\theta$ motion $T_{\theta}$. From the $\phi$ motion, we can obtain the change $\Delta\phi$ within time $T_{\theta}$. Thus, the precession angular velocity $\omega_{t}$ is given by

The numerical results in Ref. Zahrani indicate that variations in the tilt angle $\zeta$ have a negligible effect on the precession angular velocity $\omega_{t}$, despite their significant impact on the energy and angular momentum as analyzed earlier. Fig. 4 also shows minimal differences between the left and right subplots. In astronomical observations, the tilt angle is usually known, and fixing it is helpful for constraining parameters using the precession period in the subsequent analysis. We can see that the effect of changing $Q$ on $\omega_{t}$ is small compared to changing $a$. This is because the necessary condition for precession mainly comes from the spin $a$. If the angular velocity $\omega_{t}$ is expanded as a series in terms of $a$ and $Q$, the leading order contains only $a$, and $Q$ disappears (corresponding to the Kerr case). Furthermore, the metric is a function of $Q^{2}$. Therefore, the influence of $Q$ is significantly smaller than that of $a$. We can infer that for a compact object, even if its charge changes, the precession angular velocity of the surrounding spherical orbits will not experience significant changes. Interestingly, the dependence of the angular velocity $\omega_{t}$ on $a$ and $Q$ is different: $\omega_{t}$ increases with $a$, while decreases with $Q$. We also note that regardless of whether the particle is in a prograde or retrograde orbit, the precession angular velocity always aligns with the direction of the black hole’s spin, reflecting the dragging effect of the rotating black hole. Finally, it is clear that as the radius $r$ increases, $\omega_{t}$ consistently decreases. This suggests that the greater the distance from the black hole, the weaker the dragging effect becomes.

Recently, Cui et al. Cui reported a period of approximately $11$ years for the variation in the position angle of the jet based on the analysis of radio observation of galaxy M87 over $22$ years. They infered that they are seeing a spinning black hole which occurs in the Lense-Thirring precession of a misaligned accretion disk. Subsequently, this observational fact was first used to constrain the spin of the black hole by using the precession of spherical orbits Wei . In previous work Petterson ; Ostriker ; Fragile ; LodatoPrice , a clear picture has been developed for a tilted accretion disks. As the radial distance decreases, the tilt angle of the disk decreases. At a characteristic radius known as the warp radius, the disk returns to the equatorial plane. When the particles are inside the ISSO, they will rapidly fall into the black hole. Thus, the warp radius can exceed the radius of the ISSO. In complex astrophysical environments, simulating an accretion disk requires taking into account many factors, such as disk viscosity, magnetic fields, external forces or torques, and so on. This task is typically handled by magnetohydrodynamics. However, we focus on the most fundamental components of the accretion disk, using spherical orbits to model the motion of particles within the disk. Additionally, combining this with the image of a tilted disk, we propose that jets originate near the warp radius, leading to a small misalignment between the jet axis and the black hole’s spin axis. General relativistic magnetohydrodynamic (GRMHD) simulations have demonstrated that a significant portion of the accretion disk in these misaligned systems undergoes Lense-Thirring precession PCFragileOBlaes ; Liska ; White ; Chatterjee ; Ressler , and that the jet precesses in sync with the disk Liska ; JCMcKinney . Under our assumption, the precession period of the jet is naturally consistent with the precession period of the spherical orbits near the warp radius. The purpose of these assumptions is to extract as much information as possible about the black hole parameters, using the observed precession of the jet of M87* black hole. Although these assumptions are overly simplified, they provide a fast and effective way to establish relationships between black hole parameters, serving as a preliminary step for more precise simulations.

To use the precession period to constrain black hole parameters, we first summarize how to calculate the precession period for spherical orbits. The precession period after unit restoration, is given

where $M_{\odot}$ is the mass of sun. From the observation, the mass of M87^⋆ black hole is $M=6.5\times 10^{9}M_{\odot}$ Akiyama1 . From Eqs. (14) and (15), we know that the functional form of $\omega_{t}$ is given by

According to Ref. Cui , the half-opening angle of the precession cone is $1.25^{\circ}\pm 0.18^{\circ}$. However, Ref. Zahrani points out that the effect of changing $\zeta$ on $\omega_{t}$ is negligible, so we fix the tilt angle at $\zeta_{p}=1.25^{\circ}$. It is worth noting that current observations do not clearly determine the angular momentum direction of the accretion disk, making it essential to distinguish between prograde and retrograde orbits in the calculations. Given $a$, $Q$, and $r$, we first obtain the energy $E(r,a,Q,\zeta_{p})$ and angular momentum $L(r,a,Q,\zeta_{p})$ following the calculations in Sec. II. Then, we compute the angular velocity $\omega_{t}$ by following Sec. III. Finally, by substituting it into Eq. (17), we obtain the precession period $T$. The ranges of $a$ and $Q$ are determined by the existence conditions of the black hole, as described in Eq. (2). For $r$, we are particularly interested in the radius associated with the origin of the jets, namely the warp radius, since its corresponding precession is linked to the precession of the jets. In Ref. Zahrani , the range for the warp radius is set between (6$M$, 20$M$). From Fig. 2, we observe that in many cases, $r_{ISSO}$ is less than 6$M$. To comprehensively consider all spherical orbits, we set the radius range to $(r_{ISSO},20M)$. In the subsequent analysis, we will find that the maximum constrained value of the warp radius is always less than 20$M$.

In summary, the precession period $T$ of the jets relative to the black hole spin axis is determined by the black hole spin $a$, charge $Q$, and the warp radius $r$. In Ref. Cui , it is reported the precession period of the M87* jet as $11.24\pm 0.47$ years. This constrains a family of surfaces in the $\{a,Q,r\}$ parameter space, where $T$ is constant on each surface. Our task is to establish the relationship between these three parameters. The relation between $a$ and $r$ is shown in Fig. 7 of Ref. Wei , where $Q=0$ (Kerr black hole). If $Q$ takes other values, a similar relationship holds.

In this paper, we focused on the spherical orbits of test particles around a Kerr-Newman black hole and constrained the black hole parameter by using the recent observation of the precessing jet nozzle of M87*. Following the observation of the black hole’s shadow, this observation has emerged as another observational tool in strong gravitational fields, providing a promising method to constrain the properties of supermassive black holes and test gravitational theories. The periodic precession of the jet from the supermassive black hole at the center of M87*, relative to the black hole’s spin axis, suggests that the jet originates from a tilted accretion disk. The tilt of the disk induces a kinematic precession of particles within the disk relative to the spin axis. This simplified physical picture motivates us to use spherical orbits around a black hole to model the tilted accretion disk and thus determine the jet precession period closely associated with the disk.

To achieve this, we first analyzed a subclass of bound orbits, specifically spherical orbits with constant radius. In addition to the energy and angular momentum, the motion constants of these orbits include the Carter constant $\mathcal{K}$, which arises from the separability of the geodesics and is determined by fixing the tilt angle relative to the equatorial plane. We found that, for various values of spin $a$ and tilt angle $\zeta$, as the charge $Q$ increases, the absolute value of angular momentum $L$ and energy $E$ decrease. The distinction between the prograde and retrograde orbits lies in the different correlation between energy and angular momentum: for prograde orbits, they are positively correlated, while for retrograde orbits, the correlation is negative. The above results apply to ISSO as well. Significantly, near the LSO, the energy and angular momentum exhibit divergent behavior. We further investigated the radial distribution of spherical orbits, and found that it is primarily determined by the ISSO and LSO, the two special spherical orbits. As the charge $Q$ increases, both $r_{\text{ISSO}}$ and $r_{\text{LSO}}$ decrease.

After thoroughly studying the properties of spherical orbits, we solved for the energy and momentum of these orbits, which serves as the foundation for our further analysis. By numerically solving the geodesic equations in the $\theta$ and $\phi$ directions, it is found that the motion in the $\theta$ direction is periodic, while the motion in the $\phi$ direction undergoes precession relative to the $\theta$-motion. Then, we calculated the precession angular velocity $\omega_{t}$ as seen by a distant observer and found that its dependence on the charge $Q$ is much weaker compared to the spin parameter $a$. The angular velocity $\omega_{t}$ increases with $a$, while decreases with $Q$ and $r$. Furthermore, regardless of whether the orbit is prograde or retrograde, the direction of the angular velocity is always aligned with the black hole’s spin, reflecting the frame-dragging effect of the rotating black hole.

Finally, we obtained the precession period $T$ from Eq. (17). For each set of values of $r$, $a$, and $Q$, we repeated the above steps to obtain the corresponding period. Observations show that the jet from the M87* black hole forms an angle of $1.25^{\circ}\pm 0.18^{\circ}$ with the spin axis, with a precession period of $11.24\pm 0.47$ years. We assumed that the black hole’s jet originates near the warp radius, which marks the boundary between the region of the disk that is off the equatorial plane and the region that remains aligned with it. To account for all possible spherical orbits, we considered the warp radius $r$ within the range $(r_{ISSO},20M)$ and calculate the precession period for each point in the $r$, $a$, and $Q$ parameter space. Consequently, the observed precession period constrains a family of surfaces in this parameter space, with each surface corresponding to values in the range $10.77$ to $11.71$ years. We further calculated the parameter relationships on these constrained surfaces: for a fixed spin $a$, the charge $Q$ decreases as the warp radius $r$ increases, and the warp radius for prograde orbits is smaller than that for retrograde orbits. For fixed warp radius $r$, the spin $a$ increases slowly with the charge $Q$, and at larger warp radii, the difference between prograde and retrograde orbits becomes more pronounced. Although these constraints do not provide definitive parameter values, they qualitatively limit the correlations between the parameters. To further constrain the charge $Q$, we investigated the relationship between the maximum warp radius $r_{max}$ and charge $Q$, revealing a negative correlation between $r_{max}$ and $Q$. We found that the gap between the maximum warp radius for prograde and retrograde orbits decreases as the charge increases. If future observations determine that the warp radius lies between $14.12M$ and $16.11M$, the accretion disk can be confirmed to be counter-rotating relative to the black hole’s spin axis, allowing an upper limit on the charge to be set. If the warp radius falls between $8M$ and $14.12M$, it will not be possible to distinguish between prograde and retrograde orbits, but different upper limits on the charge can be obtained, with the retrograde case allowing for a larger upper limit. If the warp radius is less than $8M$, this method cannot provide strict constraints on the charge.

In summary, our calculations and analysis offer a method to constrain black hole parameters, especially providing constraints on the black hole’s charge in certain cases. While our assumptions are relatively simple, and the specific numerical values require more accurate simulations, our qualitative conclusions are significant and offer a reference for future precise calculations. As multi-messenger astronomy progresses, combining different observational methods and data may offer new opportunities for constraining black hole parameters using jet precession periods.

This work was supported by the National Natural Science Foundation of China (Grants No. 12075103, No. 12475055, and No. 12247101).