Photon emissions from Kerr equatorial geodesic orbits

In recent years, the Event Horizon Telescope (EHT) Collaboration has released the images of the supermassive black holes M87* and Sgr A* in succession [1, 2]. A bright ring and a central dark shadow have been observed in each EHT image. The colorful rings are asymmetric in brightness and are composed of the photons emitted from the equatorial accretion disks, and the dark shadows correspond to the regions where the black holes capture the photons. It is a remarkable progress in observing a black hole at the event horizon scale, which attracts broad research interests on many observational aspects of black holes, including, for example, the shadows [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], the photon rings [13, 14, 15, 16, 17, 18, 19, 20], the signatures of surrounding hot spots [21, 22, 23, 24] and the magnetospheres[25, 26, 27, 28, 29, 30]. The photon ring’s brightness and size also encode the information of the accretion disk. Therefore, it is essential to study the photons emissions from the particles in the accretion disk, and answer the following question: how many photons can escape to infinity, and how are the photon frequencies shifted at infinity?

The escaping of the photons in the Schwarzschild spacetime was first studied by Synge in [31], where it was found that the escape cone shrinks as the emitting point shifts towards the horizon. Later, the “photon escape cone” in the Kerr spacetime was studied by Semerak in [32], where both the black hole and the naked-singularity cases were discussed. The “light escape cone” in the Kerr-de Sitter spacetime was studied in [33], where the sources in both radial geodesic and circular geodesic motions were considered. Recently, the photon escaping from the light sources of different motions had been extensively studied. The photon escaping probability (PEP) of zero-angular momentum sources (ZAMS) was first studied for the Kerr-Newman spacetime in [34], where the extremal, sub-extremal and non-extremal cases were discussed separately. Soon after, the PEP of ZAMS for the Kerr-Sen spacetime was studied in [35]. The PEP and maximum observable blueshift (MOB) of the photon emissions from the sources on circular geodesics outside the ISCO of a Kerr black hole were studied in [36, 37]. The PEP and MOB of the plunging emitters starting from the ISCO of a Kerr black hole were studied in [38]. It was found that the PEP of a ZAMS approaching the event horizon tends to zero and $29\%$ for a non-extremal and extremal Kerr black hole, respectively [34], and the PEP of circular emitters at the ISCO is larger than $55\%$ [36, 37], while the PEP from plunging emitters at approximately halfway between the ISCO and horizon is about $50\%$ [38]. Moreover, it was also found that the emitter’s proper motion affects the MOB of the escaped photons [36, 37, 38]. Very recently, the condition for the photons escaping from the off-equator sources to the infinity in the Kerr spacetime was clarified in [39, 40].

For extremal and near-extremal Kerr black holes, the escaping of the photons could be investigated more clearly in the near-horizon extremal Kerr (NHEK) and near-horizon near-extremal Kerr (near-NHEK) geometries. By using the (near-)NHEK¹¹1We use “(near-)NHEK” to represent both NHEK and near-NHEK. metrics [41, 42, 43], the calculations of PEP and MOB were simplified, and some analytical results were obtained [37, 44, 45]. The PEP and the blueshift distributions of the emitters moving at the ISCO (residing in the NHEK region) were reproduced analytically in [37]. Following the method of [37], the photon emissions from ZAMS in both the NHEK and near-NHEK regions were analytically studied in [44]. Then in [45], the photon emissions from equatorial emitters following various geodesic motions in the (near-)NHEK geometry were further studied. It was found that the PEP for ZAMS in the NHEK region and for the source at the innermost photon orbit in the near-NHEK region is about $29\%$ and $13\%$ respectively; the PEP is larger than $50\%$ for the outgoing geodesic emitters that can eventually reach NHEK infinity; the PEP is less than $55\%$ for the plunging geodesic emitters that ultimately enters the horizon, and the PEP is less than $59\%$ for the bounded geodesic emitters in the (near-)NHEK region. It was also found that all escaping photons from ZAMS are redshifted due to the strong gravity, while those escaping photons from the emitters with various motions could be blueshifted when the Doppler effect overwhelms the strong gravity effect.

So far, the escaping of the photons from generic light sources for general non-extremal Kerr black holes has not been studied. This paper will study the photon emissions from the equatorial sources along all the possible geodesic orbits in the Kerr exterior, including generic plunging orbits, trapped orbits, bounded orbits, and deflected orbits. This work generalizes the results in [38], where only the marginal plunging orbits from the ISCO were considered. On the other hand, this work also generalizes the previous studies for the (near-)NHEK cases [45] to the general-spin Kerr case.

The remaining part of this paper is organized as follows. In Sec. 2, we review the timelike and null geodesics in the Kerr spacetime and then introduce a classification of the equatorial timelike geodesics. In Sec. 3, we discuss the problem of photon emissions from an equatorial emitter and obtain the formulae of the PEP and MOB. In Sec. 4, we display the numerical results for the PEP and MOB by using the figures and the tables and discuss them in detail. In Sec. 5, we conclude this work.

The Kerr spacetime metric in the Boyer-Lindquist coordinates $x^{\mu}=(t,r,\theta,\phi)$ is given by

$$ds^{2}=-\frac{\Sigma\Delta}{\Xi}dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma d\theta^{2}+\frac{\Xi\sin^{2}\theta}{\Sigma}\mathopen{}\mathclose{{}\left(d\phi-\frac{2Mar}{\Xi}dt}\right)^{2},$$

(2.1)

where

$$\Delta=r^{2}-2Mr+a^{2},\qquad\Sigma=r^{2}+a^{2}\cos^{2}\theta,\qquad\Xi=(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta.$$

(2.2)

Here, $M$ and $a$ are the mass and spin parameters of a black hole, respectively, and the spin $a$ is defined by $a=J/M$ with $J$ being the angular momentum. The outer event horizon of the black hole is located at

$$r_{H}=M+\sqrt{M^{2}-a^{2}}.$$

(2.3)

In the following, we set $M=1$ for simplicity.

There are four conserved quantities of a free particle: the mass $\mu$, the energy $E$, the axial angular momentum $l$, and the Carter constant $Q$. One can derive the four-momentum $p^{\mu}$ of this particle by using the Hamilton-Jacobi method [46], which reads

	$\displaystyle p^{r}$	$\displaystyle=$	$\displaystyle\pm_{r}\frac{1}{\Sigma}\sqrt{\mathcal{R}(r)},$		(2.4)
	$\displaystyle p^{\theta}$	$\displaystyle=$	$\displaystyle\pm_{\theta}\frac{1}{\Sigma}\sqrt{\Theta(\theta)},$		(2.5)
	$\displaystyle p^{\phi}$	$\displaystyle=$	$\displaystyle\frac{1}{\Sigma}\mathopen{}\mathclose{{}\left[\frac{a}{\Delta}[E(r^{2}+a^{2})-al]+\frac{l}{\sin^{2}\theta}-aE}\right],$		(2.6)
	$\displaystyle p^{t}$	$\displaystyle=$	$\displaystyle\frac{1}{\Sigma}\mathopen{}\mathclose{{}\left[\frac{r^{2}+a^{2}}{\Delta}[E(r^{2}+a^{2})-al]+a(l-aE\sin^{2}\theta)}\right],$		(2.7)

where

	$\displaystyle\mathcal{R}(r)$	$\displaystyle=$	$\displaystyle[E(r^{2}+a^{2})-al]^{2}-\Delta[Q+(l-aE)^{2}+\mu^{2}r^{2}],$		(2.8)
	$\displaystyle\Theta(\theta)$	$\displaystyle=$	$\displaystyle Q+a^{2}(E^{2}-\mu^{2})\cos^{2}\theta-l^{2}\cot^{2}\theta,$		(2.9)

are the radial and angular potentials, respectively, and $\pm_{r}$ and $\pm_{\theta}$ denote the signs of the radial and polar motions, respectively.

For a massive timelike particle, we have $\mu>0$ and $p^{\mu}=\mu\frac{\partial x^{\mu}}{\partial\tau}$ with $\tau$ being the proper time. For a massless particle (photon), we have $\mu=0$ and $p^{\mu}=\frac{\partial x^{\mu}}{\partial\tau}$ with $\tau$ being an affine parameter. For photons, it is convenient to express $p^{\mu}$ under a reparameterization by using the two rescaled quantities:

$$\lambda=\frac{l}{E},\qquad\eta=\frac{Q}{E^{2}}.$$

(2.10)

We now consider the equatorial timelike geodesics in the Kerr exterior. Hereafter, we set $\mu=1$ and let $s_{r}=\pm_{r}$ for timelike particles. Then $s_{r}=-1$ and $s_{r}=+1$ are for ingoing and outgoing particles, respectively. On the equatorial plane, we have $Q=0$, a geodesic is determined by the energy $E$ and angular momentum $l$. By studying the root structure of the radial potential $\mathcal{R}(r)$, the radial motions are classified in the $(E,l)$ phase space in [47]. In the following, we present a short review of the classification.

For geodesics that can reach the horizon, the energy $E$ and angular momentum $l$ is constraint by the thermodynamic bound:

$$l\leq l_{H}(E,a)=\frac{E}{\Omega_{H}}=\frac{2E}{a}(1+\sqrt{1-a^{2}}),$$

(2.11)

in which $\Omega_{H}=\frac{a}{2r_{H}}$ is the angular velocity of the event horizon. In the ergosphere, this is also the superradiation bound. When $l=l_{H}$, the potential $\mathcal{R}(r)$ has one root at the horizon. On the contrary, geodesics with $l>l_{H}$ cannot reach the horizon.

Let us consider the double root structure of $\mathcal{R}(r)$. A double root corresponds to a circular orbit, and we use the subscript “_∗” to represent a double root. For the $l=l_{H}$ case, we have one root at the horizon, and the double root requires $Y(r_{\ast})=Y^{\prime}(r_{\ast})=0$ with $Y(r)=\mathcal{R}(r)/(r-r_{H})$. The solution is given by $E=E_{c}$ with

$$E_{c}=\frac{r_{H}+2\sqrt{r_{H}-1}}{\sqrt{r_{H}(r_{H}+2+4\sqrt{r_{H}-1})}}.$$

(2.12)

Such a double root corresponds to a stable circular orbit since $Y^{\prime\prime}(r_{\ast})<0$. For $l\neq l_{H}$ case, we have $\mathcal{R}(r_{\ast})=\mathcal{R}^{\prime}(r_{\ast})=0$, then the solutions are given by [48]

	$\displaystyle E_{\pm}(r_{\ast})=\frac{r_{\ast}(r_{\ast}-2)\pm ar_{\ast}^{1/2}}{\sqrt{r_{\ast}^{3}(r_{\ast}-3)\pm 2ar_{\ast}^{5/2}}},$		(2.13)
	$\displaystyle l_{\pm}(r_{\ast})=\pm\frac{(r_{\ast}^{2}+a^{2})r_{\ast}^{1/2}\mp 2ar_{\ast}}{\sqrt{r_{\ast}^{3}(r_{\ast}-3)\pm 2ar_{\ast}^{5/2}}}.$		(2.14)

Hereafter the plus/minus sign “$\pm$” represents the prograde/retrograde orbit, respectively. The ranges of the double roots are bounded by the innermost (prograde) and outermost (retrograde) photon orbits $r_{p\pm}$, that is $r_{\ast}>r_{p\pm}$, where

$$r_{p\pm}=2\mathopen{}\mathclose{{}\left[1+\cos\mathopen{}\mathclose{{}\left(\frac{2}{3}\arccos(\mp a)}\right)}\right].$$

(2.15)

Some of these circular orbits are stable, while others are unstable. The marginal stable circular orbits have $\mathcal{R}^{\prime\prime}(r_{\ast})=0$, the solutions are the triple roots

$$r_{\ast}=r_{\text{ISCO}\pm}=3+Z_{2}\mp\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})},$$

(2.16)

where

$$Z_{1}=1+(1-a^{2})^{\frac{1}{3}}\mathopen{}\mathclose{{}\left[(1+a)^{\frac{1}{3}}+(1-a)^{\frac{1}{3}}}\right],\qquad Z_{2}=\sqrt{3a^{2}+Z_{1}^{2}}.$$

(2.17)

By substituting the triple roots (2.16) into Eqs. (2.13) and (2.14), we have

	$\displaystyle E_{\text{ISCO}\pm}=E_{\pm}(r_{\text{ISCO}\pm}),$		(2.18)
	$\displaystyle l_{\text{ISCO}\pm}=l_{\pm}(r_{\text{ISCO}\pm}).$		(2.19)

Inverting Eq. (2.13) and substituting the results into Eq. (2.14), we can formally write the angular momentum $l$ as functions of the energy $E$, that is

	$\displaystyle l^{u}(E)=l^{u}_{\pm}(E)=l^{u}_{\pm}(r_{\ast}^{u}(E)),$		(2.20)
	$\displaystyle l^{s}(E)=l^{s}_{\pm}(E)=l^{s}_{\pm}(r_{\ast}^{s}(E)).$		(2.21)

Hereafter, we use the superscripts “$u$” and “$s$” to denote the unstable and stable double roots. Note that we have $l=l^{u}$ for $r_{p\pm}<r_{\ast}<r_{\text{ISCO}\pm}$ and we have $l=l^{s}$ for $r_{\text{ISCO}\pm}<r_{\ast}$.

Based on the radial root structures, the equatorial motions are classified in the $(E,l)$ phase space, as shown in Table 1. First, we introduce the four basic classes of geodesic orbits which only involve single roots: the plunging orbits $\mathcal{P}$, the trapped orbits $\mathcal{T}$, the deflected orbits $\mathcal{D}$, and the bounded orbits $\mathcal{B}$. A plunging particle plunges into the black hole from infinity. A trapped particle emerges from the white hole and falls into the black hole after bouncing back from a turning point. A deflected particle comes from infinity and bounces back from a turning point. A bounded particle oscillates between two turning points. Note that an anti-plunging (anti$-\mathcal{P}$) particle travels on the same trajectory as its counter partner but has the opposite radial orientation. Next, we consider the classes of orbits that involve unstable double roots, that is, $l=l^{u}(E)$. We call such orbits the marginal orbits. When a particle travels across the unstable double root on a marginal orbit²²2It takes infinite amount of proper time for a marginal particle to approach the unstable double root, so that the “marginal orbits” are asymptotic orbits for the emitters having near-critical parameters, and they are not physical orbits., it either plunges toward the horizon or moves toward infinity. Depending on whether a particle could or could not potentially reach infinity, the relevant orbits are named the marginal plunging orbits $\mathcal{MP}$ or marginal trapped orbits $\mathcal{MT}$, respectively³³3A more accurate and detailed classification for these marginal geodesic motions can be found in [47], here we introduce this simplified version of classification for convenience.. Besides, the orbit involves a triple root named the marginally trapped orbit from the ISCO $\mathcal{MT}_{\text{ISCO}}$. At a double root or a triple root, the relevant circular orbits have three classes: the stable circular orbit $\mathcal{C}^{s}$, the unstable circular orbit $\mathcal{C}^{u}$ and the ISCO $\mathcal{C}_{\text{ISCO}}$.

In case we would like to specify the sign of angular momentum $l$ and (or) radial direction $s_{r}\in\{-1,+1\}$ to avoid ambiguity, we will use subscripts “$\pm$” to represent prograde/retrograde emitters and use subscripts “$,i/o$” to represent ingoing/outgoing particles. Then a specific orbit (or a quantity $O$) is labeled like “$\text{Orbit}_{\pm,i/o}$” (or “$O_{\pm,i/o}$”). Otherwise, the subscripts “$\pm$” and (or) “$,i/o$” may be dropped out for simplicity.

Now we study the PEP and MOB of photon emissions from emitters on different orbits, which depend on the black hole spin $a$, and the emitters’ parameters $(r_{s},E_{s},l_{s},s_{r},)$. Following [38], we introduce $P\geq 1/2$ and $z_{\text{mob}}\geq 0$ as indicators of an emitter’s observability. We use $r_{0}$ to denote the radii at which $P=1/2$ and use $r_{z}$ to represent the radii where $z_{\text{mob}}=0$. The results are shown in Figs. 3–6. Each pair of PEP and MOB curves can reflect the variation in the brightness of an emitter along its orbit. We can see that the general feature of the results for prograde ($l_{s}>0$) and retrograde ($l_{s}<0$) emitters are very similar. However, as the black hole spin $a$ varies, the changing trends of the results for prograde and retrograde emitters are different (see Figs. 3 and 4). We also find that for outgoing ($s_{r}=+1$) emitters, the PEP is larger than one half and the MOB is positive, that is $P_{,o}>1/2$ and $z_{\text{mob},o}>0$. This means that all outgoing emitters are well observable. Therefore, in the following, we will pay more attention to the prograde and ingoing emitters, which have $l_{s}>0$ and $s_{r}=-1$.