Electron–Positron Pair Creation Close to a Black Hole Horizon: Redshifted Annihilation Line in the Emergent X-Ray Spectra of a Black Hole. I.

As we have already discussed, the pair creation occurs very close to the horizon, of the order of 100 m from the horizon of a 10 solar mass BH. In Figure 6, we show the trajectories of the created positrons in the Schwarzschild background. It can be seen that only a few of the 5 × 10⁵ positrons simulated to produce this figure can reach 3r_S. This number is even lower if we take into account Coulomb losses and the pair-annihilation process.

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Using the results of our first Monte Carlo, we have built another Monte Carlo software with which we simulate the propagation of the created positrons in the cloud, taking into account energy losses due to Coulomb scattering with protons. This process is the dominant cooling one, considering the physical condition of the cloud (see discussion in Moskalenko & Jourdain 1997). We compute also the annihilation probability for positrons, and check whether each positron annihilates or not. If the annihilation occurs, then we store the kinetic energy and location of this positron at the moment of the annihilation. If not, these positrons can either fall into the BH, Compton scatter off background photons, escape from the system, or annihilate farther away, outside the CC, depending on the density of the surrounding material at several Schwarzschild radii. If we suppose that all positrons escaping from the CC annihilate afterward, the maximum expected 511 keV photon flux is around 10³⁶ photons s⁻¹ (i.e., 10³⁰ erg s⁻¹) for a luminosity of 10³⁸ erg s⁻¹. In a further step, we derive, using a third Monte Carlo code and the saved annihilated positrons locations, the emergent annihilation radiation. As the 511 keV annihilation emission is produced at a different radius from the BH, the line endures different redshifts, depending on the location of the annihilation, and thus, this annihilation process appears to the external observer as a specific continuum. In Figure 7, we show the resulting spectra for a CC electron temperature of 5 keV. In this set of simulations, $\dot{m}$ was set equal to 1 (blue line), 4 (red line), and 10 (green line). The spectrum become harder as $\dot{m}$ increases. Also, due to the coinciding effects of the pair-creation efficiency, which increases with $\dot{m}$, and the CC opacity to the 511 keV radiation, decreasing with $\dot{m}$, we could see in Figure 7 that the spectrum is maximum for intermediate values of $\dot{m}=4$.

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We then explored what happens if we keep the mass accretion rate constant ($\dot{m}=4$) while varying the CC electron temperature. The result can be seen in Figure 8, where we present the emergent annihilation spectrum with a CC electron temperature of 5 keV (blue line) and 50 keV (red line). One can notice that the overall spectrum shape does not change much except for the spectrum shift toward higher energies when we increase the cloud temperature, kT_e.

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In Figure 9, we show also the emergent Comptonized spectrum of the CC, with an electron temperature of kT_e = 5 keV, including the gravitationally redshifted annihilated line. In the encapsulated upper-right-hand panel, we include the redshifted annihilation line as a ratio of the resulting spectral values to the Comptonized continuum. These simulations show that the emergent annihilation spectrum is quasi-universal whatever the CC physical conditions. In Figures 7 and 8, we demonstrate also that these redshifted annihilation lines can be represented by a blackbody-like shape, as shown by the dashed lines, which give the best blackbody fit for the $\dot{m}=1$ and $\dot{m}=4$ case respectively.

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However, observationally, this feature should be transient and will not be observed in all BH spectral states. Indeed, if the resulting X-ray luminosities L_γ ≫ 10³⁷(M_bh/10M_⊙) erg s⁻¹, the dimensionless mass accretion rate (see Titarchuk & Zannias 1998) will be much greater than 1, and the produced pairs and annihilation photons generated near a BH horizon cannot escape, as they are effectively scattered off converging electrons. In the LHS, we cannot see the high-temperature blackbody (HBB) either, and consequently, the redshifted annihilation line, because τ_γ−γ ≪ 1 and pairs are not effectively generated. Moreover, in the LHS, the CF is surrounded by a hot, relatively thick CC with plasma temperature of order of 50–80 keV. Thus, the escaping HBB photons are scattered off hot electrons of the CC, and the bump is smeared out. So, we expect this feature to be seen only during the intermediate state when the mass accretion rate is sufficient to trigger the effect but not strong enough to hide it afterward.

In the previous section, we demonstrated that the pairs are created very close to the BH horizon due to photon–photon interactions. In fact, these photons reach the necessary conditions for pair creation when the product of their energies E₁E₂ > (m_e c²)². In Figure 10, we show the redshifted annihilation line and the classical bulk motion Comptonization spectrum, plus the result of the photon upscattering off these energetic pairs, which leads to an extension of the emergent spectrum up to 10 MeV. In that case, the CC temperature is 5 keV ($\dot{m}=4$). In Figure 11, we illustrate the light propagation in the Schwarzchild metric, while in Figures 12 and 13, we also demonstrate the photon and pair distributions (spectra) for different radii within the CF, respectively.

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It is worth noting that similar spectra in the HSS in quite a few BH binaries were observed by the OSSE and COMPTEL instruments (see Grove et al. 1998 and McConnell 1994, respectively). In particular, seven BH transient sources, GRO J0422+32, GX 339–4, GRS 1716–249, GRS 1009–45, 4U 1543–47, GRO J1655–40, and GRS 1915+105, observed by OSSE demonstrated two Gamma spectral states (HSS and LHS) and the transition between these two states. Grove et al. (1998) emphasize that, in the LHS, the emergent spectra are characterized by hard spectra with photon indices Γ < 2 and exponential cutoff of energies ∼100 keV. This form of the spectra is consistent with the thermal Comptonization case (see Sunyaev & Titarchuk 1980). These spectra were observed by Grove et al. (1998) for GRO J0422+32, GX 339–4, GRS 1716–249, and GRS 1009–45, while GRS 1009–45, 4U 1543–47, GRO J1655–40, and GRS 1915+105 show the HSS with a relatively soft photon index, Γ, in the range from 2.5 to 3. Furthermore, Grove et al. (1998) have found that the HSS emergent spectrum in GRO J1655–40 is extended to 690 keV without any sign of a rollover at low energies. One can compare Grove's spectrum with our Monte Carlo simulated HSS spectrum demonstrated in Figure 10.

McConnell (1994) provided details of the Cyg X-1 observations in the 0.75–30 MeV energy range using the COMPTEL instrument in CGRO. They found the Cyg X-1 spectrum in the HSS to be extended up to at least 1 MeV without any break. No physical explanation was suggested to explain this observational result besides a suggestion of incorporating a spectral component that represents the reflection of the hard X-rays from an optically thick accretion disk (see Haardt & Maraschi 1993). However, this reflection scenario is inconsistent with the soft-state emergent Cyg X-1 spectrum because spectra with photon index higher than 2 do not demonstrate the reflection effect. Indeed, the spectrum is too steep and there is not enough photons at high energies to transfer them to lower energies to make a reflection (or downscattering) bump (see Laurent & Titarchuk 2007). However, we argue using our simulations that the combined OSSE–COMPTEL spectrum for Cyg X-1 in the HSS demonstrated in McConnell (1994) can result from the Comptonization of the soft (disk) photons by pairs in the CF, as shown in Figure 10.

In flat space, the trajectory of unscattered photons is a straight line, the equation for which can be written in terms of the law of sines:

$$\begin{eqnarray}&&r{(1-{\mu }^{2})}^{1/2}=\hat{p}\end{eqnarray} \tag{ 4 }$$

where r is the length of a radius vector r at a given point of the line, μ = cos β is the cosine of the zenith angle β between the radius vector r and the straight line, and $\hat{p}$ is the impact parameter of this line. The trajectory of unscattered photons in the Schwarzschild background is just a generalization of this law of sines (TZ98):

$$\begin{eqnarray}&&\displaystyle \frac{x{(1-{\mu }^{2})}^{1/2}}{{(1-1/x)}^{1/2}}=p\end{eqnarray} \tag{ 5 }$$

where x = R/R_S and R_S = 2GM/c² is the Schwarzschild radius, $p=\hat{p}/{R}_{{\rm{S}}}$, and M is the BH mass.

Using Equation (5) we can explain all properties of the photon trajectory in the Schwarzschild background. In other words:

• (i)  
  The photon can escape from the BH horizon to infinity or vice versa if p < p₀ = (6.75)^1/2 (see TZ98).
• (ii)  
  The photon, for which p = p₀, undergoes circular rotation at x = 1.5 (at 3GM/c²).
• (iii)  
  All photons for which p > p₀ and x < 1.5 are gravitationally attracted by the BH. In this case, all photon trajectories are finite.
• (iv)  
  All photons that start at x > 1.5 and have p > p₀ escape to infinity if they are not scattered off electrons on their way out.

The photon trajectory can be presented in polar coordinates r and φ. Using the metric of the Schwarzschild background, we can write (see Figure 11)

$$\begin{eqnarray}&&{rd}\varphi =\displaystyle \frac{\tan \beta {dr}}{{(1-1/x)}^{1/2}}.\end{eqnarray} \tag{ 6 }$$

Then, it follows from Equations (5) and (6) that

$$\begin{eqnarray}&&\varphi ={\int }_{1}^{x}\displaystyle \frac{d\eta }{{\eta }^{2}\sqrt{1/{p}^{2}-(1-1/\eta )/{\eta }^{2}}}.\end{eqnarray} \tag{ 7 }$$

This formula is identical to that derived in Landau & Linfshitz (1971) using the Hamilton–Jacobi (formalism) equation. In fact, the equation of the photon trajectory in polar coordinates readily follows from the law of sines (Equation (5)) and the Schwarzschild metric. Equation (7) is valid for any nonzero p-photon trajectory except a circular one at x₀ = 3/2, for which p₀ = (6.75)^1/2. Formula (7) should be numerically calculated. From formula (5), it is evident that μ₀ = 1 at x₀ = 1, i.e., the photon always enters a BH horizon along the radial direction.

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The length of the radial trajectory (p = 0) is an integral

$$\begin{eqnarray}&&l={r}_{{\rm{S}}}{\int }_{1}^{x}\displaystyle \frac{{dx}}{{(1-1/x)}^{1/2}}.\end{eqnarray} \tag{ 8 }$$

For α = (x − 1) ≪ 1, the length from r_S to r is

$$\begin{eqnarray}&&l\approx 2{r}_{{\rm{S}}}{\alpha }^{1/2}.\end{eqnarray} \tag{ 9 }$$

For x ≫ 1, we have

$$\begin{eqnarray}&&l\approx {r}_{{\rm{S}}}x.\end{eqnarray} \tag{ 10 }$$

The Thomson optical path on the radial trajectory from r_S to r (or from x = 1 to x = r/r_S) is

$$\begin{eqnarray}&&{T}_{{\rm{T}}}(r,{r}_{{\rm{S}}})={r}_{{\rm{S}}}{\int }_{1}^{x}{\sigma }_{{\rm{T}}}{n}_{e}(\eta )d\eta ,\end{eqnarray} \tag{ 11 }$$

where σ_T is the Thomson cross-section and

$$\begin{eqnarray}&&{n}_{e}(r)=\dot{m}{({r}_{{\rm{S}}}/r)}^{1/2}/(2r{\sigma }_{T})\end{eqnarray} \tag{ 12 }$$

is the electron density (see Turolla et al. 2002 for the derivation of n_e). The optical path can be found analytically as

$$\begin{eqnarray}{T}_{{\rm{T}}}(r,{r}_{{\rm{S}}}) & = & {\tau }_{{\rm{T}}}(x,1)=\displaystyle \frac{\dot{m}}{2}{\displaystyle \int }_{1}^{x}\displaystyle \frac{{dx}}{{x}^{3/2}\sqrt{1-1/x}}\\ & = & \dot{m}[\pi /2-\arcsin (1/x)].\end{eqnarray} \tag{ 13 }$$

Hence, the total radial Thomson optical depth of the CF is

$$\begin{eqnarray}&&{\tau }_{0}={\tau }_{{\rm{T}}}(\infty ,1)=(\pi /2)\dot{m}.\end{eqnarray} \tag{ 14 }$$

For the Thomson optical paths of the finite trajectories entering a BH horizon for which α = (x − 1) ≪ 1, we have

$$\begin{eqnarray}&&{\tau }_{{\rm{T}}}(r,{r}_{{\rm{S}}})={\sigma }_{T}n({r}_{{\rm{S}}})l\approx \dot{m}{\alpha }^{1/2}.\end{eqnarray} \tag{ 15 }$$

If the energetic photons have a high number density n_ph ≫ 1/(σ_T l), where l is the characteristic scale of the region within which the Comptonization (in our case, the bulk motion Comptonization) takes place, then the process of electron pair production by two photons becomes important. Indeed, when a Compton-upscattered photon of energy E₁ interacts with another one of energy E₂, it may produce an electron–positron pair provided their energies satisfy the following relationship:

$$\begin{eqnarray}&&{E}_{1}{E}_{2}\gt {({m}_{e}{c}^{2})}^{2}.\end{eqnarray} \tag{ 16 }$$

The related cross-section for pair production γ₁ + γ₂ → e⁺ + e⁻ by two photons σ_γγ (Akhiezer & Berestetsky 1965) depends on the product y = [E₁ E₂/2(m_e c²)²](1 − Ω₁Ω₂), where (E/c)Ω is the momentum for a given photon of energy E. The cross-section is only nonzero when y > 1. The maximum of σ_γγ(y_max) ≈ 0.26σ_T takes place at y² ≈ 2.

The mean energy of the photons upscattered in the CF is (Titarchuk et al. 1997)

$$\begin{eqnarray}&&\langle E\rangle ={E}_{0}{[1+4/\dot{m}]}^{{N}_{\mathrm{sc}}}.\end{eqnarray} \tag{ 17 }$$

This formula is obtained without taking into account the gravitational redshift. It presumably works in the part of the CF, where (1 − 1/x)^1/2 is of order unity (namely, where x ≳ 1.2). The mean number of scattering ${N}_{\mathrm{sc}}\sim \dot{m}$. Then, $\langle E\rangle /{E}_{0}\sim 16$ and $\langle E\rangle =24$ keV for $\dot{m}=4$ and E₀ = 1.5 keV. These photons of energy $\langle E\rangle$, when they propagate toward the BH horizon, are blueshifted. Their energy increases by a factor

$$\begin{eqnarray}&&\langle E{\rangle }_{{bs}}/\langle E\rangle =1/{(1-1/x)}^{1/2}.\end{eqnarray} \tag{ 18 }$$

Photon energies of order m_e c² = 511 keV and higher are achieved in the narrow shell around a BH horizon,

$$\begin{eqnarray}&&0\lt \alpha =x-1\lt {(\langle E\rangle /{m}_{e}{c}^{2})}^{2}.\end{eqnarray} \tag{ 19 }$$

For $\langle E\rangle =24$ keV, we obtain that α < 2.5 × 10⁻³. For a 10 solar mass BH, the thickness of the shell where the pairs can be created due to photon–photon interactions is about 70 m. In Figure 2, we show the average photon energy of Monte Carlo simulated spectrum as a function of radius in CF. We assume in this Monte Carlo simulation that the dimensionless accretion rate of the CF $\dot{m}=4$ and the electron temperature kT_e = 5 keV. The energy of the injected soft photons is 1.5 keV. As seen from Figure 2, the average photon energy is about 25 keV at radius R = 1.2 R_S, which is very close to our estimate presented above (see Equation (17)). Also, we can see from this figure that the blueshift and Comptonization are equally important below R = 1.02 R_S.

The photon density in the converging inflow n_ph near the BH horizon can be estimated using the photon flux injected in the flow, F_inj = L_inj/E₀. Thus,

$$\begin{eqnarray}{n}_{\mathrm{ph}}\sim \displaystyle \frac{{F}_{\mathrm{inj}}}{4\pi {r}_{{\rm{S}}}^{2}c} & = & 3\times {10}^{21}({L}_{\mathrm{inj}}/{10}^{37}\ \mathrm{erg}\ {{\rm{s}}}^{-1})\\ & & \times {({\rm{m}}/10)}^{-2}{({E}_{0}/1.5\mathrm{keV})}^{-1}\quad {\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 20 }$$

and the free path for the pair creation is

$$\begin{eqnarray}{l}_{\gamma \gamma }\sim {({n}_{\mathrm{ph}}0.25{\sigma }_{{\rm{T}}})}^{-1} & = & 2\times {10}^{3}{({L}_{\mathrm{inj}}/{10}^{37}\mathrm{erg}{{\rm{s}}}^{-1})}^{-1}\\ & & \times {(m/10)}^{2}({E}_{0}/1.5\,\mathrm{keV})\quad \mathrm{cm}.\end{eqnarray} \tag{ 21 }$$

The typical length of the photon trajectories l_cross in the shell of width α can be estimated using formula (9),

$$\begin{eqnarray}&&{l}_{\mathrm{cross}}=2{\alpha }^{1/2}{R}_{{\rm{S}}}\gt 3\times {10}^{5}(m/10)\quad \mathrm{cm},\end{eqnarray} \tag{ 22 }$$

for α = 2.5 × 10⁻³. So, the shell photons have enough time to create pairs because

$$\begin{eqnarray}&&{t}_{\mathrm{cross}}/{t}_{\gamma \gamma }={l}_{\mathrm{cross}}/{l}_{\gamma \gamma }\gg 1.\end{eqnarray} \tag{ 23 }$$

The optical depth for the pair creation is

$$\begin{eqnarray}{\tau }_{\gamma \gamma } & = & {l}_{\mathrm{cross}}/{l}_{\gamma \gamma }\gt 150({L}_{\mathrm{inj}}/{10}^{37}\ \mathrm{erg}\,{{\rm{s}}}^{-1})\\ & & \times {(m/10)}^{-1}{({E}_{0}/1.5\mathrm{keV})}^{-1}.\end{eqnarray} \tag{ 24 }$$

The spectrum of the pairs created in the photon shell can be calculated using the shell photon spectrum. In Figure 12, we show the Monte Carlo photon spectra computed for three different radius ranges in the flow. The black histogram is the emergent spectrum seen by observers on Earth. The green histogram is the spectrum seen by observers staying at the radial distance R = (1 − 1.1)R_S from the BH. The blue histogram is the spectrum for observers staying at the radial distance R = (1 − 1.01)R_S.

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As we argue above, the pairs have to be created in the shell very close to the horizon. As seen from the shell photon spectrum presented in Figure 12 (blue histogram there), there are plenty of photons with energy much higher than 511 keV that along with photons with energies between 10 and 100 keV (see the bump in the blue histogram) can create a noticeable number of pairs. The photon spectrum is quite flat, with the index of the high-energy power-law part being about 1.2 (1.2 ± 0.1). It is worth pointing out that the shape of the spectrum is very close to a broken power law.

Because of the gravitational blueshift and upscattering, the peak of the spectrum is located at energies of about 25 keV, which is 17 times more than the energy of the injected soft photons (1.5 keV). The photon index of 1.2 is a typical index for the saturation Comptonization. In fact, the photon index Γ can be calculated (see Titarchuk & Lyubarskij 1995, hereafter TL95; Ebisawa et al. 1996) as

$$\begin{eqnarray}&&{\rm{\Gamma }}=1+\displaystyle \frac{\mathrm{ln}(1/P)}{\mathrm{ln}(1+\eta )},\end{eqnarray} \tag{ 25 }$$

where P is the probability of photon scattering in a cloud (shell) and $1+\eta =\langle E^{\prime} \rangle /E$ is the mean efficiency of the photon energy change at any scattering of photon off electrons. According to TL95 (in their notation, the scattering probability is λ),

$$\begin{eqnarray}&&1/P=\exp (\beta )={[\tau \mathrm{ln}(1.53/\tau )]}^{-1}.\end{eqnarray} \tag{ 26 }$$

This formula is derived for the plane geometry and τ ≪ 1.

In the photon horizon shell, the bulk velocity of the flow v is very close to the speed of light, v = c(r/r_S)^1/2. For R = 1.01R_S, the Lorentz factor γ = (1 − (v/c)²)^1/2 = 10. The mean efficiency (see, e.g., Pozdnyakov et al. 1983) is

$$\begin{eqnarray}&&\langle {E}^{{\prime} }\rangle /E=1+\eta =[1+\displaystyle \frac{4}{3}({\gamma }^{2}-1)].\end{eqnarray} \tag{ 27 }$$

For $\dot{m}=4$, the shell optical depth τ ≈ 0.4 (see Equation (15)). We assume for this estimate that α = (R − R_S)/R_S = 0.01. We obtain the photon index Γ ∼ 1.13 using formulae (25)–(27), where we put γ = 10 and τ = 0.4. This value of Γ = 1.13 is very close to that obtained in our MC simulations, Γ = 1.2 ± 0.1.

The spectrum of the electrons (positrons), the so-called differential yield of the pairs produced by the annihilation of two photons, is proportional to the rate of the number of collisions in a given volume (shell). In general terms, this rate of the number of collisions ${\dot{N}}_{\mathrm{col}}$ is calculated according to Landau & Linfshitz (1971) as follows

$$\begin{eqnarray}{\dot{N}}_{\mathrm{col}} & = & {\dot{n}}_{-}({\gamma }_{-})={C}_{f}c{\displaystyle \int }_{0}^{\infty }{n}_{\mathrm{ph}}({\epsilon }_{1})d{\epsilon }_{1}\\ & & {\displaystyle \int }_{0}^{\infty }{n}_{\mathrm{ph}}({\epsilon }_{2})\ \sigma ({\gamma }_{-},{\epsilon }_{1},{\epsilon }_{2})d{\epsilon }_{2},\end{eqnarray} \tag{ 28 }$$

where  = E/m_e c² is the dimensionless photon energy, γ₋ and γ₊ are the Lorentz factors of the electron/positron, σ(γ₋, ₁, ₂) is the differential cross-section of the pair-production cross-section, and the factor C_f is less than 1.

Formula (28) is valid for an isotropic radiation field where σ(γ₋, ₁, ₂) is the differential cross-section of the pair-production cross-section averaged over a solid angle and C_f = 1/4 (see Bottcher & Schlickeiser 1997). In our MC simulation, we implement the exact form of the cross-section where the angular dependence has been included. However, we want to demonstrate that formula (28) (formally derived for isotropic radiation) can still be applicable for the realistic situation modeled in our simulations.

The exact but quite complicated expression for this formula was derived by Bottcher & Schlickeiser (1997). Now, let us consider the simple approximations of formula (28) and compare them with that obtained in the simulations.

The first approximation of formula (28) is related to a δ-function approximation of the cross-section suggested by Zdziarski & Lightman (1985, hereafter ZL85),

$$\begin{eqnarray}&&\sigma ({\gamma }_{-},{\epsilon }_{1},{\epsilon }_{2})\approx \displaystyle \frac{1}{3}{\sigma }_{{\rm{T}}}{\epsilon }_{2}\delta (\displaystyle \frac{{\epsilon }_{1}}{2}-{\gamma }_{-})\delta \left(\displaystyle \frac{2}{{\epsilon }_{1}}-{\epsilon }_{2}\right).\end{eqnarray} \tag{ 29 }$$

To justify this cross-section approximation, ZL85 argue that the photons with ₁ ≫ 1 will produce e⁺–e⁻ pairs mostly with γ₋ ≈ γ₊ ≈ /2 (Bonometto & Rees 1971) while colliding with photons of energies of ₂ ≈ 2/₁ (Herterich 1974).

The integration in formula (28) using Equation (29) leads to

$$\begin{eqnarray}&&{\dot{n}}_{-}({\gamma }_{-})\propto c{\sigma }_{{\rm{T}}}\displaystyle \frac{1}{{\gamma }_{-}}{n}_{\mathrm{ph}}(1/{\gamma }_{-}){n}_{\mathrm{ph}}(2{\gamma }_{-}).\end{eqnarray} \tag{ 30 }$$

Γ_pair should be around 2.5 because the photon index of n_ph(1/γ₋ ) is about 0.5 for 1/γ₋ ≪ 1 and that of n_ph(2γ₋) is around 1 (see Figure 12). In Figure 13, we present the simulation results for pair energy distribution. As one can see, the simulated pair spectrum is well approximated by a power law of index 2.5.

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