HORIZON-SCALE LEPTON ACCELERATION IN JETS: EXPLAINING THE COMPACT RADIO EMISSION IN M87

2.1.1. Inverse Compton

The source of seed photons for inverse Compton scattering is not completely clear. Nevertheless, we might assume that these are related to the observed emission from the source. Thus, if the observed luminosity is synonymous with the seed photon luminosity we will have

$$\begin{eqnarray}&&{L}_{{\rm{s}}}\approx {\rm{\Omega }}{r}_{{\rm{s}}}^{2}{u}_{{\rm{s}}}c/3={\rm{\Omega }}{D}^{2}{F}_{{\rm{s}}},\end{eqnarray} \tag{ 4 }$$

where r_s is the size of the emission region, Ω is the solid angle of the emission region (i.e., beaming factor), u_s is the local energy density of seed photons, D is the source distance, and F_s is the observed seed photon flux. Thus, in terms of the observed flux,

$$\begin{eqnarray}&&{u}_{{\rm{s}}}=3\displaystyle \frac{{D}^{2}{F}_{{\rm{s}}}}{{r}_{{\rm{s}}}^{2}c}=\displaystyle \frac{3{L}_{{\rm{s}}}}{4\pi {r}_{{\rm{s}}}^{2}c},\end{eqnarray} \tag{ 5 }$$

where L_s is the seed photon luminosity. The cooling length to inverse Compton scattering is related to u_s and ${\gamma }_{\mathrm{max}}$ via

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{IC}}=\displaystyle \frac{3{m}_{{\rm{e}}}{c}^{2}}{4{\sigma }_{{\rm{T}}}{u}_{{\rm{s}}}{\gamma }_{\mathrm{max}}},\end{eqnarray} \tag{ 6 }$$

where ${\sigma }_{{\rm{T}}}$ is the Thomson cross-section.⁶ This defines the typical distance over which particles can accelerate within the gap, thus

$$\begin{eqnarray}&&{\gamma }_{\mathrm{max},\mathrm{IC}}{m}_{{\rm{e}}}{c}^{2}\approx {eE}{{\ell }}_{\mathrm{IC}},\end{eqnarray} \tag{ 7 }$$

which implies a limiting Lorentz factor of

$$\begin{eqnarray}&&{\gamma }_{\mathrm{max},\mathrm{IC}}={(\displaystyle \frac{3{eR}{{\rm{\Omega }}}_{{\rm{F}}}B}{4{\sigma }_{{\rm{T}}}{u}_{{\rm{s}}}c})}^{1/2}=4.2\times {10}^{8}{R}_{15}^{1/2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{1/2}{B}_{2}^{1/2}{u}_{{\rm{s}},0}^{-1/2},\end{eqnarray} \tag{ 8 }$$

after inserting the expression for E in which we have adopted the standard notation, $R\equiv {R}_{15}{10}^{15}\;\mathrm{cm}$, ${{\rm{\Omega }}}_{{\rm{F}}}\equiv {{\rm{\Omega }}}_{{\rm{F}},-4}{10}^{-4}\;{{\rm{s}}}^{-1}$, $B\equiv {B}_{2}{10}^{2}\;{\rm{G}}$, and ${u}_{{\rm{s}}}\equiv {u}_{{\rm{s}},0}\;\mathrm{erg}\;{\mathrm{cm}}^{-3}$. The associated cooling length is ${{\ell }}_{\mathrm{IC}}=2.2\times {10}^{9}{R}_{15}^{-1/2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{-1/2}{B}_{2}^{-1/2}{u}_{{\rm{s}},0}^{-1/2}\;\mathrm{cm}$.

2.1.2. Curvature

Curvature radiation losses provide a second, seed-photon-independent limit upon ${\gamma }_{\mathrm{max}}$. The power emitted depends upon the curvature radius of the magnetic field, ${r}_{{\rm{c}}}\approx R$, and is given by (Levinson & Rieger 2011)

$$\begin{eqnarray}&&{P}_{{\rm{C}}}=\displaystyle \frac{2{e}^{2}c{\gamma }_{\mathrm{max}}^{4}}{3{R}^{2}}.\end{eqnarray} \tag{ 9 }$$

The corresponding cooling length is

$$\begin{eqnarray}&&{{\ell }}_{{\rm{C}}}=\displaystyle \frac{3{R}^{2}{m}_{{\rm{e}}}{c}^{2}}{2{e}^{2}{\gamma }_{\mathrm{max}}^{3}},\end{eqnarray} \tag{ 10 }$$

implying a limit of

$$\begin{eqnarray}&&{\gamma }_{\mathrm{max},{\rm{C}}}={(\displaystyle \frac{3{R}^{3}{{\rm{\Omega }}}_{{\rm{F}}}B}{2{ec}})}^{1/4}=3.2\times {10}^{10}{R}_{15}^{3/4}{{\rm{\Omega }}}_{{\rm{F}},-4}^{1/4}{B}_{2}^{1/4}.\end{eqnarray} \tag{ 11 }$$

In practice, the maximum particle Lorentz factor is set by the minimum radiative loss limit. As we will see in Section 4 the limit arising due to inverse Compton losses is typically more severe. This is true if

$$\begin{eqnarray}&&{u}_{{\rm{s}}}\gt {(\displaystyle \frac{3{e}^{3}{{\rm{\Omega }}}_{{\rm{F}}}B}{8{\sigma }_{{\rm{T}}}^{2}{Rc}})}^{1/2}=1.8\times {10}^{-4}{{\rm{\Omega }}}_{{\rm{F}},-4}^{1/2}{B}_{2}^{1/2}{R}_{15}^{-1/2}\;\mathrm{erg}\;{\mathrm{cm}}^{-3}\\ &&\quad \Rightarrow {F}_{{\rm{s}}}\gtrsim 1.8\;{r}_{{\rm{s}},16}^{2}{D}_{26}^{-2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{1/2}{B}_{2}^{1/2}{R}_{15}^{-1/2}\;\mathrm{mJy}\;\mathrm{THz},\end{eqnarray} \tag{ 12 }$$

where ${r}_{{\rm{s}}}\equiv {r}_{{\rm{s}},16}{10}^{16}\;\mathrm{cm}$ and $D\equiv {D}_{26}{10}^{26}\;\mathrm{cm}$. That is, as long as the seed photon density is sufficiently high, the gap dynamics is set by inverse Compton losses. Even for the highly sub-Eddington sources (e.g., M87 and Sgr A^*) this is often satisfied.

2.1.3. Synchrotron

2.1.4. Collective Processes

Finally, we consider the importance of collective effects due to the counter-streaming electron–positron plasmas. Due to their opposite charges, electrons and positrons will be accelerated in opposite directions, producing a pair of cold, counter-propagating lepton beams. This is unstable to the well-known two-stream instability, modified slightly due to the opposite charges in the two beams (which we refer to as the "counter-streaming instability" and presented briefly in Appendix A). The instability growth rate is generally dependent upon the stream densities, corresponding to a cooling length of

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{CSI}}\approx {(\displaystyle \frac{2{m}_{{\rm{e}}}{\gamma }_{\mathrm{max}}^{3}}{\pi {e}^{2}{n}_{{\rm{g}}}})}^{1/2}c,\end{eqnarray} \tag{ 13 }$$

and thus we cannot fully evaluate the importance of plasma instabilities without an estimate of the pair densities. However, for the estimate presented below (Equation (18)), ${{\ell }}_{\mathrm{CSI}}\approx 7.1\times {10}^{18}{u}_{{\rm{s}},0}^{-1}{\eta }_{\mathrm{th}}^{1/4}\;\mathrm{cm}$, and therefore this is never relevant.

As the limiting radiative process, inverse Compton cooling also provides a limit upon the gap thickness. The up-scattered photons have energies of roughly $\mathrm{min}(2{\gamma }_{\mathrm{max}}^{2}{\epsilon }_{{\rm{s}}},{\gamma }_{\mathrm{max}}{m}_{{\rm{e}}}{c}^{2})$. Thus, at meV energies, comparable to the photon energy near the peak of the spectral energy distribution (SED) of the sources of primary interest here (see Sections 4.1.4 and 5.1.3), the up-scattered photon energy is similar to that of the incident electron. Thus, the up-scattered photons will pair produce off photons with energies above

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{th}}\approx \displaystyle \frac{4{m}_{{\rm{e}}}{c}^{2}}{{\gamma }_{\mathrm{max}}}=4.8\;{R}_{15}^{-1/2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{-1/2}{B}_{2}^{-1/2}{u}_{{\rm{s}},0}^{1/2}\;\mathrm{meV},\end{eqnarray} \tag{ 14 }$$

and thus just above the SED peak for our fiducial parameters. Hence, the inverse Compton cooling of the gap-accelerated pairs initiates a pair-production catastrophe.

The pair-production cross section peaks near $2{\epsilon }_{\mathrm{th}}$ at ${\sigma }_{\gamma \gamma }\approx 0.35{\sigma }_{{\rm{T}}}$ (Gould & Schréder 1967). Thus, the mean free path of the up-scattered seed photons is approximately

$$\begin{eqnarray}{{\ell }}_{\gamma \gamma } & \approx & \displaystyle \frac{2{\epsilon }_{\mathrm{th}}}{{u}_{{\rm{s}},\mathrm{th}}{\sigma }_{\gamma \gamma }}\\ & \approx & 6.6\times {10}^{10}{R}_{15}^{-1/2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{-1/2}{B}_{2}^{-1/2}{u}_{{\rm{s}},0}^{-1/2}{\eta }_{\mathrm{th}}\;\mathrm{cm},\end{eqnarray} \tag{ 15 }$$

where ${u}_{{\rm{s}},\mathrm{th}}\equiv {u}_{{\rm{s}},0}/{\eta }_{\mathrm{th}}$ is the energy density of seed photons with energies above ${\epsilon }_{\mathrm{th}}$. For a source with infrared spectral index α (i.e., ${F}_{\nu }\propto {\nu }^{-\alpha }$, in M87 $\alpha \approx 1.2$), the number of seed photons above an energy ${\epsilon }_{{\rm{s}}}$ is $\propto {\epsilon }_{{\rm{s}}}^{-\alpha }$. Thus, if there is a spectral break at ${\epsilon }_{b}$, below which the seed photon energy density may be neglected,

$$\begin{eqnarray}{\eta }_{\mathrm{th}}=\{\begin{array}{lc}1 & {\epsilon }_{\mathrm{th}}\lt {\epsilon }_{b}\\ {({\epsilon }_{\mathrm{th}}/{\epsilon }_{b})}^{\alpha } & {\epsilon }_{\mathrm{th}}\geqslant {\epsilon }_{b}.\end{array}\end{eqnarray} \tag{ 16 }$$

For sufficiently high ${\epsilon }_{\mathrm{th}}$, it is possible for ${\eta }_{\mathrm{th}}\gg 1$, implying that ${{\ell }}_{\gamma \gamma }\gg {{\ell }}_{\mathrm{IC}}$. When ${\eta }_{\mathrm{th}}=1$, this is larger than ${{\ell }}_{\mathrm{IC}}$ by roughly a factor of three, implying in this case that the gap scale height is comparable to either. A more quantitative model of the gap lepton population may be obtained in one dimension by assuming that the pairs are accelerated instantaneously to their asymptotic velocities. As shown in Appendix B.1, this implies a gap thickness of

$$\begin{eqnarray}{\rm{\Delta }} & \approx & \sqrt{2{{\ell }}_{\mathrm{IC}}{{\ell }}_{\gamma \gamma }}\\ & \approx & 1.7\times {10}^{10}{R}_{15}^{-1/2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{-1/2}{B}_{2}^{-1/2}{u}_{{\rm{s}},0}^{-1/2}{\eta }_{\mathrm{th}}^{1/2}\;\mathrm{cm},\end{eqnarray} \tag{ 17 }$$

consistent with this. Generally, ${{\ell }}_{\gamma \gamma }$ is larger than ${{\ell }}_{\mathrm{IC}}$ and therefore the asymptotic Lorentz factor is gained on a length scale ${{\ell }}_{\mathrm{IC}}\ll {\rm{\Delta }}$, thereby justifying our assumption that the pairs are instantaneously accelerated.

Counter-propagating positrons or electrons constantly reinitiate the cascade, which continues until sufficient charges are produced to screen the electric fields. Thus we would anticipate that the gap can be evacuated only over a single scale height. The top of the gap is then roughly defined when the charge density is sufficient to generate the gap electric field gradients, and thus the lepton density in the lab frame can be expressed as (see Appendix B.1):

$$\begin{eqnarray}{n}_{{\rm{g}}} & = & \displaystyle \frac{{\boldsymbol{\nabla }}\cdot {\boldsymbol{E}}}{4\pi e}\approx \displaystyle \frac{E}{4\pi e{\rm{\Delta }}}\\ & = & 3.3\;{R}_{15}^{3/2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{3/2}{B}_{2}^{3/2}{u}_{{\rm{s}},0}^{1/2}{\eta }_{\mathrm{th}}^{-1/2}\;{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 18 }$$

Note that this exceeds the Goldreich–Julian density by a multiplicity factor given by

$$\begin{eqnarray}&&\eta =\displaystyle \frac{R}{{\rm{\Delta }}},\end{eqnarray} \tag{ 19 }$$

resulting from the smaller typical scale set by the gap thickness.⁷ Here we have assumed that only a single species is present at the top of the gap, an assumption that is justified by the rapid acceleration of electrons and positrons in opposite directions. However, the nonlinear response of the magnetosphere will supply the charges of the opposite sign to screen the gap electric field and ensure that the microphysical current matches the global magnetospheric current (which is generically of the order of the Goldreich–Julian current, ${j}_{\mathrm{GJ}}={{en}}_{\mathrm{GJ}}c$).

We note that first-principles particle-in-cell (PIC) simulations are needed in order to simulate the cascade in detail and properly compute the flow of charges and connection between the pair cascade and global MHD currents. Fortunately, our simplified, steady-state consideration of the gap structure appears to capture the most crucial aspects of the gap structure to within an order of magnitude. For instance, in PIC simulations of polar cascades in pulsar magnetosphere (Timokhin & Arons 2013, hereafter TA13), the simulated pair multiplicity $\eta \sim 100$ (for their polar gap thickness of ${\rm{\Delta }}\sim 1.7\times {10}^{4}$ cm and a characteristic length scale $R={10}^{6}$ cm; see, e.g., Figure 22 in TA13) agrees with our estimate (19), which gives $\eta =R/{\rm{\Delta }}\approx 60$. Furthermore, in agreement with the findings of TA13, our gap thickness is much larger than the skin depth,

$$\begin{eqnarray*}&&\displaystyle \frac{{\rm{\Delta }}}{{d}_{{\rm{e}}}}=5.6\times {10}^{5}{\eta }_{\mathrm{th}}^{1/4}{R}_{15}^{1/4}{{\rm{\Omega }}}_{{\rm{F}},-4}^{1/4}{B}_{2}^{1/4}{u}_{{\rm{s}},0}^{-1/4},\end{eqnarray*}$$

with the skin depth given by

$$\begin{eqnarray*}&&{d}_{{\rm{e}}}=\sqrt{\displaystyle \frac{{{mc}}^{2}}{4\pi {e}^{2}{n}_{{\rm{g}}}}}.\end{eqnarray*}$$

Encouraged by this agreement, we will adopt the results of our steady-state gap model and consider astrophysical consequences.

The relativistic outflow of particles at the top of the gap can carry a substantial total kinetic luminosity. Given the above estimates for the number density and typical Lorentz factor, the kinetic flux in the outflowing leptons is

$$\begin{eqnarray}{F}_{\mathrm{lep}} & = & {\gamma }_{\mathrm{max},\mathrm{IC}}{m}_{{\rm{e}}}{c}^{3}{n}_{{\rm{g}}}\\ & = & 3.4\times {10}^{13}{R}_{15}^{2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{2}{B}_{2}^{2}{\eta }_{\mathrm{th}}^{-1/2}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 20 }$$

Integrating this across the entire stagnation surface gives a total kinetic luminosity of

$$\begin{eqnarray}{L}_{\mathrm{cascade}} & \approx & {\displaystyle \int }_{0}^{R}4\pi R^{\prime} {dR}^{\prime} {F}_{\mathrm{lep}}\\ & = & 1.1\times {10}^{44}{R}_{15}^{4}{{\rm{\Omega }}}_{{\rm{F}},-4}^{2}{B}_{2}^{2}{\eta }_{\mathrm{th}}^{-1/2}\;\mathrm{erg}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 21 }$$

Note that typically ${{\rm{\Omega }}}_{{\rm{F}}}\propto {M}^{-1}$ and $R\propto M$, where M is the mass of the black hole. Hence, at fixed magnetic field strength the pair luminosity scales as M², implying that ${L}_{{{\rm{e}}}^{\mathrm{lep}}}/{L}_{\mathrm{Edd}}\propto M$. Our fiducial numbers correspond roughly to M87, and thus a black hole mass of order ${10}^{10}\;{M}_{\odot }$, for which the above luminosity is about $3\times {10}^{-4}{L}_{\mathrm{Edd}}$. This suppresses the cascade luminosity for less massive black holes, as we will see in the example of Sgr A^* in Section 5.

The lepton structure in the gap is generally quite unstable, both due to the large-scale GRMHD processes that lead to its formation and the particle acceleration that fills it. In the case of its generic structure, the typical variability timescale is on the order of the light crossing time of the radial position of the gap, typically $\approx 10{r}_{{\rm{g}}}$. The radial position of the gap varies by $\lesssim 10$% on this timescale, as can be seen in Figure 2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The particle acceleration processes will induce local variability on much shorter timescales, comparable to the light crossing time across the gap (see Appendix B.2). In our fiducial model, this is roughly $0.7{R}_{15}^{-1/2}{{\rm{\Omega }}}_{{\rm{F}},-4}^{-1/2}{B}_{2}^{-1/2}{u}_{{\rm{s}},0}^{-1/2}\;{\rm{s}}$. For M87, this is roughly six orders of magnitude smaller than ${GM}/{c}^{3}$! However, in practice, the gap has a transverse extent that greatly exceeds Δ, and therefore usually many independently fluctuating regions are visible at once. Assuming that each independent region has a local radius of Δ, if the visible extent of the stagnation surface is R_V, the net degree of variability on the particle variation timescale is roughly

$$\begin{eqnarray}&&\displaystyle \frac{\delta {n}_{{\rm{g}}}}{\langle {n}_{{\rm{g}}}\rangle }\approx \displaystyle \frac{{\rm{\Delta }}}{{R}_{{\rm{V}}}}.\end{eqnarray} \tag{ 22 }$$

For low frequencies, ${R}_{{\rm{V}}}\approx R$, and this reduction is enormous. However, for the very-high-energy gamma rays produced directly by inverse Compton scattering, the intrinsic relativistic beaming will substantially reduce R_V. For emission associated with electrons with Lorentz factor γ, a patch of the surface that will be visible has a typical size of $R/\gamma$, implying that for $\gamma \gtrsim {10}^{6}$ the observed particle density variations, and therefore the flux variations, will be of order unity on the particle acceleration timescale. In our fiducial model, where the soft seed photon energy density peaks around 1 meV, this corresponds to a gamma-ray energy above a few GeV.

As the energy ${\epsilon }_{\gamma }$ of a gamma ray increases, the number density of soft photons that lie above the pair creation threshold increases, and so does the optical depth ${\tau }_{\gamma \gamma }$ to pair production. The existence of the pair catastrophe implies that at sufficiently high ${\epsilon }_{\gamma }$ the optical depth to pair production is high, i.e., ${\tau }_{\gamma \gamma }\gg 1$. Since the energy threshold for annihilation of a gamma ray is $\propto {\epsilon }_{\gamma }^{-1}$, the number density of soft photons above the pair creation threshold (and therefore the optical depth to annihilation) scales as $\propto {\epsilon }_{\gamma }^{\alpha }$. The normalization of the total gamma ray optical depth can be obtained in terms of ${{\ell }}_{\gamma \gamma }$, set by ${\tau }_{\gamma \gamma }=1$ for gamma rays with energy ${\gamma }_{\mathrm{max},\mathrm{IC}}{m}_{{\rm{e}}}{c}^{2}$:

$$\begin{eqnarray}{\tau }_{\gamma \gamma }({\epsilon }_{\gamma }) & \approx & \displaystyle \frac{r}{{{\ell }}_{\gamma \gamma }}\displaystyle \frac{{n}_{{\rm{s}}}({\epsilon }_{{\rm{s}}}\gt 4{m}_{{\rm{e}}}^{2}{c}^{4}/{\epsilon }_{\gamma })}{{n}_{{\rm{s}}}({\epsilon }_{{\rm{s}}}\gt 4{m}_{{\rm{e}}}{c}^{2}/{\gamma }_{\mathrm{max},\mathrm{IC}})}\\ & \approx & \displaystyle \frac{r}{{{\ell }}_{\gamma \gamma }}{(\displaystyle \frac{\epsilon /{m}_{{\rm{e}}}{c}^{2}}{{\gamma }_{\mathrm{max},\mathrm{IC}}})}^{\alpha }.\end{eqnarray} \tag{ 23 }$$

Hence, for gamma rays with energies

$$\begin{eqnarray}&&{\epsilon }_{\gamma }\lesssim {\epsilon }_{\gamma ,\mathrm{min}}\equiv {\gamma }_{\mathrm{max},\mathrm{IC}}{m}_{{\rm{e}}}{c}^{2}{(\displaystyle \frac{{{\ell }}_{\gamma \gamma }}{r})}^{1/\alpha },\end{eqnarray} \tag{ 24 }$$

the seed photon distribution will be optically thin to the resulting inverse Compton gamma rays, allowing the latter to escape. For values typical of M87, this corresponds to energies of order ${\epsilon }_{\gamma }\approx 630$ GeV, above which significant attenuation can be expected.

We have already noted that for gamma rays with energies below a lower limit determined by the seed photon SED the ambient soft photon population will be transparent, halting the pair production cascade. Explicitly, for a power-law seed photon distribution above a low-energy cut-off ${\epsilon }_{b}$ (as described above Equation (16)) the fraction of gamma-rays with energies above ${\epsilon }_{\gamma ,\mathrm{min}}$ following a single scattering is

$$\begin{eqnarray}{f}_{\gt {\epsilon }_{\gamma ,\mathrm{min}}}(\gamma )\approx \{\begin{array}{ll}{({\epsilon }_{\gamma ,\mathrm{min}}/2{\gamma }^{2}{\epsilon }_{b})}^{-\alpha } & \gamma \leqslant \sqrt{{\epsilon }_{\gamma ,\mathrm{min}}/2{\epsilon }_{b}}\\ 1 & \mathrm{otherwise},\end{array}\end{eqnarray} \tag{ 25 }$$

where we have assumed that ${\epsilon }_{b}$ is sufficiently small. This is necessarily a strong function of the initial lepton Lorentz factor, dropping rapidly for Lorentz factors below $\sqrt{{\epsilon }_{\gamma ,\mathrm{min}}/2{\epsilon }_{b}}$, and thus ${\epsilon }_{\gamma ,\mathrm{min}}$ implies a limit upon γ.

However, since the energies of the up-scattered gamma rays are typically small in comparison to the lepton energy near this limit, i.e., $2{\gamma }^{2}{\epsilon }_{b}\ll \gamma {{mc}}^{2}$, in practice the lepton will undergo many scatterings before cooling appreciably. That is, there will be roughly

$$\begin{eqnarray}&&{N}_{\mathrm{scat}}\approx \displaystyle \frac{\gamma {{{m}}_{\text{e}}c}^{2}}{2{\gamma }^{2}{\epsilon }_{b}}=\displaystyle \frac{{{{m}}_{\text{e}}c}^{2}}{2\gamma {\epsilon }_{b}}\end{eqnarray} \tag{ 26 }$$

opportunities to produce a gamma ray with energy above ${\epsilon }_{\gamma ,\mathrm{min}}$. This increases with decreasing γ and ${\epsilon }_{b}$, ameliorating the decrease in ${f}_{\gt {\epsilon }_{\gamma ,\mathrm{min}}}$ due to the latter. Thus the number of pair-producing gamma rays is expected to be approximately ${N}_{\mathrm{scat}}{f}_{\gt {\epsilon }_{\gamma ,\mathrm{min}}}$. Setting this number to unity provides the desired limit upon the lepton Lorentz factor, below which electrons cease to typically have sufficient energy to continue the pair cascade:

$$\begin{eqnarray}&&{\gamma }_{\gamma \gamma }\approx {(\displaystyle \frac{{2}^{1-\alpha }{\epsilon }_{b}^{1-\alpha }{\epsilon }_{\gamma ,\mathrm{min}}^{\alpha }}{{{{m}}_{\text{e}}c}^{2}})}^{1/(2\alpha -1)}.\end{eqnarray} \tag{ 27 }$$

For our fiducial model, ${\gamma }_{\gamma \gamma }\approx 7.7\times {10}^{4}$. This limit on the redistribution of energy implies an associated limit on increase in the particle density. An even redistribution, maximizing the number of leptons produced, results in

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{\infty }}{{n}_{{\rm{g}}}}\approx \displaystyle \frac{{\gamma }_{\mathrm{max},\mathrm{IC}}}{{\gamma }_{\gamma \gamma }},\end{eqnarray} \tag{ 28 }$$

and is roughly 6000 daughter leptons for each lepton produced by the gap for our fiducial model.

The absence of the generation of energetic gamma rays does not imply the absence of additional inverse Compton cooling. Hence, a second, typically more stringent limit arises from inverse Compton cooling directly. For the cooling length to become comparable to the scale height of the seed photon distribution requires

$$\begin{eqnarray}&&\displaystyle \frac{3{m}_{{\rm{e}}}{c}^{2}}{4{\sigma }_{{\rm{T}}}{u}_{{\rm{s}}}{\gamma }_{\infty }}\gtrsim r\\ &&\quad \Rightarrow {\gamma }_{\infty }\lesssim {\gamma }_{\mathrm{max},\mathrm{IC}}\displaystyle \frac{{{\ell }}_{\mathrm{IC}}}{r}\approx {10}^{2}{u}_{{\rm{s}},0}^{-1}{r}_{16}^{-1}.\end{eqnarray} \tag{ 29 }$$

We note that in our fiducial model the cooling of the leptons from ${\gamma }_{\gamma \gamma }$ to ${\gamma }_{\infty }$ is unable to produce gamma rays energetic enough to pair produce. Therefore, the resulting radiation can escape and can be a substantial source of emission.

Since ${\gamma }_{\infty }{n}_{\infty }\ll {\gamma }_{\mathrm{max},\mathrm{IC}}{n}_{{\rm{g}}}$ most of the energy imparted to the leptons by the accelerating electric field across the gap has been radiated via the Comptonization of the soft seed photons. Typical energies range from the optical to the GeV, with an SED that depends upon that of the underlying soft photon population. However, this emission will be highly beamed along the magnetic field lines that govern the particle motions.

In practice the estimates obtained in the previous section depend critically upon assumptions regarding the final electron spectrum, which is itself dependent upon the soft photon SED. For this reason we also computed the asymptotic number and energy distribution of electrons produced due to inverse Compton scattering off of a given seed photon SED from a nearly monoenergetic injection at high Lorentz factor. Throughout this section, for compactness we will measure energies in units of the electron rest mass energy.

We make two simplifying assumptions, both of which are almost certainly well justified. The first is that we may ignore any subsequent pair annihilation due the comparatively high lepton energies and low lepton densities. The second is that the number density of soft seed photons vastly exceeds the number densities of up-scattered gamma rays, a consequence of the short mean free path to pair production and the low lepton densities. These remove the nonlinear terms in the coupled Boltzmann equations for the leptons and the gamma rays. Thus, it is sufficient to consider the evolution of a single lepton through multiple Compton generations. That is, we compute the energy probability distribution of a single lepton as a function of the number of scatterings instead of time. The asymptotic form of this probability distribution then describes the asymptotic spectrum of a full population of leptons.

Explicit inputs are the seed photon spectrum and electron injection energy, which we leave as a parameter, ${\epsilon }_{0}$. For the former we assume a power-law model between some minimum (${\epsilon }_{{\rm{m}}}$) and maximum (${\epsilon }_{{\rm{M}}}$) energies, consistent with the observed radio and infrared emission from the sources of interest, i.e.,

$$\begin{eqnarray}{f}_{{\rm{s}}}(\epsilon )\equiv \displaystyle \frac{{{dN}}_{{\rm{s}}}}{d\epsilon }\propto \{\begin{array}{ll}{\epsilon }^{-\alpha -1} & {\epsilon }_{{\rm{m}}}\lt \epsilon \lt {\epsilon }_{{\rm{M}}}\\ 0 & \mathrm{otherwise}.\end{array}\end{eqnarray} \tag{ 30 }$$

The values of ${\epsilon }_{{\rm{m}}}$ and ${\epsilon }_{{\rm{M}}}$ are determined by the particular source under consideration, though typically when a large enough dynamic range is chosen the resulting evolution becomes insensitive to their specific choice.⁸ The linearity in the gamma-ray distribution is synonymous with assuming that f_s remains fixed throughout the post-gap pair cascade.

With the above we may now compute the update in the lepton energy distribution with each scattering. We do this in two steps, first computing the implied up-scattered gamma-ray distribution associated with a monoenergetic electron distribution, and then from that inferring the changes to the electron distribution arising from inverse Compton losses and pair production. To compute the former, we begin with the relationship between the original seed photon energy ${\epsilon }_{{\rm{s}}}$, up-scattered gamma-ray energy ${\epsilon }_{\gamma }$, and initial electron energy ${\epsilon }_{{\rm{e}},{\rm{i}}}$:

$$\begin{eqnarray}{\epsilon }_{{\rm{s}}}({\epsilon }_{\gamma },{\epsilon }_{{\rm{e}},{\rm{i}}})=\{\begin{array}{ll}\displaystyle \frac{{\epsilon }_{\gamma }}{2{\epsilon }_{{\rm{e}},{\rm{i}}}({\epsilon }_{{\rm{e}},{\rm{i}}}-{\epsilon }_{\gamma })} & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{e}},{\rm{i}}}\\ 0 & \mathrm{otherwise},\end{array}\end{eqnarray} \tag{ 31 }$$

where we assumed ${\epsilon }_{{\rm{s}}}\ll {\epsilon }_{{\rm{e}},{\rm{i}}}$ and approximated the energy of the electron post-collision as ${\epsilon }_{{\rm{e}},f}={\epsilon }_{{\rm{e}},{\rm{i}}}-{\epsilon }_{\gamma }$. Equation (31) may be obtained from the standard Compton scattering formula in the high-energy electron and oblique seed photon limits. As a result, for a monoenergetic electron distribution, the up-scattered gamma-ray distribution is

$$\begin{eqnarray}&&g({\epsilon }_{\gamma },{\epsilon }_{{\rm{e}},{\rm{i}}})=\displaystyle \frac{d{\epsilon }_{{\rm{s}}}}{d{\epsilon }_{\gamma }}{f}_{{\rm{s}}}[{\epsilon }_{{\rm{s}}}({\epsilon }_{\gamma },{\epsilon }_{{\rm{e}},{\rm{i}}})]=\displaystyle \frac{{f}_{{\rm{s}}}[{\epsilon }_{{\rm{s}}}({\epsilon }_{\gamma },{\epsilon }_{{\rm{e}},{\rm{i}}})]}{2{({\epsilon }_{{\rm{e}},{\rm{i}}}-{\epsilon }_{\gamma })}^{2}}.\end{eqnarray} \tag{ 32 }$$

As mentioned above, since ${\epsilon }_{{\rm{s}}}\ll {\epsilon }_{{\rm{e}},{\rm{i}}},{\epsilon }_{\gamma }$, we approximate the electron energy following scattering by ${\epsilon }_{{\rm{e}},f}\approx {\epsilon }_{{\rm{e}},{\rm{i}}}-{\epsilon }_{\gamma }$. Therefore, if ${f}_{j}({\epsilon }_{{\rm{e}}})$ is the electron distribution after the jth scatter, the electron distribution is rearranged by inverse Compton losses to

$$\begin{eqnarray}&&{f}_{\mathrm{IC},j+1}^{\mathrm{lep}}({\epsilon }_{{\rm{e}}})={\displaystyle \int }_{0}^{\infty }d{\epsilon }_{{\rm{e}},{\rm{i}}}g({\epsilon }_{{\rm{e}},{\rm{i}}}-{\epsilon }_{{\rm{e}}},{\epsilon }_{{\rm{e}},{\rm{i}}}){f}_{j}^{\mathrm{lep}}({\epsilon }_{{\rm{e}},{\rm{i}}}).\end{eqnarray} \tag{ 33 }$$

In practice the integral need only be computed up to the initial maximum energy of the electron distribution.

Similarly, since ${\epsilon }_{\gamma }\gg 2\gg {\epsilon }_{{\rm{s}}}$, i.e., much greater than the rest mass of the produced pair, which is itself much greater than that of the seed photons, we may approximate the energy of each lepton in the resulting pair by ${\epsilon }_{{\rm{e}}}={\epsilon }_{\gamma }/2$. Hence, the distribution of new pairs is

$$\begin{eqnarray}{f}_{\gamma \gamma ,j+1}^{\mathrm{lep}}({\epsilon }_{{\rm{e}}}) & = & 4{\displaystyle \int }_{0}^{\infty }d{\epsilon }_{{\rm{e}},{\rm{i}}}g(2{\epsilon }_{{\rm{e}}},{\epsilon }_{{\rm{e}},{\rm{i}}}){f}_{j}^{\mathrm{lep}}\\ & & \times ({\epsilon }_{{\rm{e}},{\rm{i}}}){\rm{\Theta }}(2{\epsilon }_{{\rm{e}}}-{\epsilon }_{\gamma ,\mathrm{min}}),\end{eqnarray} \tag{ 34 }$$

where ${\rm{\Theta }}(x)$ is the Heaviside function and ensures that the integral is only over regions in which the gamma ray can pair produce on the soft background. Note that we assume that all gamma rays will pair produce as long as their energy exceeds the threshold ${\epsilon }_{\gamma ,\mathrm{min}}$, above which the seed photon distribution is optically thin to pair production.

In principle, we should then set ${\epsilon }_{{\rm{M}}}$ by the pair production optical depth; from Equation (24) this gives a typical

$$\begin{eqnarray}&&{\epsilon }_{{\rm{M}}}\approx 4{m}_{{\rm{e}}}^{2}{c}^{4}/(630\;\mathrm{GeV}),\end{eqnarray} \tag{ 35 }$$

which for our fiducial parameters gives ${\epsilon }_{{\rm{M}}}\approx 1.6\;\mathrm{eV}$. In practice, we find the simulation results are insensitive to the particular value we choose for ${\epsilon }_{{\rm{M}}}$. In all of our simulations, unless specified otherwise, we choose ${\epsilon }_{{\rm{M}}}=0.8$ eV (as we discuss below, results for ${\epsilon }_{{\rm{M}}}=1.6$ eV are essentially identical; see Figure 6). In a similar fashion to Equation (34), we can write down the distribution function of photons that are below the pair production threshold, ${\epsilon }_{\gamma }\lt {\epsilon }_{\gamma ,\mathrm{min}}$: these photons escape the system without undergoing scattering and thus form the high-energy emission spectrum,

$$\begin{eqnarray}&&{f}_{\gamma \gamma ,j+1}^{\mathrm{ph}}({\epsilon }_{\gamma })\\ &&=4{\displaystyle \int }_{0}^{\infty }d{\epsilon }_{{\rm{e}},{\rm{i}}}g({\epsilon }_{\gamma },{\epsilon }_{{\rm{e}},{\rm{i}}}){f}_{j}^{\mathrm{lep}}({\epsilon }_{{\rm{e}},{\rm{i}}}){\rm{\Theta }}({\epsilon }_{\gamma ,\mathrm{min}}-{\epsilon }_{\gamma }).\end{eqnarray} \tag{ 36 }$$

Combining the above, we obtain the energy distribution of the leptons and photons in the $(j+1)\mathrm{th}$ generation via

$$\begin{eqnarray}&&{f}_{j+1}^{\mathrm{lep}}({\epsilon }_{{\rm{e}}})={f}_{\mathrm{IC},j+1}^{\mathrm{lep}}({\epsilon }_{{\rm{e}}})+{f}_{\gamma \gamma ,j+1}^{\mathrm{lep}}({\epsilon }_{{\rm{e}}}),\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&{f}_{j+1}^{\mathrm{ph}}({\epsilon }_{\gamma })={f}_{j}^{\mathrm{ph}}({\epsilon }_{\gamma })+{f}_{\gamma \gamma ,j+1}^{\mathrm{ph}}({\epsilon }_{\gamma }).\end{eqnarray} \tag{ 38 }$$

Beginning with some initial injection distribution, we can compute the asymptotic distribution, in which generations correspond in a loose sense to time or height from the injection point.

For this, we integrate Equations (37) and (38) numerically. We start the integration at the top of the gap, and inject a single electron at an energy

$$\begin{eqnarray}&&{E}_{0}={\gamma }_{\mathrm{max},\mathrm{IC}}{m}_{{\rm{e}}}{c}^{2},\end{eqnarray} \tag{ 39 }$$

which is the energy gained by an electron between its consequent encounters of seed photons in the gap and is therefore the characteristic energy with which the electrons emerge from the gap. We discretize Equations (37) and (38) on a logarithmic grid in energy, which extends from ${E}_{\mathrm{min}}={10}^{-6}{m}_{{\rm{e}}}{c}^{2}$ to ${E}_{\mathrm{max}}=2{E}_{0}$. For numerical reasons, we smooth out the initial energy distribution by 1.5% so it is properly represented on our numerical grid, by choosing the following initial injection distribution,

$$\begin{eqnarray}&&{f}_{0}^{\mathrm{lep}}({\epsilon }_{{\rm{e}}})\equiv \displaystyle \frac{{{dN}}_{{\rm{e}}}}{d{\epsilon }_{{\rm{e}}}}=\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{E}^{2}}}\mathrm{exp}[-\displaystyle \frac{{({\epsilon }_{{\rm{e}}}-{E}_{0})}^{2}}{2{\sigma }_{E}^{2}}],\end{eqnarray} \tag{ 40 }$$

with ${\sigma }_{E}=0.015{E}_{0}$.

In order to ensure robust evolution of pair cascade numerically, we found it to be important to ensure lepton number conservation under Inverse Compton cooling (Equation (33)). To do this, we compute the discretization of the integral in Equation (33) on two energy grids that contain the same number of grid cells but are shifted relative to each other by half a cell. If the changes in the number of leptons due to these two discretizations are different by more than 50% relative to each other, then we use their weighted sum that preserves the total number of leptons exactly. Otherwise, we use the discretization that results in the smallest change in lepton number. We implemented the numerical code in a Cython-based Python library. In order to speed up the integrations, we parallelized the code using OpenMP. We describe the numerical results of cascade evolution in Section 4.3.

Observations of stellar dynamics within M87's sphere of influence produce an inferred mass of $6.6\times {10}^{9}\;{M}_{\odot }$, assuming a distance of $17.9\;\mathrm{Mpc}$ (Gebhardt et al. 2011).⁹ The energy output of M87 peaks at 1 mm (300 GHz or 1 meV), at which point the flux at Earth is roughly $1\;\mathrm{Jy}$. The typical isotropic-equivalent total luminosity is $L\approx {10}^{42}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\approx {10}^{-6}{L}_{\mathrm{Edd}}$, where ${L}_{\mathrm{Edd}}$ is the Eddington luminosity. This makes M87 quite underluminous and dominated by emission at wavelengths about the peak frequency, as we discuss below. The typical scale near the black hole is the gravitational radius of the black hole, ${r}_{{\rm{g}}}\simeq {10}^{15}\;\mathrm{cm}$. Here we discuss the empirical implications of various additional observations for the physical properties near the stagnation surface.

4.1.1. Jet Velocity and Orientation

4.1.2. Estimates of Magnetic Field Strength

The observational implications for the magnetic field in M87 depend upon the structure and dynamics of the material in the vicinity of the black hole. Here we present two estimates, assuming in the first that the observed mm-wavelength emission is due to a RIAF (Narayan & Yi 1994, 1995), and in the second that the jet is Poynting-dominated and responsible for the power observed in the radio lobes. Both reach similar conclusions regarding the field strength.

As suggested by their name, RIAFs are characterized by their extraordinarily low radiative efficiency, i.e., $L={\eta }_{d}\dot{M}{c}^{2}$ with ${\eta }_{d}\approx {10}^{-3}$–10⁻⁴. Since most of the gravitational binding energy is released, and therefore most of the luminosity is produced, near the black hole, this implies an accretion flow density of

$$\begin{eqnarray}&&\rho \approx \displaystyle \frac{\dot{M}}{4\pi {r}_{{\rm{g}}}^{2}\beta c}=\displaystyle \frac{L}{4\pi {\eta }_{d}{r}_{{\rm{g}}}^{2}\beta {c}^{3}}\end{eqnarray} \tag{ 41 }$$

where we have assumed a disk height of r and an inflow velocity near the horizon of $\beta c$. The jet magnetic field will accumulate until it balances the ram pressure of the accretion flow, and thus we arrive at the first magnetic field strength estimate of

$$\begin{eqnarray}B & \approx & \sqrt{8\pi \rho {\beta }^{2}{c}^{2}}\approx {(\displaystyle \frac{2L\beta }{{\eta }_{d}{r}_{{\rm{g}}}^{2}c})}^{1/2}\\ & \approx & 3\times {10}^{2}{\eta }_{d,-3}^{-1/2}{L}_{42}^{1/2}{\beta }^{1/2}\;{\rm{G}}.\end{eqnarray} \tag{ 42 }$$

It is not clear that the mm-wavelength emission is associated with an accretion flow. At 7 mm the jet dominates the source morphology (Walker et al. 2008). Thus, we present a second estimate based upon the assumption that the jet is Poynting-dominated near the black hole and relating the mechanical power at small and large radii. The Poynting luminosity associated with black hole driven jets is (Tchekhovskoy et al. 2010):

$$\begin{eqnarray}&&{L}_{\mathrm{EM}}=\displaystyle \frac{k}{4\pi c}{{\rm{\Omega }}}_{{\rm{H}}}^{2}{{\rm{\Phi }}}^{2},\end{eqnarray} \tag{ 43 }$$

where $k\approx 0.045$ for parabolic jet geometry, Φ is the poloidal magnetic flux threading the black hole, and ${{\rm{\Omega }}}_{{\rm{H}}}$ is the angular velocity of the horizon of the black hole, which encodes the dependence upon the black hole spin.¹¹

A variety of estimates of the total jet power in M87 have been obtained across a wide range of scales. Radio lobe measurements probe emission extending over 30 kpc, and have resulted in estimates of ${L}_{\mathrm{jet}}\approx {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ (de Gasperin et al. 2012). This is similar to estimates arising from the modeling of the complex of knot features on kpc scales (Owen et al. 2000), and on 60 pc scales from interpreting the HST-1 complex as a recollimation shock (Stawarz et al. 2006). Thus, we adopt this estimate as a conservative estimate of the total jet power. For a characteristic black hole spin, a = 0.9, equating these then gives a poloidal magnetic field strength estimate at the event horizon:

$$\begin{eqnarray}&&{B}_{{\rm{P}}}=\displaystyle \frac{{\rm{\Phi }}}{A}\approx 2.5\times {10}^{2}\;{\rm{G}},\end{eqnarray} \tag{ 44 }$$

where $A=2\pi ({a}^{2}{r}_{{\rm{g}}}^{2}+3{r}_{{\rm{H}}}^{2})/3$ is the area of one of the two hemispheres of the event horizon, where ${r}_{{\rm{H}}}={r}_{{\rm{g}}}(1+\sqrt{1-{a}^{2}})$ is the radius of the black hole event horizon.

Despite the significantly different physics being invoked, the two estimates are quite similar. While this is not accidental (thick disks are believed to be a necessary component of jet formation), it does critically depend upon the low radiative efficiency generally implicated in vastly sub-Eddington sources (such as M87).

At the distances of the stagnation surface, which lies inside the light cylinder, the magnetic field is poloidally dominated. If we now assume that the jet is roughly parabolic, i.e., ${R}_{j}/{r}_{{\rm{g}}}\approx {(r/{r}_{{\rm{g}}})}^{1/2}$, within the gap region the above estimates imply

$$\begin{eqnarray}&&B\approx \displaystyle \frac{{\rm{\Phi }}}{\pi {R}_{j}^{2}}\approx 35\;{\rm{G}}\end{eqnarray} \tag{ 45 }$$

for a typical stagnation surface distance of $r\sim 10{r}_{{\rm{g}}}$.

4.1.3. Synchrotron Cooling Times

In the jet frame, the observed synchrotron cooling timescale is

$$\begin{eqnarray}{t}_{\mathrm{sync}}^{\prime } & = & \displaystyle \frac{3{{{m}}_{{\rm{e}}}{c}}^{2}}{4{\sigma }_{{\rm{T}}}{u}_{B}^{\prime }c{\gamma }^{\prime }}=\displaystyle \frac{6\pi {{{m}}_{{\rm{e}}}{c}}^{2}}{{\sigma }_{{\rm{T}}}B{^{\prime} }^{2}c}\sqrt{\displaystyle \frac{{\nu }_{B}^{\prime }}{{\nu }^{\prime }}}\\ & = & \displaystyle \frac{3}{{\sigma }_{{\rm{T}}}}{(\displaystyle \frac{2\pi {e}{{m}}_{{\rm{e}}}{c}}{B{^{\prime} }^{3}{\nu }^{\prime }})}^{1/2},\end{eqnarray} \tag{ 46 }$$

where $\nu ^{\prime}$ and ${\nu }_{B}^{\prime }\equiv {eB}^{\prime} /2\pi {m}_{{\rm{e}}}c$ are the observation and cyclotron frequencies in the jet frame, and $\gamma ^{\prime}$ is the Lorentz factor of nonthermal particles, again measured in the jet frame. To relate this to emission in the lab frame, note that ${\nu }^{\prime }=\nu {\rm{\Gamma }}(1-\beta \mathrm{cos}{\rm{\Theta }})$ and since the magnetic field is likely to be dominantly toroidal, $B^{\prime} \approx B/{\rm{\Gamma }}$. Hence, as measured in the lab frame,

$$\begin{eqnarray}{t}_{\mathrm{sync}} & = & {t}_{\mathrm{sync}}^{\prime }{\rm{\Gamma }}\approx \displaystyle \frac{3}{{\sigma }_{{\rm{T}}}}{[\displaystyle \frac{2\pi {emc}{{\rm{\Gamma }}}^{4}}{{B}^{3}\nu (1-\beta \mathrm{cos}{\rm{\Theta }})}]}^{1/2}\\ & \approx & 1.8\times {10}^{5}{B}_{1.5}^{-3/2}{\nu }_{11.5}^{-1/2}\;{\rm{s}},\end{eqnarray} \tag{ 47 }$$

where $B=30{B}_{1.5}\;{\rm{G}}$ and $\nu =300{\nu }_{11.5}\;\mathrm{GHz}$ and the remainder of the quantities are given by our fiducial M87 values.

Evident in Equation (47) is the strong dependence of ${t}_{\mathrm{sync}}$ on B and Γ, and therefore position. As the jet propagates, B decreases and Γ increases in a fashion that depends on the specific jet structure, resulting in a corresponding rapid increase in ${t}_{\mathrm{sync}}$. Generally, magnetic flux conservation implies that the poloidal and toroidal components of the magnetic field within the jet scale ∝R⁻² and ∝R⁻¹, respectively (it is for this reason that, despite having similar strengths near the black hole, jet magnetic fields quickly become strongly toroidally dominated). For a parabolic magnetic field, $R\propto {z}^{1/2}$, which we assumed when obtaining Equation (47). Similarly, Γ is expected to rise $\propto R$. However, in practice the Lorentz factor of the jet in M87 appears to asymptote to $\approx 5$ at large radii, limiting its impact. Regardless, near the stagnation surface these imply that ${t}_{\mathrm{sync}}$ will initially grow $\propto {z}^{7/4}$, eventually slowing to $\propto {z}^{3/4}$.

The implication for the image structure is obtained by comparing ${t}_{\mathrm{sync}}$ to the outflow timescale,

$$\begin{eqnarray}&&{t}_{\mathrm{outflow}}=\displaystyle \frac{z}{\beta c}\approx 3.3\times {10}^{5}(\displaystyle \frac{r}{10{r}_{{\rm{g}}}})\;{\rm{s}},\end{eqnarray} \tag{ 48 }$$

which grows linearly with jet height. When ${t}_{\mathrm{sync}}\gt {t}_{\mathrm{app}}$ the relativistic leptons cannot efficiently cool, and therefore the nonthermal particle density is effectively conserved. At moderate distances from jet base ($z\gtrsim 15{r}_{{\rm{g}}}$) the above estimates suggest that this is the case. At much larger distances where the Lorentz factor saturates ($z\gtrsim {10}^{3}{r}_{{\rm{g}}}$) the synchrotron cooling becomes efficient.

Near the stagnation surface ${t}_{\mathrm{sync}}$ is comparable to ${t}_{\mathrm{outflow}}$. However, this is complicated by the large initial bulk Lorentz factors resulting from the acceleration across the gap. As a result, a substantial fraction of leptons still resides in the ordered nonthermal relativistic component that emerged from the polar cascade propagating along magnetic field lines and is therefore geometrically immune to synchrotron cooling. The study of how these nonthermal electrons isotropize and cool is beyond the scope of this paper. Thus, for simplicity we will ignore synchrotron cooling in the estimation of the energetic lepton population of the jet.

4.1.4. Estimates of Seed Photon Density

The source of the soft seed photons in M87 is not immediately clear due to potential contamination from the jet itself. Nonetheless, it is now certain that the millimeter-wavelength emission is associated with compact features with scales comparable to that of the horizon (Doeleman et al. 2012). This provides circumstantial evidence that the observed millimeter, infrared, and optical emission arises very near the black hole itself. Thus, even if the emission is from within the jet itself, at the jet heights of relevance the bulk Lorentz factors are expected to be small (of order unity), and hence not strongly beamed. As a consequence, we estimate the seed photon density from the observed emission directly.

The SED of M87, shown in Figure 3, is reasonably well fit by a broken power law, i.e., ${F}_{\nu }\propto {\nu }^{-\alpha }$ where α evolves rapidly as the source becomes optically thin (see Figure 3):

$$\begin{eqnarray}\alpha \approx \{\begin{array}{cc}0 & \nu \lt 300\;\mathrm{GHz}\\ 1.2 & \nu \geqslant 300\;\mathrm{GHz}.\end{array}\end{eqnarray} \tag{ 49 }$$

The flux density at the break is roughly 1 Jy. If all of the emission above 300 GHz arises in a radiatively inefficient accretion disk, then the soft photon energy density is roughly

$$\begin{eqnarray}&&{u}_{{\rm{s}}}\approx 0.01\;\mathrm{erg}\;{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 50 }$$

where we have now fixed the location of the stagnation surface to $10{r}_{{\rm{g}}}$. This is well above the densities necessary for inverse Compton losses to dominate the limits upon the acceleration of the leptons within the gap.

Zoom In Zoom Out Reset image size

4.1.5. Implied Nonthermal Particle Properties

If the submillimeter luminosity arises due to optically thin emission of the jet launching region, as suggested by the recent mm-VLBI observations (Doeleman et al. 2012), it implies a rather soft nonthermal particle density, with ${dn}/d\gamma \approx {\gamma }^{-3.4}$. The source size is roughly $40\;\mu \mathrm{as}$, implying that the associated millimeter flux is related to the particle density by

$$\begin{eqnarray}{F}_{\mathrm{mm}} & \approx & \displaystyle \frac{\sqrt{27}{e}^{2}{\nu }_{B}}{8{\pi }^{2}c}{(\displaystyle \frac{3{\nu }_{B}}{\nu })}^{\alpha }\displaystyle \frac{2\alpha {\gamma }_{\mathrm{min}}^{2\alpha }}{\alpha +1}{\rm{\Gamma }}\\ & & \times (\displaystyle \frac{\alpha }{2}+\displaystyle \frac{11}{6}){\rm{\Gamma }}(\displaystyle \frac{\alpha }{2}+\displaystyle \frac{1}{6}){\rm{\Omega }}{rn}\\ & = & 4.9\times {10}^{-2}{(\displaystyle \frac{\theta }{40\;\mu \mathrm{as}})}^{2}(\displaystyle \frac{r}{10{r}_{{\rm{g}}}}){{nB}}_{1.5}^{1+\alpha }\;\mathrm{Jy},\end{eqnarray} \tag{ 51 }$$

where ${\nu }_{B}\equiv {eB}/2\pi {m}_{{\rm{e}}}c$ is the cyclotron frequency, ${\rm{\Gamma }}(x)$ is the standard gamma function, $\nu =300$ GHz, and we have assumed a typical lower cut-off on the accelerated particle distribution of ${\gamma }_{\mathrm{min}}\approx {10}^{2}$. Comparing this to an observed flux of roughly 1 Jy and assuming the magnetic field estimate in Equation (45) gives an approximate nonthermal particle density estimate of

$$\begin{eqnarray}&&n\approx 15\;{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 52 }$$

consistent with efforts to quantitatively model the emission region of M87 on horizon scales (A. E. Broderick et al. 2015, in preparation).

We can now estimate the properties of the gap, and the associated accelerated particles, from M87's parameters given above. This gives our fiducial model for M87. The asymptotic lepton Lorentz factor is determined by inverse Compton losses and is roughly

$$\begin{eqnarray}&&{\gamma }_{\mathrm{max}}\equiv {\gamma }_{\mathrm{max},\mathrm{IC}}\approx 1.6\times {10}^{9},\end{eqnarray} \tag{ 53 }$$

and this sets our fiducial value of electron energy, E₀. Whereas its numerical value is known only to a factor of few, we choose not to round this value in order to get a self-consistent answer. This corresponds to a seed photon pair-production threshold of

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{th}}\approx 1.2\;\mathrm{meV}\quad \Rightarrow \quad {\lambda }_{\mathrm{th}}\approx 1.1\;\mathrm{mm}.\end{eqnarray} \tag{ 54 }$$

Therefore, photons at and just below the spectral break in M87's SED will contribute to pair production within the gap.

The length scale over which the asymptotic lepton Lorentz factors are obtained is (see Equation (6))

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{IC}}\approx 5.5\times {10}^{10}\;\mathrm{cm}=5.6\times {10}^{-5}{r}_{{\rm{g}}},\end{eqnarray} \tag{ 55 }$$

and the typical mean free path of the up-scattered photons to pair production is (see Equation (15))

$$\begin{eqnarray}&&{{\ell }}_{\gamma \gamma }\approx 1.7\times {10}^{12}\;\mathrm{cm}\approx 1.7\times {10}^{-3}{r}_{{\rm{g}}}.\end{eqnarray} \tag{ 56 }$$

Hence, as anticipated, both the acceleration and subsequent pair production within the gap occur on scales much smaller than the typical gap thickness that might be expected on global considerations (i.e., ${r}_{{\rm{g}}}$). The resulting gap thickness is then roughly (see Equation (17))

$$\begin{eqnarray}&&{\rm{\Delta }}\approx 4.3\times {10}^{11}\;\mathrm{cm},\end{eqnarray} \tag{ 57 }$$

implying a corresponding density at the top of the gap of (see Equation (18))

$$\begin{eqnarray}&&{n}_{{\rm{g}}}\approx 2.2\times {10}^{-2}\;{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 58 }$$

We note that since Δ exceeds by about an order of magnitude ${{\ell }}_{\mathrm{IC}}$, over which the pairs attain their terminal Lorentz factor, our earlier assumption that the pairs instantaneously attain their terminal Lorentz factor is justified.

Given the threshold seed photon energy of 1.2 meV, the minimum gamma-ray energy for which the bath of seed photons is optically thick to pair production is ${\epsilon }_{\gamma ,\mathrm{min}}=640\;\mathrm{GeV}$, hence the asymptotic Lorentz factor for M87 is (see Equation (27))

$$\begin{eqnarray}&&{\gamma }_{\gamma \gamma }=2.6\times {10}^{6},\end{eqnarray} \tag{ 59 }$$

implying enhancement in the number density due to the post-gap cascade of roughly ${n}_{\infty }/{n}_{{\rm{g}}}=670$ (see Equation (28)).

To verify this analytical estimate, we carried out numerical integration of the pair cascade Equations (37) and (38), as described in Section 3.2. We choose the following parameters to describe the seed photon spectrum in our fiducial model of M87: $\alpha =1.2$, ${\epsilon }_{{\rm{m}}}=1.2$ meV, ${\epsilon }_{{\rm{M}}}=0.8$ eV. We start with an initial distribution that corresponds to an electron emerging from the top of the gap at an energy E₀. For numerical convenience, we represent its energy distribution with a narrow Gaussian distribution shown in Figures 4(e)–(h) with a thin solid dark red line. Different rows show distributions for different values of ${E}_{0}/{m}_{{\rm{e}}}{c}^{2}$: 10⁶, 10⁷, $1.6\times {10}^{9}$ (fiducial value), and 10¹⁰. We then evolve the system of Equations (37) and (38) numerically, as described in Section 3.2, and show subsequent distributions in Figure 4 with colored lines (see the legend).

Zoom In Zoom Out Reset image size

We adopt as our fiducial model pair cascade for M87 the case when ${E}_{0}=1.6\times {10}^{9}{m}_{{\rm{e}}}{c}^{2}$, shown in Figure 4(g). The electron cools (see Equation (33)) due to the up-scattering of a seed photon into a γ-ray photon, which in turn pair produces on an additional seed photon (see Equation (34)). These two contributions to lepton energy distribution are clearly visible in Figure 4(h): after the first scattering, i.e., in generation 1, the IC-cooled electron distribution is a power law extending from ${\gamma }_{\mathrm{lep}}\approx 0.5{\epsilon }_{\gamma ,\mathrm{min}}=6\times {10}^{5}$ to ${\gamma }_{\mathrm{lep}}=2\times {10}^{8}$ and the new pairs form a Gaussian distribution centered around ${\gamma }_{\mathrm{lep}}\approx 7\times {10}^{8}$. Note that the lower energy cut-off in the power law emerges because the seed photon bath is transparent to gamma-rays with energy less than ${\epsilon }_{\gamma ,\mathrm{min}}$.

Numerically obtained values of density enhancement versus generation number are shown in Figure 5(a), for different values of E₀. It is clear that the density enhancement saturates around generation ${N}_{\mathrm{gen}}\gtrsim \mathrm{few}\times 10$. The numerical result for density enhancement in our fiducial model of M87, ${n}_{\infty }/{n}_{{\rm{g}}}=640$, is shown with a long-dashed magenta line and is in excellent agreement with the analytic expectation, discussed above and given by Equation (28), ${n}_{\infty }/{n}_{{\rm{g}}}\approx 670$. This yields a jet lepton density near the stagnation surface of

$$\begin{eqnarray}&&{n}_{\infty }\approx 15\;{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 60 }$$

In the absence of nonlinear plasma phenomena, these energic pairs will inverse Compton cool over roughly a single jet scale height, i.e., within a distance r, at which point they will have undergone large-angle deflections. These leptons, now misaligned with the background magnetic field, will subsequently synchrotron cool and isotropize. The rate of this cooling depends on the microscopic Lorentz factors and becomes comparable to the outflow timescale only when $\gamma \approx {10}^{2-3}$ on scales of the gap height. Thus the resulting population is in excellent agreement with that required to produce the observed submillimeter emission.

Zoom In Zoom Out Reset image size

As is also clear from Figure 5(a), higher initial energy leads to a higher density enhancement. Similarly, Figure 5(b) shows that the total energy and the energy in leptons are higher for higher E₀. The flatness of the solid lines in Figure 5(b) demonstrates that the total energy of the cascade (carried by leptons and photons) is conserved at high accuracy by our numerical scheme. Note that, as seen in Figure 4, there exist asymptotic properties of the pair cascade that are independent of E₀. For example, it is explicitly apparent in Figure 5(c) that the average lepton Lorentz factor, $\langle {\gamma }_{\mathrm{lep}}\rangle$, is independent of E₀ for ${N}_{\mathrm{gen}}\gtrsim 100$ and is controlled completely by ${\tau }_{\gamma \gamma }$.

We investigated the dependence of density enhancement factor on key parameters of our model, and the results are shown in Figure 6. The blue dots connected by solid lines show the numerical results; they are in excellent agreement with the analytical scaling given by Equation (28). The density enhancement is most sensitive to the initial electron energy, E₀: ${N}_{\mathrm{lep},\infty }\propto {E}_{0}$ (see Figure 6(a)), and the gamma-ray energy below which the seed photon bath is optically thin to pair production, ${\epsilon }_{\gamma ,\mathrm{min}}$: ${N}_{\mathrm{lep},\infty }\propto {\epsilon }_{\gamma ,\mathrm{min}}^{\alpha /(2\alpha -1)}\approx {\epsilon }_{\gamma ,\mathrm{min}}^{0.86}$ (see Figure 6(c)). However, if we now recall that both ${E}_{0}={\gamma }_{\mathrm{max},\mathrm{IC}}{m}_{{\rm{e}}}{c}^{2}$ and ${\epsilon }_{\gamma ,\mathrm{min}}\propto {\gamma }_{\mathrm{max},\mathrm{IC}}$ depend on ${\gamma }_{\mathrm{max},\mathrm{IC}}$ (see Equation (24)), we find that these dependencies nearly cancel out, and that density enhancement is most sensitive to the density of seed photons n_s, or source luminosity:

$$\begin{eqnarray}\displaystyle \frac{{n}_{\infty }}{{n}_{{\rm{g}}}} & \propto & {({\gamma }_{\mathrm{max},\mathrm{IC}}{\epsilon }_{b})}^{(\alpha -1)/(2\alpha -1)}{n}_{{\rm{s}}}^{1/(2\alpha -1)}\\ & & \approx {({\gamma }_{\mathrm{max},\mathrm{IC}}{\epsilon }_{b})}^{0.14}{n}_{{\rm{s}}}^{0.71}.\end{eqnarray} \tag{ 61 }$$

Importantly, the numerical results are insensitive to the upper energy cut-off of the seed photon spectrum, as shown in Figure 6(d).

Zoom In Zoom Out Reset image size

We ensured that our simulation results are numerically converged. For this, we varied the number of energy bins for more than three orders of magnitude and computed the error relative to the highest-resolution simulation carried out using 2 × 10⁵ energy bins. As seen in Figure 7, the simulation results converge at first order with increasing energy resolution. For the simulations described here, we employed a fiducial resolution of 10⁴ energy bins, for which the relative error is less than 0.3% in all quantities.

Zoom In Zoom Out Reset image size

The asymptotic Lorentz factor after the pair cascade saturates, set by when the inverse Compton cooling length is comparable to the jet scale height, is roughly 10⁴. This implies direct synchrotron emission up to $\approx 10$–100 eV, well into the ultraviolet.

The direct inverse Compton signal from the stagnation surface itself should be visible up to ${\epsilon }_{\mathrm{VHEGR}}\approx 640$ GeV, above which it will be absorbed by the pair cascade. As is clear from Figure 4, the direct IC signal from the cascade has a flat energy spectrum in ${E}^{2}{dN}/{dE}$, with every decade in energy carrying approximately the same amount of energy. This is consistent with the analytical expectation given in Appendix C. Thus, it is possible that the observed very-high-energy gamma-ray emission is associated with the stagnation surface (see, e.g., Levinson & Rieger 2011). Emission in the band pass of Fermi/LAT, around ${\epsilon }_{\gamma }\approx 10$ GeV, is mostly due to Compton up-scattering of seed photons near the peak of the SED, ${\epsilon }_{{\rm{s}}}\approx 1.2$ meV, by electrons with Lorentz factors $\gamma \approx {({\epsilon }_{\gamma }/2{\epsilon }_{{\rm{s}}})}^{1/2}=2\times {10}^{6}$, which is comparable to ${\gamma }_{\gamma \gamma }$. The associated power per lepton is

$$\begin{eqnarray}&&{P}_{10\;\mathrm{GeV}}\approx 4{\sigma }_{{\rm{T}}}{u}_{{\rm{s}}}c{\gamma }^{2}/3.\end{eqnarray} \tag{ 62 }$$

The highly relativistic nature of the lepton distribution produces a large relativistic aberration, beaming the up-scattered gamma-rays within a cone with an opening angle $1/\gamma$ around the original electron momentum vector, and thus concentrating the emission in a solid angle of ${{\rm{\Omega }}}_{\gamma }=\pi /{\gamma }^{2}$. However, it also limits the region of the stagnation surface that can be viewed to an area ${{\rm{\Omega }}}_{\gamma }{r}^{2}$. Finally, inverse Compton cooling also sets a length scale over which we expect to find particles that can produce sufficiently energetic up-scattered gamma-rays, ${{\ell }}_{\mathrm{IC},10\;\mathrm{GeV}}$ typically much smaller than ${r}_{{\rm{g}}}$, and here roughly $0.05{r}_{{\rm{g}}}$ (see Equation (6)). Thus, the typical anticipated 10 GeV flux produced near the stagnation surface is independent of the beaming, assuming that the viewing angle is within the jet opening angle:

$$\begin{eqnarray}{{F}_{\epsilon }| }_{10\;\mathrm{GeV}} & \approx & \displaystyle \frac{{P}_{10\;\mathrm{Gev}}}{{{\rm{\Omega }}}_{\gamma }}{n}_{\infty }\displaystyle \frac{{{\rm{\Omega }}}_{\gamma }{R}_{j}^{2}}{{D}^{2}}{{\ell }}_{\mathrm{IC},10\;\mathrm{GeV}}\\ & = & \gamma {{{m}}_{{\rm{e}}}{c}}^{3}{n}_{\infty }\displaystyle \frac{\pi {R}_{j}^{2}}{{D}^{2}}\\ & = & 7.1\times {10}^{-9}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 63 }$$

In principle, all leptons with Lorentz factors above γ will contribute to the energy flux. In practice, this increases the flux near 10 GeV by a factor of less than two (see Appendix C), to

$$\begin{eqnarray}{\epsilon {F}_{\epsilon }| }_{10\;\mathrm{GeV}} & \approx & \displaystyle \frac{12(\alpha -1)}{2\alpha -1}\gamma {{{m}}_{{\rm{e}}}{c}}^{3}{n}_{\infty }\displaystyle \frac{\pi {R}_{j}^{2}}{{D}^{2}}\\ & = & 1.2\times {10}^{-8}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 64 }$$

This is considerably larger than the $(7.5\pm 5.9)\times {10}^{-12}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}{{\rm{s}}}^{-1}$ reported for M87 by Fermi in the first Fermi/LAT catalog of $\gt 10$ GeV sources (1FHL, Ackermann et al. 2013). It is, however, consistent with the distance-adjusted fluxes of other gamma-ray-bright blazars in the Fermi sample (see, e.g., Ackermann et al. 2011).

The disparity is almost certainly due to the high degree of beaming anticipated. At 10 GeV, the high-energy emission is beamed within 01 of the original electron direction. Beyond this angle, the energy of the up-scattered photons drops dramatically; at 10° the typical up-scattered photon energy is only roughly 70 times larger than that of the original seed photon, well below the energies of interest. Thus, the gamma-ray emission from the stagnation surface would be expected to drop precipitously outside of the jet half-opening angle. This is a strong function of height, scaling $\propto {z}^{-1/2}$ near the black hole, though at our fiducial height of the stagnation surface this is ${\theta }_{j}\equiv R/z\approx 18^\circ$. Because the jet is collimating, the tangent to the field lines, which correspond to the range of angles over which the relativistic leptons are directed and subsequent gamma-rays emitted, makes a considerably smaller angle with the jet axis, ${\theta }_{b}\approx 0.5{\theta }_{j}=9^\circ$ for a parabolic jet. Both of these are well within the inclination implied by radio observations of 25°, suggesting that in this case it would be rare to find substantial gamma-ray emission in M87.

Using Equation (21), we obtain the total power dissipated by the cascade,

$$\begin{eqnarray}&&{L}_{\mathrm{cascade}}\approx {10}^{43}\ \mathrm{erg}\;{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 65 }$$

corresponding to roughly 10% of the total jet power, ${L}_{\mathrm{jet}}\approx {10}^{44}$ $\mathrm{erg}\;{{\rm{s}}}^{-1}$. Thus, most of the jet power flows out in the form of Poynting flux and only 10% of it gets converted into particle energy at the stagnation surface and the cascade. Most of this power is radiated in the form of gamma-rays and is not observable due to beaming away from our line of sight, as we discussed above. A small fraction of this power,

$$\begin{eqnarray}{P}_{\infty } & = & \displaystyle \frac{{\gamma }_{\infty }{n}_{\infty }}{{\gamma }_{\mathrm{max},\mathrm{IC}}{n}_{{\rm{g}}}}{L}_{\mathrm{cascade}}=\displaystyle \frac{{\gamma }_{\infty }}{{\gamma }_{\gamma \gamma }}{L}_{\mathrm{cascade}}\\ & \approx & 4\times {10}^{-3}{L}_{\mathrm{cascade}}=5\times {10}^{40}\ \mathrm{erg}\;{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 66 }$$

is carried asymptotically by the cascade and will be radiated via synchrotron emission. This is roughly a factor of 20 smaller than the isotropic-equivalent luminosity, consistent with the beaming correction inferred from detailed source modeling (Broderick & Loeb 2009).¹² This is not surprising since the asymptotic values of density and Lorentz factor produced by the cascade, ${n}_{\infty }$ and ${\gamma }_{\infty }$, are consistent with those necessary to explain M87's radio emission, as discussed in Section 6.

We note that in addition to total energy budget our model has the potential to address an important morphological feature of M87's jet that is pronounced in radio images at a frequency of 43 GHz (Walker et al. 2008): edge-brightening of the jet. The electric field in the gap, given by Equation (1), increases away from the rotational axis. This leads to density enhancement peaked toward the edge of the jet (see Equations (8) and (61)) and therefore has the potential to lead to stronger jet emission near the jet edges. More detailed study of this issue is warranted.

Particle acceleration within the stagnation surface has been further implicated by very-high-energy gamma-ray observations of a handful of other blazars as well. Recently very short-timescale TeV gamma-ray variability was reported in radio galaxy IC 310. Ackermann et al. (2014) interpreted this as emission from the stagnation surface. The main reason for this interpretation is the variability timescale that is much shorter than the light crossing time of the black hole event horizon, suggesting a very compact region. While this is a tantalizing possibility, we note that other physical processes, such as jets in a jet, can also lead to such a rapid variability (e.g., Giannios et al. 2009, 2010). Additionally, microlensing was used to place stringent constraints on the size of gamma-ray emission region for blazars PKS 1830–211 (Neronov et al. 2015, r ≲ 2 × 10¹⁵ cm) and B0218+357 (Vovk & Neronov 2015, r ≲ 10¹⁴ cm), suggesting that the emission region is located in the immediate vicinity of the central supermassive black hole.

The mass of and distance to Sgr A^* are the best known of any black hole candidate, obtained from observations of the orbits of individual stars, yielding $M=4.3\pm 0.4\times {10}^{6}\;{M}_{\odot }$ and 8.3 ± 0.4 kpc, respectively (Ghez et al. 2008; Gillessen et al. 2009). As with M87, the SED peaks near 1 mm, where the typical observed flux is roughly 3 Jy. However, the isotropic-equivalent luminosity is much lower, roughly $L\approx {10}^{36}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\approx {10}^{-9}{L}_{\mathrm{Edd}}$. Hence, even in Eddington units, Sgr A^* is considerably more underluminous than M87, for which $L\approx {10}^{42}\ \mathrm{erg}\;{{\rm{s}}}^{-1}\approx {10}^{-6}{L}_{\mathrm{Edd}}$. Here we collect various parameter estimates relevant for the putative jet launching region near Sgr A^*.

We will find that Sgr A^* and M87 have roughly similar photon energy densities, the frequency of the spectral peak, and magnetic field strengths. This leads to comparable (to within a factor of few) physical (in cm) gap thicknesses in both cases. However, the mass of the black hole in Sgr A^* is about 1500 times smaller than that in M87. This means that in Sgr A^* the gap thickness is comparable to the large-scale size of the system, and this leads to the suppression of the cascade.

5.1.1. Jet Velocity and Orientation

5.1.2. Estimates of Magnetic Field Strength and Synchrotron Cooling Times

The detection of polarized emission at millimeter wavelengths, and thus the measurement of the Faraday rotation measure of $\approx 6\times {10}^{5}\;\mathrm{rad}\;{{\rm{m}}}^{-2}$, produces an estimate for the accretion rate of $\dot{M}=2\times {10}^{-9}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}-2\times {10}^{-7}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$, depending on the details of the accretion model assumed (Agol 2000; Marrone et al. 2007). Following Equations (41) and (42) gives a corresponding estimate of the possible confined jet magnetic field of up to

$$\begin{eqnarray}&&B\approx \sqrt{\displaystyle \frac{2\beta c\dot{M}}{{r}_{{\rm{g}}}^{2}}}\approx {10}^{2-3}\;{\rm{G}}.\end{eqnarray} \tag{ 67 }$$

In both cases, the accretion power, ranging from 10³⁸–10⁴⁰ erg s⁻¹, exceeds the potential jet power by two orders of magnitude, and implies the need for low radiative efficiencies, i.e., ${\eta }_{d}\approx 0.01-{10}^{-4}$. These field strengths are similar to those inferred from spectral fitting (e.g., Yuan et al. 2003; Broderick & Loeb 2006). For concreteness we will assume the lower field strength, corresponding to the comparatively higher radiative efficiency, i.e., we will assume 100 G on the horizon and thus roughly 10 G near the stagnation surface. As a consequence, the synchrotron cooling timescale is similar to that of M87:

$$\begin{eqnarray}&&{t}_{\mathrm{sync}}\approx 4.0\times {10}^{5}{B}_{1}^{-3/2}{\nu }_{11.5}^{-1/2}\;{\rm{s}}.\end{eqnarray} \tag{ 68 }$$

Since the gravitational radius of Sgr A^* is much smaller, so is the apparent outflow timescale, comparable to

$$\begin{eqnarray}&&{t}_{\mathrm{outflow}}\approx 2.4\times {10}^{2}(\displaystyle \frac{r}{10{r}_{{\rm{g}}}})\;{\rm{s}},\end{eqnarray} \tag{ 69 }$$

and thus for Sgr A^* synchrotron cooling is completely irrelevant to the evolution of the nonthermal particle population on horizon scales.

5.1.3. Seed Photon Density Estimates

Unlike M87, Sgr A^* exhibits a clear submillimeter bump, often modeled as a thermal disk component (Yuan et al. 2003), though also potentially produced by the jet component associated with the jet launching region (Falcke & Markoff 2000; Yuan et al. 2002; Mos´cibrodzka & Falcke 2013). Nevertheless, above 1 mm the SED is well modeled by a power law with $\alpha \approx 1.25$ (see Figure 8):

$$\begin{eqnarray}\alpha \approx \{\begin{array}{ll}0 & \nu \lt 300\;\mathrm{GHz}\\ 1.25 & \nu \geqslant 300\;\mathrm{GHz}.\end{array}\end{eqnarray} \tag{ 70 }$$

Furthermore, short-timescale variability from the millimeter through the infrared implies that this emission arises near the black hole, and thus may be identified with the soft seed photons relevant for particle production and acceleration at the putative jet stagnation surface.

Zoom In Zoom Out Reset image size

The implied seed photon energy density above 1 mm is

$$\begin{eqnarray}&&{u}_{{\rm{s}}}\approx 1.5\times {10}^{-2}\;\mathrm{erg}\;{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 71 }$$

which is comparable to that implied in M87 (see Equation (50)). We will find, however, that the relevant seed photon population lies above ${10}^{12}\;\mathrm{Hz}$ (wavelengths shorter than $0.26\;\mathrm{mm}$), for which the energy density is reduced by roughly a factor of 5. As we will see below, this leads to about a factor of 3 increase in the physical thickness of the gap (in cm) in comparison to M87. This turns out to lead to a qualitative change in the cascade operation: since the black hole mass of Sgr A^* is ∼1500 times smaller than in M87, the gap thickness of Sgr A^* comes out to be comparable to the size of the black hole, and this suppresses the cascade.

5.1.4. Implied Nonthermal Particle Properties

The roughly power-law, declining SED of Sgr A^* in the submillimeter region again justifies assuming that the emission is well characterized by optically thin emission from a soft population of nonthermal electrons. In this case, ${dn}/d\gamma \propto {\gamma }^{-3.5}$, quite similar to M87. With a source size of $37\;\mu \mathrm{as}$, equating the expression in Equation (51) to the observed millimeter flux of 3 Jy yields a nonthermal particle density estimate of

$$\begin{eqnarray}&&n\approx {10}^{6}\;{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 72 }$$

This is much higher than for M87 as a direct result of the much smaller mass, and therefore more compact emission region.

Given the previous magnetic field estimate, the typical Lorentz factors are expected to be of order 10². While the infrared emission requires considerably larger γ's, it is also highly variable and thus potentially associated with additional dissipative events.

As might be expected given the similarities in the soft photon densities, the inverse Compton-limited Lorentz factor is about a factor of 2 smaller than that found in M87 (see Equation (3); see Equation (53) for M87 value):

$$\begin{eqnarray}&&{\gamma }_{\mathrm{max}}\approx 7.6\times {10}^{8}.\end{eqnarray} \tag{ 73 }$$

The length scale over which this is attained is (see Equation (6))

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{IC}}\approx 8.4\times {10}^{10}\;\mathrm{cm}\approx 0.13{r}_{{\rm{g}}}.\end{eqnarray} \tag{ 74 }$$

This value in cm is larger by about a factor of 2 and in units of gravitational radius by about a factor of 2300 than the M87 value (see Equation (55)). As a consequence of the moderately lower ${\gamma }_{\mathrm{max}}$ the threshold energy for seed photons that are able to pair produce on the subsequently up-scattered inverse Compton gamma is moderately higher,

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{th}}=2.7\;\mathrm{meV}\quad \Rightarrow \quad {\lambda }_{\mathrm{th}}\approx 0.46\;\mathrm{mm},\end{eqnarray} \tag{ 75 }$$

which is pushing into the far infrared. Hence, unlike in M87, for Sgr A^* the soft photons responsible for pair production in the gap are significantly above the spectral break.

Due to the larger ${\epsilon }_{\mathrm{th}}$, the corresponding mean free path to pair production is comparatively large (see Equation (15)),

$$\begin{eqnarray}&&{{\ell }}_{\gamma \gamma }\approx 6.6\times {10}^{12}\;\mathrm{cm}\approx 10{r}_{{\rm{g}}}.\end{eqnarray} \tag{ 76 }$$

this value exceeds that for M87 by a factor of $\simeq 4$ in cm and by a factor of 5900 in units of r_g (see Equation (56)). This implies that, at most, the gap pair catastrophe is limited to a single generation and therefore the global jet dynamics will play a significant role in determining the structure of the gap. Ignoring this potential complication, the equilibrium gap thickness is on the order of (see Equation (17); see Equation (57) for M87 value)

$$\begin{eqnarray}&&{\rm{\Delta }}\approx 1.1\times {10}^{12}\;\mathrm{cm}\approx 1.7{r}_{{\rm{g}}},\end{eqnarray} \tag{ 77 }$$

itself larger than a gravitational radius (though given the wide disparity between ${{\ell }}_{\mathrm{IC}}$ and ${{\ell }}_{\gamma \gamma }$ this may be highly variable). The associated density at the top of the gap is (see Equation (18))

$$\begin{eqnarray}&&{n}_{{\rm{g}}}\approx 2.5\times {10}^{-3}\;{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 78 }$$

about an order of magnitude smaller than for M87 (Equation (58)).

In summary, the gap densities for Sgr A^* (Equation (78)) and M87 (Equation (58)) are within an order of magnitude of each other because of the similarity of seed photon densities, magnetic field strengths, and the peak frequency of the spectrum. The order-of-magnitude smaller density in Sgr A^* comes from values of magnetic field strength that are a factor of few smaller and an above-threshold photon density for Sgr A^*, falling well short of the ${n}_{\infty }\simeq {10}^{6}\;{\mathrm{cm}}^{-3}$ inferred from radio observations (see Section 5.1.4).

Since ${{\ell }}_{\gamma \gamma }$ is much larger than the system scale height at the gap, i.e., ${{\ell }}_{\gamma \gamma }\gg r\approx 10{r}_{{\rm{g}}}$, there is effectively no post-gap cascade. This is because for compact emission regions the pair-production optical depth falls significantly at large radii for two reasons. First, the seed photon density drops $\propto {r}^{-2}$ for sufficiently large r. Second, the direction of propagation of the seed photons and gamma-rays become increasingly aligned as they travel far from their origin, reducing their center-of-mass energy and increasing the energy threshold for pair production ∝r². Hence a gamma-ray that fails to annihilate in the first mean free path is unlikely to ever do so.

Thus, ${n}_{\infty }\approx {n}_{{\rm{g}}}\approx 2.5\times {10}^{-3}\;{\mathrm{cm}}^{-3}$, implying that in this case it is likely that Sgr A^* emission is coming from the accretion disk or from the disk outflow just outside the highly magnetized jets (e.g., Mos´cibrodzka & Falcke 2013). If a substantial fraction of Sgr A^* arises from jet emission (but note that this does not have to be the case: in fact, Sgr A^* might not have any jet at all), some other particle loading and acceleration mechanisms must be occurring, such as gas entrainment from the accretion disk and the acceleration of electrons via reconnection and/or shocks.

According to Equations (20) and (21), the pair cascade luminosity scales as $\propto {M}^{2}{n}_{{\rm{g}}}{\gamma }_{\mathrm{max},\mathrm{IC}}$. Since black hole mass is smaller by three orders of magnitude, density is smaller by an order of magnitude, and ${\gamma }_{\mathrm{max},\mathrm{IC}}$ is smaller by a factor of two, the pair cascade luminosity is roughly ∼7 orders of magnitude smaller than in M87 (Equation (65)), or $\approx 3\times {10}^{35}\ \mathrm{erg}\;{{\rm{s}}}^{-1}$. Since ${\gamma }_{\infty }\approx {10}^{7}$ the overwhelming majority of this is deposited in gamma-rays beamed along the jet with typical energies of ${\gamma }_{\mathrm{max}}{m}_{{\rm{e}}}{c}^{2}\approx 400\;\mathrm{TeV}$.

Based on the overall gamma-ray luminosity it is tempting to relate this to the power in the 1–100 GeV gamma-ray features of the 10 kpc scale jet found by Su & Finkbeiner (2012) (though disputed by Ackermann et al. 2014). Any such identification, however, requires at least two additional mechanisms. First, the gamma-rays must be reprocessed down in energy by between three and five orders of magnitude. Second, the emission must be scattered in a distributed fashion along the jet to produce the observed extended feature. Accordingly, we leave the discussion of potential signatures of gamma-ray emission from Sgr A^*'s putative jet for future work.

We finish our discussion of Sgr A^* with a consideration of potentially self-consistent jet solutions. While it has been found above that a jet confined by an observationally motivated RIAF model is expected to produce little intrinsic emission, this need not be the case if the RIAF picture is relaxed.

Increasing the magnetic field increases the efficiency of the inverse Compton pair catastrophe. For $B\gtrsim 200\;{\rm{G}}$, ${{\ell }}_{\gamma \gamma }$ becomes small in comparison to the gap height, and thus multiple generations of pairs can be sustained, resulting in a well-defined inverse Compton pair catastrophe. Simultaneously, the number of pairs required to reproduce the observed submillimeter emission decreases. For $B\approx {10}^{4}\;{\rm{G}}$ the two are approximately equal, suggesting the existence of a self-consistent jet model similar to that found for M87. As with weaker fields, synchrotron cooling may continue to be ignored (this is true for $B\lesssim 1.5\times {10}^{4}\;{\rm{G}}$).

However, note that ${10}^{4}\;{\rm{G}}$ would be much larger than even optimistic estimates for the magnetic field strength in Sgr A^*. The implied jet power would be roughly ${L}_{\mathrm{EM}}\gtrsim {10}^{41}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, five orders of magnitude larger than Sgr A^*'s bolometric luminosity. While such large luminosities have been implicated in Sgr A^*'s past (e.g., the Fermi bubbles, Su et al. 2010), it is difficult to imagine sufficiently low radiative efficiencies to hide such jet power in the present epoch.

Perhaps more problematic is the need to confine such a large magnetic flux. This is ostensibly done via an accretion flow, which implies a mass accretion rate of

$$\begin{eqnarray}&&\dot{M}\approx \displaystyle \frac{{B}^{2}{r}_{{\rm{g}}}^{2}}{2\beta c}\gtrsim {10}^{-5}\;{M}_{\odot }\;{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 79 }$$

comparable to the Bondi rate for Sgr A^*. As discussed in Section 5.1.2, such high accretion rates are already excluded by polarimetric observations. Moreover, the corresponding radiative efficiency of the accretion flow would need to be exceedingly small: ${\eta }_{d}\lesssim {10}^{-6}$, much lower than the already small efficiencies implicated in RIAF models.

For these reasons we consider a self-consistent jet model driven by a gap-powered inverse Compton pair catastrophe to be highly disfavored as an explanation of Sgr A^*'s current submillimeter emission. Nevertheless, this may not be the case for Sgr A^*'s recent past, where the luminosity of Sgr A^* has been inferred to be considerably higher.

Our model relies on the presence of seed photons with energies ${\epsilon }_{{\rm{s}}}\approx 1$ meV, or wavelengths $\lambda \approx 1$ mm, to sustain the cascade. The spectra of Sgr A^* and M87 above the peak of the SED give us an estimate of the seed photon density at the source. However, so far we have been agnostic about the origin of these photons: whereas in the case of Sgr A^* they could plausibly only come from the accretion disk, in M87 they could come from the disk, the jet, or both (see, e.g., Broderick & Loeb 2009).

In our model of M87, the synchrotron luminosity of the jet, which is lit up by the leptons accelerated in the cascade, naturally explains the SED of M87 above its peak at ${\epsilon }_{{\rm{s}}}\gtrsim 1$ meV. This is because the pair cascade catastrophe naturally leads to a pair density of ${n}_{\infty }\approx 15\ {\mathrm{cm}}^{-3}$ and Lorentz factor ${\gamma }_{\infty }\approx {10}^{4}$ (see Sections 4.3 and 4.5): both of these values are comparable to those required to account for M87's radio emission above the peak of the SED (Broderick & Loeb 2009). This means that in principle no external source of seed photons is required: the cascade is self-sustaining, similar to polar cascades in pulsar magnetospheres (see, e.g., Timokhin & Arons 2013). That is, given a stray seed photon (e.g., from an accretion disk), the cascade fires up and produces energetic electrons that then cool and produce submillimeter photons that can serve as seed photons for the cascade.

An alternative process of pair production in jets is annihilation of gamma-rays from the disk (Levinson & Rieger 2011; Mos´cibrodzka et al. 2011). In this scenario, a submillimeter photon, emitted by the disk, undergoes a sequence of inverse Compton scattering events with the disk's own hot electrons. These interactions up-scatter the photon to gamma-ray energies, and eventually above the pair production threshold. Collisions of these gamma-rays in the funnel mass-load the jet with pairs.

In order for this process to lead to an interestingly large number density of pairs (larger than the Goldreich–Julian density, ${n}_{\mathrm{GJ}}$), the following two conditions must be met: (i) the millimeter emission of the disk must be compact and marginally optically thick to Thompson scattering; (ii) disk electrons must be very hot, so that on average no more than $\approx 1.5$ Compton scatterings are needed to reach pair-production threshold. To satisfy these requirements, disk radiative efficiency is assumed to be high, ${\eta }_{d}\approx 0.3$ (Mos´cibrodzka et al. 2011).

In contrast, our model allows for standard interpretation of a RIAF in M87, ${\eta }_{d}\ll 1$, which we believe is a more natural assumption. The seed photons around the SED peak are produced in the jet, outside the disk proper, and thus they have a lower probability of being up-scattered to gamma-ray energies by disk electrons. Both of these factors combined render ineffective the process of jet pair loading due to γ–γ collisions in the context of our model.

Levinson & Rieger (2011) considered pair cascade at the stagnation surface of the black hole magnetosphere and focused on the gamma-ray emission from the resulting leptons. In our work we take a broader approach and construct quantitative 1D numerical simulations of the post-cascade evolution. We consider both the direct gamma-ray signal as well the synchrotron emission of the leptons produced in the cascade. In fact, we argue that the seed photons are produced in the cascade itself and not the accretion flow. We use the observed spectrum of M87 to determine the seed photon spectrum and eliminate the sensitivity of the model to the uncertain details of the accretion flow.

Assuming stationarity, all the time derivatives vanish, and Equation (90) is a set of coupled ordinary differential equations for the gap structure:

$$\begin{eqnarray}{\partial }_{z}{n}_{{{\rm{e}}}^{+}} & = & {\alpha }_{\gamma \gamma }({n}_{{\gamma }^{+}}+{n}_{{\gamma }^{-}})\\ {-\partial }_{z}{n}_{{{\rm{e}}}^{-}} & = & {\alpha }_{\gamma \gamma }({n}_{{\gamma }^{+}}+{n}_{{\gamma }^{-}})\\ {\partial }_{z}{n}_{{\gamma }^{+}} & = & {\alpha }_{\mathrm{IC}}{n}_{{{\rm{e}}}^{+}}{-\alpha }_{\gamma \gamma }{n}_{{\gamma }^{+}}\\ {-\partial }_{z}{n}_{{\gamma }^{-}} & = & {\alpha }_{\mathrm{IC}}{n}_{{{\rm{e}}}^{-}}-{\alpha }_{\gamma \gamma }{n}_{{\gamma }^{-}}.\end{eqnarray} \tag{ 91 }$$

Choosing z = 0 to lie in the gap center, symmetry requires that

$$\begin{eqnarray}&&{n}_{{{\rm{e}}}^{+}}(z)={n}_{{{\rm{e}}}^{-}}(-z)\equiv f(z)\end{eqnarray} \tag{ 92 }$$

and similarly

$$\begin{eqnarray}&&{n}_{{\gamma }^{+}}(z)={n}_{{\gamma }^{-}}(-z)\equiv g(z).\end{eqnarray} \tag{ 93 }$$

Thus, Equations (91) reduce to

$$\begin{eqnarray}&&f^{\prime} ={\alpha }_{\gamma \gamma }[g(z)+g(-z)]\end{eqnarray} \tag{ 94 }$$

and

$$\begin{eqnarray}&&g^{\prime} ={\alpha }_{\mathrm{IC}}f-{\alpha }_{\gamma \gamma }g,\end{eqnarray} \tag{ 95 }$$

where for compactness we have denoted ${\partial }_{z}$ with primes. Taking the second derivative of the second equation and inserting the first, we obtain the following for g:

$$\begin{eqnarray}&&g^{\prime\prime} +{\alpha }_{\gamma \gamma }g^{\prime} -{\alpha }_{\mathrm{IC}}{\alpha }_{\gamma \gamma }[g(z)+g(-z)]=0.\end{eqnarray} \tag{ 96 }$$

Noting that $g^{\prime} (-z)=-[g(-z)]^{\prime}$ and $g^{\prime\prime} (-z)=[g(-z)]^{\prime\prime}$, this implies

$$\begin{eqnarray}&&[g(z)+g(-z)]^{\prime\prime} -2{\alpha }_{\mathrm{IC}}{\alpha }_{\gamma \gamma }[g(z)+g(-z)]=0\\ &&\quad {\alpha }_{\gamma \gamma }[g(z)-g(-z)]^{\prime} =0,\end{eqnarray} \tag{ 97 }$$

from which, with the required symmetry, we obtain the solutions

$$\begin{eqnarray}&&g(z)=A\mathrm{cosh}(\sqrt{2{\alpha }_{\mathrm{IC}}{\alpha }_{\gamma \gamma }}\;z).\end{eqnarray} \tag{ 98 }$$

We may now construct f by integration:

$$\begin{eqnarray}&&f=A\sqrt{\displaystyle \frac{{\alpha }_{\gamma \gamma }}{2{\alpha }_{\mathrm{IC}}}}\mathrm{sinh}(\sqrt{2{\alpha }_{\mathrm{IC}}{\alpha }_{\gamma \gamma }}\;z)+B.\end{eqnarray} \tag{ 99 }$$

The contribution to the electric field from the pairs within the gap may be obtained directly by integrating Gauss's law,

$$\begin{eqnarray}{E}_{\parallel }^{\prime } & = & 4\pi e({n}_{{{\rm{e}}}^{+}}-{n}_{{{\rm{e}}}^{-}})=4\pi e[f(z)-f(-z)]\\ \Rightarrow {E}_{\parallel } & = & \displaystyle \frac{4\pi {eA}}{{\alpha }_{\mathrm{IC}}}\mathrm{cosh}(\sqrt{2{\alpha }_{\mathrm{IC}}{\alpha }_{\gamma \gamma }}\;z)+C.\end{eqnarray} \tag{ 100 }$$

When A = 0, i.e., the charge density vanishes, the total electric field is simply that due to the gap, E_g. The particle cascade saturates when the gap is closed, i.e., ${E}_{\parallel }\approx 0$ at the center, and ${E}_{\parallel }={E}_{{\rm{g}}}$ at the gap boundary, where it is presumably screened by MHD processes, taken to lie at a height z_g. The first condition gives

$$\begin{eqnarray}&&C=-\displaystyle \frac{4\pi {eA}}{{\alpha }_{\mathrm{IC}}}.\end{eqnarray} \tag{ 101 }$$

The second then sets A in terms of the gap size:

$$\begin{eqnarray}&&A=\displaystyle \frac{{\alpha }_{\mathrm{IC}}{E}_{{\rm{g}}}}{4\pi e(\mathrm{cosh}\delta -1)},\end{eqnarray} \tag{ 102 }$$

where $\delta \equiv \sqrt{2{\alpha }_{\mathrm{IC}}{\alpha }_{\gamma \gamma }}{z}_{{\rm{g}}}=2{z}_{{\rm{g}}}/{\rm{\Delta }}$, where ${\rm{\Delta }}\equiv \sqrt{2{{\ell }}_{\mathrm{IC}}{{\ell }}_{\gamma \gamma }}$ is the characteristic gap length scale.

Because all particle production is assumed to occur within the gap, only outgoing particles exist above and below the gap boundaries. The condition that $f\geqslant 0$ everywhere within the gap then requires that the density of each species vanishes at a corresponding boundary, set by their respective directions of motion, i.e., the positron and electron densities vanish at the bottom and top of the gap, respectively. This implies

$$\begin{eqnarray}&&B=A\sqrt{\displaystyle \frac{{\alpha }_{\gamma \gamma }}{2{\alpha }_{\mathrm{IC}}}}\mathrm{sinh}\delta ,\end{eqnarray} \tag{ 103 }$$

from which we infer the particle densities at the top of the gap,

$$\begin{eqnarray}&&{n}_{{{\rm{e}}}^{+}}({z}_{{\rm{g}}})\approx \displaystyle \frac{\sqrt{2{\alpha }_{\mathrm{IC}}{\alpha }_{\gamma \gamma }}\;{E}_{{\rm{g}}}}{4\pi e}\displaystyle \frac{\mathrm{sinh}\delta }{\mathrm{cosh}\delta -1}.\end{eqnarray} \tag{ 104 }$$

It is clear from the above that the natural gap scale is Δ. However, the density at the top of the gap is a weak function of δ, with $\mathrm{sinh}\delta /(\mathrm{cosh}\delta -1)$ dropping to 2 at $\delta \simeq 1$ and falling below 1.3 for $\delta \gt 2$. Thus, independent of the precise gap thickness, the density at the top of the gap is

$$\begin{eqnarray}&&{n}_{{{\rm{e}}}^{+}}\approx 2\displaystyle \frac{{E}_{{\rm{g}}}}{4\pi e{\rm{\Delta }}},\end{eqnarray} \tag{ 105 }$$

which is directly comparable to Equation (18). Kinetic plasma simulations are desirable to verify this linearized model but are beyond the scope of this work.

Equation (105) gives a value of charge density that is a factor $R/{\rm{\Delta }}=10{r}_{{\rm{g}}}/{\rm{\Delta }}\sim {10}^{5}$ times greater than the Goldreich–Julian density, ${\rho }_{\mathrm{GJ}}\sim {\rm{\Omega }}B/2\pi c$. Consequently, the gap structure implies a large pair multiplicity that is, however, not unusual for magnetospheric cascades, e.g., in the context of pulsars (TA13). In fact, the multiplicity $\eta =R/{\rm{\Delta }}$ implied by Equation (105) is in good agreement with the simulation results of TA13 (see the discussion after Equation (18)).

The gap model described above is already linearized, simplifying the discussion of its dynamics considerably. Here we look at small, harmonic perturbations about the steady-state configuration. We have some freedom to measure time and distance in convenient units, which to simplify the following expressions, we do in ${{\ell }}_{\gamma \gamma }=1/{\alpha }_{\gamma \gamma }c$. Then, defining $\varpi \equiv \omega /{\alpha }_{\gamma \gamma }c$, $\kappa \equiv k/{\alpha }_{\gamma \gamma }c$, and $\zeta \equiv {\alpha }_{\mathrm{IC}}/{\alpha }_{\gamma \gamma }$, the equation for the perturbations becomes

$$\begin{eqnarray}-i(\varpi -\kappa )\delta {n}_{{{\rm{e}}}^{+}} & = & \delta {n}_{{\gamma }^{+}}+\delta {n}_{{\gamma }^{-}}\\ -i(\varpi +\kappa )\delta {n}_{{{\rm{e}}}^{+}} & = & \delta {n}_{{\gamma }^{+}}+\delta {n}_{{\gamma }^{-}}\\ -i(\varpi -\kappa )\delta {n}_{{\gamma }^{+}} & = & \zeta \delta {n}_{{{\rm{e}}}^{+}}-\delta {n}_{{\gamma }^{+}}\\ -i(\varpi +\kappa )\delta {n}_{{\gamma }^{-}} & = & \zeta \delta {n}_{{{\rm{e}}}^{-}}-\delta {n}_{{\gamma }^{-}}.\end{eqnarray} \tag{ 106 }$$

This may be trivially rearranged to define the modes as a standard eigenmode problem:

$$\begin{eqnarray}(\begin{array}{cccc}\kappa & 0 & i & i\\ 0 & -\kappa & i & i\\ i\zeta & 0 & \kappa -i & 0\\ 0 & i\zeta & 0 & -\kappa -i\end{array})(\begin{array}{c}\delta {n}_{{{\rm{e}}}^{+}}\\ \delta {n}_{{{\rm{e}}}^{-}}\\ \delta {n}_{{\gamma }^{+}}\\ \delta {n}_{{\gamma }^{-}}\end{array})=\varpi (\begin{array}{c}\delta {n}_{{{\rm{e}}}^{+}}\\ \delta {n}_{{{\rm{e}}}^{-}}\\ \delta {n}_{{\gamma }^{+}}\\ \delta {n}_{{\gamma }^{-}}\end{array}).\end{eqnarray} \tag{ 107 }$$

The associated four mode-specific dispersion relations are

$$\begin{eqnarray}&&\varpi =-\displaystyle \frac{i}{2}[1\pm \sqrt{1+4\zeta -4{\kappa }^{2}\pm 4\sqrt{{\zeta }^{2}-(1+4\zeta ){\kappa }^{2}}}],\end{eqnarray} \tag{ 108 }$$

where every combination of the two ± may be chosen. Two of the four modes are unstable (−− and $-+$), both of which correspond to the pair catastrophe (related to the symmetric and antisymmetric components). The maximum growth rate is

$$\begin{eqnarray}{{{\rm{\Gamma }}}_{--}| }_{\kappa =0} & \equiv & {\alpha }_{\gamma \gamma }c{\mathfrak{I}}({\varpi }_{--})\\ & = & \displaystyle \frac{\sqrt{{\alpha }_{\gamma \gamma }^{2}+8{\alpha }_{\gamma \gamma }{\alpha }_{\mathrm{IC}}}-{\alpha }_{\gamma \gamma }}{2}c.\end{eqnarray} \tag{ 109 }$$

Thus, the growth time, ${{\rm{\Gamma }}}_{--}^{-1}$, is roughly the light crossing time of the open gap. The physical origin of the instability is easy to understand: additional leptons induce additional inverse Compton scattered photons which, in turn, produce additional electrons, causing the process to exponentiate.