Pair production/annihilation

1. Get photon density of upscattered photons at the base from the model (do not consider synchrotron because the synchrotron
spectrum from the thermal population of electrons/positrons cuts off sharply at energies well below 511 keV, thus they cannot produce any pairs). The figure below shows the photon spectrum for the best fits of J1118 and GX339 reported in the paper.

The relevant model parameters in these two cases were:

 J1118 GX339 $D_{\rm src}$ [kpc] 1.7 6 $r_0/c$ [s] 5e-4 1e-3 $T_{\rm thermal\ particles}$ [K] 4.0e+10 6.75e+10 $n_0$ [#/$cm^{-3}$ ] 0.6e+14 1.8e+14

2. Estimate the pair annihilation rate for a given fit.

3. Estimate the pair production rate for a given fit.

4. Results and interpretation: The total rates of pp, pa [Unit: #/cm^3/s] and the fractional rates of pp ($\dot n_{pp}/n_0$) and pa ($\dot n_{pa}/n_0$), calculated using the above procedure gives the following numbers. The fractional rates $\dot n/n_0$ give an estimate of the rate at which particles are gained/lost at the jet-base.

 J1118 GX339 Pair production rate $\dot n_{pp}$ [#/cm^3/s] $\sim 10^7$ $\sim 10^{17}$ Pair annihilation rate $\dot n_{pa}$ [#/cm^3/s] $\sim 10^{11}$ $\sim 10^{12}$ Fractional pair production rate $\dot n_{pp}/n_0$ [1/s] $\sim 10^{-6}$ $\sim 10^3$ Fractional pair annihilation rate $\dot n_{pa}/n_0$ [1/s] $\sim 10^{-2}$ $\sim 10^{-2}$

For J1118 $\dot n_{pp} \ll \dot n_{pa}$, suggesting that any pair that is created will be quickly annihilated. Also the fact that
$\dot n_{pa}/n_0$ is small implies that the total number density is not affected very much by pair processes. Thus for J1118 (I think) we are safe.

However for GX339 $\dot n_{pp} \gg \dot n_{pa}$ suggesting that pp dominates over pa. Also the fact that $\dot n_{pa}/n_0 \gg 1$ implies that particles produced by pp will quickly dominate the overall number density at the base. Even taking into account the fact that the residence time of the plasma near the jet base $\tau_r \sim R/(\beta_jc) \sim 10^{-3}$s, it looks like pp cannot be ignored for GX339. While I am exploring the model parameter space to see if we can find another minima with lower pp rate, it would be good to be sure that the way these numbers we calculated are reasonable.
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