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.

+(show how we get the spectrum)

The energy distribution of IC scattered photons $dn/dt/dE$ [Unit: #/cm^3/erg/s] of SSC emission at the base is dumped for the best fit model. This number ($dn/dt/dE$) is multiplied by $dt=r_0/c$ ($r_0$ is the size of the base and dt is an estimate of the photon residence time in the base) to get $dn/dE$. Finally, since calculation of pair production is convenient if the photon spectrum is in the dimensionless unit $\epsilon=E/m_ec^2$, we multiply $dn/dE$ by $m_ec^2$ to get $dn/d\epsilon$. The figure below shows $dn/d\epsilon$ as a function of $\epsilon$ for the best fit cases of J1118 and GX339.

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.

+(show formula)

For the thermal plasma, pair annihilation rate is estimated using eq. 68 of Svensson (1982). The pair annihilation rate is (eq. 15 of Svensson 1982)

(1)

\begin{align} \dot{n}_- = n_+ n_- < \overline{v\sigma} > = n^2 < \overline{v\sigma} > \end{align}

where

(2)

\begin{align} n_+ = n_- = n = \frac{dn}{d\gamma} = n_0 \frac{\beta \gamma^2}{\theta K_2(1/\theta)} exp(-\gamma/\theta) \end{align}

is the electron/positron number-density distribution, $\theta=kT/(m_ec^2)$, $n_0$ the total number density [Unit: #/cm^3], and $\dot{n}$ is the total number of pair annihilations per unit volume per unit time [Unit: #/cm^3/s ]. Here

(3)

\begin{align} < \overline{v\sigma} > = c r_e^2 \frac {\pi}{1+2\theta^2/[ln(2\eta_E\theta + 1.3)]} \end{align}

with $\eta_E = 0.5616$.

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

+(show formula)

The $\gamma-\gamma$pair production rate is estimated using eq. 26 of Boettcher & Schlickeiser (1997). The pair production rate of particles having Lorentz factor in the range of $\gamma$ and $\gamma+d\gamma$ is

(4)

\begin{align} {d n_{-} \over d \gamma d t} = \frac{3}{4}\sigma_T c \int_0^\infty d\epsilon_1 \frac{n_{ph}(\epsilon_1)}{\epsilon_1^2} \int_{\max\left\{\frac{1}{\epsilon_1},\gamma+1-\epsilon_1\right\}}^\infty d\epsilon_2 \frac{n_{ph}(\epsilon_2)}{\epsilon_2^2} f(\epsilon_1,\epsilon_2,\gamma) \end{align}

where
$\epsilon_{1,2} = h\nu_{1,2}/(m_ec^2)$ are the dimensionless photon energies in lab frame,
$n_{ph}(\epsilon)=dn_{ph}/d\epsilon$ is the energy distribution of the photon number densities [Unit: #/cm^3/$d\epsilon$ ] and $f(\epsilon_1,\epsilon_2,\gamma)$ is a rather nasty function of $\epsilon_1,\epsilon_2$ and $\gamma$. The total number of pairs produced per unit volume per unit time [Unit: #/cm^3/s ] is obtained by an integration of the above spectrum over $\gamma$, i.e.

(5)

\begin{align} \dot{n}_- = \int_1^{2\epsilon_{max}-1} d\gamma {d n_{-} \over d \gamma d t} \end{align}

The full integral is rather nasty, and numerically tricky (that's why it took me some time to get it working properly). However, since the pp cross-section sharply peaks near $\epsilon_1\epsilon_2\sim 2-3$ (see e.g. Fig. 5 of Coppi & Blandford 1990), a useful order-of-magnitude estimate of the total pp rate is

(6)

\begin{align} \dot{n} \sim \sigma_T \times c \times n_{ph}^2(\epsilon\sim1) \sim 10^{-15}n_{ph}^2(\epsilon\sim1) \end{align}

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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Added on 2009 Apr 02

Taking the values of the model parameters from 'fit 1' of the submitted paper, froze everything but T_e and r_0. The plots below (click on the plot to see full size) show the variation of number density, pair production rate and pair annihilation rate at the base of the jet.

Here's the Dove et al. paper.
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