}{\sqrt{2\pi ^{2}T^{2}+\omega ^{2}}}+\frac{\sqrt{m_{e}^{2}+k_{\perp }^{2}}}{\sqrt{2\pi ^{2}T^{2}+\omega
^{2}}}\times  \nonumber\\
&&\left( \frac{\omega ^{2}}{2\pi ^{2}T^{2}+\omega ^{2}}+1\right)
\eta +O\left( \eta ^{2}\right).\end{aligned}$$

Therefore in the first order of approximation $\sigma $ is given by $$\sigma \approx \sqrt{\frac{3\pi }{\alpha }}\frac{m_{e}^{2}+k_{\perp }^{2}}{\omega \sqrt{\frac{\omega ^{2}}{2}+\pi
^{2}T^{2}}}.$$

By fixing the electron energy so that $\omega =V(z,T)$ and taking into account the expression for the electric field in the electrosphere, given by Eq. (\[el\]), it follows that the electron-positron pair production rate can be written as $$\begin{aligned}
\dot{n}_{\pm }\left( z,T\right)  &\approx &\frac{2}{4\pi
^{2}}eE\left(
z,T\right) \int_{0}^{\infty }\exp \left( -2\sqrt{\frac{3\pi }{\alpha }}\frac{m_{e}^{2}+k^{2}}{V\sqrt{\frac{V^{2}}{2}+\pi ^{2}T^{2}}}\right)
kdk  \nonumber\\
&&\approx\frac{2}{4\pi ^{2}}eE\left( z,T\right) \int_{0}^{\infty
}\exp
\left[ -\pi \frac{m_{e}^{2}+k^{2}}{eE(z,T)}\right] kdk  \nonumber\\
&\approx &\frac{1}{4\pi ^{3}}e^{2}E^{2}(z,T)\exp \left[ -\pi \frac{E_{cr}}{E(z,T)}\right] .\end{aligned}$$

This is exactly the leading term in the Schwinger formula for the pair creation in a constant electric field $E_0$, but with the constant electric field substituted with the inhomogeneous, $z$-dependent electric field in the electrosphere, $E_0\rightarrow
E(z,T)$. Therefore, in the first approximation, it is possible to study electron-positron pair production in arbitrary electric fields in the electrosphere by simply substituting the constant electric field in the Schwinger formula with the inhomogeneous electric field.

In the second order of approximation we obtain $$\begin{aligned}
\sigma  &\approx &\sqrt{\frac{3\pi }{\alpha }}\frac{5}{6}\frac{m_{e}^{2}+k^{2}}{\omega \sqrt{\frac{\omega ^{2}}{2}+\pi ^{2}T^{2}}}+\frac{1}{3}\sqrt{\frac{3\pi 