of Eq. (\[prod\]), describing the electron-positron pair production in a constant electric field. In both cases the pair production rate is proportional to the intensity of the electric field, multiplied by an integral over the transverse momentum of the particles. However, since in the electrosphere $E(z,T)\rightarrow 0$ for large $z$, the particle production rate also naturally tends to zero and outside the electrosphere or in the regions with low electric field the particle production ceases. The variation of the pair production rate in the electrosphere, as a function of the distance $z$ is represented for different values of the temperature in Fig. 3.

By introducing a new variable $\zeta
=\sqrt{m_{e}^{2}+k^{2}}/V\left( z,T\right) $, the pair production rate can also be represented as $$\begin{aligned}
\dot{n}\left( z,T\right)  &=&\frac{s}{4\pi ^{2}}eE\left(
z,T\right) V^{2}\left( z,T\right) \times  \nonumber\\
&&\int_{\frac{m_{e}}{V(z,T)}}^{\infty }\zeta \exp \left[
-2\sqrt{\frac{3\pi }{2\alpha }}\zeta
^{2}\int_{-1}^{1}\frac{\sqrt{1-\eta ^{2}}\left( \zeta \eta
-1\right) ^{-1}d\eta }{\sqrt{\frac{2\pi ^{2}T^{2}}{V\left(
z,T\right) ^{2}}+\left( \zeta \eta -1\right) ^{2}}}\right] d\zeta
.\end{aligned}$$

In order to find an approximate representation of the pair production rate we power expand the integrand in Eq. (\[sigma\]). In the first order in $\eta $ we obtain $$\begin{aligned}
\frac{\sqrt{1-\eta ^{2}}}{\left( \frac{\sqrt{m_{e}^{2}+k_{\perp }^{2}}}{\omega }\eta -1\right) \sqrt{\frac{2\pi ^{2}T^{2}}{\omega ^{2}}+\left( \frac{\sqrt{m_{e}^{2}+k_{\perp }^{2}}}{\omega }\eta -1\right) ^{2}}}
&\approx
&\frac{\omega 