solution of the equation $\omega =V(z)$, as the location in the vicinity of which a pair of particles is produced. With this approximate association, the energy of the produced particle is then identified by an approximate location, and the energy interval also can be approximately related to a spatial interval at which the pair of particles is produced via the relation $d\omega =\left( \partial V/\partial z\right) dz=eE\left(
z,T\right) dz$.

Moreover, we shall introduce polar coordinates in the momentum space so that $k_{x}=k\cos \theta $ and $k_{y}=k\sin \theta $. Therefore the electron-positron pair production rate $\dot{n_{\pm
}}$, giving the number of electron-positron pairs created per unit time and per unit volume by the electric field of the electrosphere at the surface of quark stars, $\left\langle N\right\rangle /\Delta t\Delta x\Delta y\Delta z$, can be written as $$\begin{aligned}
\label{prode}
\dot{n_{\pm }}\left( z,T\right)  &=&\frac{s}{4\pi ^{2}}eE\left(
z,T\right) \times \nonumber\\
&&\int_{0}^{\infty }k\exp \left[ -2\sqrt{\frac{3\pi }{2\alpha }}\frac{m_{e}^{2}+k^{2}}{V^{2}\left( z,T\right) }\int_{-1}^{1}\frac{\sqrt{1-\eta ^{2}}\left( \frac{\sqrt{m_{e}^{2}+k^{2}}}{V\left( z,T\right) }\eta
-1\right)
^{-1}d\eta }{\sqrt{\frac{2\pi ^{2}T^{2}}{V ^{2}\left( z,T\right)}+\left( \frac{\sqrt{m_{e}^{2}+k^{2}}}{V\left( z,T\right) }\eta -1\right) ^{2}}}\right] dk,\end{aligned}$$ where $s$ describes the spin degrees of freedom of the produced particles ($s=1$ for bosons and $s=2$ for fermions).

Eq. (\[prode\]) has a general formal structure which is very similar to that 