rate of $0.98$ at $z=197.5$ fm and $1 $ at $z=987.362$ fm.

However, the case of the electric field parallel to the boundary is not relevant for the quark star electrosphere, in which the electric field, which is oriented outward, is perpendicular on the surface of the star (the boundary). In the case of the electric field perpendicular to the boundary one can find the correction terms by using a method based on approximating the Green functions in terms of classical paths. The first order correction to the effective Lagrangian for particle production is [@Ma89] $$L_{eff}^{(1)}\left( z\right) =-\frac{1}{8\pi }\int_{0}^{\infty }\frac{ds}{s^{2}}\left[ eE\coth \left( eEs\right) -\frac{1}{s}\right] \exp
\left[ -im_{e}^{2}s+ie\frac{E}{2}z^{2}\coth \left(
\frac{eEs}{2}\right) \right] .$$

The first order correction $w_{1}(z)$ to the pair production rate is given by $w_{1}\left( z\right) =2{\rm Im}L_{eff}^{(1)}(z)$, and for $m_e\approx 0$ can be written in the form [@Ma89] $$w_{1}(z)=-\frac{1}{4\pi ^{2}}e^{2}E^{2}\int_{1}^{\infty }\Phi
\left( s\right) \sin \left( \frac{eE}{2}z^{2}s\right) ds,$$ where $$\Phi \left( s\right) =\frac{1}{\left( s^{2}-1\right) \left( \ln \frac{s+1}{s-1}\right) ^{2}}\left( s+\frac{1}{s}-\frac{2}{\ln
\frac{s+1}{s-1}}\right) .$$

The ratio of the total production rate $w=w_{0}+w_{1}$ and of the Schwinger production rate $w_{0}=\left( e^{2}E^{2}/4\pi
^{3}\right) \exp \left( -\pi E_{cr}/E\right) $ is represented, for different values of the electric field, in Fig. 4.

For large $z$ the function 