is given by $$E\left( z,T\right) =\sqrt{\frac{4}{3\pi
}}V\sqrt{\frac{V^{2}}{2}+\pi ^{2}T^{2}},$$ and can be expressed as $$\label{el}
E\left( z,T\right) =\sqrt{\frac{8\pi ^{3}}{3}}\frac{\cosh \left[ 2\sqrt{\frac{\alpha \pi }{3}}T\left( z+z_{0}\right) \right] }{\sinh
^{2}\left[ 2\sqrt{\frac{\alpha \pi }{3}}T\left( z+z_{0}\right)
\right] }T^{2}.$$

The ratio of the electric field of the electrosphere of quark stars and of the critical electric field $E_{cr}=m_e^2/e$ is represented, as a function of the distance from the quark star surface and for different values of the temperature, in Fig. 1.

Electron-positron pair production rates in the electrosphere
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Let us consider a quantized Dirac field coupled to a classical external electromagnetic field described by the potential $A_{\mu
}$. The probability $P$ to remain in the ground state, i.e., the probability of emitting no pairs, is given by $P=\left|
\left\langle 0\left| S\right| 0\right\rangle
\right| ^{2}$, where $S$ is the $S$ matrix defined as $S=\hat{T}\exp \left[ i\int L_{I}d^{4}x\right] $, $\hat{T}$ is the time ordering operator and $L_{I}$ is the interaction Lagrangian [@So82]. The probability $P$ can also be written as $P=\exp \left[ -\int w(x)d^{4}x\right] $, where $w\left( x\right) =2{\rm Im}L_{eff}\left( x\right) $, where $L_{eff}\left( x\right) $ is the one-loop effective Lagrangian density, which includes all orders in the external field, but neglects self-interactions of the matter fields. The 