the barrier. In the case of the energy loss via the production of $e^{-}e^{+}$ pairs, the energy flux is given by $F_{\pm }=\varepsilon _{\pm }\dot{n}$, with $\varepsilon _{\pm }=m_{e}+T$ and $$\label{usov}
\dot{n }=\left( \frac{9T}{2\pi \varepsilon _{F}^{2}}\right)
\sqrt{\frac{\alpha }{\pi }}\exp \left( -2m_{e}/T\right)
n_{e}T^{2}J\left( \xi \right) \Delta r_E,$$ where $\varepsilon _{F}=\left( \pi ^{2}n_{e}\right) ^{1/3}$ is the Fermi energy of the electrons, $\xi =2\sqrt{\alpha /\pi }\left(
\varepsilon _{F}/T\right) $, $$\label{j}
J\left( \xi \right) =\frac{1}{3}\frac{ \xi ^{3}\ln \left( 1+2\xi
^{-1}\right)} {\left( 1+0.074\xi \right) ^{3}}+\frac{ \pi
^{5}}{6}\frac{ \xi ^{4}}{\left( 13.9+\xi \right) ^{4}},$$ and $\Delta r_E$ is the thickness of the region with the strong electric field [@Us98a; @Us01].

At surface temperatures of around $10^{11}$ K, the luminosity of the outflowing plasma may be of the order $\sim 10^{51}$ ergs$^{-1}$ [@Us98a; @Us98b]. Moreover, as shown by @PaUs02, for about one day for normal quark matter and for up to a hundred years for superconducting quark matter, the thermal luminosity from the star surface, due to both photon emission and $e^{+}e^{-}$ pair production may be orders of magnitude higher than the Eddington limit. Hence, electron-positron pair creation due to the electric field of the electrosphere could be one of the main observational signatures of quark stars.

It 