We consider the bremsstrahlung surface photon emissivity of strange quark stars, by systematically taking into account the effects of the multiple scatterings of highly relativistic quarks in a dense medium (the Landau-Pomeranchuk-Migdal effect). Because of interference between amplitudes of nearby interactions, the bremsstrahlung emissivity from a strange star surface is suppressed for frequencies smaller than a critical frequency. The range of the suppressed frequencies is a function of the quark matter density at the star's surface and of the QCD coupling constant. For temperatures much smaller than the Fermi energy of the quarks, the bremsstrahlung spectrum has the same temperature dependence as equilibrium blackbody radiation. Multiple collisions could reduce by an order of magnitude the intensity of the bremsstrahlung radiation. The effect of the thin electron layer at the surface of the quark star on the bremsstrahlung spectrum is also analyzed in detail. It is shown that absorption in the semidegenerate electron gas can also significantly reduce the intensity of the quark-quark bremsstrahlung radiation and, consequently, the surface radiation of the quark star. Hence, the combined effects of multiple collisions and absorption in the electron layer could make the soft photon surface radiation of quark stars 6 orders of magnitude smaller than equilibrium blackbody radiation.
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