Luminosity of a quark star undergoing torsional oscillations and the problem of gamma-ray bursts

International audienceAims: We discuss whether the winding-up of the magnetic field by differential rotation in a new-born quark star can produce a sufficiently-high, energy, emission rate of sufficiently long duration to explain long gamma-ray bursts. Methods: In the context of magnetohydrodynamics, we study the torsional oscillations and energy extraction from a new-born, hot, differentially-rotating quark star. Results: The new-born compact star is a rapid rotator that produces a relativistic, leptonic wind. The star's torsional oscillation modulates this wind emission considerably when it is odd and of sufficient amplitude, which is relatively easy to reach. Odd oscillations may occur just after the formation of a quark star. Other asymmetries can cause similar effects. The buoyancy of wound-up magnetic fields is inhibited, or its effects are limited, by a variety of different mechanisms. Direct electromagnetic emission by the torsional oscillation in either an outside vacuum or the leptonic wind surrounding the compact object is found to be insignificant. In contrast, the twist given to the outer magnetic field by an odd torsional oscillation is generally sufficient to open the star's magnetosphere. The Poynting emission of the star in its leptonic environment is then radiated from all of its surface and is enhanced considerably during these open episodes, tapping at the bulk rotational energy of the star. This results in intense energy shedding in the first tens of minutes after the collapse of magnetized quark stars with an initial poloidal field of order of 1014 Gauss, sufficient to explain long gamma-ray bursts