Aims. We describe the so-called low magnetic field magnetars, SGR 0418+5729, Swift J1822.3-1606, and the AXP prototype 1E 2259+586 as massive, fast-rotating, highly magnetized white dwarfs. Methods. We give bounds for the mass, radius, moment of inertia, and magnetic field for these sources by requesting the stability of realistic, general relativistic, uniformly rotating white dwarfs. We also present the theoretical expectation of the infrared, optical, and ultraviolet emission of these objects and show their consistency with the current available observational data. Results. We improve the theoretical prediction of the lower limit of the spindown rate of SGR 0418+5729; for a white dwarf close to its maximum stable mass we obtain the very stringent interval for the spindown rate of 4.1 × 10 −16 <Ṗ < 6 × 10 −15 , where the upper value is the known observational limit. A lower limit has been also set for Swift J1822.3-1606, whose spindown rate is not yet fully confirmed. Our model provides for the sourceṖ ≥ 2.13 × 10 −15 if the star is close to its maximum stable mass. We give in addition the frequencies at which absorption features could be present in the spectrum of these sources as the result of the scattering of photons with the quantized electrons by the surface magnetic field.Key words. stars: magnetic field -stars: rotation -white dwarfs -stars: magnetars
IntroductionSoft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs) are a class of compact objects that show interesting observational properties (see, e.g., Mereghetti 2008): rotational periods in the range P ∼ (2−12) s, spindown ratesṖ ∼ (10 −13 -10 −10 ), strong outburst of energies ∼(10 41 −10 43 ) erg, and in the case of SGRs, giant flares of even large energies ∼(10 44 -10 47 ) erg.The most popular model for the description of SGRs and AXPs, the magnetar model, is based on a neutron star of fiducial parameters M = 1.4 M , R = 10 km, and a moment of inertia I = 10 45 g cm 2 . It needs a neutron star magnetic field larger than the critical field for vacuum polarization B c = m 2 e c 3 /(e ) = 4.4 × 10 13 G in order to explain the observed X-ray luminosity in terms of the release of magnetic energy (Duncan & Thompson 1992; Thompson & Duncan 1995). There exist in the literature other models still based on neutron stars but of ordinary fields B ∼ 10 12 G; these models involve either the generation of drift waves in the magnetosphere or the accretion of fallback material via a circumstellar disk (see Malov 2010;Trümper et al. 2013, respectively, and references therein).Turning to the experimental point of view, the observation of SGR 0418+5729 with a rotational period of P = 9.08 s, an upper limit of the first time derivative of the rotational perioḋ P < 6.0 × 10 −15 (Rea et al. 2010), and an X-ray luminosity of L X = 6.2 × 10 31 erg s −1 can be considered as the Rosetta Stone for alternative models of SGRs and AXPs. The inferred upper limit of the surface magnetic field of SGR 0418+5729, B < 7.5 × 10 12 G; which describes it as a neutr...