magnetic field decay of magnetars in supernova remnants

in this paper, we modify our previous research carefully, and derive a new expression of electron energy density in superhigh magnetic fields. based on our improved model, we re-compute the electron capture rates and the magnetic fields’ evolutionary timescales t of magnetars. according to the calculated results, the superhigh magnetic fields may evolve on timescales ∼(10⁶−10⁷) yrs for common magnetars, and the maximum timescale of the field decay, t ≈ 2.9507 × 10⁶ yrs, corresponding to an initial internal magnetic field b₀ = 3.0 × 10¹⁵ g and an initial inner temperature t₀ = 2.6 × 10⁸ k. motivated by the results of the neutron star-supernova remnant (snr) association of zhang and xie (2011), we calculate the maximum b₀ of magnetar progenitors, b_max ∼ (2.0 × 10¹⁴−2.93 × 10¹⁵) g when t₀ = 2.6 × 10⁸ k. when t₀ ∼ 2.75 × 10⁸−1.75 × 10⁸ k, the maximum b₀ will also be in the range of ∼10¹⁴−10¹⁵ g, not exceeding the upper limit of magnetic field of a magnetar under our magnetar model. we also investigate the relationship between the spin-down ages of magnetars and the ages of their snrs, and explain why all axps associated with snrslook older than their real ages, whereas all sgrs associated with snrs appear younger than they are.