tokens: of the crustal currents and the polar magnetic field. We compute the consequent X-ray luminosities and compare them with the observed values for magnetars. In Section 3 , we examine the dependence of conductivity tensor and Ohmic dissipation time-scale for high magnetic fields. In Sections 4 and 5 , we look at the X-ray and dipolar radiation luminosities, periods, and decay time-scales in the dynamo model and the screened core model. We find that in the screened core model, the ratio of the dipole radiation loss to the X-ray luminosity ( L X ) is a ‘U’ curve as a function of increasing spin-down age and period. Finally, in Section 6 , we try and understand the observed cut-off in the period of magnetars and end with a discussion in Section 7 . 2 CRUSTAL CURRENT AND POLAR MAGNETIC FIELD DISSIPATION AND X-RAY LUMINOSITY FOR SOME MODELS 2.1 Estimation of crustal magnetic field energy decay rate from the magnetar spin-down age Let us first make a simple estimate of the rate of magnetic energy dissipation in the crust. We assume the crust to be a shell of thickness, Δ R cr , in which the magnetic fields diminish with time as the crustal currents dissipate. We shall give some estimates which, though qualitative, capture some of the physics. The amount of magnetic energy stored in the crust is given roughly by the expression where B is the observed surface polar field and V is the volume of the crust. V = 4 R 2 · Δ R cr ∼ 4 × 10 17 cm 3 , where we have taken R ∼ 10 6 cm as the radius of the star and Δ R cr ∼ 1 km as the thickness of the crust. We shall use this definition of the crust for the rest of the paper. As the field drops to a fraction of its original value, we assume this energy will be dissipated and released as X-ray luminosity. We further assume that the typical time-scale in which this happens is given by the spin-down age of the magnetars, Thus, a crude estimate for the ‘average’ X-ray luminosity is given by dividing the stored magnetic energy by the spin-down age, This equation can be found in Dass Soni ( 2012 ). Hence, For , we use the relation below which is based on the dipole formula, As a standard (average) value of the polar field for magnetars we take, B p ∼ 4 × 10 14 G (which is the value of the initial field in the following Section 2.2 ). For such fields, the magnetar data indicates average periods of ∼8 s. On substitution, this yields an average crustal X-ray luminosity: The instantaneous luminosity when the periods reach ∼8 s will, of course, be less than the average. The dipolar spin-down radiation energy loss rate for magnetars with such magnetic fields and periods is then given by: where we have taken, I = 10 45 gm/cc, here and in what follows. 2.2 Estimation of magnetic field energy decay rate due to Ohmic resistance and the Hall effect The work of Gourgouliatos Cumming ( 2014 ) and Wood Hollerbach ( 2015 ) computes the magnetic field evolution in the crust via the Hall effect. In their work, they start with polar fields
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