Magnetars are neutron stars (NSs) with extreme magnetic fields1 of strength 5 × 1013−1015 G. These fields are generated by dynamo action during the proto-NS phase, and are expected to have both poloidal and toroidal components2,3,4,5,6, although the energy of the toroidal component could be ten times larger7. Only the poloidal dipolar field can be measured directly, via NS spin-down8. The magnetic field provides heating and governs how this heat flows through the crust9. Magnetar thermal X-ray emission in quiescence is modulated with the rotational period of the NS, with a typical pulsed fraction 10–58%, implying that the surface temperature is substantially non-uniform despite the high thermal conductivity of the star’s crust. Poloidal dipolar fields cannot explain this large pulsed fraction10,11. Previous two-dimensional simulations12,13 have shown that a strong, large-scale toroidal magnetic field pushes a hot region into one hemisphere and increases the pulsed fraction. Here, we report three-dimensional magneto-thermal simulations of magnetars with strong, large-scale toroidal magnetic fields. These models, combined with ray propagation in curved spacetime, accurately describe the observed light curves of 10 out of 19 magnetars in quiescence and allow us to further constrain their rotational orientation. We find that the presence of a strong toroidal magnetic field is enough to explain the strong modulation of thermal X-ray emission in quiescence.
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The data that support the plots within the paper and other findings are provided as Source data.
The codes that were used to prepare our models within the paper are available from the corresponding authors upon reasonable request.
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We thank F. Verbunt for valuable comments that helped to improve our work notably. This work was supported by STFC grant no. ST/S000275/1. The numerical simulations were carried out on the STFC-funded DiRAC I UKMHD Science Consortia machine, hosted as part of and enabled through the ARC3 HPC resources and support team at the University of Leeds.
The authors declare no competing interests.
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Extended Data Fig. 1 Surface temperature, Ts, maps obtained for models with no initial toroidal magnetic field.
a, model C for NS with initial dipolar poloidal magnetic field B0 = 7 × 1013 G at age 10 kyr. b, model D for NS with initial poloidal dipolar magnetic field B0 = 5 × 1014 G age 10 kyr. The surface temperatures are in units of MK.
Extended Data Fig. 2 Results for model E with equal initial energy in toroidal and poloidal magnetic fields.
Panel A: surface temperature, Ts, maps obtained in 3D magneto-thermal simulations. We show NS with initial dipolar poloidal magnetic field B0 = 1 × 1014 G at age 10 kyr. Panels B, C, D: the soft X-ray light-curves expected for this thermal map at 10 kyr. Panel B shows k = 30∘ panel C shows k = 60∘ and panel D shows k = 90∘. Dotted black lines correspond to i = 30∘, solid blue lines correspond to i = 60∘ and red dashed lines correspond to i = 90∘.
Extended Data Fig. 3 The soft X-ray light-curves expected for models with no initial toroidal magnetic field.
Panels A, B, C are for model C, panels D,E,F are for model D. Panels A and D show k = 30∘, panels B and E show k = 60∘ and panels C and F show k = 90∘. Dotted black lines correspond to i = 30∘, solid blue lines correspond to i = 60∘ and red dashed lines correspond to i = 90∘.
The instrument modes are as following: CC is the Continuous Clocking mode (2.85 msec time resolution), TE is the timed exposure (3.2 sec time resolution), PN has time resolution 73.4 msec in full frame mode and MOS has time resolution of 2.6 sec in full frame mode.
A panel: PSR J1119-6127, B panel CXOU J164710.0-455216. Dashed blue lines show observations, and the theoretical light-curve for the most favourable orientation is shown with black solid lines. Red error bars are 1σ confidence intervals.
A panel: XTE J1810-197, B panel Swift J1822.3-1606. Dashed blue lines show observations, and the theoretical light-curve for the most favourable orientation is shown with black solid lines. Red error bars are 1σ confidence intervals.
A panel: CXOU J171405.7-381031, B panel SGR 1900+14. Dashed blue lines show observations, and the theoretical light-curve for the most favourable orientation is shown with black solid lines. Red error bars are 1σ confidence intervals.
A panel: 4U 0142+61, B panel 1E 1841-045. Dashed blue lines show observations, and the theoretical light-curve for the most favourable orientation is shown with black solid lines. Red error bars are 1σ confidence intervals.
A panel: SGR 0501+4516, B panel 3XMM J185246.6+003317. Dashed blue lines show observations, and the theoretical light-curve for the most favourable orientation is shown with black solid lines. Red error bars are 1σ confidence intervals.
The dashed blue lines and red error bars are observations and 1σ confidence intervals. The solid black lines are the theoretical light-curve for the most favourable orientation.
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Igoshev, A.P., Hollerbach, R., Wood, T. et al. Strong toroidal magnetic fields required by quiescent X-ray emission of magnetars. Nat Astron 5, 145–149 (2021). https://doi.org/10.1038/s41550-020-01220-z
Nature Astronomy (2021)