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 components^2,3,4,5,6, 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 fraction^10,11. Previous two-dimensional simulations^12,13 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.

磁场是具有5×10 ¹³ -10 ¹⁵ G强度的极端磁场¹的中子星（NSs）。这些磁场是由原始NS阶段的发电机作用产生的，并且有望同时具有极向和超环向分量^{2,3 ，4,5,6}，尽管环形分量的能量可以大十倍⁷。只能通过NS旋转⁸来测量极向偶极场。磁场提供热量并控制热量如何流过地壳⁹。静止状态下的磁热X射线发射随NS的旋转周期而调制，典型的脉冲分数为10-58％，这意味着尽管恒星地壳具有很高的导热率，但表面温度基本上是不均匀的。极地偶极场不能解释这个大的脉冲分数^10,11。先前的二维模拟^12,13已有研究表明，强大的，大范围的环形磁场将热区域推入一个半球，并增加了脉冲分数。在这里，我们报告具有强大的大规模环形磁场的磁星的三维磁热模拟。这些模型与弯曲时空中的射线传播相结合，准确地描述了静止状态下19个磁星中有10个的观测光曲线，并允许我们进一步限制它们的旋转方向。我们发现强环形磁场的存在足以说明静态X射线发射的强调制。