Does GD 356 have a terrestrial planetary companion?

GD 356 is unique among magnetic white dwarfs because it shows Zeeman-split Balmer lines in pure emission. The lines originate from a region of nearly uniform field strength (δB/B≈ 0.1) that covers 10 per cent of the stellar surface in which there is a temperature inversion. The energy source that heats the photosphere remains a mystery but it is likely to be associated with the presence of a companion. Based on current models, we use archival Spitzer Infrared Array Camera (IRAC) observations to place a new and stringent upper limit of 12 M_J for the mass of such a companion. In the light of this result and the recent discovery of a 115-min photometric period for GD 356, we exclude previous models that invoke accretion and revisit the unipolar inductor model that has been proposed for this system. In this model, a highly conducting planet with a metallic core orbits the magnetic white dwarf and, as it cuts through field lines, a current is set flowing between the two bodies. This current dissipates in the photosphere of the white dwarf and causes a temperature inversion. Such a planet is unlikely to have survived both the red and asymptotic giant branch phases of evolution so we argue that it may have formed from the circumstellar disc of a disrupted He or CO core during a rare merger of two white dwarfs. GD 356 would then be a white dwarf counterpart of the millisecond binary pulsar PSR 1257+12 which is known to host a planetary system.