Magnetic fields are detected in a few percent of white dwarfs. The number of such magnetic white dwarfs known is now some hundreds. Fields range in strength from a few kG to several hundred MG. Almost all the known magnetic white dwarfs have a mean field modulus >= 1 MG. We are trying to fill a major gap in observational knowledge at the low field limit (<= 200 kG) using circular spectro-polarimetry. In this paper we report the discovery and monitoring of strong, periodic magnetic variability in two previously discovered super-weak field magnetic white dwarfs, WD2047+372 and WD2359-434. WD2047+372 has a mean longitudinal field that reverses between about -12 and +15 kG, with a period of 0.243 d, while its mean field modulus appears nearly constant at 60 kG. The observations can be intepreted in terms of a dipolar field tilted with respect to the stellar rotation axis. WD2359-434 always shows a weak positive longitudinal field with values between about 0 and +12 kG, varying only weakly with stellar rotation, while the mean field modulus varies between about 50 and 100 kG. The rotation period is found to be 0.112 d using the variable shape of the Halpha line core, consistent with available photometry. The field of this star appears to be much more complex than a dipole, and is probably not axisymmetric. Available photometry shows that WD2359-434 is a light variable with an amplitude of only 0.005 mag, our own photometry shows that if WD2047+372 is photometrically variable, the amplitude is below about 0.01 mag. These are the first models for magnetic white dwarfs with fields below about 100 kG based on magnetic measurements through the full stellar rotation. They reveal two very different magnetic surface configurations, and that, contrary to simple ohmic decay theory, WD2359-434 has a much more complex surface field than the much younger WD2047+372.
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