Modelling the hidden magnetic field of low-mass stars

Zeeman-Doppler imaging is a spectropolarimetric technique that is used to map the large-scale surface magnetic fields of stars. These maps in turn are used to study the structure of the stars' coronae and winds. This method, however, misses any small-scale magnetic flux whose polarization signatures cancel out. Measurements of Zeeman broadening show that a large percentage of the surface magnetic flux may be neglected in this way. In this paper we assess the impact of this 'missing flux' on the predicted coronal structure and the possible rates of spin-down due to the stellar wind. To do this we create a model for the small-scale field and add this to the Zeeman-Doppler maps of the magnetic fields of a sample of 12 M dwarfs. We extrapolate this combined field and determine the structure of a hydrostatic, isothermal corona. The addition of small-scale surface field produces a carpet of low-lying magnetic loops that covers most of the surface, including the stellar equivalent of solar 'coronal holes' where the large-scale field is opened up by the stellar wind and hence would be X-ray dark. We show that the trend of the X-ray emission measure with rotation rate (the so-called 'activity-rotation relation') is unaffected by the addition of small-scale field, when scaled with respect to the large-scale field of each star. The addition of small-scale field increases the surface flux; however, the large-scale open flux that governs the loss of mass and angular momentum in the wind remains unaffected. We conclude that spin-down times and mass-loss rates calculated from surface magnetograms are unlikely to be significantly influenced by the neglect of small-scale field.