2014MNRAS.439.2122L

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.