The magnetic field vector of the Sun-as-a-star

Direct comparison between stellar and solar magnetic maps are hampered by their dramatic differences in resolution. Here, we present a method to filter out the small-scale component of vector fields, in such a way that comparison between solar and stellar (large-scale) magnetic field vector maps can be directly made. Our approach extends the technique widely used to decompose the radial component of the solar magnetic field to the azimuthal and meridional components as well. For that, we self-consistently decompose the three-components of the vector field using spherical harmonics of different $l$ degrees. By retaining the low $l$ degrees in the decomposition, we are able to calculate the large-scale magnetic field vector. Using a synoptic map of the solar vector field at Carrington Rotation CR2109, we derive the solar magnetic field vector at a similar resolution level as that from stellar magnetic images. We demonstrate that the large-scale field of the Sun is not purely radial, as often assumed -- at CR2109, $83\%$ of the magnetic energy is in the radial component, while $10\%$ is in the azimuthal and $7\%$ is in the meridional components. By separating the vector field into poloidal and toroidal components, we show that the solar magnetic energy at CR2109 is mainly ($>90\%$) poloidal. Our description is entirely consistent with the description adopted in several stellar studies. Our formalism can also be used to confront synoptic maps synthesised in numerical simulations of dynamo and magnetic flux transport studies to those derived from stellar observations.