Straight outta photosphere: Open solar flux without coronal modeling

The open solar flux, that is, the total magnetic flux escaping the Sun, is one of the most important parameters connecting solar activity to the Earth. The open solar flux is commonly estimated from photospheric magnetic field measurements by making model assumptions about the solar corona. However, the question in which way the open solar flux is directly related to the distribution of the photospheric magnetic field is still partly unknown. We aim to reconstruct the open solar flux directly from the photospheric magnetic fields without making any assumptions about the corona and without using coronal hole observations, for instance. We modified an earlier vector sum method by taking magnetic field polarities into account and applied the method to the synoptic magnetograms of six instruments to determine the open solar flux from solar cycles 21-24. Results. The modified vector sum method produces a vector of the global solar magnetic field whose magnitude closely matches the open solar flux from the potential field source surface (PFSS) model both by the absolute scale and the overall time evolution for each of the six magnetograms. The latitude of this vector follows the Hale cycle by always pointing toward the dominantly positive-polarity hemisphere, and its longitude coincides with the location of the main coronal holes of the McIntosh Archive. We find multi-year periods during which the longitude of the vector slowly drifts or stays rather stationary in the Carrington frame. These periods are punctuated by times when the longitude moves rapidly in the Carrington frame. By comparing the magnitude of this vector to the open solar flux calculated from the PFSS model with different source surface heights, we find that the best match is produced with a source surface height $R_{ss} = 2.4 - 2.5R_\odot$.