  We estimate disc-averaged RV variations of the Sun over the last magnetic
cycle, from the single Fe I line observed by SDO/HMI, using a physical model
for rotationally modulated magnetic activity that was previously validated
against HARPS-N solar observations. We estimate the disc-averaged, unsigned
magnetic flux and show that a simple linear fit to it reduces the RMS of RV
variations by 62%, i.e. a factor of 2.6. We additionally apply the FF' method,
which predicts RV variations based on a star's photometric variations. At cycle
maximum, we find that additional physical processes must be at play beyond
suppression of convective blueshift and velocity imablances resulting from
brightness inhomogeneities, in agreement with recent studies of solar RV
variations. By modelling RV variations over the magnetic cycle using a linear
fit to the unsigned magnetic flux, we recover injected planets at an orbital
period of about 300 days with RV semi-amplitudes down to 0.3 m/s. To reach
semi-amplitudes of 0.1 m/s, we will need to identify and model additional
physical phenomena that are not well traced by the unsigned magnetic flux or
FF'. The unsigned magnetic flux is an excellent proxy for rotationally
modulated, activity-induced RV variations, and could become a key tool in
confirming and characterising Earth analogs orbiting Sun-like stars. The
present study motivates ongoing and future efforts to develop observation and
analysis techniques to measure the unsigned magnetic flux at high precision in
slowly rotating, relatively inactive stars like the Sun.
