  The fine structure constant, $\alpha$, sets the strength of the
electromagnetic force. The Standard Model of particle physics provides no
explanation for its value, which could potentially vary. The wavelengths of
stellar absorption lines depend on $\alpha$, but are subject to systematic
effects owing to astrophysical processes in stellar atmospheres. We measured
precise line wavelengths using 17 stars, selected to have almost identical
atmospheric properties to those of the Sun (solar twins), which reduces those
systematic effects. We found that $\alpha$ varies by $\lesssim$50
parts-per-billion (ppb) within 50 parsecs from Earth. Combining the results
from all 17 stars provides an empirical, local reference for stellar
measurements of $\alpha$ with an ensemble precision of 12 ppb.
