  Sun-like stars are a new probe of variations in the fine-structure constant,
$\alpha$, via the solar twins approach: velocity separations of close pairs of
absorption lines are compared between stars with very similar stellar
parameters, i.e. effective temperature, metallicity and surface gravity within
100K, 0.1 dex and 0.2 dex of the Sun's values. Here we assess possible
systematic errors in this approach by analysing $\gtrsim$10,000 archival
exposures from the High-Accuracy Radial velocity Planetary Searcher (HARPS) of
130 stars covering a much broader range of stellar parameters. We find that
each transition pair's separation shows broad, low-order variations with
stellar parameters which can be accurately modelled, leaving only a small
residual, intrinsic star-to-star scatter of 0-33 m/s (average $\approx$7 m/s,
$\approx$10$^{-4}$\r{A} at 5000\r{A}). This limits the precision available from
a single pair in one star. We consider potential systematic errors from a range
of instrumental and astrophysical sources (e.g. wavelength calibration, charge
transfer inefficiency, stellar magnetic activity, line blending) and conclude
that variations in elemental abundances, isotope ratios and stellar rotational
velocities may explain this star-to-star scatter. Finally, we find that the
solar twins approach can be extended to solar analogues - within 300K, 0.3 dex
and 0.4 dex of the Sun's parameters - without significant additional systematic
errors, allowing a much larger number of stars to be used as probes of
variation in $\alpha$, including at much larger distances.
