  We consider the effect of $\alpha$ variation on the vibrational spectrum of
Sr$_2$ in the context of a planned experiment to test the stability of
$\mu\equiv m_e/m_p$ using optically trapped Sr$_2$ molecules [Zelevinsky et
al., Phys. Rev. Lett. {\bf 100}, 043201; Kotochigova et al., Phys. Rev. A {\bf
79}, 012504]. We find the prospective experiment to be 3 to 4 times less
sensitive to fractional variation in $\alpha$ as it is to fractional variation
in $\mu$. Depending on the precision ultimately achieved by the experiment,
this result may give justification for the neglect of $\alpha$ variation or,
alternatively, may call for its explicit consideration in the interpretation of
experimental results.
