  Peik and Tamm [Europhys. Lett. 61, 181 (2003)] proposed a nuclear clock based
on the isomeric transition between the ground state and the first excited state
of thorium-229. This transition was recognized as a potentially sensitive probe
of possible temporal variation of the fine-structure constant, $\alpha$. The
sensitivity to such a variation can be determined from measurements of the
mean-square charge radius and quadrupole moment of the different isomers.
However, current measurements of the quadrupole moment are yet to achieve an
accuracy high enough to resolve non-zero sensitivity. Here we determine this
sensitivity using existing measurements of the change in the mean-square charge
radius, coupled with the ansatz of constant nuclear density. The enhancement
factor for $\alpha$ variation is $K = -(0.82 \pm 0.25) \times 10^4$. For the
current experimental limit, $\delta\alpha/\alpha \lesssim 10^{-17}$ per year,
the corresponding frequency shift is $\sim\!200$ Hz per year. This shift is six
orders of magnitude larger than the projected accuracy of the nuclear clock,
paving the way for increased accuracy of the determination of $\delta \alpha$
and interaction strength with low-mass scalar dark matter. We verify that the
constant-nuclear-density ansatz is supported by nuclear theory and propose how
to verify it experimentally. We also consider a possible effect of the octupole
deformation on the sensitivity to $\alpha$ variation, and calculate the effects
of $\alpha$ variation in a number of M\"ossbauer transitions.
