  The recent demonstration of laser excitation of the $\approx 8$ eV isomeric
state of thorium-229 is a significant step towards a nuclear clock. The low
excitation energy likely results from a cancellation between the contributions
of the electromagnetic and strong forces. Physics beyond the Standard Model
could disrupt this cancellation, highlighting nuclear clocks' sensitivity to
new physics. Accurate predictions of the different contributions to nuclear
transition energies, and therefore of the quantitative sensitivity of a nuclear
clock, are challenging. We improve upon previous sensitivity estimates and
assess a ''nightmare scenario'', where all binding energy differences are small
and the new physics sensitivity is poor. A classical geometric model of
thorium-229 suggests that fine-tuning is needed for such a scenario. We also
propose a $d$-wave halo model, inspired by effective field theory. We show that
it reproduces observations and suggests the ''nightmare scenario'' is unlikely.
We find that the nuclear clock's sensitivity to variations in the effective
fine structure constant is enhanced by a factor of order $10^4$. We finally
propose auxiliary nuclear measurements to reduce uncertainties and further test
the validity of the halo model.
