  The spin-down of a millisecond pulsar in a compact binary system leads to
self-energy losses, that cause the binary's orbital period to increase (the
effect being of first post-Newtonian (1PN) order). If the pulsar's
period-derivative is not exceedingly small, this effect can become measurable
in long-term high-precision timing measurements of the binary's orbital motion.
We use rotating compressible spheroids to obtain an approximate, explicit
expression for the orbital period derivative, due to spin-down, valid to 1PN
order. This expression can be used to observationally constrain the pulsar's
moment of inertia, and, combined with other observations, the high-density
equation of state of compact stars. We apply our expression to representative
models of millisecond pulsars in binary systems and demonstrate the importance
of including higher-order rotational effects as well as the importance of the
choice of equation of state, which both have been neglected, so far, in the
literature. The computed increase in orbital period is larger, by as much as
32%, when the higher-order rotational effects are included, while it can change
by more than a factor of two between different plausible equations of state.
