  We consider the general class of theories in which there is a new ultralight
scalar field that mediates an equivalence principle violating, long-range
force. In such a framework, the sun and the earth act as sources of the scalar
field, leading to potentially observable location dependent effects on atomic
and nuclear spectra. We determine the sensitivity of current and
next-generation atomic and nuclear clocks to these effects and compare the
results against the existing laboratory and astrophysical constraints on
equivalence principle violating fifth forces. We show that in the future, the
annual modulation in the frequencies of atomic and nuclear clocks in the
laboratory caused by the eccentricity of the earth's orbit around the sun may
offer the most sensitive probe of this general class of equivalence principle
violating theories. Even greater sensitivity can be obtained by placing a
precision clock in an eccentric orbit around the earth and searching for time
variation in the frequency, as is done in anomalous redshift experiments. In
particular, an anomalous redshift experiment based on current clock technology
would already have a sensitivity to fifth forces that couple primarily to
electrons at about the same level as the existing limits. Our study provides
well-defined sensitivity targets to aim for when designing future versions of
these experiments.
