  When applying the barycentric correction to a precise radial velocity
measurement, it is common practice to calculate its value only at the
photon-weighted midpoint time of the observation instead of integrating over
the entire exposure. However, since the barycentric correction does not change
linearly with time, this leads to systematic errors in the derived radial
velocities. The typical magnitude of this second-order effect is of order 10 cm
s$^{-1}$, but it depends on several parameters, e.g. the latitude of the
observatory, the position of the target on the sky, and the exposure time. We
show that there are realistic observing scenarios, where the errors can amount
to more than 1 ms$^{-1}$. We therefore recommend that instruments operating in
this regime always record and store the exposure meter flux curve (or a similar
measure) to be used as photon-weights for the barycentric correction. In
existing data, if the flux curve is no longer available, we argue that
second-order errors in the barycentric correction can be mitigated by adding a
correction term assuming constant flux.
