  Gyrochronology allows the derivation of ages for cool main sequence stars
based on their observed rotation periods and masses, or a suitable proxy
thereof. It is increasingly well-explored for FGK stars, but requires further
measurements for older ages and K-M-type stars. We study the nearby, 3 Gyr-old
open cluster Ruprecht 147 to compare it with the previously-studied, but far
more distant, NGC 6819 cluster, and especially to measure cooler stars than was
previously possible there. We constructed an inclusive list of 102 cluster
members from prior work, including Gaia DR2, and for which light curves were
also obtained during Campaign 7 of the Kepler/K2 space mission. [...] Periodic
signals are found for 32 stars, 21 of which are considered to be both highly
reliable and to represent single, or effectively single, Ru147 stars. These
stars cover the spectral types from late-F to mid-M stars, and they have
periods ranging from 6d-32d, allowing for a comparison of Ruprecht 147 to both
of the other open clusters and to models of rotational spindown. The derived
rotation periods connect reasonably to, overlap with, and extend to lower
masses the known rotation period distribution of the 2.5 Gyr-old cluster NGC
6819. The data confirm that cool stars lie on a single surface in rotation
period-mass-age space, and they simultaneously challenge its commonly assumed
shape. The shape at the low mass region of the color-period diagram at the age
of Ru147 favors a recently-proposed model, which requires a third
mass-dependent timescale in addition to the two timescales required by a former
model, suggesting that a third physical process is required to model rotating
stars effectively.
