  We present results from a 5-month photometric survey for stellar rotation
periods combined with a 4-year radial-velocity survey for membership and
binarity in the 220Myr open cluster M34. We report surface rotation periods for
120 stars, 83 of which are late-type cluster members. A comparison to previous
work serves to illustrate the importance of high cadence long baseline
photometric observations and membership information. The new M34 periods are
less biased against slow rotation and cleaned for non-members. The rotation
periods of the cluster members span more than an order of magnitude from 0.5
day up to 11.5 days, and trace two distinct rotational sequences - fast (C) and
moderate-to-slow (I) - in the color-period diagram. The sequences represent two
different states in the rotational evolution of the late-type cluster members.
We use the color-period diagrams for M34 and for younger and older clusters to
estimate the timescale for the transition from the C to the I sequence and find
~<150Myr, ~150-300Myr, and ~300-600Myr for G, early-mid K, and late K dwarfs,
respectively. The small number of stars in the gap between C and I suggest a
quick transition. We estimate a lower limit on the maximum spin-down rate
(dP/dt) during this transition to be ~0.06 days/Myr and ~0.08 days/Myr for
early and late K dwarfs, respectively. We compare the I sequence rotation
periods in M34 and the Hyades for G and K dwarfs and find that K dwarfs spin
down slower than the Skumanich rate. We determine a gyrochronology age of
240Myr for M34. We measure the effect of cluster age uncertainties on the
gyrochronology age for M34 and find the resulting error to be consistent with
the error estimate for the technique. We use the M34 I sequence to redetermine
the coefficients in the expression for rotational dependence on color used in
gyrochronology (abridged).
