Extended Main Sequence Turnoffs in Intermediate-Age Star Clusters: A
  Correlation Between Turnoff Width and Early Escape Velocity
  We present color-magnitude diagram analysis of deep Hubble Space Telescope
imaging of a mass-limited sample of 18 intermediate-age (1 - 2 Gyr old) star
clusters in the Magellanic Clouds, including 8 clusters for which new data was
obtained. We find that ${\it all}$ star clusters in our sample feature extended
main sequence turnoff (eMSTO) regions that are wider than can be accounted for
by a simple stellar population (including unresolved binary stars). FWHM widths
of the MSTOs indicate age spreads of 200-550 Myr. We evaluate dynamical
evolution of clusters with and without initial mass segregation. Our main
results are: (1) the fraction of red clump (RC) stars in secondary RCs in eMSTO
clusters scales with the fraction of MSTO stars having pseudo-ages $\leq 1.35$
Gyr; (2) the width of the pseudo-age distributions of eMSTO clusters is
correlated with their central escape velocity $v_{\rm esc}$, both currently and
at an age of 10 Myr. We find that these two results are unlikely to be
reproduced by the effects of interactive binary stars or a range of stellar
rotation velocities. We therefore argue that the eMSTO phenomenon is mainly
caused by extended star formation within the clusters; (3) we find that $v_{\rm
esc} \geq 15$ km/s out to ages of at least 100 Myr for ${\it all}$ clusters
featuring eMSTOs, while $v_{\rm esc} \leq 12$ km/s at all ages for two
lower-mass clusters in the same age range that do ${\it not}$ show eMSTOs. We
argue that eMSTOs only occur for clusters whose early escape velocities are
higher than the wind velocities of stars that provide material from which
second-generation stars can form. The threshold of 12-15 km/s is consistent
with wind velocities of intermediate-mass AGB stars in the literature.
