Star Cluster Formation from Turbulent Clumps. II. Gradual Star Cluster
  Formation
  We investigate the dynamical evolution of star clusters during their
formation, assuming that they are born from a turbulent starless clump of a
given mass that is embedded within a parent self-gravitating molecular cloud
characterized by a particular mass surface density. In contrast to the standard
practice of most $N$-body studies, we do not assume that all stars are formed
at once. Rather, we explore the effects of different star formation rates on
the global structure and evolution of young embedded star clusters, also
considering various primordial binary fractions and mass segregation levels.
Our fiducial clumps studied in this paper have initial masses of $M_{\rm cl} =
3000\,M_\odot$, are embedded in ambient cloud environments of $\Sigma_{\rm
cloud} = 0.1$ and 1 g cm$^{-2}$, and gradually form stars with an overall
efficiency of 50% until the gas is exhausted. We investigate star formation
efficiencies per free-fall time in the range $\epsilon_{\rm ff}=0.01$ to 1, and
also compare to the instantaneous case ($\epsilon_{\rm ff}=\infty$) of Paper I.
We show that most of the interesting dynamical processes that determine the
future of the cluster, happen during the early formation phase. In particular,
the ejected stellar population is sensitive to the duration of star cluster
formation: for example, clusters with longer formation times produce more
runaway stars, since these clusters remain in a dense state for longer, thus
favouring occurrence of dynamical ejections. We also show that the presence of
radial age gradients in star clusters depends sensitively on the star formation
efficiency per free fall time, with observed values being matched best by our
slowest forming clusters with $\epsilon_{\rm ff}\lesssim0.03$.
