  We examine the origin of clustercentric gradients in the star formation rates
and colors of rich cluster galaxies within the context of a simple model where
clusters are built through the ongoing accretion of field galaxies. The model
assumes that after galaxies enter the cluster their star formation rates
decline on a timescale of a few Gyrs, the typical gas consumption timescale of
disk galaxies in the field. Such behaviour might be expected if tides and ram
pressure strip off the gaseous envelopes that normally fuel star formation in
spirals over a Hubble time. Combining these timescales with mass accretion
histories derived from N-body simulations of cluster formation in a Lambda-CDM
universe, we reproduce the systematic differences observed in the color
distribution of cluster and field galaxies, as well as the strong suppression
of star formation in cluster galaxies and its dependence on clustercentric
radius. The simulations also indicate that a significant fraction of galaxies
beyond the virial radius of the cluster may have been within the main body of
the cluster in the past, a result that explains naturally why star formation in
the outskirts of clusters (and as far out as two virial radii) is
systematically suppressed relative to the field. The agreement with the data
beyond the cluster virial radius is also improved if we assume that stripping
happens within lower mass systems, before the galaxy is accreted into the main
body of the cluster. We conclude that the star formation rates of cluster
galaxies depend primarily on the time elapsed since their accretion onto
massive virialized systems, and that the cessation of star formation may have
taken place gradually over a few Gyrs.
