We study the star formation rates (SFRs) of galaxies as a function of local galaxy density at 0.6<z<0.9. We used a low-dispersion prism in IMACS on the 6.5-m Baade (Magellan I) telescope to obtain spectra and measured redshifts to a precision of sigma_z/(1+z)=1% for galaxies with z<23.3 AB mag. We utilized a stellar mass-limited sample of 977 galaxies above M>1.8x10^{10} Msun to conduct our main analysis. With three different SFR indicators, (1) Spitzer MIPS 24-micron imaging, (2) SED fitting, and (3) [OII]3727 emission, we find the median specific SFR (SSFR) and SFR to decline from the low-density field to the cores of groups and a rich cluster. For the SED and [OII] based SFRs, the decline in SSFR is roughly an order of magnitude while for the MIPS based SFRs, the decline is a factor of ~4. We find approximately the same magnitude of decline in SSFR even after removing the sample of galaxies near the cluster. Galaxies in groups and a cluster at these redshifts therefore have lower star formation (SF) activity than galaxies in the field, as is the case at z~0. We investigated whether the decline in SFR with increasing density is caused by a change in the proportion of quiescent and star forming galaxies (SFGs) or by a decline in the SFRs of SFGs. Using the rest-frame U-V and V-J colors to distinguish quiescent galaxies from SFGs we find the fraction of quiescent galaxies increases from ~32% to 79% from low to high density. In addition, we find the SSFRs of SFGs, selected based on U-V and V-J colors, to decline with increasing density by factors of ~5-6 for the SED and [OII] based SFRs. The MIPS based SSFRs for SFGs decline with a shallower slope. The order of magnitude decline in the SSFR-density relation at 0.6<z<0.9 is therefore driven by both a combination of declining SFRs of SFGs as well as a changing mix of SFGs and quiescent galaxies [ABRIDGED].
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