The lifetime of FR II sources in groups and clusters: Implications for radio-mode feedback

We determine the maximum lifetime t(max) of 52 Fanaroff-Riley type II (FR II) radio sources found in 26 central group galaxies from cross-correlation of the Berlind SDSS group catalog with the VLA FIRST survey. Mock catalogs of FR II sources were produced to match the selection criteria of FIRST and the redshift distribution of our parent sample, while an analytical model was used to calculate source sizes and luminosities. The maximum lifetime of FR II sources was then determined via a comparison of the observed and model-projected length distributions. We estimate that the average FR II lifetime is 1.5 x 10(7) yr and the duty cycle is similar to 8; 10(8) yr. Degeneracies between tmax and the model parameters - jet power distribution, axial ratio, energy injection index, and ambient density - introduce at most a factor of 2 uncertainty in our lifetime estimate. In addition, we calculate the radio AGN fraction in central group galaxies as a function of several group and host galaxy properties. The lifetime of radio sources recorded here is consistent with the quasar lifetime, even though these FR IIs have substantially sub-Eddington accretion. These results suggest a fiducial time frame for energy injection from AGNs in feedback models. If the morphology of a given extended radio source is set by large-scale environment, while the lifetime is determined by the details of the accretion physics, this FR II lifetime is relevant for all extended radio sources.