In a previous study, we analysed the spectra of 230 cool ($T_mathrm{eff}$ < 9000 K) white dwarfs exhibiting strong metal contamination, measuring abundances for Ca, Mg, Fe and in some cases Na, Cr, Ti, or Ni. Here we interpret these abundances in terms of the accretion of debris from extrasolar planetesimals, and infer parent body compositions ranging from crust-like (rich in Ca and Ti) to core-like (rich in Fe and Ni). In particular, two white dwarfs, SDSSJ0823+0546 and SDSSJ0741+3146, which show log[Fe/Ca] > 1.9 dex, and Fe to Ni ratios similar to the bulk Earth, have accreted by far the most core-like exoplanetesimals discovered to date. With cooling ages in the range 1-8 Gyr, these white dwarfs are among the oldest stellar remnants in the Milky Way, making it possible to probe the long-term evolution of their ancient planetary systems. From the decrease in maximum abundances as a function of cooling age, we find evidence that the arrival rate of material on to the white dwarfs decreases by 3 orders of magnitude over a $simeq$6.5 Gyr span in white dwarf cooling ages, indicating that the mass-reservoirs of post-main sequence planetary systems are depleted on a $simeq$1 Gyr e-folding time-scale. Finally, we find that two white dwarfs in our sample are members of wide binaries, and both exhibit atypically high abundances, thus providing strong evidence that distant binary companions can dynamically perturb white dwarf planetary systems.
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