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tokens: fragments. Our analysis finds that of the Ca-rich systems outlined by Hollands et al. ( 2018 ), only one can be attributed to being crust-rich (SDSSJ0744+4649 to 5σ significance), whilst for two other systems (SDSSJ1033+1809 and SDSSJ1351+2645) there is insufficient information available for a crustal model to be the best fit. SDSSJ1055+3725 was no longer included in the sample (see § 2.1 ). The three Fe-rich systems (SDSS1043+3516 (5.2σ), SDSSJ0741+3146 (2.8σ) and SDSSJ0823+0546 (10σ) are indeed found to be core-rich. Both the two two Mg-rich systems, SDSSJ0956+5912 and SDSSJ1158+1845 actually have fairly typical Mg/Fe ratios and due to the combination of sinking time-scales, the best-fitting models suggest that both systems are core-rich to 2σ. 4.3.3 The mass of white dwarf pollutants For those systems in the declining phase, accretion has finished and the mass in the atmosphere, combined with the sinking time-scales can be converted into the total mass of the polluting body. This calculation no longer yields a lower limit as there cannot be any material left in a circumstellar orbit which is yet to accrete. Such systems uniquely probe the mass of the polluting bodies. As shown in Fig. 11 , the bodies polluting the white dwarfs considered here are found to have masses between just half that of Vesta and the mass of the Moon. The bodies polluting white dwarfs are likely to be similar to large Solar System asteroids, as previously concluded in the literature (Farihi et al. 2010a ; Girven et al. 2012 ), although at least two systems have accreted planetary bodies approaching the mass of the Moon. Figure 11. The posterior distributions for the total mass accreted on to the thirteen polluted white dwarfs whose abundances suggest that they are in the declining phase. During the declining phase, accretion has ceased and therefore, the mass currently in the atmosphere, alongside the sinking time-scales, can be used to calculate the total mass accreted. The masses range from Vesta to the Moon. 4.3.4 Constraints on the accretion event lifetimes The accretion event lifetime for a system can be constrained from the abundances because the duration of accretion affects the probability that the system will be observed in build-up, steady state, or declining phase. The accretion event lifetimes from the 202 systems analysed here suggests that the average accretion events last yr. This result is consistent with the result derived in Girven et al. ( 2012 ) and is consistent with the theoretical predictions of Rafikov ( 2011a ) and Rafikov ( 2011b ), but is in stark contrast to the short accretion event (or disc) lifetimes predicted by Wyatt et al. ( 2014 ) to explain the difference between observed accretion rates of DA and DB white dwarfs. 5 CONCLUSIONS In this work, we present a Bayesian model to explain the abundances of planetary material observed in the atmospheres of some white dwarfs. Our method reproduces the abundance patterns observed in polluted