  Polluted white dwarfs that have accreted planetary material provide a unique
opportunity to probe the geology of exoplanetary systems. However, the nature
of the bodies which pollute white dwarfs is not well understood: are they small
asteroids, minor planets, or even terrestrial planets? We present a novel
method to infer pollutant masses from detections of Ni, Cr and Si. During
core--mantle differentiation, these elements exhibit variable preference for
metal and silicate at different pressures (i.e., object masses), affecting
their abundances in the core and mantle. We model core--mantle differentiation
self-consistently using data from metal--silicate partitioning experiments. We
place statistical constraints on the differentiation pressures, and hence
masses, of bodies which pollute white dwarfs by incorporating this calculation
into a Bayesian framework. We show that Ni observations are best suited to
constraining pressure when pollution is mantle-like, while Cr and Si are better
for core-like pollution. We find 3 systems (WD0449-259, WD1350-162 and
WD2105-820) whose abundances are best explained by the accretion of fragments
of small parent bodies ($<0.2M_\oplus$). For 2 systems (GD61 and WD0446-255),
the best model suggests the accretion of fragments of Earth-sized bodies,
although the observed abundances remain consistent ($<3\sigma$) with the
accretion of undifferentiated material. This suggests that polluted white
dwarfs potentially accrete planetary bodies of a range of masses. However, our
results are subject to inevitable degeneracies and limitations given current
data. To constrain pressure more confidently, we require serendipitous
observation of (nearly) pure core and/or mantle material.
