  The structure and composition of the crust of neutron stars plays an
important role in their thermal and magnetic evolution, hence in setting their
observational properties. One way to study the crust properties is to measure
how it cools after it has been heated during an accretion outburst in a
low-mass X-ray binary (LMXB). Such studies have shown that there is a
tantalizing source of heat, of currently unknown origin, that is located in the
outer layers of the crust and has a strength that varies between different
sources and different outbursts. With the aim of understanding the mechanism
behind this "shallow heating", we present Chandra and Swift observations of the
neutron star LMXB Aql X-1, obtained after its bright 2016 outburst. We find
that the neutron star temperature was initially much lower, and started to
decrease at much later time, than observed after the 2013 outburst of the
source, despite the fact that the properties of the two outbursts were very
similar. Comparing our data to thermal evolution simulations, we infer that the
depth and magnitude of shallow heating must have been much larger during the
2016 outburst than during the 2013 one. This implies that basic neutron star
parameters that do not change between outbursts, do not play a strong role in
shallow heating. Furthermore, it suggests that outbursts with a similar
accretion morphology can give rise to very different shallow heating. We also
discuss alternative explanations for the difference in quiescent evolution
after the 2016 outburst.
