Aql X-1 is a prolific transient neutron star low-mass X-ray binary that exhibits an accretion outburst approximately once every year. Whether the thermal X-rays detected in intervening quiescent episodes are the result of cooling of the neutron star or due to continued low-level accretion remains unclear. In this work we use Swift data obtained after the long and bright 2011 and 2013 outbursts, as well as the short and faint 2015 outburst, to investigate the hypothesis that cooling of the accretion-heated neutron star crust dominates the quiescent thermal emission in Aql X-1. We demonstrate that the X-ray light curves and measured neutron star surface temperatures are consistent with the expectations of the crust cooling paradigm. By using a thermal evolution code, we find that ≃1.2 − 3.2 MeV nucleon −1 of shallow heat release describes the observational data well, depending on the assumed mass-accretion rate and temperature of the stellar core. We find no evidence for varying strengths of this shallow heating after different outbursts, but this could be due to limitations of the data. We argue that monitoring Aql X-1 for up to ≃1 year after future outbursts can be a powerful tool to break model degeneracies and solve open questions about the magnitude, depth and origin of shallow heating in neutron star crusts.

We establish an explicit correspondence between two-dimensional projective structures admitting a projective vector field, and a class of solutions to the SU (∞) Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini-twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle T * N of any projective structure (N, [∇]). We show that there is a canonical Einstein of metric on an R * -bundle over T * N , with a connection whose curvature is the pull-back of the natural symplectic structure from T * N .

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi-Hanson instanton admits a projective compactification which is non-metric, and that a metric cone over any (pseudo)-Riemannian manifolds admits a metric projective compactification. We construct a para-c-projective compactification of neutral signature Einstein metrics canonically defined on certain rank-n affine bundles M over n-dimensional manifolds endowed with projective structures.

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