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Thermal X-ray emission from rotation-powered pulsars is believed to originatefrom localized "hotspots" on the stellar surface occurring where large-scalecurrents from the magnetosphere return to heat the atmosphere. Lightcurvemodeling has primarily been limited to simple models, such as circularantipodal emitting regions with constant temperature. We calculate morerealistic temperature distributions within the polar caps, taking advantage ofrecent advances in magnetospheric theory, and we consider their effect on thepredicted lightcurves. The emitting regions are non-circular even for a puredipole magnetic field, and the inclusion of an aligned magnetic quadrupolemoment introduces a north-south asymmetry. As the aligned quadrupole moment isincreased, one hotspot grows in size before becoming a thin ring surroundingthe star. For the pure dipole case, moving to the more realistic model changesthe lightcurves by $5-10\%$ for millisecond pulsars, helping to quantify thesystematic uncertainty present in current dipolar models. Including thequadrupole gives considerable freedom in generating more complex lightcurves.We explore whether these simple dipole+quadrupole models can account for thequalitative features of the lightcurve of PSR J0437$-$4715.

X-ray Lightcurves from Realistic Polar Cap Models: Inclined Pulsar Magnetospheres and Multipole Fields