  We report on Nuclear Spectroscopic Telescope Array (NuSTAR) hard X-ray
observations of the young rotation-powered radio pulsar PSR B1509$-$58 in the
supernova remnant MSH 15$-$52. We confirm the previously reported curvature in
the hard X-ray spectrum, showing that a log parabolic model provides a
statistically superior fit to the spectrum compared with the standard power
law. The log parabolic model describes the NuSTAR data, as well as previously
published gamma-ray data obtained with COMPTEL and AGILE, all together spanning
3 keV through 500 MeV. Our spectral modelling allows us to constrain the peak
of the broadband high energy spectrum to be at 2.6$\pm$0.8 MeV, an improvement
of nearly an order of magnitude in precision over previous measurements. In
addition, we calculate NuSTAR spectra in 26 pulse phase bins and confirm
previously reported variations of photon indices with phase. Finally, we
measure the pulsed fraction of PSR B1509$-$58 in the hard X-ray energy band for
the first time. Using the energy resolved pulsed fraction results, we estimate
that the pulsar's off-pulse emission has a photon index value between 1.26 and
1.96. Our results support a model in which the pulsar's lack of GeV emission is
due to viewing geometry, with the X-rays originating from synchrotron emission
from secondary pairs in the magnetosphere.

High Energy Astrophysical Phenomena