Title: A Hard Look at the Neutron Stars and Accretion Disks in 4U 1636-53, GX 17+2, and 4U 1705-44 with $\emph{NuSTAR}$,
Abstract: We present $\emph{NuSTAR}$ observations of neutron star (NS) low-mass X-ray
binaries: 4U 1636-53, GX 17+2, and 4U 1705-44. We observed 4U 1636-53 in the
hard state, with an Eddington fraction, $F_{\mathrm{Edd}}$, of 0.01; GX 17+2
and 4U 1705-44 were in the soft state with fractions of 0.57 and 0.10,
respectively. Each spectrum shows evidence for a relativistically broadened Fe
K$_{\alpha}$ line. Through accretion disk reflection modeling, we constrain the
radius of the inner disk in 4U 1636-53 to be $R_{in}=1.03\pm0.03$ ISCO
(innermost stable circular orbit) assuming a dimensionless spin parameter
$a_{*}=cJ/GM^{2}=0.0$, and $R_{in}=1.08\pm0.06$ ISCO for $a_{*}=0.3$ (errors
quoted at 1 $\sigma$). This value proves to be model independent. For
$a_{*}=0.3$ and $M=1.4\ M_{\odot}$, for example, $1.08\pm0.06$ ISCO translates
to a physical radius of $R=10.8\pm0.6$ km, and the neutron star would have to
be smaller than this radius (other outcomes are possible for allowed spin
parameters and masses). For GX 17+2, $R_{in}=1.00-1.04$ ISCO for $a_{*}=0.0$
and $R_{in}=1.03-1.30$ ISCO for $a_{*}=0.3$. For $a_{*}=0.3$ and $M=1.4\
M_{\odot}$, $R_{in}=1.03-1.30$ ISCO translates to $R=10.3-13.0$ km. The inner
accretion disk in 4U 1705-44 may be truncated just above the stellar surface,
perhaps by a boundary layer or magnetosphere; reflection models give a radius
of 1.46-1.64 ISCO for $a_{*}=0.0$ and 1.69-1.93 ISCO for $a_{*}=0.3$. We
discuss the implications that our results may have on the equation of state of
ultradense, cold matter and our understanding of the innermost accretion flow
onto neutron stars with low surface magnetic fields, and systematic errors
related to the reflection models and spacetime metric around less idealized
neutron stars.