  Recent observations of neutron stars (NS) with radio, X-rays and
gravitational waves have begun to constrain the equation of state (EoS) for
nuclear matter beyond the nuclear saturation density. To one's surprise, there
exist approximate universal relations connecting certain bulk properties of NSs
that are insensitive to the underlying EoS and having important applications on
probing fundamental physics including nuclear and gravitational physics. To
date, analytic works on universal relations for realistic NSs are lacking,
which may lead to a better understanding of the origin of the universality.
Here, we focus on the universal relations between the compactness(C), moment of
inertia (I), and tidal deformability(related to the Love number), and derive
analytic, approximate I-Love-C relations. To achieve this, we construct
slowly-rotating/ tidally-deformed NS solutions analytically starting from an
extended Tolman VII model that accurately describes non-rotating realistic NSs,
allowing us to extract the moment of inertia and the tidal deformability on top
of the compactness. We solve the field equations analytically by expanding them
about the Newtonian limit and keeping up to 6th order in the stellar
compactness. Based on these analytic solutions, we can mathematically
demonstrate the O(10%) EoS variation in the I-C and Love-C relations and the
O(1%) variation in the I-Love relation that have previously been found
numerically. Our new analytic relations agree more accurately with numerical
results for realistic NSs(especially the I-C and Love-C ones) than the analytic
relations for constant density stars derived in previous work. Based on these
analytic findings, we attribute a possible origin of the universality for the
I-C and Love-C relations to the fact that the energy density of realistic NSs
can be approximated as a quadratic function, as is the case for the Tolman VII
solution.
