  The gravitational properties of astrophysical objects depend sensitively on
their internal structure. In Newtonian theory, the gravitational potential of a
rotating star can be fully described by an infinite number of multipole moments
of its mass distribution. Recently, this infinite number of moments for
uniformly-rotating stars were shown semi-analytically to be expressible in
terms of just the first three: the mass, the spin, and the quadrupole moment of
the star. The relations between the various lower multipole moments were
additionally shown to depend weakly on the equation of state, when considering
neutron stars and assuming single polytropic equations of state. Here we extend
this result in two ways. First, we show that the universality also holds for
realistic equations of state, thus relaxing the need to use single polytropes.
Second, we derive purely analytical universal relations by perturbing the
equations of structure about an $n=0$ polytrope that reproduce semi-analytic
results to $\mathcal{O}(1\%)$. We also find that the linear-order perturbation
vanishes in some cases, which provides further evidence and a deeper
understanding of the universality.
