  Neutron stars are ideal astrophysical sources to probe general relativity due
to their large compactnesses and strong gravitational fields. For example,
binary pulsar and gravitational wave observations have placed stringent bounds
on certain scalar-tensor theories in which a massless scalar field is coupled
to the metric through matter. A remarkable phenomenon of neutron stars in such
scalar-tensor theories is spontaneous scalarization, where a normalized scalar
charge remains order unity even if the matter-scalar coupling vanishes
asymptotically far from the neutron star. While most works on scalarization of
neutron stars focus on numerical analysis, this paper aims to derive accurate
scalar charges analytically. To achieve this, we consider a simple energy
density profile of the Tolman VII form and work in a weak-field expansion. We
solve the modified Tolman-Oppenheimer-Volkoff equations order by order and
apply Pad\'e resummation to account for higher-order effects. We find that our
analytic scalar charges in terms of the stellar compactness beautifully model
those computed numerically. We also find a quasi-universal relation between the
scalar charge and stellar binding energy that is insensitive to the underlying
equations of state. Comparison of analytic scalar charges for Tolman VII and
constant density stars mathematically support this quasi-universal relation.
The analytic results found here provide physically motivated, ready-to-use
accurate expressions for scalar charges.
