  In this paper we study how a Yukawa coupling of the Standard Model fermions
to a light scalar field effects the stellar structure of cold stellar remnants
such as neutron stars. We elucidate the stellar structure phenomenology using a
simple model of a massive scalar coupled to a single dominant fermion with no
other interactions. For a broad scalar mass range ($10^{-10}\,\mathrm{eV}\ll
m_\phi\ll10^3\,\mathrm{eV}$ for neutron stars) we show that the
equation-of-state and stellar structure depends only the effective coupling
$\mathfrak{g}=\frac{g_f\,m_f}{m_\phi}$, where $g_f$ is the Yukawa coupling,
$m_f$ the fermion mass and $m_\phi$ is the scalar kinematic mass at nuclear
densities. If $\mathfrak{g}>\mathcal{O}(1)$ the Yukawa coupled matter exhibits
various anomalous behaviors including hydrodynamic instability, negative
pressure, distinct phases (soft and hard) of matter with sharp phase boundaries
between them and with vacuum. These anomalies can lead to stars consisting of
only soft, only hard or hybrid of soft and hard matter. These stars can have
varying signs of the slope of the mass-radius relation, anomalously large and
small masses, gaps in allowed radii, multiple radii for the same mass, very
thin crusts and radiate anomalously large amounts of energy when they form (in
the form of neutrinos for neutron stars). To the extent that these anomalies
have not and/or will not be observed limits the effective coupling to
$\mathfrak{g}<\mathcal{O}(1)$. We argue this phenomenology is generic to
realistic models of stars with Yukawa coupled matter.
