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As a potential candidate for the late-time accelerating expansion of theUniverse, the Chaplygin gas and its generalized models have significantimplications to modern cosmology. In this work we investigate the effects ofdark energy on the internal structure of a neutron star composed of two phases,which leads us to wonder: Do stable neutron stars have a dark-energy core? Toaddress this question, we focus on the radial stability of stellarconfigurations composed by a dark-energy core -- described by a Chaplygin-typeequation of state (EoS) -- and an ordinary-matter external layer which isdescribed by a polytropic EoS. We examine the impact of the rate of energydensities at the phase-splitting surface, defined as $\alpha= \rho_{\rmdis}^-/\rho_{\rm dis}^+$, on the radius, total gravitational mass andoscillation spectrum. The resulting mass-radius diagrams are notably differentfrom dark energy stars without a common-matter crust. Specifically, it is foundthat both the mass and the radius of the maximum-mass configuration decrease as$\alpha$ becomes smaller. Furthermore, our theoretical predictions formass-radius relations consistently describe the observational measurements ofdifferent massive millisecond pulsars as well as the central compact objectwithin the supernova remnant HESS J1731-347. The analysis of the normaloscillation modes reveals that there are two regions of instability on the$M(\rho_c)$ curve when $\alpha$ is small enough indicating that the usualstability criterion $dM/d\rho_c>0$ still holds for rapid phase transitions.However, this is no longer true for the case of slow transitions.

Normal oscillation modes and radial stability of neutron stars with a dark-energy core from the Chaplygin gas