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Gravitational wave astronomy is expected to provide independent constraintson neutron star properties, such as their dense matter equation of state. Thisis possible with the measurements of binary components' tidal deformability,which alter the point-particle gravitational waveforms of the late inspiralphase of neutron-star binaries. Although current gravitational wave detectorsare not sensitive enough for a precise determination of the individual tidaldeformations of the components, a large number of combined observations withfuture detectors will decrease uncertainties in this quantity. Here we providea first study of the tidal deformability effects due to the elasticity/solidityof the crust (hadronic phase) in a hybrid neutron star, as well as theinfluence of a quark-hadronic phase density jump on tidal deformations. Weemploy the framework of nonradial perturbations with zero frequency and studyhadronic phases presenting elastic aspects when perturbed (with the shearmodulus approximately $1\%$ of the pressure). We find that the relative tidaldeformation change in a hybrid star with a perfect-fluid quark phase and ahadronic phase presenting an elastic part is never larger than about $2-4\%$(with respect to a perfect-fluid counterpart). These maximum changes occur whenthe elastic region of a hybrid star is larger than approximately $60\%$ of thestar's radius, which may happen when its quark phase is small and the densityjump is large enough, or even when a hybrid star has an elastic mixed phase.For other cases, the relative tidal deformation changes due to an elastic crustare negligible ($10^{-5}-10^{-1}\%$), therefore unlikely to be measured evenwith third generation detectors. Thus, only when the size of the elastichadronic region of a hybrid star is over half of its radius, the effects ofelasticity could have a noticeable impact on tidal deformations.

Tidal deformations of hybrid stars with sharp phase transitions and elastic crusts