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Gravitational quadrupole deformation and the tidal deformability for stellar systems: (The number of) Love for undergraduates

This article is intended for undergraduate students with the aim to provide a pedagogical introduction to the physics of stellar tidal deformations. The spherically symmetric shape of any star is deformed via rotation around an arbitrary axis or by the presence of an external tidal field. We compute the ellipticity of such a stellar object and show that rotation can be treated analogously to tidal effects caused by an external field, which induces a quadrupole moment. The detection of gravitational waves from a binary neutron star merger in 2017 set constraints on the tidal deformability parameter and on equations of state for compact stars. We derive the corresponding formalism classically and show that the Newtonian limit taken via general relativity is justified. The compressibility and the compactness of matter are discussed classically and compared to the relativistic approach. We find that relativistically the main influence on tidal deformability is related to the star's compactness.