Abstract
Though individual stellar parameters of compact stars usually demonstrate obvious dependence on the equation of state (EOS), EOS-insensitive universal formulas relating these parameters remarkably exist. In the present paper, we explore the interrelationship between two such formulas, namely the $f$-$I$ relation connecting the $f$-mode quadrupole oscillation frequency $\omega_2$ and the moment of inertia $I$, and the $I$-Love-$Q$ relations relating $I$, the quadrupole tidal deformability $\lambda_2$, and the quadrupole moment $Q$, which have been proposed by Lau, Leung, and Lin [Astrophys. J. {\bf 714}, 1234 (2010)] and Yagi and Yunes [Science {\bf 341}, 365 (2013)], respectively. A relativistic universal relation between $\omega_l$ and $\lambda_l$ with the same angular momentum $l=2,3,\ldots$, the so-called "diagonal $f$-Love relation" that holds for realistic compact stars and stiff polytropic stars, is unveiled here. An in-depth investigation in the Newtonian limit is further carried out to pinpoint its underlying physical mechanism and hence leads to a unified $f$-$I$-Love relation. We reach the conclusion that these EOS-insensitive formulas stem from a common physical origin --- compact stars can be considered as quasiincompressible when they react to slow time variations introduced by $f$-mode oscillations, tidal forces and rotations.
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