Tidal Love numbers and moment-Love relations of polytropic stars

Abstract: The physical significance of tidal deformation in astronomical systems has long been known. The recently discovered universal I-Love-Q relations, which connect moment of inertia, quadrupole tidal Love number, and spin-induced quadrupole moment of compact stars, also underscore the special role of tidal deformation in gravitational wave astronomy. Motivated by the observation that such relations also prevail in Newtonian stars and crucially depend on the stiffness of a star, we consider the tidal Love numbers of Newtonian polytropic stars whose stiffness is characterised by a polytropic index $n$. We first perturbatively solve the Lane-Emden equation governing the profile of polytropic stars through the application of the scaled delta expansion method and then formulate perturbation series for the multipolar tidal Love number about the two exactly solvable cases with $n=0$ and $n=1$, respectively. Making use of these two series to form a two-point Pad\'e approximant, we find an approximate expression of the quadrupole tidal Love number, whose error is less than $2.5 \times 10^{-5}$ per cent (0.39 per cent) for $n\in[0,1]$ ($n\in[0,3]$). Similarly, we also determine the mass moments for polytropic stars accurately. Based on these findings, we are able to show that the I-Love-Q relations are in general stationary about the incompressible limit irrespective of the equation of state (EOS) of a star. Moreover, for the I-Love-Q relations, there is a secondary stationary point near $n \approx 0.4444$, thus showing the insensitivity to $n$ for $n\in[0,1]$. Our investigation clearly tracks the universality of the I-Love-Q relations from their validity for stiff stars such as neutron stars to their breakdown for soft stars.