Under suitable scaling, the structure of self-gravitating polytropes is described by the standard Lane-Emden equation (LEE), which is characterised by the polytropic index n. Here we use the known exact solutions of the LEE at n = 0 and 1 to solve the equation perturbatively. We first introduce a scaled LEE (SLEE) where polytropes with different polytropic indices all share a common scaled radius. The SLEE is then solved perturbatively as an eigenvalue problem. Analytical approximants of the polytrope function, the radius and the mass of polytropes as a function of n are derived. The approximant of the polytrope function is well-defined and uniformly accurate from the origin down to the surface of a polytrope. The percentage errors of the radius and the mass are bounded by 8.1 × 10 −7 per cent and 8.5 × 10 −5 per cent, respectively, for n ∈ [0, 1]. Even for n ∈ [1, 5), both percentage errors are still less than 2 per cent.

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 Padé approximant, we find an approximate expression of the quadrupole tidal Love number, whose error is less than 2.5 × 10 −5 per cent (0.39 per cent) for n ∈ [0, 1] (n ∈ [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 ≈ 0.4444, thus showing the insensitivity to n for n ∈ [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.

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