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Under suitable scaling, the structure of self-gravitating polytropes isdescribed by the standard Lane-Emden equation (LEE), which is characterised bythe 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 scaledLEE (SLEE) where polytropes with different polytropic indices all share acommon scaled radius. The SLEE is then solved perturbatively as an eigenvalueproblem. Analytical approximants of the polytrope function, the radius and themass of polytropes as a function of $n$ are derived. The approximant of thepolytrope function is well-defined and uniformly accurate from the origin downto the surface of a polytrope. The percentage errors of the radius and the massare bounded by $8.1 \times 10^{-7}$ per cent and $8.5 \times 10^{-5}$ per cent,respectively, for $n\in[0,1]$. Even for $n\in[1,5)$, both percentage errors arestill less than $2$ per cent.

Perturbative solution to the Lane-Emden equation: An eigenvalue approach