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 \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 are still less than 2 per cent .