Linear stability of the elliptic relative equilibrium with $ (1 + 7) $-gon central configuration in planar $ N $-body problem

The $ (1+n) $-gon elliptic relative equilibrium (ERE for short) is the planar central configuration solution consisting of $ n $ unit masses at the vertices of a regular $ n $-gon with a body of mass $ m $ at the center, and each particle moves on a Keplerian orbit with a common eccentricity $ e\in [0,1) $. Maxwell first considered this model in his study on the stability of the Saturn's rings. Moeckel [10] proves that for $ e = 0 $, the $ (1+n) $-gon is linearly stable for sufficiently large $ m $ for $ n\geq7 $. A natural question is that whether Moeckel's stability result holds for $ e>0 $. In the recent paper [2], Hu, Long and Ou give an affirmative answer for $ n\geq8 $, but the case $ n = 7 $ is difficult and still open. In this paper, we show that it is also true for $ n = 7 $, that is the $ (1+7) $-gon ERE is still linearly stable for any $ e\in(0,1) $ when $ m $ is large enough.