Behavior of Fr\'echet mean and Central Limit Theorems on spheres

We compute higher derivatives of the Fr\'{e}chet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i)~a practical condition to test if the mode of the symmetric distribution is a local Fr\'{e}chet mean; (ii)~a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii)~an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least~$4$.