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We give a definition of tropical orbit spaces and their morphisms. We showthat, under certain conditions, the weighted number of preimages of a point inthe target of such a morphism does not depend on the choice of this point. Weequip the moduli spaces of elliptic tropical curves with a structure oftropical orbit space and, using our results on tropical orbit spaces, simplifythe known proof of the fact that the weighted number of plane elliptic tropicalcurves of degree d with fixed j-invariant which pass through 3d-1 points ingeneral position in $\RR^2$ is independent of the choice of a configuration ofpoints.

Tropical orbit spaces and the moduli spaces of elliptic tropical curves