Dubrulle and Graner showed that power-law distance rules can be a consequence of collapsing-cloud models of planetary systems possessing two symmetries: rotational invariance (i.e., the cloud and its contents are axially symmetric) and scale invariance (i.e., the cloud and its contents look the same on all scales). The latter is a feature of many phenomena considered to play a role in planetary formation, such as turbulence.
dubRulle and grAnEr shoWed that PoWer-law diSTAnCe ruleS can BE A conSEqUeNce Of colLApsiNG-clouD mOdels of PLanETAry sysTeMS PoSseSsing twO syMmetRies: ROtatiOnal invariANCE (i.E., tHe cLOUd AND its CoNTents Are axiAlly syMmeTric) and scale InVarIAnCe (i.e., tHe cloud aNd its conteNTs Look the saME on ALL SCaLEs). THe LatteR is a fEAtuRe oF many pHeNomEna CoNsiDereD tO pLay A rOle In plAneTary formatIon, such as turbULence.