Acyclic　coloring　problem　is　a　specialized　problem　that　arises　in　the　efficient　computation　of　Hessians.　A　proper　edge　coloring　of　a　graph　G　is　called　acyclic　if　there　is　no　2-colored　cycle　in　G.　The　acyclic　chromatic　index　of　G,　denoted　by　χa′(G),　is　the　least　number　of　colors　in　an　acyclic　edge　coloring　of　G.　Let　G　be　planar　graphs　with　girth　g　and　maximum　degree　Δ.　In　this　paper,　it　is　shown　that　if　g