The　sun　gear　in　a　planetary　gearbox　is　prone　to　generate　tooth　crack　due　to　its　much　more　severe　load　conditions　than　other　gears.　The　cracked　sun　gear　will　evolve　into　distinct　tooth　breakages　and　thus　may　deteriorate　the　operation　status　and　even　cause　catastrophic　loss　of　the　whole　transmission　system.　To　avoid　tooth　crack　induced　malfunction　of　the　planetary　gearbox,　the　tooth　crack　of　sun　gear　should　be　accurately　identified　during　the　early　stage.　Therefore,　a　novelty　quantitative　assessment　methodology　for　tooth　crack　damage　is　proposed,　which　is　a　mathematical　model　for　crack　propagation　of　the　sun　gear　in　planetary　gearboxes　based　on　vibration　acceleration　signal.　Firstly,　the　quantitative　relationship　between　the　crack　propagation　parameters　along　the　gear　thickness　and　width　directions　and　the　time-varying　mesh　stiffness　is　analytically　formulated.　The　analytical　time-varying　mesh　stiffness　formulation　is　incorporated　into　a　translational-torsional　dynamic　model　to　reveal　the　effects　of　crack　propagation　on　the　vibration　signal　of　internal　and　external　meshing　pairs　in　the　planetary　gearbox.　Secondly,　a　quantitative　mapping　function　is　established　by　taking　the　cracked　section　area　as　the　variable　and　the　residual　of　the　summed　envelope　spectrum　amplitude　of　the　vibration　acceleration　signal　as　the　assessment　index.　Finally,　a　numerical　simulation　as　well　as　an　experimental　test　is　conducted　and　compared　to　verify　the　correctness　of　the　proposed　assessment　methodology.　We　believe　that　the　established　model　and　the　corresponding　index　provide　a　physically　intuitive　yet　mathematically　quantitative　reference　for　diagnosing　tooth　crack　in　planetary　gearboxes.　©　2023　IOP　Publishing　Ltd.