Implications of a spatially resolved main sequence for the size evolution of star forming galaxies

Two currently debated problems in galaxy evolution, the fundamentally local or global nature of main sequence star formation and evolution mass-size relation forming galaxies (SFGs), are shown to be intimately related each other. As a preliminary step, growth function $g$ is defined, which quantifies differential change half-mass radius per unit increase stellar mass ($g = d \log R_{1/2}/d M_\star$) due formation. A general derivation shows that $g K \Delta(sSFR)/sSFR$, meaning proportional relative difference specific rate between outer inner half galaxy, with $K$ dimensionless structural factor for handy expressions provided. an application, it obeying also obey, good approximation, \simeq \gamma n$, where $\gamma$ slope normalized ($sSFR \propto \Sigma_\star^{-\gamma}$) $n$ Sersic index. An exact expression Quantitatively, consistent SFGs growing along stationary relation, but inconsistent continuation at $z=0$ evolutionary laws derived higher $z$. This demonstrates either not local, has converged equilibrium state some finite time past, both.