# Airy Isostacy

Local Airy isostacy requires that the weight of every column down at the compensation depth is equal to the reference column. If we stretch the crust by a factor $\beta$ then we can write the following balance between reference column and the stretched configuration where we have less crust, some infill on top, and modify the amount of mantle material. ${\rho _c}{h_c} + {\rho _m}\left( {{h_{tot}} - {h_c}} \right) = {\rho _c}{{{h_c}} \over \beta } + {\rho _m}\left( {{h_{tot}} - {{{h_c}} \over \beta } - {h_i}} \right) + {\rho _i}{h_i}$

Rearranging yields the following expression for the thickness of the infill. ${h_i} = {{\left( {{\rho _m} - {\rho _c}} \right)} \over {\left( {{\rho _m} - {\rho _i}} \right)}}\left( {1 - {1 \over \beta }} \right){h_c}$

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