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}