How to calculate the thickness of the ice crust on a frozen ocean planet/moon?

Question

How do I calculate the thickness of the upper ice layer on cold ocean worlds like Europa, Enceladus, Ganymede, ...? I'm asking this for a programm I'm currently writing.

Given/Known is: mass, radius, average density of metal (assume Fe) core, rock (assume MgSiO3) layer, ice (assume H2O) layer; internal heat from tidal heating, radioactive decay and residual formation heat; external heating from solar radiation (assume equal constant heating distributed over the entire surface or ignore it if it is irrelevant/minuscule); pressure of the atmosphere above (assume pure N2 with variable pressure or no atmosphere)

Wanted is: thickness frozen (crust) and molten layer

Bonus stuff: If easy adaptation of the formula for the calculation of the rocky layers crust thickness is possible I would appreciate it. Should it be possible, an adaptation to find the rocky crust thickness and the ice crust thickness on wolds where both are needed would be helpful.

Thanks for any answer and time spent of solving this.

Answer 1

Based on assumptions, if you are ok with approx solution then you can solve heat transfer equation in steady state.

Heat transfer equation

Ignoring the time - long time has passed to attain equilibrium.

Heat flux assuming outward and hence internal ocean would be sum of all internal heating mechanism.

Easiest would be if somehow you can come up with surface temperature value. Then you can solve conduction equation through ice for temperature and weight for pressure and just see at what temperature and pressure ice will go into liquid. Simply asking what is ice layer thickness to maintain a internal heat flux and given surface temperature.

You need one boundary condition. Other way would be little harder, starting at centre with outward heatflux. You need to know both phase and composition since density and cp both changes with these conditions.