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I am trying to work through this problem so that I can understand how to convert from pressure values to radius values inside a planetary core in a code. The core has variable density depending on whether the radius is in a solid inner core region or in a liquid outer core.

I was able to obtain two different pressures (those accounting for pressures of Fe-snow in the core). Given Fe-snow is developing in the core, the snow region has a different density than pure Fe or pure FeS. This density changes as the core changes. I am having trouble converting these from pressure values to radius values given the non-uniform density profile in the core.

I tried to break the core up into small discrete shells but was having trouble trying to figure out how to deal with the gravity issue.

Could someone point me in the correct direction?

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  • $\begingroup$ For spherical shells the gravity at a given radius is given by the total mass inside that shell. The problem with numerical calculations for gravity acting on compressible matter is that it has to be self-consistent. The gravity depends on the density function rho(r), but the density function rho(r) also depends on the gravity. In astronomy the problem of hydrostatic equilibrium of compressible bodies is associated with the term "polytrope". I found e.g. web.gps.caltech.edu/classes/ge131/notes2016/Ch9.pdf discussing this. $\endgroup$
    – FlatterMann
    Commented Jan 16 at 21:50
  • $\begingroup$ See e.g. vikdhillon.staff.shef.ac.uk/teaching/phy213/phy213_le.html for the numerical solution of the Lane-Emden equation. I am not an astronomer, so I didn't know that this is what this equation is called. There is something new to learn every day. $\endgroup$
    – FlatterMann
    Commented Jan 16 at 21:58
  • $\begingroup$ Thank you! I'll try looking into these sources. I appreciate the help! $\endgroup$
    – Priya Bose
    Commented Jan 17 at 23:23

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