- Small planet candidate in the
*Kepler*sample - 4.2-hour orbital period
- Discovery paper: Rappaport et al. (2013)
- Used interpolated polytrope models
- Assumed a fluid-like planet filling its Roche lobe
- Measured $0.61 R_\oplus$ radius and $0.44 M_\oplus$ mass
- Inferred an iron-rich composition — exo-Mercury!

- Polytrope models $P \propto \rho^\gamma$ do
**not**accurately represent rocky material, which has nonzero density at zero pressure - We can do better than Rappaport et al. (2013) by using a simple modified polytrope model, $\rho = c P^n + \rho_0$ (Seager et al. 2007)

- We are interested in the full, three-dimensional shape of planets in extreme environments, because the shape influences the transit
- Relevant forces:
- Extreme gravitational force from the star
- Gravitational force of the planet on itself
- Centrifugal force from rotational motion
- There is
**no**analytic way to compute the shape of the planet!

- Hachisu (1986a,b) presents an elegant method for computing self-consistent structures of stars in multiple systems
- Cycles through enthalpy and density calculations until convergence is reached
- Requires computing the gravitational potential at all points (slow)
- Also requires an analytic or numerical approximation to the enthalpy as a function of density

- Important update to the Hachisu method: Adding a point-mass star at fixed distance from the planet
- We have found that the most stable numerical procedure is as follows:
- Assume values for the core-mantle boundary pressure, central pressure, and distance from the star
- Solve for constants, including stellar mass, that meet boundary conditions
- Update density, compute potential, and loop

This method is “backwards” — we find the physical, measured parameters (radius, mass, etc.) only at the end of the simulation