# Introduction

## Representative case: KOI 1843.03

• 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!

## The problem with rocky materials

• 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)

## A circular problem

• 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!

# Methods

## Hachisu method for multiple stars

• 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

## Applying Hachisu method

• 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

## Applying Hachisu method: Caveat

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