Tidally-distorted, Rocky Planets

Ellen M. Price

Öberg Group Meeting

1 February 2019

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

Results

Revised Roche limit for Fe cores

Revised Roche limit for FeS cores

How distorted is KOI 1843.03?

Questions?