DDA 2015 – Measurement of planet masses with transit timing variations due to synodic “chopping” effects

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Katherine Deck (CalTech)


Gravitational interactions between planets in transiting exoplanetary systems lead to variations in the times of transit (TTVs) that are diagnostic of the planetary masses and the dynamical state of the system. I will present analytic formulae for TTVs which can be applied to pairs of planets on nearly circular orbits which are not caught in a mean motion resonance. For a number of Kepler systems with TTVs, I will show that synodic “chopping” contributions to the TTVs can be used to uniquely measure the masses of planets without full dynamical analyses involving direct integration of the equations of motion. This demonstrates how mass measurements from TTVs may primarily arise from an observable chopping signal. I will also explain our extension of these formulae to first order in eccentricity, which allows us to apply the formulae to pairs of planets closer to mean motion resonances and with larger eccentricities.


  • Still don’t know much about formation and evolution of exoplanet systems
  • Use TTVs to measure planet masses?
  • e.g. Kepler 36
    • TTV amplitude ~2 hr p-p
    • mass constraints: Carter et al. 2012
    • composition constraints: Rogers et al. in prep
  • TTVs largest nearMMRs
    • Lithwick et al. 2012
    • $\dfrac{\delta t}{P} \propto \dfrac{M_{pert}}{M_{star}}$
    • short-period components and res components
  • Derive formula for synodicTTVs
    • sums of sinusoids, linear in mass ratios and periods [duh]
    • constrain masses
      • measure harmonic component period $\rightarrow$ mass ratio
    • Near first order MMR, degeneracy between mass and eccentricity breaks
    • Schmidt et al. 2015
    • Can use to set upper bounds, even in absence of TTVs
  • see also Algol et al. 2005, Nesvorny & Vokrouhlicky 2014