DDA 2015 – Dialing the Love Number of Hot Jupiter HAT-P-13b

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.

Peter Buhler (CalTech) (Duncombe award winner)


HAT-P-13b is Jupiter-mass transiting planet in a 0.04 AU orbit around its host star. It has an outer companion, HAT-P-13c, with a minimum mass of 14.7 $M_{Jup}$ in a highly eccentric 1.2 AU orbit. These two companions form an isolated dynamical system with their host star [1]. The nature of this system allows the two bodies to settle into a fixed eccentricity state where the eccentricity of HAT-P-13b is directly related to its oblateness as described by the Love number, $k_2$ [2]. In order to constrain the eccentricity, and therefore $k_2$, of HAT-P-13b, we use the Spitzer Space Telescope to measure the timing of its secondary eclipses at 3.6 and 4.5 μm. We then simultaneously fit our secondary eclipse data in conjunction with previously measured radial velocity and transit data. Finally, we apply the fact that, if the orbits of HAT-P-13b and HAT-P-13c are coplanar, then their apsides are aligned [3]. The apsidal orientation of HAT-P-13c is much better constrained because of its high eccentricity, which helps break the degeneracy between the eccentricity and apsidal orientation in interpreting the measured secondary eclipse time. Our analysis allows us to measure the eccentricity of HAT-P-13b’s orbit with a precision approximately ten times better than that of previously published values, in the coplanar case, and allows us to place the first meaningful constraints on the core mass of HAT-P-13b. [1] Becker & Batygin 2013, ApJ 778, 100 [2] Wu & Goldreich 2002, ApJ 564, 1024 [3] Batygin+ 2009, ApJ 704, L49


  • Trying to understand interior mass distribution ofHAT-P-13b
    • data from Spitzer Space Telescope, 2010
    • measure secondary eclipse timing
    • constrain $e$
    • constrain tidal Love number $k_2$ and interior
  • HAT-P-13: 5 Gyr G-type, 1.2 $M_\odot$, ~5650K
  • 13b: ~0.9$M_J$
  • 13d: driver of the dynamics
  • Secondary eclipse:
    • difference in timing from circular $\rightarrow e$
    • signal ~1% of noise
      • fit jitter model
      • fit eclipse model (Mandel & Agol 2002)
      • bin data after noise removal
    • depth: ~0.05%
    • 3.6 μm: ~24 min early eclipse time
    • secondary eclipse constrains $e \cos \omega_b$
    • RV measurements constrain $e \sin \omega_b$
    • eccentricity result: $e \sim 0.01$ at $3 \sigma$ level
  • tidal Love number:
    • tidal friction extracts energy
    • system quickly finds fixed point under tidal friction
    • fixed point implies aligned apsides and identical precession rates
    • system maintainsconfig over long timescales
      • $k_{2,b} = f(e_b)$
    • apsidal alignment helps constrain $e$ by constraining $e\cos\omega$ and $e\sin\omega$ since $\omega_b=\omega_c$ (if coplanar)
    • apsidal alignment increases precision
    • use to connect $e$ to $k_2$
  • result:
    • ~10$\times$ tighter constraint
    • core mass of 13b has to be very small
    • problems:
      • noncoplanarity
      • EoS not known

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