## 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)

#### Abstract

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

#### Notes

• 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

## DDA 2015 – Measurement of planet masses with transit timing variations 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)

#### Abstract

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.

#### Notes

• 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

## DDA 2015 – Dynamical stability of imaged planetary systems in formation – Applicaon to HL Tau

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.

Daniel Tamayo (U. Toronto)

#### Abstract

A recent ALMA image revealed several concentric gaps in the protoplanetary disk surrounding the young star HL Tau. We consider the hypothesis that these gaps are carved by planets, and present a general framework for understanding the dynamical stability of such systems over typical disk lifetimes, providing estimates for the maximum planetary masses. We argue that the locations of resonances should be significantly shifted in disks as massive as estimated for HL Tau, and that theoretical uncertainties in the exact offset, together with observational errors, imply a large uncertainty in the dynamical state and stability in such disks. An important observational avenue to breaking this degeneracy is to search for eccentric gaps, which could implicate resonantly interacting planets. Unfortunately, massive disks should also induce swift pericenter precession that would smear out any such eccentric features of planetary origin. This motivates pushing toward more typical, less massive disks. For a nominal non-resonant model of the HL Tau system with five planets, we find a maximum mass for the outer three bodies of approximately 2 Neptune masses. In a resonant configuration, these planets can reach at least the mass of Saturn. The inner two planets’ masses are unconstrained by dynamical stability arguments.

#### Notes

• Manyexoplanetary systems are highly eccentric
• Can we back out what the ICs might have been?
• HL Tau
• age ~1 Myr
• Outer gaps are too close to contain giant planets
• but if planet-cleared, must be giants, not smaller
• dynamically unstable for larger planets
• But outer 3 gaps are near 4:3MMR chain
• can put planets there (at least for 1 Myr)
• Solution(?)
• Grow the planets in situ in resonance
• Conclusions
• Giant planets could be possible explanation for the gaps
• Precession from massive disks can significantly alter locations of resonances
• $\phi = \lambda_1 – \lambda_2 – \varpi_{12}$
• $\dot{\phi} = n_1 – n_2 – \dot{\varpi}_{12}$
• Hal Levison: can’t grow planets that fast, so something else must be going on here.

## DDA 2015 – Dynamical Evolution of planets in α Centauri AB

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.

### Session: Dynamical Constraints from Exoplanet Observations I

Billy L. Quarles (NASA Ames Research Center)

#### Abstract

Circumstellar planets within α Centauri AB have been suggested through formation models (Quintana et al. 2002) and recent observations (Demusque et al. 2012). Driven by a new mission concept that will aRempt to directly image Earth-sized planets, ACESat (Belikov et al. 2015), we revisit their possible existence through simulations of orbital stability that are far more comprehensive than were feasible by Wiegert and Holman (1997). We evaluate the stability boundary of Earth-like planets within α Centauri AB and elucidate some of the necessary observational constraints relative to the sky plane to directly image Earth-like planets orbiting either stellar component. We confirm the qualitative results of Wiegert and Holman regarding the approximate size of the regions of stable orbits and find that mean motion resonances with the stellar companion leave an imprint on the limits of orbital stability. Additionally, we discuss the differences in the extent of the imprint when considering both prograde and retrograde motions relative to the binary plane.

#### Notes

• Why $\alpha$Cen?
• solar-like stars separated by 10s of AU
• planet formation
• astrobiology
• Dumusque et al,Demory et al 2015
• 3.2-day planet, ~1.1 $M_E$
• HST: transit observed
• RedoofWiegert & Holman 1997 numerical sims
• 10 Myr, 100 Myr, 1 Gyr
• circular inclined case
• planar eccentric case
• stability @100Myr:
• $a_{max} \sim 2.5$ AU