## DDA 2015 – Saturn’s F ring – A decade of perturbations and collisions

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.

Carl D Murray (Queen Mary University of London)

#### Abstract

We present an overview of the gravitational and collisional processes at work in Saturn’s F ring deduced from images obtained by the Imaging Science Subsystem (ISS) on the Cassini spacecraft since 2004. The moon Prometheus exerts the dominant gravitational perturbation on the ring. As well as creating the observed periodic tistreamer-channelti structures in the ring, there is evidence that Prometheus also causes the formation and orbital evolution of clumps that can, in turn, perturb local ring particles. We show how Prometheus’ effect can be understood in terms of a simple epicyclic model. Jets of material seen emanating from the F ring are produced when objects orbiting nearby collide with material in the core. We show that there are fundamental differences between the larger and smaller jets even though both are caused by collisions. A comparison between the morphology seen in ISS observations and the results of simulations suggests that both the impactors and the core material are in the form of aggregates of material. We present the results of a study of one particular sheared jet and its associated clumps over a two-month interval in early 2008, deriving orbits for the clumps and showing how they change as they encounter Prometheus.

#### Notes

• F ring:
• 16,150 images
• FWHM is $16 \pm 9$km
• eccentric
• Clear evidence of grav. effect of Prometheus, collisions with smaller bodies
• Jets & strands are the result of collisions
• “streamer channels” from both Prometheus and Pandora
• Evidence for embedded eccentric objects
• “Fan” structures (Beurle et al. 2010)
• Evidence for collisions in F ring core
• “mini-jets”
• $\Delta a = a \Delta e$
• ~1 m/s impacts
• Appearsto be clusters of objects colliding with clusters of objects
• from collisional simulations
• best agreement with observations
• Clumps in strands
• $\Delta a > a \Delta e$
• $\rightarrow$ suppression of $\Delta e$ by apse alignment?

## DDA 2015 – Saturn Ring Seismology – How ring dynamics reveal the internal structure of the planet

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.

Jim Fuller (CalTech)

#### Abstract

Seismology allows for direct observational constraints on the interior structures of stars and planets. Recent observations of Saturn’s ring system have revealed the presence of density waves within the rings excited by oscillation modes within Saturn, allowing for precise measurements of a limited set of the planet’s mode frequencies. Additional ring waves are created at Lindblad resonances with density inhomogeneities in the planet, allowing for measurements of internal differential rotation. I construct interior structure models of Saturn, compute the corresponding mode frequencies, and compare them with the observed mode frequencies. The observed modes, some of which are finely split in frequency, can only be reproduced in models containing gravity modes that propagate in a stably stratified region of the planet. The stable stratification must exist deep within the planet near the large density gradients between the core and envelope. The planetary oscillation modes may in turn influence the evolution of the rings by depositing angular momentum at Lindblad resonances. In particular, the Maxwell gap is likely opened due to a resonance with Saturn’s $l=m=2$ fundamental mode.

#### Notes

• Internal structures of giant planets  poorly constrained
• Haven’t been able to do seismology…until Cassini @ Saturn
• Consider just the C ring spiral density waves.
• Pattern speed & pattern number: diagnostics.
• $m(\Omega – \Omega_p) = \kappa$
• $\Omega_p = -\sigma_\alpha/m$
• Very tiny perturbations cause these density waves.
• mode periods: ~hours
• mode amplitudes (inside Saturn): ~1 m
• Planet model:
• inner core, stable outer core, g-mode cavity, f-mode cavity, convective outer envelope
• resonances with $l=m$ f modes
• unexpected: frequency fine-splitting! (Maxwell Gap)
• new: implies stable stratification region
• generates families of g modes ($2^{nd}$ order)
• fast rotation $\rightarrow$ mode mixing
• mess!
• analogous to hydrogen atom in strong electromagnetic field
• strongest mixing near f-mode freq’s
• $\rightarrow$ lots of modes generated in the rings that are currently to “faint” to see
• Conclusions:
• Evidence for stable stratification (non-adiabatic interior) of Saturn
• Helium rain, core erosion, both, something else?
• Missing ingredient: differential rotation?
• Some evidence for density inhomogeneities within Saturn

## DDA 2015 – The Titan -1:0 bending wave in Saturn’s C ring

Philip D. Nicholson (Cornell)

#### Abstract

In 1988 Rosen & Lissauer identified an unusual wavelike feature in Saturn’s inner C ring as a bending wave driven by a nodal resonance with Titan (Science 241, 690) This is sometimes referred to as the -1:0 resonance since it occurs where the local nodal regression rate is approximately equal to $-n_T$, where $n_T = 22.577$ deg/day is Titan’s orbital mean motion. We have used a series of 44 stellar occultation profiles of this wave observed by the Cassini VIMS instrument to test their hypothesis. We find that, as predicted, this wave is an outward-propagating m=1 spiral with a leading orientation and a retrograde paRern speed equal to $-n_T$. Applying the standard linear dispersion relation (Shu 1984), we find a mean background surface mass density of $0.7\ g/cm^2$, similar to previous estimates for the inner C ring.

But the most intriguing feature of the wave is a narrow, incomplete gap which lies ~7 km outside the resonance. This gap varies noticeably in width and is seen in roughly 3/4 of the occultation profiles, appearing to rotate with the wave in a retrograde direction. We have developed a simple, kinematical model which accounts for the observations and consists of a continuous but very narrow gap (radial width = 0.5 km), the edges of which are vertically distorted by the propagating bending wave as it crosses the region. Differences in viewing geometry then largely account for the apparent width variations. We find a vertical amplitude of 3.8 km for the inner edge and 1.2 km for the outer edge, with nodes misaligned by ~110 deg. Moreover, both edges of the gap are slightly eccentric, with pericenters aligned with Titan, suggesting that the eccentricities are forced by the nearby Titan apsidal resonance. We hypothesize that the gap forms because the local slope of the ring becomes so great that nonlinear effects result in the physical disruption of the ring within the first wavelength of the bending wave. However, the vertical relief on the gap edges is ~10 times the predicted amplitude of the bending wave, so this story may be incomplete.

#### Notes

• Stellar occultation with VIMS
• Small region of interest:
• resolution ~2 km
• bending wave
• nodal precession = rate of Titan’s motion: -1:0 MMR
• Colombo ringlet (in Colombo Gap)
• Titan 1:0 MMR
• pericenter of ring locked to position of Titan
• That -1:0 bending wave:
• wave amplitude varies occultation to occultation
• (angle of view)
• resonance location just inside of wave
• episodic appearance of a ~1-5 km gap!
• about half the time, there’s a density peak instead of a gap!
• variation appears to be due to viewing geometry
• $\rightarrow$ leading spiral density wave
• Adjust for viewing geometry, and regular pattern emerges
• gap features associated with bending wave
• $W(\lambda,t) = W_0\, – \Delta z(\theta) \cos (\lambda\, – \lambda_{star})/\tan (B_{star})$
• (B = star-ring plane angle)
• pretty decent fit to peaklets & gaplets
• Allow each gap edge to be eccentric:
• 10 parameters to fit
• eccentric at ~1 km amplitude
• vertical displacements: about $110^{\circ}$ out of phase
• So what’s going on?
• bending wave propagating outward
• gap forms when local slope of wave first exceeds unity
• beyond gap, wave re-establishes itself with a smaller amplitude
• Don’t know why
• gap is probably a nonlinear response of ring to the steep local slope, leading to vertical ‘tearing’ of the ring surface

## DDA 2015 – Irregular Structure in Saturn’s Huygens Ringlet

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: Ring Dynamics

Joseph Spitale (PSI)

#### Abstract

Saturn’s Huygens ringlet is a narrow eccentric ringlet located ~250 km exterior to the outer edge of Saturn’s B ring. Based on about 5 years of Cassini observations, the ringlet contains multiple wavenumber-2 patterns superimposed on its edges (Spitale et al., in prep). Additional higher-order modes may be present, but a few km of radial variation on the edge of the ringlet likely cannot be explained by normal modes with pattern speeds appropriate for those modes. Instead, there is an irregular component to the ringlet’s shape that moves at a speed near the local Keplerian rate and is recognizable for multiple years. The pattern sometimes appears inverted, suggesting that the shape arises from a perturbation in eccentricity rather than semimajor axis. The synodic period between the inner and outer edges of the ring is ~5 years, so a significant evolution of the pattern would be expected if the shape were driven by multiple embedded perturbers distributed across the ring. The relatively static shape of the pattern may indicate that only perturbers with semimajor axes in a narrow region close to the edges of the ringlet play a role. A better understanding of the effect of embedded bodies on ring edges is needed.

#### Notes

• Broad trend: $m=1$
• Other normal modes present (Spitale & Hahn 2015)
• $r(\theta,t) = a\{\sum_{i=0}^n e_i \cos m_i \left[\theta\, – \varpi_0^i – \Omega_p^i (t-t_0)\right]\}$
• width-radius relation: $W(r) = \delta a \left[1\, – \left(e + \frac{q}{e}\right)\left(1-\frac{r}{a}\right)\right]$
• Features track embedded massive objects
• Persist for at least 3.5 yr
• Synogic periods much longer than 3.5 yr
• Wakes?
• Wake-like structures originate at two points on the inner edge
• $\rightarrow$ two dominant masses
• eccentricity perturbations clues to dynamiics
• Occupy narrow band near inner edge

## DDA 2015 – The Fate of Debris from a Giant Impact on Mars

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.

David Minton (Arizona State)

#### Abstract

We use published models for the formation of the $\sim1 \times 10$ km Borealis Basin on Mars from a ~2000 km impactor to investigate the fate of ejected debris. We use an n-body integrator to show that debris from this event could have been an important contributor to the cratering history of the Earth, Moon, and Mars well aOer the basin formed. We investigate whether this event could have been responsible for the Late Heavy Bombardment (LHB) on these planets. We show that the giant impact debris model has a number of features that are more favorable for explaining the LHB compared with giant planet instability models, such as the Nice model.

#### Notes

• Craters
• fossil record of small bodies
• previously thought:
• Strom et al. 2005: Heavily cratered terrains of Moon, Mars, Mercurywere dominated byMBAs ejected in a size-dependent way.
• resonant sweeping of asteroid belt
• Gomes et al. 2005: classic Nice model
• Kring & Cohen 2002: impactors had asteroidal geochemistry
• But…
• Nice model only works if Jupiter jumps
• Only ~1-5 percent of simulations produce required jump.
• Cratered terrain evolution model
• Input impactor size & velocity distributions.
• Constraints:
• must reach observed crater density in Lunar highlands
• cannot make more Lunar basins than seen
• Results:
• MBA is not a good model for the Lunar highlands
• So, what was the highlands impactorSFD?
• Size distribution primordial “bump” around ~100 km is missing in the model
• SPH codes: not very good at these scales
• N-body sims:
• Mars sucks as a scatterer.
• Collisional evolution then produces the bump.
• Gets about the right number of basins on Moon and Mars.
• Bodies collect in theHungarias, kind of no matter what.
• Thus, we can’t use Hungarias as a constraint.

## DDA 2015 – Implications of Resonant and Near-Resonant Planetary Systems for Planet Formation

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.

Eric Ford (Penn State)

#### Abstract

Observations of strongly interacting planetary systems in or near a mean motion resonance are unusually sensitive to planet masses and orbital properties, including dynamical properties that can help illuminate planet formation. Having developed a powerful toolbox for translating Doppler and/or transit timing observations into physics parameters, now we are able to characterize the resonant and secular behaviour of several strongly interacting planetary systems. I will present recent results for selected resonant and near-resonant planetary systems and discuss implications for planet formation. In particular, I will address implications for the nature and extent of orbital migration for giant and low-mass planets.

#### Notes

• How didSTIPs form?
• Three strawman models:
• In situ formation: wrong
• Large-scale disk formation: wrong
• Nearly in situ formation plus modest early radial drift
• STIP examples:
• GJ 876: 4 planets
• 55 Cnc: 5 planets, MMR
• … Meh.

## DDA 2015 – The Öpik Approximation and Giant Planet Shielding of the Inner Solar System

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.

William Newman (UCLA)

#### Abstract

Öpik (1976) proposed that close-range gravitational interactions between planetesimal material and planets could be approximated by a two-step integration scheme: (1) while the planetesimal was outside the gravitational sphere of influence of the planet, its orbit would be described by a heliocentric Keplerian orbit; and (2) once its orbit entered the sphere of influence of the planet, its trajectory would then become a planetocentric Keplerian orbit until it exited the sphere of influence and resumed a heliocentric path. This approximation, however, was also limited by the requirement noted by Öpik that the perihelion or aphelion distance of the planetesimal differ from the orbital distance of the planet from the sun. This approximation proved to be a useful tool during early solar system dynamical investigations but this process was often employed as a numerical integration method without checking Öpik’s requirements, as well as establishing whether the orbital passage through the sphere of influence was sufficiently accurate. Öpik’s scheme was used to establish many features of solar system evolution, including the commonly-held belief that the giant planets serve as a shield preventing substantial numbers of planetesimals from entering the inner solar system. Wetherill (1994) in a pioneering work that exploited the Öpik approximation as an integration scheme estimated that present-day Jupiter could prevent 99.9% of planetesimals from entering the inner solar system. Here, we employ high precision first principles calculations of the orbits of swarms of planetesimals emerging from the Jupiter-Saturn, Saturn-Uranus, and Uranus- Neptune zones and have shown (1) the conditions necessary for Öpik’s approximation to be valid fail for a substantial fraction of the planetesimal population during their lifetimes, and (2) approximately 44% of the planetesimal swarm originating in the Jupiter-Saturn zone alone are injected into the inner Solar System while 18% ultimately become Earth-crossers.

#### Notes

• Does Jupiter shield the inner solar system?
• Impact history
• Öpik:
• novel scheme for solar system integrations
• but identified a useful criterion for valid (numerical) results
• exploits near-Keplerian orbits of inner SS
• Öpik’s method:
• Keplerian time step
• Criterion: aphelion or perihelion must be different from mean distance. [um…duh]
• Öpik’s result: 99.95% of outer solar system planetesimals could not have entered inner solar system!
• Newman: let’s check, using an extremelyaccurate numerical integrator.
• counted fraction of particles in each planet-planet outer zone fail Öpik’s criterion
• found: ~20% get through
• thus: Öpik’s criterion gets violated a lot

## DDA 2015 – Tidal Effects in Late Stage Accretion

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: Planet Formation II

Kevin Graves (Purdue)

#### Abstract

We model the effects of tidal dissipation in the late stages of planetary accretion. We investigate the tidal dissipation during close encounters between embryos and nearly-formed planets using a modified version of the N-body integrator SyMBA. We calculate a total energy lost due to tides per close encounter and estimate the change in velocities of the bodies at each encounter. We measure the effects on the dynamics, evolution, and final outcome of the planets. Our initial results show a clear separation between the tidal and non-tidal case for a relatively strong tidal dissipation factor. We compare these results to traditional late stage simulations both with and without fragmentation.

#### Notes

• Overview of late-stage terrestrial planet accretion
• a few dozen embryos
• a few thousand planetesimals
• Morby 2012
• giant plant migration?
• increases AMD of inner solar system
• AMD: Jacobson & Morbidelli 2014
• Tidal effects on planetary embryos
• Lots of heat generation from various processes $\rightarrow$ magma oceans
• Simple model for energy loss during a close encounter (Kaula & Harris 1973): tides
• free parameters: tidal Love numbers, dissipation param
• combine to a “tidal parameter”: $\frac{h_2 (k_2 + 1)}{Q}$
• Tidal effects in an n-body integrator
• no tides vs. strong tides:
• plot: mass concentration (Chambers 2013) vs AMD
• strong tides: higher mass concentration with AMD
• weak tides: inverse

## DDA 2015 – The Formation of Terrestrial Planets from the Direct Accretion of Pebbles

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.

Hal Levison (SwRI)

#### Abstract

Building the terrestrial planets has been a challenge for planeVormation models. In particular, classical theories have been unable to reproduce the small mass of Mars and instead predict that a planet near 1.5 AU should roughly be the same mass as the Earth (Chambers 2001, icarus 152,205). Recently, a new model, known as ‘slow pebble accretion’, has been developed that can explain the formation of the gas giants (Levison+ 2015, Nature submitted). This model envisions that the cores of the giant planets formed from 100 to 1000 km bodies that directly accreted a population of pebbles (Lambrechts & Johansen 2012, A&A 544, A32) – centimeter- to meter-sized objects that slowly grew in the protoplanetary disk. Here we apply this model to the terrestrial planet region and find that it can reproduce the basic structure of the inner Solar System, including a small Mars and a low-mass asteroid belt. In particular, our models show that for an initial population of planetesimals with sizes similar to those of the main belt asteroids, slow pebble accretion becomes inefficient beyond ~1.5 AU. As a result, Mars’s growth is stunted and nothing large in the asteroid belt can accumulate.

#### Notes

• Standard view:
• disk forms, dust settles to midplanet
• dust accumulates, ~1-10 km
• runaway growth
• oligarchic growth
• late-stage
• violent endgame for terrestrial planets
• main problem: Mars is way to small
• possible solution: pebble accretion
• dust
• settling dust creates turbulence
• ~10 mm – 1 m pebbles
• large planetesimals can accrete pebbles very effectively
• strong gas drag $\rightarrow$ huge collision cross section (~Hill sphere)
• Can this explain the low mass of Mars?
• low-pebble-mass exponential cutoff
• encounter time too short
• A Ceres can grow if $r \lt \sim 1$ AU, but it can’t grow if $r \gt \sim 1$ AU.
• $\rightarrow$ leaves ~20 planets inside of ~1 AU
• subsequently very unstable and < 1 AU largely clears out
• leaves behind essentially the Solar System architecture

## DDA 2015 – Did our Solar System once have a STIP?

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.

#### Abstract

Continuing the established tradition in the field of speculative “fairy tales”, we postulate that our Solar System once had a set of several additional Earth-scale planets interior to the orbit of Venus. This would resolve a known issue that the energy and angular momentum of our inner-planet system is best explained by accreting the current terrestrial planets from a disk limited to 0.7-1.1 AU; in our picture the disk material closer to the Sun also formed planets, but they have since been destroyed. By studying the orbital stability of systems like the known Kepler systems, Volk and Gladman (companion abstract) demonstrate that orbital excitation and collisional destruction could be confined to just the inner parts of the system. In this scenario, our Mercury is the final remnant of the inner system’s destruction via a violent multi-collision (and/or hit-and-run disruption) process.This would provide a natural explanation for Mercury’s unusually high eccentricity and orbital inclination; it also fits into the general picture of long-timescale secular orbital instability, with Mercury’s current orbit being unstable on 5 Gyr time scales. The common decade spacing of instability time scales raises the intriguing possibility that this destruction occurred roughly 0.6 Gyr after the formation of our Solar System and that the lunar cataclysm is a preserved record of this apocalyptic event that began when slow secular chaos generated orbital instability in our former super-Earth system.

#### Notes

• Motivation
• inner edge of terrestrial planet zone
• Mercury is weird.
• Why don’t we have a STIP (system of tightly-packed inner planets)?
• Mercury:
• surfing the edge of secular chaos
• not clear how it got to $e^2 + i^2 \sim (0.25)^2$
• tough to strip mantle without it quickly falling right back
• Ausphaug & Reiner (2014): Mercury is the end state of a sequence of collisions.
• Why is there an inner edge?
• Wetherill 1978 (Protostars & Planets): E and L of terrestrial planets requires an inner edge ~0.6 AU.
• Historical way out: it’s too hot.
• But modern studies indicate $T < 1500$K until much later.
• If there is (collision) debris, where does it go?
• PR drag: kyr
• meteoritic transfer: kyr-Myr
• planetary interactions: ~10 Myr
• $\rightarrow$ disappears quickly
• if self-collisional, it will still disappear quickly
• Secular architecture rearrangement
• pump up to large $e$
• fast collisions (~50 km/s)
• vapor production
• “bullet factory” — erosion of remnants
• Meng et al. 2014 (Science)
• spike of hot dust around young star
• decay ~1 yr

## DDA 2015 – On the Robust Production of Super Earths and Suppressed Emergence of Gas Giants in Dynamically Evolving Protostellar Disks

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: Planet Formation I

Doug Lin (UC Santa Cruz) (Brouwer award winner)

#### Abstract

Radial velocity and transit surveys indicate the presence of super Earth around half of the main sequence stars regardless of their mass and metallicity. In contrast, the frequency of gas giants is much lower and increases with stellar mass and metallicity. I will show how the emergence of super-Earth is a robust process whereas the formation of gas giant planets is a threshold phenomena. The topics to be discussed include physical barriers in the planet building process, the role of migration in their evolving natal disks, planets’ interaction with each other and with their host stars. I will also discuss some key observations which may provide quantitative tests for planet formation theories.

#### Notes

• Observed properties of exoplanets: Howard 2013 (Science)
• Showstoppers:
• disk formation
• grain growth: the “meter barrier”
• Trapping of refractory grains beyond the magnetospheric cavity
• Tends to pile up at boundary
• grain growth: the “kilometer barrier”
• collisional fragmentation vs. grav.
• oligarchic barrier: isolation mass
• typically very small
• embryo retention barrier — Type I migration
• planet-disk tidal interaction
• get to high mass $\rightarrow$ migrate outward
• resonant sweeping $\rightarrow 2^{nd}$ generation
• core barrier: embryo resonant trapping
• bypass the resonant barrier
• inner scattered outward, outer scattered inward $\rightarrow$ collisions $\rightarrow$ impacts of super Earths
• gas accretion barrier
• Is there a threshold mass for gas accretion?
• runaway accretion
• Why didn’t this happen for observed super Earths?
• plenty of material left over: why didn’t they turn into gas giants?
• Measured disk accretion rate…?
• metal rich stars: no observed dependence, despite theory
• But metallicity of star and disk need not be the same.
• Rapid growth of proto gas giants
• grand design barrier: dynamical instability
• How did gas giants acquire their eccentricities?
• Type II migration
• provides constraint on growth process
• Why did hot Jupiters stop their inward migration?
• Close in planets
• e.g. Kepler-78
• 8-hour period
• Star is magnetic
• ~15 g
• analogous to Jupiter-Io
• induced EMF (unipolar induction) $\rightarrow$ energy dissipation at expense of planet’s orbit
• Planet surface cannot be iron; must be silicates.
• Flux tube footprints on star should move at period of planetary orbit, not stellar rotation.
• Many other issues!

## DDA 2015 – Inclination Excitation in Compact Extrasolar Planetary Systems

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.

Juliette Becker (U. Michigan) (Duncombe prize winner)

#### Abstract

The Kepler Mission has detected dozens of compact planetary systems with more than four transiting planets. This sample provides a collection of close-packed planetary systems with relatively liRle spread in the inclination angles of the inferred orbits. We have explored the effectiveness of dynamical mechanisms in exciting orbital inclination in this class of solar systems. The two mechanisms we discuss are self-excitation of orbital inclination in initially (nearly) coplanar planetary systems and perturbations by additional unseen larger bodies in the outer regions of the solar systems. For both of these scenarios, we determine the regimes of parameter space for which orbital inclination can be effectively excited. For compact planetary systems with the observed architectures, we find that the orbital inclination angles are not spread out appreciably through self-excitation, resulting in a negligible scaRer in impact parameter and a subsequently stable transiting system. In contrast, companions in the outer solar system can be effective in driving variations of the inclination angles of the inner planetary orbits, leading to significant scatter in impact parameter and resultantly non-transiting systems. We present the results of our study, the regimes in which each excitation method – self-excitation of inclination and excitation by a perturbing secondary – are relevant, and the magnitude of the effects.

#### Notes

• Why so many multi-planet transiting system?
• Ballard & Johnson 2014, Ballard 2014, Morton 2014, Morton & Winn 2014
• Seems to be a “Kepler dichotomy”
• $\rightarrow$ inclination excitation important
• $2^{nd}$ order secular Laplace-Lagrange theory (Murray &Dermott)
• inc. & ecc. decoupled
• Inclination as function of time (analytical)
• Use Kepler 4+ planets as model systems
• Conclusions:
• Self-excitation in compact solar system planets does not appear to be a significant mechanism
• Current Kepler systems with non-transiting planets could have started out transiting but driven out of transit by self-excitation
• Excitation by compact solar system planets themselves (smear their mass into a disk) does notappearto be a significant mechanism
• It might be possible to see multi-transiting systems with Jovian masses (if they exist)
• Dynamical transit duration variations due to secular interactions will be small ($10^{-4}$ to $10^{-7}$ sec) but potentially observable (via statistics on long time series)

## DDA 2015 – Secular Star-Disk Coupling and the Origin of Exoplanetary Spin-Orbit Misalignments

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.

Christopher Spalding (CalTech) (Duncombe prize winner)

#### Abstract

A recent paradigm shift in exoplanetary astronomy has come with the detection of a substantial number of planets possessing orbits that are misaligned with respect to the spin axes of their host stars. Moreover, observations of misalignments now include coplanar, multi-transiting systems, suggesting that these planets inherited their orbital planes from a protoplanetary disk which was once itself inclined with respect to the star. It has been proposed that mutual star-disk inclination may arise as a consequence of turbulence within the collapsing molecular cloud core, out of which both the star and its disk form. Alternatively, misalignments may be aRained later on, through secular interactions between the disk and companion stars. In this work, we examine the secular dynamics of the stellar spin axis arising in response to the gravitational and accretional torques communicated between the star and its disk throughout the epoch of star and planet formation. Our analysis shows that even though the disk forms from turbulent material, and is thus expected to exhibit a stochastic variation in its orientation with time during the star formation process, gravitational disk-star coupling adiabatically suppresses the excitation of mutual star-disk inclination under all reasonable parameter regimes. As such, the excitation of mutual star-protoplanetary disk inclination must occur later on in the disk’s lifetime, by way of an encounter with a secular resonance between stellar precession and the gravitational perturbations arising from an external potential, such as a binary companion.

#### Notes

• Motivation: our solar system, Laplace 1796
• Ecliptic disk oriented approx perp to Sun’s spin axis
• Goldreich & Tremaine 1980:
• disk-driven migration
• Jupiters eaten by stars
• Why aren’t observed hot Jupiters eaten?
• $\rightarrow$ hot Jupiters should be aligned with their disks
• But significant fraction is seriously misaligned!
• Tends to be more massive planets
• How to getmisalignments?
• Disk-driven migration doesn’t work
• High-eccentricity + tidal?
• Cannot explain multi-transiting misaligned systems (Huber et al. 2013)
• $\rightarrow$ Are disks really aligned with their stars?
• Hypothesis 1: misalignment during formation
• Spalding et al. 2014 (ApJ)
• Cores are turbulent
• Spin dir varies randomly by $\approx30^{\circ}$ every ~0.01 pc
• Shell infall time $\approx 10^4$ yr
• Disk adopts plane of whatever shell falls last (Bate et al. 2010)
• Star-disk system forms misaligned
• BUT: disk-star coupling?
• Young stars spin rapidly $\rightarrow$ oblate
• Dynamically equivalent to massive wire around point mass
• $\rightarrow$ disk precession
• Use Laplace-Lagrange secular theory
• Disk annuli act as outer perturbers upon stellar irientation
• $\rightarrow$ precession period ~100 years(!)
• Numerical simulation — will star spin axis follow motion of disk?
• Star trails disk, even though motion stochastic
• Hypothesis 2: binary companion in orbit around star+disk — disktorquing
• Companion causes $\gg 10^4$ yr precession
• Star-disk coupling weakens with time
• mass loss
• stellar contraction
• Spalding & Batygin 2014 (ApJ)
• Eventually, disk-binary precession ~ star-disk precession
• hits secular resonance, catapulting disk/star into retrograde orbits
• Final inclination only depends upon initial binary inclination
• Summary:
• Gravitational star-disk coupling prevents misalignment early on.
• Neighboring stars excite misalignments by was of a secular resonance.
• Misalignments are consistent with disk-driven migration.

## DDA 2015 – MMRs and the Origins of Extrasolar Orbital Architectures

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.

Konstantin Batygin (CalTech)

#### Abstract

The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this talk, we will show how an integrable Hamiltonian formalism for planetary resonances that allows both secondary bodies to have finite masses and eccentricities, can be used to construct a comprehensive theory for resonant capture. Employing the developed analytical model, we shall examine the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets, and demonstrate that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean motion commensurabilities on slightly eccentric (e ~ 0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary disks.

#### Notes

• SeeMécaniqueCéleste, Laplace 1805!
• But origins not really understood until Roy & Ovenden 1954, Goldreich 1964 (MNRAS)
• Disk-satellite interactions (Goldreich & Tremaine)
• But what about more than one planet?
• All tend to migrateinward then lock intoMMRs (Pierens 2013 A&A)
• $\Rightarrow$ numerical models predict MMR lock
• BUT only ~15% of observed planet pairs are in resonance
• The real Hamiltonian (planet-planet interactions) is actually probably a mess.
• See Poincare’s book, vol. 2(!)
• Define a canonical rotation that gives an integral of the motion (“generalized reducing transformation” –Poincare)
• Basically, a generalized Tisserand parameter
• Batygin & Morbidelli 2013(?)
• An analytical theory for resonant capture: unrestricted ETB problem.
• Batygin 2015 (MNRAS, submitted)
• Capture prob. only depends on total mass of the planets, NOT the mass ratio
• phase space area occupied by planet is small
• Kepler sample: critical eccentricity is ~0.02 — very small!
• Larger than this, capture fails
• Matches observed value!
• Explains Jupiter-Saturn MMR lock
• Perhaps slight deviations from axial symmetry in protoplanetary disks are responsible for the orbital architecture we observe today.

## DDA 2015 – Consolidating and Crushing Exoplanet Systems

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.

Kathryn Volk (U. British Columbia)

#### Abstract

Kepler revealed the common existence of tightly-packed planetary systems around solar-type stars, existing entirely on orbits with periods shorter than ~200 days. Those systems must have survived for the ages of their host stars (~5 Gyr), so their formation mechanism must provide inter-planet spacings that permit long-term stability. If one postulates that most planetary systems form with tightly-packed inner planets, their current absence in some systems could be explained by the collisional destruction of the inner system after a period of meta-stability. The signatures of such intense collisional environments may have been observed around stars in the form of rapidly varying debris disks; in these observed disks, collisional products are being disposed of via drag down onto the star or grinding to the nearly instantaneous dust blow-out limit. We use the orbital spacings and planet masses of the observed Kepler multi-planet systems to investigate the stability and long-term behavior of the systems. We find that many of our Kepler system analogs are unstable on 100 Myr timescales, even for initially small eccentricities (0-0.05); the instability timescales in these systems are distributed such that equal fractions of the systems experience planetary collisions in each decade in time. We discuss the likely outcomes of collisions in these systems based on the typical collision speeds from our numerical integrations and what implications this has for interpreting the observed Kepler multi-planet systems. The possible implications for our Solar System are discussed in a companion abstract (Gladman and Volk).

#### Notes

• Architectures of close-in (closely packed) planetary systems (from Kepler)
• Fabrycky 2014
• ~5-10% ofFGK field stars
• These systems must be stable on Gyr timescales
• Are all stars formed tightly packed?
• Modeled 13 such Kepler systems
• Preserved $a$ and masses, orbital angles randomized
• Allowed $e_0$ to vary $0 < e_0 < 0.05$
• Sudden onset of instability in 11 of these 13 after tens to ~100 Myr
• [why is she surprised?]
• These eccentricities are in range of observed values
• Decay rates consistent with e.g. Holman & Wisdom (1992 AJ)
• Why sudden onset?
• History is very sensitive to ICs [duh]
• Consolidation (low-speed collisions) vs. Destruction (high-speed collisions)
• First collision is often near the accretion/erosion boundary — i.e., low-speed
• Masses in 4-5 planet systems tend to be lower, while individual masses in ~3-planet systems are higher: mergers?
• Tracked collision speeds during integrations.
• Second collision often goes into erosion regime (i.e., high-speed)
• Observing debris should be rare (but see Meng et al. 2012)
• Ergodicity allows large variety of outcomes
• $\Rightarrow$ tightly packed systems could be ubiquitous initially
• Young stars should show higher fraction
• The remaining ~95% should be 0-2 planet systems