## 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

## DDA 2015 – Dynamical Analysis of the 6:1 Resonance of the Brown Dwarfs Orbiting the K Giant Star ν Ophiuchi

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ons II

Man Hoi Lee (University of Hong Kong)

#### Abstract

The K giant star ν Oph has two brown dwarf companions (with minimum masses of about 22 and 25 times the mass of Jupiter), whose orbital periods are about 530 and 3200 days and close to 6:1 in ratio. We present a dynamical analysis of this system, using 150 precise radial velocities obtained at the Lick Observatory in combination with data already available in the literature. We investigate a large set of orbital fits by applying systematic $\chi^2$ grid-search techniques coupled with self-consistent dynamical fitting. We find that the brown dwarfs are indeed locked in an aligned 6:1 resonant configuration, with all six mean-motion resonance angles librating around 0°, but the inclination of the orbits is poorly constrained. As with resonant planet pairs, the brown dwarfs in this system were most likely captured into resonance through disk-induced convergent migration. Thus the ν Oph system shows that brown dwarfs can form like planets in disks around stars.

#### Notes

• Lick G & K giants RV survey
• 373 bright G & K giant stars
• 0.6-m Coude
• ~1999-2012
• RV precision ~5 m/s
• $\nu$Oph
• K0III HB star, 2.73 $M_\odot$
• brown dwarf companion, P = 530 d
• 150 Lick RV measurements
• Fitting codes: Tan et al. 2013
• Grid search to minimize $\chi^2$
• SyMBA 10 Myr integrations
• Best fit:
• $M_1 = 22 M_J$, $P_1 = 530$ d, $a_1 = 1.79$ AU, $e_1 = 0.124$
• $M_2 = 25 M_J$, $P_2 = xxx$ d, $a_2 = 6.02$ AU, $e_2 = 0.1xx$
• 6:1 MMR at $3\sigma$
• Stability: all fits stable (numerically) to 10 Myr
• No constraints on inclination
• Origin
• Resonant capture via migration
• Type II (Ward 1997)
• $\left|\dfrac{\dot{a}}{a}\right| = \dfrac{3\nu}{2a^2}$
• Conclusions
• 2 brown dwarf companions
• minimum mass $22 M_J$ and $25 M_J$
• 6:1 MMR
• 6:1MMR couldindicate formation & migration in a disk
• But resonant capture requires slow migration and nonzero eccentricities

## 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

## DDA 2015 – End-State Relative Equilibria in the Sphere-Restricted Full Three-Body Problem

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.

Travis SJ Gabriel (UC Boulder)

#### Abstract

The Sphere-Restricted Full Three-Body Problem studies the motion of three finite density spheres as they interact under surface and gravitational forces. When accounting for the dissipation of energy, full-body systems may achieve minimum energy states that are unatainable in the classic treatment of the N-Body Problem. This serves as a simple model for the mechanics of rubble pile asteroids, interacting grains in a protoplanetary disk, and potentially the interactions of planetary ring particles. Previous studies of this problem have been performed in the case where the three spheres are of equal size and mass, with all possible relative equilibria and their stability having been identified as a function of the total angular momentum of the system. These studies uncovered that at certain levels of angular momentum there exists more than one stable relative equilibrium state. Thus a question of interest is which of these states a dissipative system would preferentially settle in provided some domain of initial conditions, and whether this would be a function of the dissipation parameters. Using perfectly-rigid dynamics, three-equal-sphere systems are simulated in a purpose-written C-based code to uncover these details. Results from this study are relevant to the mechanics and dynamics in small solar system bodies where relative forces are not great enough to compromise the rigidity of the constituents.

#### Notes

• Sphere-restrictedTBP:
• $U = -G \dfrac{m_1m_2}{r_{12}}$ singularity
• $E \ge U + \dfrac{H^2}{2 I_H}$
• For $N=3$ equal spheres, normalized min. energy function
• Scheeres 2012: 9 relative equilibria for planar motion case
• 3 stable
• $\rightarrow$ 2 min. energy solutions
• Which solution will the system land on?
• Numerical simulations
• randomized ICs in 2-solution regime, vary dissipation
• brute force statistics
• Results:
• More Euler resting states as H increases, regardless of dissipation
• End state depends heavily on dissipation
• Hence knowledge of restitution is key

## DDA 2015 – The family of Quasi-satellite periodic orbits in the co-planar RTBP

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: New Approaches to Classical Dynamical Problems II

Alexandre Pousse (IMCCE – Observatoire de Paris)

#### Abstract

In the framework of the Restricted Three-body Problem (RTBP), we consider a primary whose mass is equal to one, a secondary on circular or eccentric motion with a mass ε and a massless third body. The three bodies are in coplanar motion and in co-orbital resonance. We actually know three classes of regular co-orbital motions: in rotating frame with the secondary, the tadpole orbits (TP) librate around Lagrangian equilibria $L_4$ or $L_5$; the horseshoe orbits (HS) encompass the three equilibrium points $L_3$, $L_4$ and $L_5$; the quasi-satellite orbits (QS) are remote retrograde satellite around the secondary, but outside of its Hill sphere.

Contrarily to TP orbits which emerge from a fixed point in rotating frame, QS orbits emanate from a one-parameter family of periodic orbits, denoted family-f by Henon (1969). In the averaged problem, this family can be understood as a family of fixed points. However, the eccentricity of these orbits can reach high values. Consequently a development in eccentricity will not be efficient. Using the method developed by Nesvorný et al. (2002) which is valid for every values of eccentricity, we study the QS periodic orbits family with a numerical averaging.

In the circular case, I will present the validity domain of the average approximation and a particular orbit. Then, I will highlight an unexpected result for very high eccentricity on families of periodic orbits that originate from $L_3$, $L_4$ and $L_5$. Finally, I will sketch out an analytic method adapted to QS motion and exhibit associated results in the eccentric case.

#### Notes

• Quasi-satellite motion (QS):
• retrograde motion outside Hill sphere
• co-orbital motion with the secondary libration around $\theta = \lambda\, – \lambda_{pl}$ (heliocentric coords)
• Sidorenko et al 2013, Christou 2000, Kinoshita & Nakai 2007, etc.
• Model:coplanarRTBP + averaging (to $2^{nd}$ order)
• 2 DoF, in co-orbital resonance config
• Circular case:
• rotation symmetry $\rightarrow \Gamma = (1+u)(1-\sqrt{1-e^2}) = const.$
• projection in $u-\theta$ plane then captures the dynamics
• classic hyperbolic & elliptic fixed points, stable & unstable separatrices, chaotic regions
• plot: frequencies vs $e_0$
• “frozen ellipse”: $e_0 = 0.8352$
• bifurcation, appearance of libration around $\theta = \pi$
• Man Hoi Lee: 2 exoplanets exhibit this (Laughlin & Chambers)

## DDA 2015 – Instabilies in Near-Keplerian 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.

#### Abstract

Closed orbits drive secular gravitational instabilities, and Kepler potentials are one of only two potentials in which bound orbits are closed. Though the Kepler potential is common in astrophysics — relevant for stars orbiting massive black holes in the centers of galaxies, for planets orbiting stars, and for moons orbiting planets — few instabilities have been explored beyond the linear regime in this potential. I will present two new instabilities which grow exponentially from small initial perturbations and act to reorient eccentric orbits in near-Keplerian disks. The first results from forces in the plane of the disk and acts to spread orbits in eccentricity. The second instability results from forces out of the disk plane and drives orbits to high inclination. I will explain the dynamical mechanism behind each and make observational predictions for both planetary systems and galactic nuclei.

#### Notes

• Why Kepler potentials?
• Only two potentials yield closed orbits: $\psi \sim -\frac{1}{r}$, $\psi \sim r^2$
• More general, richer dynamics (than quadratic potentials)
• Eccentric disk instability
• Galactic center vs. Andromeda nucleus
• single peak vs. double peak (in luminosity)
• Presence/absence of nuclear star cluster changes direction of apsidal precession.
• Andromeda: apsidally aligned orbits $\rightarrow$ double luminosity peak
• torque from disk grav. reducesang. momentum, increasing eccentricity.
• produces oscillations
• but stable disks
• Andromeda
• Inclination instability
• Thick disk of stars in Galactic center
• Dwarf planets (inner Oort Cloud)are clustered in $\omega$. How?
• $\cos \omega = \dfrac{\sin i_a}{\sin i_b}$ (inclinations wrt major and minor axes)
• Dwarf planets are clustered in $\omega$ because high eccentricity orbits in a disk are unstable.
• In Galactic center, ~80% of young stars are not in disk plane (Yelda 2014). How did they get there?
• Set by initial inclinations.
• Two-body diffusion stage, then ~sudden instability.
• Instability grows exponentially.

## DDA 2015 – Modeling relativistic orbits and gravitational waves

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: New Approaches to Classical Dynamical Problems I

Marc Favata (Montclair State University) (invited)

#### Abstract

Solving the relativistic two-body problem is difficult. Motivated by the construction, operation, and recent upgrades of interferometric gravitational-wave detectors, significant progress on this problem has been achieved over the past two decades. I will provide a summary of techniques that have been developed to solve the relativistic two-body problem, with an emphasis on semi-analytic approaches, their relevance to gravitational-wave astronomy, and remaining unsolved issues.

#### Notes

• Gravitational wave (GW) detector networks:
• Upgrades complete as of 1 April 2015!
• ~3 yr to get to final design sensitivity
• Upgrade: ~10 times more sensitive
• Kagra (~2018)
• LIGO-India (~2022)
• Pulsar timing arrays (~now)
• NANOgrav, EPTA, PPTA
• Future: third-gen LIGO
• GW sources
• Merging stellar-mass compact-object binaries (NS or BH)
• measure masses and spins
• determine merger rates
• core-collapse SN
• isolated neutron stars
• cosmic strings, stochastic bg
• unexpected
• Low-freq sources (LISA):
• merging SMBHs
• extreme-mass ratio ???
• ???
• Coalescing binaries
• phases: inspiral (periodic, long), merger (frequency chirp and peak amplitude, short), and ringdown (damping)
• During merger and ringdown, the two holes merge and the remnant undergoes damped oscillations
• Why two-body GR is hard
• Einstein’s eqs. are just a lot more complicated
• Newton: only mass density
• E: density, vel., kinetic energy, etc.
• Highly nonlinear
• Solutions to E equations
• Exact solutions: Kerr and FrW
• Perturbation theory: PN theory, BH pert. theory
• Numerical relativity: finite resolution, inexact ICs, cpu time
• Numerical Relativity
• Not really viable until ~2005, despite efforts from the 1960s
• Mergers now routine
• Future: detailed exploration of BH/BH param space
• NS+BH, NS+NS: realistic EOS, mag. fields, neutrinos…
• Computationally expensive beyond ~10 orbits
• NS+NS: 8000 orbits, NS+BH: 1800 orbits, BH+BH: 300 orbits
• small mass ratios < 1/10 very costly
• Current best achievement: 176 orbits
• Need for phase accuracy
• LIGO data is noisy $\rightarrow$ need good signal template
• integral of an oscillating function
• phase evol. of signal needs to be accurate to fraction of a cycle
• Templates: >10 parameters
• PN approx.
• write E eqs as perturbation on flat-space wave eqn
• series expansions
• plug expansions back into E eqs
• iterate
• gets very messy very quickly
• orbital phasing is where the information lives — need to get to as high an order as possible
• need to get to 3.5PN ($v^7$)
• high-order harmonics can be important
• “memory modes”: non-oscillatory but time-varying modes (secular effects)
• nonlinear effect
• GWs themselves produce GWs(!)
• Spin effects
• aligned: minor correction
• non-aligned: mess
• eqs to describe spin evolution must also be solved
• Eccentricity effects
• GWs damp eccentricity, so often ignored
• But eccentric signals possible from binaries
• periastron precession
• eccentricity-induced modulations to orbital phase & amplitude
• corrections also need to be high-order
• Tidal interactions
• near end of inspiral
• tidal distortionparameterized in terms of tidal Love number
• Measuring tidal Love number provides useful constraints
• types:
• electric
• magnetic
• shape
• electric Love number most observationally relevant
• BH pert. theory
• EMRI orbits
• very complicated — rich structure, resonant effects
• produces interesting “jumps” in phasing and orbital elements
• self-force approach
• Conclusion: $2^{nd}$ gen network of GW detectors is coming online now
• Need good modeling
• Need good control over systematic errors (hence high-order PN work)

## DDA 2015 – How massive is Saturn's B ring – Clues from cryptic density waves

Matthew M. Hedman (Cornell)

#### Abstract

The B ring is the brightest and most opaque of Saturn’s rings, but it is also amongst the least well understood because basic parameters like its surface mass density are still poorly constrained. Elsewhere in the rings, spiral density waves driven by resonances with Saturn’s various moons provide precise and robust mass density estimates, but for most the B ring extremely high opacities and strong stochastic optical depth variations obscure the signal from these wave patterns. We have developed a new wavelet-based technique that combines data from multiple stellar occultations (observed by the Visual and Infrared Mapping Spectrometer (VIMS) instrument onboard the Cassini spacecraft) that has allowed us to identify signals that may be due to waves generated by three of the strongest resonances in the central and outer B ring. These wave signatures yield new estimates of the B-ring’s mass density and indicate that the B-ring’s total mass could be quite low, perhaps a fraction of the mass of Saturn’s moon Mimas.

#### Notes

• B ring long assumed to be the most massive ring structure
• essentially opaque
• Density (and bending) waves
• $k(r) = dfrac{3(m-1)M(r-r_L)}{2 pi sigma_0 r^4_L}$
• Wavenumbers can be quantified using wavelets
• frequency chirping at moon (e.g., Prometheus, Pandora, Enceladus) MMRs
• Few waves have been identified in Saturn’s B ring(!)
• $rightarrow$ mass density poorly constrained
• Expect to see density waves, but…
• resonances in opaque region
• a lot of the structure in the rings is of unknown origin
• some are likely density waves, some not
• Wave-like signatures not obvious in wavelet transforms
• Solution? Include phase information in wavelet analysis
• Different occultations cut through the spiral pattern at different places
• Noise fluctuations confuse the signal
• Normally ignored
• Calculate what phase shifts ought to have been and remove them
• can average components
• Ideally, noise averages to zero
• $rightarrow$ suppresses background mess
• Can now measure wave number of resonances!
• even in region where opacity is ~3
• $rightarrow$ mass density
• Regions with same mass density can have very different optical depths
• (from scatter in the data)
• Don’t know why
• Indications: B ring mass density lower than expected

## DDA 2015 – How massive is Saturn’s B ring – Clues from cryptic density waves

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.

Matthew M. Hedman (Cornell)

#### Abstract

The B ring is the brightest and most opaque of Saturn’s rings, but it is also amongst the least well understood because basic parameters like its surface mass density are still poorly constrained. Elsewhere in the rings, spiral density waves driven by resonances with Saturn’s various moons provide precise and robust mass density estimates, but for most the B ring extremely high opacities and strong stochastic optical depth variations obscure the signal from these wave patterns. We have developed a new wavelet-based technique that combines data from multiple stellar occultations (observed by the Visual and Infrared Mapping Spectrometer (VIMS) instrument onboard the Cassini spacecraft) that has allowed us to identify signals that may be due to waves generated by three of the strongest resonances in the central and outer B ring. These wave signatures yield new estimates of the B-ring’s mass density and indicate that the B-ring’s total mass could be quite low, perhaps a fraction of the mass of Saturn’s moon Mimas.

#### Notes

• B ring long assumedto be the most massive ring structure
• essentially opaque
• Density (and bending) waves
• $k(r) = \dfrac{3(m-1)M(r-r_L)}{2 \pi \sigma_0 r^4_L}$
• Wavenumbers can be quantified using wavelets
• frequency chirping at moon (e.g., Prometheus, Pandora, Enceladus) MMRs
• Few waveshave been identified in Saturn’s B ring(!)
• $\rightarrow$ mass density poorly constrained
• Expect to see density waves, but…
• resonances in opaque region
• a lot of the structure in the rings is of unknown origin
• some are likely density waves, some not
• Wave-like signatures not obvious in wavelet transforms
• Solution? Include phase information inwavelet analysis
• Different occultations cut through the spiral pattern at different places
• Noise fluctuations confuse the signal
• Normally ignored
• Calculate what phase shifts ought to have been and remove them
• can average components
• Ideally, noise averages to zero
• $\rightarrow$ suppresses background mess
• Cannow measure wavenumber of resonances!
• even in region where opacity is ~3
• $\rightarrow$ mass density
• Regions with same mass density can have very different optical depths
• (from scatter in the data)
• Don’t know why
• Indications: B ring mass density lower than expected

## 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