## DDA 2015 – Rotational and interior models for Enceladus II

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

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

#### Notes

• Libration measurement
• 3D reconstruction of coords of a network of control point (fiducial satellite surface points — e.g. craters)
• most of Enceladus’s orbit was covered
• Thomas et al. 2015
• minimize RMS residual $\rightarrow 0.120 \pm 0.014$ deg
• Solid models
• core plus two-layer in hydro.equilib. plus south polar sea
• measured libration amplitude rules this out
• decoupled shell from the core (indep.librations)
• consistent with observed libration amplitude if shell thickness 21-26 km and ocean thickness 26-31 km
• Gravity data
• suggests a local mass anomaly — interpreted as ocean thicker under south pole

## DDA 2015 – Rotational and interior models for Enceladus I

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: Moon Formation and Dynamics II

Matthew S. Tiscareno (Cornell)

#### Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

#### Notes

• 2nd largest Saturnian moon
• Plumes — salty jets — observed by Cassini
• What is under the surface?
• Rotational parameters $\rightarrow$ interior structure
• Forcedlibrations
• same period as orbital
• nat. freq. $\omega_0 \approx n \sqrt{3 (B-A)/C}$
• near-spherical: moon always points at empty focus (synchronous)
• elongated: moon would always point at Saturn
• Enceladus axis oscillates around empty focus (synchronous rotation)
• as $\dfrac{B-A}{C} \rightarrow \dfrac{1}{3}$, resonance (Tiscareno et al. 2009)
• but Enceladus $\dfrac{B-A}{C} \ll \dfrac{1}{3}$
• Enceladus libration $0.120\pm0.014$ deg
• rules out rigid connection between surface and core
• hence, some kind of global subsurface ocean
• Mean motion variations
• Enceladus resonant arguments from interaction with Dione:
$ILR_D = \lambda_E\, – 2 \lambda_D + \varpi_E$ (librating)
$CIR_D = \lambda_E\, – 2 \lambda_D + \Omega_D$ (circulating)
$CER_D = \lambda_E\, – 2 \lambda_D + \varpi_D$ (circulating)
• As long as damping is sufficiently strong, synchronous rotation maintained
• damping must be $\gamma_{\pi/2} = \dfrac{2 e}{1\, – \left(\dfrac{n}{\omega_0}\right)^2} \Rightarrow \tau \approx 1.0\,Q\ \mathrm{days}$
• but $10 \lt Q \lt 100$ days
• rot. rate varies with the CER and ILR freqs
• not really “librations”
• maintaining synch. rot., while the mean motion varies quasiperiodically
• Rotational models
• Global Fourier components have limited usefulness
• MM variation more complex than a few periodic terms
• Define rot.statewrt Saturn
• base state: synch rot (expected for low triaxiality)
• accounts for MM variation
• easy to generate a range of kernels for many vals of $\gamma$
• Tiscareno 2015
• deflect $\psi(t) = (2 e+\gamma)\sin M$
• generate kernels of $\psi(t)$ for a wide range of $\gamma$ values, check for best control-point resids
• dissipation?

## DDA 2015 – Forced libration of tidally synchronized planets and moons

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: Moon Formation and Dynamics I

Valeri Makarov (for Bryan Dorland) (USNO)

#### Abstract

Tidal dissipation of kinetic energy, when it is strong enough, tends to synchronize the rotation of planets and moons with the mean orbital motion, or drive it into long-term stable spin-orbit resonances. As the actual orbital motion undergoes periodic acceleration due to a finite orbital eccentricity, the spin rate oscillates around the equilibrium mean value too, giving rise to the forced, or eccentricity-driven, librations. We contend that both the shape and amplitude of forced librations of synchronous viscoelastic planets and moons are defined by a combination of two different types of perturbative torque, the tidal torque and the triaxial torque, the latter related to a permanent deformation of the distribution of mass from a perfect rotational symmetry. Consequently, forced librations can be tidally dominated (e.g., Io and possibly Titan) or deformation-dominated (e.g., the Moon) depending on a set of orbital, rheological, and other physical parameters. With small eccentricities, for the former kind, the largest term in the libration angle can be minus cosine of the mean anomaly, whereas for the latter kind, it is minus sine of the mean anomaly. The shape and the amplitude of tidal forced librations determine the rate of orbital evolution of synchronous planets and moons, i.e., the rate of dissipative damping of the semimajor axis and eccentricity. The known super-Earth exoplanets can exhibit both kinds of libration, or a mixture of thereof, depending on, for example, the effective Maxwell time of their rigid mantles. Our approach can be extended to estimate the amplitudes of other libration harmonics, as well as the forced libration in non-synchronous spin-orbit resonances.

#### Notes

• Numerical sim of Lunarspinlibrations in longitude with polar torques exerted by Earth
• spectrum of harmonics
• dominant term: $\dfrac{\dot{\theta}}{n}\, – 1 \propto -\cos M$
• Geometry offorcedlibrations
• longest axis of planet tries to align with line of centers (but can’t)
• triaxial torque (Eckhardt 1981) $\ddot{\theta}\approx 6 n^2 \dfrac{B-A}{C^3}\left[e\, – \dfrac{31}{16}e^2 + O\left(e^5\right)\right] \sin M$
• Problem: Io
• striking difference with moon
• dominant term now $\dfrac{\dot{\theta}}{n}\, – 1 \propto +\sin M$
• Tidal torques
• torque = triaxial + tidal
• tidal = secular + periodic
• secular can dominate in the “linear” regime
• plot: $\ddot{\theta}$ vs. $\dfrac{\dot{\theta}}{n}\, – 1$
• It’s not the amplitude that’s important, but the gradient, which provides a subtle but powerful feedback.
• tiny interval of pert freqs where secular torque can be important
• Problem: doesn’t work for Moon
• because Moon is not sitting on that linear portion of $\dfrac{\dot{\theta}}{n}\, – 1$
• $\therefore$ need further harmonic terms

## DDA 2015 – Solving the Mystery of the Fermi Bubbles?

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 F. Bartlett (UC Boulder)

#### Abstract

The Fermi Bubbles are large structures that stretch symmetrically between galactic latitudes of -55 degrees and +55 degrees and between galactic longitudes of -45 degrees and +45 degrees. For almost a decade they have been under the intense scrutiny of the Fermi-Large Area Telescope, a gamma-ray detector in orbit about the earth. The Bubbles remain mysterious: are the gamma-rays – with energies up to a few hundred GeV – produced by hadrons or do they come from inverse Compton scattering of galactic electrons with the low energy interstellar radiation field? Why are the edges of the bubbles only 3 degree wide? How old are the bubbles? For some time we have been considering a non-Newtonian cosinusoidal potential $U=-\dfrac{G M}{r} \cos(k_0 r)$, and its complement, a non-Coulombic electric potential $U=Q \exp(-k_0 r)$. In both cases, $k_0 = 2 \pi/400$ pc. In this talk we present evidence that our putative potentials acting in concert can help answer the mysteries of the Bubbles.

• Oh my…

## DDA 2015 – Bringing Black Holes Together: Plunging through the Final Parsec

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.

Kelly Holley-Bockelmann (Vanderbilt)

[None]

#### Notes

• How to bringBHs together: three phases
• Galaxy mergers sinkBHs through dynamical friction
• $\sim 10^5$ pc
• Sink closer via 3-body scattering
• Quinlan 1997
• dynamical friction no longer operating
• $\sim 10$ pc, $\sim 10^{10}$ yr(!?)
• Finally, GW complete the merger
• $\sim 10^{-5}$ pc
• Problem: once loss coneis depleted by 3-body scattering, it can only be refilled by 2-body relaxation
• Merger stalls at $\sim 1$ pc
• hence the “final parsec problem”
• Begelman, Blandford, & Rees 1980, Makino 1997, Merritt & Milos 2005
• Solution: galaxies are not idealized gas-free, stable, equilibrium systems!
• Mayer et al. 2007: galaxies have gas; gas drives theBHs closer
• drag(?)
• spiral wave torques
• problem: AGN feedback
• Ostriker,Binney, & Silk 1989: galaxies aretriaxial
• triaxiality introduces new mechanisms for phase space transport
• chaos
• Berczik et al. 2005
• gets you to grav radiation regime
• Khan,KHB,Berczik, and Just 2013: test limits of 3-body scattering
• N-body sims
• gas-poor, non-rotating, axisymmetric potential
• still takes too long (2.5 Gyr)
• found special orbits to fill loss cone
• because near BH the potential istriaxial
• even closer, is spherical
• $\rightarrow$ changes in shape are important
• add rotation $\rightarrow$ merger goes faster (yay)
• axisymmetric, counter-rotating: ~100 Myr
• but eccentricities very high

## DDA 2015 – p-ellipse Orbit Approximations, Lindblad Zones, and Resonant Waves in Galaxy 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.

Curtis Struck (Iowa State)

#### Abstract

p-ellipses are simple, yet very accurate formulae for orbits in power-law potentials, like those approximating galaxy disks. These precessing elliptical orbits reveal important systematics of orbits in such potentials, including simple expressions for the dependence of apsidal precession on eccentricity, and the fact that very few terms (or parameters) are needed for the approximation of even nearly radial orbits. The orbit approximations are also useful tools for addressing problems in galaxy dynamics. In particular, they indicate the existence of a range of eccentric resonances associated with the usual, near-circular Lindblad resonances. Collectively these change an isolated Lindblad resonance to a Lindblad Zone of eccentric resonances. A range of these resonances could be excited at a common paRern speed, aiding the formation of a variety of bars and spirals, out of eccentric orbits. Such waves would be persistent, and not wind up or disperse, since differences in their precession frequencies offset differences in the circular velocities at the radii of their parent orbits. The p-ellipse approximation further reveals how a non-axisymmetric component of the gravitational potential (e.g., due to bar self-gravity) significantly modifies precession frequencies, and similarly modifies the Lindblad Zones.

#### Notes

• Precessing ellipses
$\dfrac{1}{r} = \dfrac{1}{p}\left[1+e \cos (m \phi)\right]^{\frac{1}{2}+\delta}$
yield good fits for orbits in different potentials
• Apsidal precession:
$\Delta \phi = \dfrac{\pi}{\sqrt{2(1-\delta)}}$
$\rightarrow$ kinematic waves, bars
• Nearly radial orbits: p-ellipse fits not so good
• But can tweak to get good fits:
• Fit to the extremal radii, not the position
• Note that apsidal precession is not constant but depends on eccentricity
$\dfrac{1}{r} = \dfrac{1}{p}\left[1+e_1 \cos (m \phi)+e_2 \cos (2 m \phi)\right]^{\frac{1}{2}+\delta}$
• Example of (possibly) kinematic counter-rotating waves: NGC 4622
• Summary
• Accurate, simple approximations for orbits in a range of potentials
• Can be extended to radial orbits
• Allows formation of persistent kinematic waves of various types
• but usually requires fine tuning

## DDA 2015 – The Relative Influence of Dynamical Nature and Nurture on the Formation of Disk Galaxies

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.

Jonathan Bird (Vanderbilt) (invited)

[None]

#### Notes

• Are disk galaxies formed by nature or nurture?
• e.g. NGC 891
• Thick disk and thin disk (Gilmore & Reid 1983)
• Extragalactic thick disks are ubiquitous (Dalcanton & Bernstein 2002, Yoachim & Dalcanton 2006)
• Nature:
• Stellar kinematics dominated by those of gas from which stars formed
• Subsequent dynamics are second-order
• Planetary disk: core accretion; static
• $\alpha$ abundance is a tracer for stellar age
• plot: [$\alpha$/Fe] vs [Fe/H]
• Thick disk is old, $\alpha$-rich, kinematically hot
• Thin disk is young, (relatively to Fe) $\alpha$-poor, dynamically cold
• Smooth correlation between chemistry and kinematics
• APOGEE survey: velocity dispersion increases with stellar age
• power law
• $\rightarrow$ disk grows over time
• Nurture:
• Stellar kinematics dominated by dynamical interactions after birth
• Most stars born in dynamical cold gas (level playing field)
• Resonances play huge role; pebble accretion
• Scattering processes heat stellar velocity distributions
• Sellwood & Binney 2002: can redistribute stars without globally heating the disk
• Can outwardly migrating stars create the thick disk?
• No: vertical action is conserved. (Tolfree et al. 2014)

## DDA 2015 – Cross Sections for Planetary Systems Interacting with Passing Stars and Binaries

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.

### Fred C. Adams (U. Michigan)

#### Abstract

Numerous spectroscopic and photometric studies have provided strong evidence of the presence of multiple stellar populations in globular clusters and raised many fundamental questions concerning the formation and dynamical evolution of these stellar systems. After a brief review of the main observational studies, I will present the results of theoretical investigations exploring a number of aspects of the internal dynamics of multiple-population clusters and their formation history. Most planetary systems are formed within stellar clusters, and these environments can shape their properties. This talk considers scattering encounters between solar systems and passing cluster members, and calculates the corresponding interaction cross sections. The target solar systems are generally assumed to have four giant planets, with a variety of starting states, including circular orbits with the semimajor axes of our planets, a more compact configuration, an ultracompact state with multiple mean motion resonances, and systems with massive planets. We then consider the effects of varying the cluster velocity dispersion, the relative importance of binaries versus single stars, different stellar host masses, and finite starting eccentricities of the planetary orbits. For each state of the initial system, we perform an ensemble of numerical scaRering experiments and determine the cross sections for eccentricity increase, inclination angle increase, planet ejection, and capture. This talk reports results from over 2 million individual scattering simulations. Using supporting analytic considerations, and fibng functions to the numerical results, we find a universal formula that gives the cross sections as a function of stellar host mass, cluster velocity dispersion, starting planetary orbital radius, and final eccentricity. The resulting cross sections can be used in a wide variety of applications. As one example, we revisit constraints on the birth aggregate of our Solar System due to dynamical scattering and find N < 10,000 (consistent with previous estimates).

#### Notes

• Most stars form in clusters
• particle fluxes
• dynamical interactions
• need to know cross sections and rates at which things fly by
• closest approach distribution = power law
• Simulationstodetermine cross sections
• many Monte Carlo simulations
• 2 million runs
• many parameters + chaotic behavior
• do planetary eccentricities get pumped up?
• yes
• Results:
• by and large, $\sigma \gg A$
• $\dfrac{\sigma}{a} = A v^{-\frac{7}{5}} \exp\left[b(1-e)\right]$
• $\sigma = \sigma_0 \exp\left[b(1-\sin \Delta i)\right]$
• $\Delta i \propto \Delta e$
• size of birth cluster constrained to $< 10^4$ stars
• G. Li & Adams 2015 (MNRAS 448:344)

## DDA 2015 – Dynamical Evolution of Multiple-Population Globular Clusters

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: Star Cluster and Galaxy Dynamics

Enrico Vesperini (Indiana University) (invited) [withdrawn]

#### Abstract

Numerous spectroscopic and photometric studies have provided strong evidence of the presence of multiple stellar populations in globular clusters and raised many fundamental questions concerning the formation and dynamical evolution of these stellar systems. After a brief review of the main observational studies, I will present the results of theoretical investigations exploring a number of aspects of the internal dynamics of multiple-population clusters and their formation history.

## DDA 2015 – Lense-Thirring Effect Measurement from LAGEOS Node: Limitation from Radiation Forces

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: Dynamics of Small Solar System Bodies III

Victor J. Slabinski (USNO)

#### Abstract

The Lense‑Thirring (L‑T) effect from General Relativity predicts a small secular increase to the node right ascension for close Earth satellites. For the LAGEOS 1 satellite, the predicted node increase is 31 mas/y. There is a current effort to observationally evaluate L‑T to 1 percent accuracy through an orbit analysis of the laser‑ranged LAGEOS 1, LAGEOS 2, and LARES satellites. Uncertainty in the computed gravitational perturbations to the satellite nodes, due to parameter uncertainties, is largely eliminated by taking a linear combination of the node positions which eliminates the uncertainty due to the major terms. One then looks for the L‑T effect on this composite node.

But there remains uncertainty in the computed perturbations due to two radiation (non‑gravitational) forces: the solar radiation (SR) force and thermal thrust (Yarkovsky effects). This paper treats LAGEOS 1 perturbations. For simplicity in discussion, we treat perturbations to its node rather than perturbations to the composite node.

Uncertainty in the perturbation rates arises from ignorance of parameter values for the LAGEOS 1 exterior aluminum surface, specifically, the solar absorbtance and thermal emiRance. The LAGEOS 1 Phase B design study proposed three different sets of aluminum surface parameters without recommending a particular set. The LAGEOS 1 as-built surface parameters were not measured prior to spacecraft launch.

The possible spread in LAGEOS 1 solar absorbtance values gives a spread of ±0.42 mas/y in the SR force contribution to its node rate. This results in a ±1.3 percent uncertainty to the L‑T determination. But because of its long‑period perturbation to the eccentricity vector, evaluating the SR force parameter as a solved‑for parameter in the orbit analysis should significantly reduce the uncertainty in the corresponding node motion. The possible spread in LAGEOS 1 surface values gives a spread of ±0.16 mas/y in the thermal thrust contribution to its node rate. This represents a ±0.53 percent uncertainty in the L‑T determination which leaves little room for other error sources. Ground-based satellite brightness measurements could improve knowledge of the surface absorbtance and reduce the uncertainty from thermal thrust.

#### Notes

• Lense-Thirring
• gravitomagnetic effect
• spinning Earth:
• $\rightarrow$ frame-dragging
• $\rightarrow$ precession of $\Omega$ and $\omega$
• LAGEOS 1 & 2: linear motion of $\Omega \approx 1.8$ m/yr
• Goal: 1% measurement of L-T effect
• Other perturbing forces
• requires knowledge of satellite surface material properties
• notably: aging
• Thermal thrust
• IR from Earth
• fused silica of corner-cube reflectors is an excellent absorber of IR
• Oops
• thermal phase lag: max recoil force not at local midnight but somewhat past
• $\rightarrow \sim 3 \mathrm{pm/s^2}$ acceleration component along orbit track
• $\rightarrow$ also a component perpendicular to orbital plane
• affects nodal precession rate
• Satellite surface properties
• Corner-cube reflectors: no problem. We know fused silica.
• Aluminum frame: uh oh…
• Not measured beforeLAGEOS 1 launch!
• thermal absorptance
• thermal emittance
• Node precession from solar radiation term: ~1/4 L-T effect
• But radiation force also changes eccentricity vector, from which you can get diffuse reflection coefficient
• but not specular
• One solution: brightness measurements from the ground
• Magnitude range: 11.5-14

## DDA 2015 – New Trans-Neptunian Objects in the Dark Energy Survey Supernova Fields

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 W. Gerdes (U. Michigan)

#### Abstract

The Dark Energy Survey (DES) observes ten separate 3 sq. deg. fields approximately weekly for six months each year. Although intended primarily to detect Type Ia supernovae, this data set provides a rich time series that is well suited for the detection of objects in the outer solar system, which move slowly enough that they can remain in the same field of view for weeks, months, or even across multiple DES observing seasons. Because the supernova fields have ecliptic latitudes ranging from -15 to -45 degrees, DES is particularly sensitive to the dynamically hot population of Kuiper Belt objects, as well as detached/inner Oort cloud objects. Here I report the results of a search for new trans-Neptunian objects in the first two seasons of DES data, to limiting magnitudes of r~23.8 in the eight shallow fields and ~24.5 in the two deep fields. The 22 objects discovered to date include two new Neptune trojans, a number of objects in mean motion resonances with Neptune, two objects with orbital inclinations above 45 degrees, a Uranian resonator, and several distant scaRered disk objects including one with an orbital period of nearly 6000 years. This latter object is among the half-dozen longest-period trans-Neptunian objects known, and like the other such objects has an argument of perihelion near zero degrees. I will discuss the properties and orbital dynamics of objects discovered to date, and will also discuss prospects for extending the search to the full 5000 sq. deg. DES wide survey.

#### Notes

• Piggy back on DES to find and characterizeTNOs
• Will surpass all previous TNO surveys
• DECam:
• 570 Mpix imager
• CTIO 4-meter
• 3 deg fov
• first light Sep. 2012
• first two of five seasons complete
• 125 nights/yr, 5 optical bands
• 60 2k$\times$4k CCDs (two died)
• Biased towards high inclination objects
• Sensitive to hot population
• New objects identified via difference imaging
• Confusion an issue
• But KBOs move slowly
• Once you find a 3-visit orbit consistent with KB motion, it’s easy
• Should be able to discover a ~600 km object at 80 AU
• 23 new objects discovered in first two seasons (~10% of hot population)
• Case studies
• 2013RG98
• 3:4 Uranian MMR (temporary)
• Likely to become a Jupiter family comet, or maybe ejected
• 2014QO441, 2014QP441
• Neptune Trojans
• libration period ~9100 yr
• stable on Gyr timescales
• 2013RF98
• An extreme TNO
• $a = 325$ AU, $i = 30^\circ$, $e = 0.89$, $P = 5682$ yr
• Clustering of $\omega$

## DDA 2015 – Stochastic YORP On Real Asteroid Shapes

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.

Jay W. McMahon (UC Boulder)

#### Abstract

Since its theoretical foundation and subsequent observational verification, the YORP effect has been understood to be a fundamental process that controls the evolution of small asteroids in the inner solar system. In particular, the coupling of the YORP and Yarkovsky effects are hypothesized to be largely responsible for the transport of asteroids from the main belt to the inner solar system populations. Furthermore, the YORP effect is thought to lead to rotational fission of small asteroids, which leads to the creation of multiple asteroid systems, contact binary asteroids, and asteroid pairs. However recent studies have called into question the ability of YORP to produce these results. In particular, the high sensitivity of the YORP coefficients to variations in the shape of an asteroid, combined with the possibility of a changing shape due to YORP accelerated spin rates can combine to create a stochastic YORP coefficient which can arrest or change the evolution of a small asteroid’s spin state. In this talk, initial results are presented from new simulations which comprehensively model the stochastic YORP process. Shape change is governed by the surface slopes on radar based asteroid shape models, where the highest slope regions change first. The investigation of the modification of YORP coefficients and subsequent spin state evolution as a result of this dynamically influenced shape change is presented and discussed.

#### Notes

• Background
• YORP controls small asteroid spin evolution
• YORP highly sensitive to location of features on surface (Statler 2009)
• “stochastic YORP” (Cotto-Figueroa 2013)
• “stochastic YORP” $\rightarrow$ evolution of asteroid families (Bottke et al. 2015)
• Motivation
• Do shapes change as spin increases?
• How does shape evolution map to YORP coefficients?
• Shape evolution
• Regolith will flow “downhill”
• Body will reshape to relax to some slope limit (Scheeres 2015)
• This study: use actual radar-derived asteroid shapes instead of idealized sphere/ellipsoid
• Use (101955)Bennu
• Apollo asteroid
• OSIRIS-REx sample return target
• Results
• 5-m boulder (as spin limitis approached):
• effect on obliquity very small
• larger effects on spin rate
• shape of boulder matters
• Much future work to do

## DDA 2015 – Contact Binary Asteroids

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.

Samantha Rieger (UC Boulder)

#### Abstract

Recent observations have found that some contact binaries are oriented such that the secondary impacts with the primary at a high inclination. This research investigates the evolution of how such contact binaries came to exist. This process begins with an asteroid pair, where the secondary lies on the Laplace plane. The Laplace plane is a plane normal to the axis about which the pole of a satellite’s orbit precesses, causing a near constant inclination for such an orbit. For the study of the classical Laplace plane, the secondary asteroid is in circular orbit around an oblate primary with axial tilt. This system is also orbiting the Sun. Thus, there are two perturbations on the secondary’s orbit: J2 and third body Sun perturbations. The Laplace surface is defined as the group of orbits that lie on the Laplace plane at varying distances from the primary. If the secondary is very close to the primary, the inclination of the Laplace plane will be near the equator of the asteroid, while further from the primary the inclination will be similar to the asteroid-Sun plane. The secondary will lie on the Laplace plane because near the asteroid the Laplace plane is stable to large deviations in motion, causing the asteroid to come to rest in this orbit. Assuming the secondary is asymmetrical in shape and the body’s rotation is synchronous with its orbit, the secondary will experience the BYORP effect. BYORP can cause secular motion such as the semi-major axis of the secondary expanding or contracting. Assuming the secondary expands due to BYORP, the secondary will eventually reach the unstable region of the Laplace plane. The unstable region exists if the primary has an obliquity of 68.875 degrees or greater. The unstable region exists at 0.9 Laplace radius to 1.25 Laplace radius, where the Laplace radius is defined as the distance from the central body where the inclination of the Laplace plane orbit is half the obliquity. In the unstable region, the eccentricity of the orbit increases. Once the eccentricity becomes very large or approaching 1, the orbit of the secondary intersects with the primary and will eventually collide and becomes a contact binary.

#### Notes

• Motivation
• contact binaries exist with high obliquity, ~90 deg
• Does Laplace plane have a role?
• Resonances between binary orbit and solar perturbations?
• Laplace plane
• $\omega_2 \sin 2 \phi + \omega_s \sin 2(\phi – \epsilon) = 0$
• $\phi$ = incl. orbit relative to equator
• Near asteroid, orbit lies close to equator. Further, orbit lies near orbit plane.
• LP unstable in $e$ for obliquity above 68.875 deg and $a$ between 0.9 and 1.25 Laplace radii (Tremaine et al. 2009)
• Evolution of contact binary
• Fission occurs. Jacobson & Scheeres 2011
• Dissipation $\rightarrow$ stable circ. orbit in LP
• Model: simple model — secular expansion of $a$ from BYORP and tides
• Const. accel. perp. to radial vector
• Use first Fourier coefficient for BYORP accel.
• Results
• Verify instability region
• Unstable region: eccentric instability causes deviation from LP, collision
• New (eccentricity) instability mode
• cf Cuk & Nesvorny 2010
• reimpact
• regardless of obliquity
• $\rightarrow$ contact binary
• Evection resonance could also play a role

## DDA 2015 – Gravity and Tide Parameters Determined from Satellite and Spacecraft Orbits

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: Dynamics of Small Solar System Bodies II

Robert A. Jacobson (JPL)

#### Abstract

As part of our work on the development of the Jovian and Saturnian satellite ephemerides to support the Juno and Cassini missions, we determined a number of planetary system gravity parameters. This work did not take into account tidal forces. In fact, we saw no obvious observational evidence of tidal effects on the satellite or spacecraft orbits. However, Lainey et al. (2009 Nature 459, 957) and Lainey et. al (2012 Astrophys. J. 752, 14) have published investigations of tidal effects in the Jovian and Saturnian systems, respectively. Consequently, we have begun a re-examination of our ephemeris work that includes a model for tides raised on the planet by the satellites as well as tides raised on the satellites by the planet. In this paper we briefly review the observations used in our ephemeris production; they include astrometry from the late 1800s to 2014, mutual events, eclipses, occultatons, and data acquired by the Pioneer, Voyager, Ulysses, Cassini, Galileo, and New Horizons spacecraft. We summarize the gravity parameter values found from our original analyses. Next we discuss our tidal acceleration model and its impact on the gravity parameter determination. We conclude with preliminary results found when the reprocessing of the observations includes tidal forces acting on the satellites and spacecraft.

#### Notes

• Jupiter and Saturn gravity fields program at JPL
• started with Pioneer
• probably end with Juno (or proposed Europa) mission
• also Earth-based
• 1874-2014
• Saturnrigh stellar occultations
• pole orientation
• Saturn ring plane crossing times
• pole orientation
• spacecraft:
• imaging
• VLBI
• Saturn ring occultations
• But no tidal forces used in any analysis so far.
• But tidal effects are not zero
• Lainey et al. 2009, 2012
• Efroimsky & Lainey 2007 (JGR 112)
• $U_{jk} = k_2^k \left(\dfrac{\mu_j}{R_k}\right)^3 \left(\dfrac{R_k}{r}\right)^3 \left(\dfrac{R_k}{r^*_{jk}}\right)^3 P_2\left(\hat{r} \cdot \hat{r}^*_{jk}\right)$
• $r^*_{jk} = r_{jk} – \Delta t_j \left[\dot{r}_{jk} + \dot{W}_k\left(\hat{r}_{jk}\times\hat{h}_k\right)\right]$
• Tidal lag effects
• Put tides in fitting model
• $\rightarrow k_2$
• $\rightarrow$ gravity harmonic coefficients
• tidal lags: indeterminate from existing data
• tidal dissipation function $Q = \dfrac{2 \pi E}{\Delta E} = f(\Delta t)$
• $E$ = max energy stored in one tidal cycle
• $\Delta E$ = energy dissipated during that cycle
• $f(\Delta t) = \dfrac{1}{\omega^{\alpha} \Delta t}$
• comparison to Lainey for Jupiter:
• indeterminate
• comparison to Lainey for Saturn (common $Q$):
• $\Delta t$ and $\dfrac{k_2}{Q}$ successfully detected for Mimas, Enceladus, Tethys, Dione, and Rhea, $k_2 = 0.379 \pm 0.011$
• Lainey: $\dfrac{k_2}{Q} = 2.3\pm0.7 \times 10^{-4}$, $k_2 = 0.341$
• JPL:$\dfrac{k_2}{Q} = 1.0\pm0.2 \times 10^{-4}$, $k_2 = 0.381 \pm 0.011$

## DDA 2015 – The Evidence for Slow Migration of Neptune from the Inclination Distribution of Kuiper Belt Objects

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 Nesvorny (SWRI)

#### Abstract

Much of the dynamical structure of the Kuiper Belt can be explained if Neptune migrated over several AU, and/or if Neptune was scattered to an eccentric orbit during planetary instability. An outstanding problem with the existing formation models is that the distribution of orbital inclinations predicted by them is narrower than the one inferred from observations. Here we perform numerical simulations of the Kuiper belt formation starting from an initial state with Neptune at $20\lt a^{N,0} \lt 30$ AU and a dynamically cold outer disk extending from beyond $a^{N,0}$ to 30 AU. Neptune’s orbit is migrated into the disk on an e-folding timescale $1 \le \tau \le 100$ Myr. A small fraction ($\sim10^{-3}$) of disk planetesimals become implanted into the Kuiper belt in the simulations. By analyzing the orbital distribution of the implanted bodies in different cases we find that the inclination constraint implies that $\tau \ge 10$ Myr and $a^{N,0} \le 26$ AU.The models with $\tau \lt 10$ Myr do not satisfy the inclination constraint, because there is not enough time for various dynamical processes to raise inclinations. The slow migration of Neptune is consistent with other Kuiper belt constraints, and with the recently developed models of planetary instability/migration. Neptune’s eccentricity and inclination are never large in these models ($e^N \lt 0.1$, $i^N \lt 2$ deg), as required to avoid excessive orbital excitation in the $\gt 40$ AU region, where the Cold Classicals presumably formed.

#### Notes

• Early SS evolution
• giant planets emerged from dispersing protopl disk on compact orbits (inside massive belt)
• planetesimal driven migration?
• dynamical instability?
• giant planets now spread from 5 to 30 AU
• Kuiper Belt is the best clue to evolution of Neptune’s orbit
• KB structure is complex (plot: $e$ vs $a$)
• between 3:2 and 2:1 MMRs: a mess, but hot and cold populations
• where did hot population come from (including high-$i$ 3:2 objects)?
• model: too many Plutinos compared to observations
• New model
• 4 outer planets
• ICs:
• Neptune starting points: 22, 24, 26, 28 AU
• Neptune migration e-folding timescales 1, 3, 10, 30, 100 Myr
• 1e6 particles, Rayleigh initial distribution
• swift_rmvs3 integrator
• 500 cores of Pleiades supercomputer
• 20 jobs total, most stopped 1 Gyr, interesting ones to 4 Gyr
• $\rightarrow$ result matches observed distribution
• 24 AU, 30 Myr
• but too manyPlutinos(?)
• observational bias?
• cf Petit et al. 2012
• CFEPS detection simulator
• agreement (of hot population) is actually pretty good
• Gomes capture mechanism:Gomes 2003
• 2:1 MMR secular structure is complex
• Conclusions:
• Neptune migrated into a massive cometary disk at $\lt 30$ AU
• Neptune’s migration hadto be slow
• need time to increase inclinations
• Model also explains other KB properties
• Initial disk had to be $\sim 20 M_\oplus$