## DDA 2015 – Constraints on Titan’s rotation from Cassini mission radar data

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.

Bruce Bills (JPL)

#### Abstract

We present results of a new analysis of the rotational kinematics of Titan, as constrained by Cassini radar data, extending over the entire currently available set of flyby encounters. Our analysis provides a good constraint on the current orientation of the spin pole, but does not have sufficient accuracy and duration to clearly see the expected spin pole precession. In contrast, we do clearly see temporal variations in the spin rate, which are driven by gravitational torques which attempt to keep the prime meridian oriented toward Saturn.

Titan is a synchronous rotator. At lowest order, that means that the rotational and orbital motions are synchronized. At the level of accuracy required to fit the Cassini radar data, we can see that synchronous rotation and uniform rotation are not quite the same thing. Our best fibng model has a fixed pole, and a rotation rate which varies with time, so as to keep Titan’s prime meridian oriented towards Saturn, as the orbit varies.

A gravitational torque on the tri-axial figure of Titan attempts to keep the axis of least inertia oriented toward Saturn. The main effect is to synchronize the orbit and rotation periods, as seen in inertial space. The response of the rotation angle, to periodic changes in orbital mean longitude, is modeled as a damped, forced harmonic oscillator. This acts as a low-pass filter. The rotation angle accurately tracks orbital variations at periods longer than the free libration period, but is unable to follow higher frequency variations.

The mean longitude of Titan’s orbit varies on a wide range of time scales. The largest variations are at Saturn’s orbital period (29.46 years), and are due to solar torques. There are also variations at periods of 640 and 5800 days, due to resonant interaction with Hyperion.

For a rigid body, with moments of inertia estimated from observed gravity, the free libration period for Titan would be 850 days. The best fit to the radar data is obtained with a libration period of 645 days, and a damping time of 1000 years.

The principal deviation of Titan’s rotation from uniform angular rate, as seen in the Cassini radar data, is a periodic signal resonantly forced by Hyperion.

#### Notes

• Titan:
• hard to see surface
• Cassini’s radar intended for mapping surface
• didn’t get much by way of repeat observations (“tie points”), which are needed to constrain rotation
• most data near poles — not terribly helpful
• Rotation model from tie-point observations
• Stiles et al. 2008: 50 tie points over 2.8 yr
• Now: 2602 tie points over 10 yr
• solve for 3 params (RA & DEC of spin pole, angular rate)
• $P = 15.94547727 \pm 6.03 \times 10^{-7}$ d
• spin pole precession
• gravity model: ~250 yr
• not clearly seen in data
• spin rate variations
• seen in data
• dynamical model
• assume Titan in synch. rotation
• gravity torque
• dissipation
• $\rightarrow$ libration period ~850 d
• Hyperion has nontrivial influence
• fit: libration period = 645.4 d, damping time = 430 yr, rotation period slightly changed

## DDA 2015 – Recent Formation of Saturnian Moons: Constraints from Their Cratering Records

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 Formation and Dynamics I

Henry C. (Luke) Dones (SWRI)

#### Abstract

Charnoz et al. (2010) proposed that Saturn’s small “ring moons” out to Janus and Epimetheus consist of ring material that viscously spread beyond the Roche limit and coagulated into moonlets. The moonlets evolve outward due to the torques they exert at resonances in the rings. More massive moonlets migrate faster; orbits can cross and bodies can merge, resulting in a steep trend of mass vs. distance from the planet. Canup (2010) theorized that Saturn’s rings are primordial and originated when a differentiated, Titan-like moon migrated inward when the planet was still surrounded by a gas disk. The satellite’s icy shell could have been tidally stripped, and would have given rise to today’s rings and the mid-sized moons out to Tethys. Charnoz et al. (2011) investigated the formation of satellites out to Rhea from a spreading massive ring, and Crida and Charnoz (2012) extended this scenario to other planets. Once the mid-sized moons recede far from the rings, tidal interaction with the planet determines the rate at which the satellites migrate. Charnoz et al. (2011) found that Mimas would have formed about 1 billion years more recently than Rhea. The cratering records of these moons (Kirchoff and Schenk 2010; Robbins et al. 2015) provide a test of this scenario. If the mid-sized moons are primordial, most of their craters were created through hypervelocity impacts by ecliptic comets from the Kuiper Belt/Scattered Disk (Zahnle et al. 2003; Dones et al. 2009). In the Charnoz et al. scenario, the oldest craters on the moons would result from low-speed accretionary impacts. We thank the Cassini Data Analysis program for support.

References
Canup, R. M. (2010). Nature 468, 943
Charnoz, S.; Salmon, J., Crida, A. (2010). Nature 465, 752
Charnoz, S., et al. (2011). Icarus 216, 535
Crida, A.; Charnoz, S. (2012). Science 338, 1196
Dones, L., et al. (2009). In Saturn from Cassini-Huygens, p. 613
Kirchoff, M. R.; Schenk, P. (2010). Icarus 206, 485
Robbins, S. J.; Bierhaus, E. B.; Dones, L. (2015). Lunar and Planetary Science Conference 46, abstract 1654
(http://www.hou.usra.edu/meetings/lpsc2015/eposter/1654.pdf)
Zahnle, K.; Schenk, P.; Levison, H.; Dones, L. (2003). Icarus 163, 263

#### Notes

• Can cratering records constrain moon ages?
• see http://space.jpl.nasa.gov
• small inner moons (and Mimas) interact strongly with rings — the so-called “ring moons”
• migrated from outer edge of rings ~100 Myr
• regular moons (Mimas-Iapetus) are (assumed?) primordial
• transition is abrupt where tidal forces prevent formation
• formation of moons from spreading rings:Charnoz et al. 2010,Canup 2010,Charnoz et al. 2011,Crida &Charnoz 2012
• outside Roche limit, formation
• Lainey et al. 2012: dissipation stronger than thought
• decreases timescale considerably
• Impact rates
• $R_{moon} = R_J \dfrac{R_S}{R_J} \dfrac{R_{moon}}{R_S}$
• Crater scaling: diameter vs. velocity
• impacts/$10^9$ yr: Mimas 8.5, Rhea 48
• Mimas & Rhea counts: Robbins et al. 2015 (LPSC)
• plot: #craters larger than D vs. D
• Mimas: saturated up to $D \sim 20-45$ km
• Rhea: saturated up to $D \sim 25$ km
• Summary
• Mimas: Craters are near saturation for diameters < 20 km
• Rhea: saturation < 25 km
• Ages may be underestimated

## DDA 2015 – On the Spin-axis Dynamics of the Earth

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.

Gongjie Li (Harvard) (Duncombe award winner)

#### Abstract

The variation of a planet’s obliquity is influenced by the existence of satellites with a high mass ratio. For instance, the Earth’s obliquity is stabilized by the Moon, and would undergo chaotic variations in the Moon’s absence. In turn, such variations can lead to large-scale changes in the atmospheric circulation, rendering spin-axis dynamics a central issue for understanding climate. The relevant quantity for dynamically-forced climate change is the rate of chaotic diffusion. Accordingly, here we reexamine the spin-axis evolution of a Moonless Earth within the context of a simplified perturbative framework. We present analytical estimates of the characteristic Lyapunov coefficient as well as the chaotic diffusion rate and demonstrate that even in absence of the Moon, the stochastic change in the Earth’s obliquity is sufficiently slow to not preclude long-term habitability. Our calculations are consistent with published numerical experiments and illustrate the putative system’s underlying dynamical structure in a simple and intuitive manner. In addition, we examine if at any point in the Earth’s evolutionary history, the obliquity varied significantly. We find that even though the orbital perturbations were different in the past, the system nevertheless avoided resonant encounters throughout its evolution. This indicates that the Earth obtained its current obliquity during the formation of the Moon.

#### Notes

• Obliquity $\cos \epsilon$ affects climate
• Mars obliquity variations caused collapse of Martian atmosphere
• Obliquity variations of a Moonless Earth
• without Moon, $\epsilon$ is chaotic (Laskar et al. 1993)
• geostrophic winds
• but N-body sims: $\epsilon$ constrained to $\epsilon \lesssim 45$ deg — why?
• Sun and planetary torques: spin precession rate, inclination variation
• model as superposition of linear modes
• resonance overlap: two connected chaotic zones — Laskar 1993, Morby 2000, Laskar 1996
• average over primary resonances $\rightarrow$ secondary resonances
• overlap of secondary resonances creates the chaotic bridge (Chirikov 1979)
• Results
• regular at $\ge 85$ deg
• less chaotic in bridge
• analytic and numerical are consistent
• Li & Batygin 2014a
• diffusion timescale 10 Myr in primary chaotic zones, 2 Gyr in the bridge
• Pre-late heavy bombardment evolution of Earth’s obliquity
• Li & Batygin 2014b
• solar system starts more compact (Nice model)
• study evolution of mode freqs and effects on Earth’s inclination
• also, Moon was closer
• two freqs match prior to LHB only if $\epsilon \ge 85$ deg
• $\therefore$ Earth’s obliquity arose during the formation of the Moon

## DDA 2015 – Recent dynamical evolution of Mimas and Enceladus

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 Formation and Dynamics III

Maja Cuk (SETI Institute)

#### Abstract

Mimas and Enceladus are the smallest and innermost mid-sized icy moons of Saturn. They are each caught in a 2:1 orbital resonance with an outer, larger moon: Mimas with Tethys, Enceladus with Dione. This is where the similarities end. Mimas is heavily cratered and appears geologically inactive, while Enceladus has a young surface and high tidal heat flow. Large free eccentricity of Mimas implies low tidal dissipation, while Enceladus appears very dissipative, likely due to an internal ocean. Their resonances are very different too. Mimas is caught in a 4:2 inclination type resonance with Tethys which involves inclinations of both moons. Enceladus is in a 2:1 resonance with Dione which affects only Enceladus’s eccentricity. The well-known controversy over the heat flow of Enceladus can be solved by invoking a faster tidal evolution rate than previously expected (Lainey et al. 2012), but other mysteries remain. It has been long known that Mimas has very low probability of being captured into the present resonance, assuming that the large resonant libration amplitude reflects sizable pre-capture inclination of Mimas. Furthermore, Enceladus seems to have avoided capture into a number of sub-resonances that should have preceded the present one. An order of magnitude increase in the rate of tidal evolution does not solve these problems. It may be the time to reconsider the dominance of tides in the establishment of these resonances, especially if the moons themselves may be relatively young. An even faster orbital evolution due to ring/disk torques can help avoid capture into smaller resonances. Additionally, past interaction of Mimas with Janus and Epimetheus produce some of the peculiarities of Mimas’ current orbit. At the meeting I will present numerical integrations that confirm the the existence of these problems, and demonstrate the proposed solutions.

#### Notes

• tidal rates $\dfrac{1}{a}\dfrac{d a}{d t}$: Mimas = 59, Enceladus = 23
• numerical integrations — brute force
• artificial migration
• slow
• the trouble with Mimas
• Mimas and Tethys in inclination-type 4:2 MMR
• inclination of both moons affected by the resonance
• libration amp. of resonance is large, ~100 deg $\rightarrow$ primordial Mimas inclination — doesn’t work
• eccentricity of Tethys has complex effects
• Mimas-Tethys evolution rate: $\dfrac{da_{moon}}{dt} \propto \dfrac{R^5_{planet}}{a^{3/2}}$
• introduce ad hoc ring torques — artificial torque on Prometheus
• gives Tethys resonance a kick
• $\therefore$ don’t take Mimas-Tethys resonance too seriously
• rings-Janus-Mimas-Enceladus-Dione system evolution is very complex

## DDA 2015 – On the in situ formation of Pluto’s small satellites

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.

Man Yin Woo (University of Hong Kong)

#### Abstract

The formation of Pluto’s small satellites – Styx, Nix, Keberos and Hydra remains a mystery. Their orbits are nearly circular (eccentricity $e = 0.0055$ or less) and near resonances and coplanar with respect to Charon. One scenario suggests that they all formed close to their current locations from a disk of debris, which was ejected from the Charon-forming impact. We test the validity of this scenario by performing N-body simulations with Pluto-Charon evolving tidally from an initial orbit at a few Pluto radii. The small satellites are modeled as test particles with initial orbital distances within the range of the current small satellites and damped to their coldest orbits by collisional damping. It is found that if Charon is formed from a debris disk and has low initial eccentricity, all test particles survive to the end of the tidal evolution, but there is no preference for resonances and the test particles’ final $e$ is typically > 0.01. If Charon is formed in the preferred intact capture scenario and has initial orbital eccentricity ~ 0.2, the outcome depends on the relative rate of tidal dissipation in Charon and Pluto, $A$. If $A$ is large and Charon’s orbit circularizes quickly, a significant fraction of the test particles survives outside resonances with $e \gtrsim 0.01$. Others are ejected by resonance or survive in resonance with very large $e$ (> 0.1). If $A$ is small and Charon’s orbit remains eccentric throughout most of the tidal evolution, most of the test particles are ejected. The test particles that survive have $e \gtrsim 0.01$, including some with $e \gt 0.1$. None of the above cases results in test particles with sufficiently low final $e$.

This work is supported in part by Hong Kong RGC grant HKU 7030/11P.

#### Notes

• Pluto satellite system
• 5 known
• Charon dominant
• all nearly coplanar
• all nearly circular
• all near MMR with Charon
• Brozovic et al. 2015
• Formation scenarios
• forced resonant migration
• Ward & Canup 2006
• Nix & Hydra formed in same giant impact that formed Charon
• ruled out by Lithwick & Wu
• multi-resonance capture
• unlikely (Cheng et al. 2014)
• collisional capture of planetesimals
• Lithwick & Wu 2008, Dos Santos et al. 2012
• ruled out: capture time << collisional timescale, also Walsh & Levison 2015
• in situ formation
• Kenyon & Bromley 2014
• giant impact produced debris ring
• problem: outward tidal evolution of Charon
• Solving the migration problem
• forced eccentricity — Leung and Lee 2013
• for $e_C = 0.24$, $e_f \sim 0.01$ to $0.02$ for test particles (small moons)
• integrate two tidal models, constant $\Delta t$ and constant $Q$
• For constant $\Delta t$, no preference for resonances and $e \gt 0.01$
• Conclusion: it is unlikely that all the small satellites formed close to their current position

## 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