DDA 2015 – The Evidence for Slow Migration of Neptune from the Inclination Distribution 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 $\sim 20 M_\oplus$

DDA 2015 – The Evolution of the Grand Tack’s Main Belt through the Solar System’s Age

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.

Rogerio Deienno (National Institute for Space Research)

Abstract

The Asteroid Belt is marked by the mixture of physical properties among its members, as well as its peculiar distribution of orbital eccentricities and inclinations. Formation models of the Asteroid Belt show that its formation is strongly linked to the process of terrestrial planet formation. The Grand Tack model presents a possible solution to the conundrum of reconciling the small mass of Mars with the properties of the Asteroid Belt, providing also a scenario for understanding the mixture of physical properties of the Belt objects. Regarding the orbital distribution of these objects, the Grand Tack model achieved good agreement with the observed inclination distribution, but failed in relation to the eccentricities, which are systematically skewed towards too large values at the end of the dynamical phase described by the Grand Tack model. Here, we evaluate the evolution of the orbital characteristics of the Asteroid Belt from the end of the phase described by the Grand Tack model, throughout the subsequent evolution of the Solar System. Our results show the concrete possibility that the eccentricity distribution after the Grand Tack phase is consistent with the current distribution. Finally, favorable and unfavorable issues faced by the Grand Tack model will be discussed, together with the influence of the primordial eccentricities of Jupiter and Saturn. Acknowledgement: FAPESP.

Notes

  • Asteroid belt:
    • formation process halted before formation of a planet due to Jupiter
    • so-called “Grand Tack” model
      • Walsh et al. 2011
      • Jup &  Sat migrate inwards, Saturn faster
      • inward stops, outward begins
      • but fails to explain a lot
        • current MB structure different from Grand Tack predictions
        • especially dist. in $e$, also $a$ ($i$ not bad)
  • This work
    • Num int 5 planets & 10,000 test particles, 4.5 Gyr
    • ICs: Grand Tack
    • Mercury integrator, 10-day time step — expensive
    • E-belt (Bottke et al. 2012) results @ 0.4 Gyr (planetary instability)
    • at 0.4 Gyr, reset planets to their current orbits
    • $\rightarrow$ asteroid belt of today — almost
      • much better match to observed $a$-$e$-$i$ distributions
      • lost somewhat more asteroids than observed
    • Influence of primordial eccentricities ofJup & Saturn
      • destabilizes MMRs
      • $\rightarrow$ constraints on primordial eccentricities
    • see also: Nesvorny & Morbidelli 2012 AJ 144:117

DDA 2015 – The onset of dynamical instability and chaos in navigation satellite 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.

Aaron Jay Rosengren (IFAC-CNR)

Abstract

Orbital resonances are ubiquitous in the Solar System and are harbingers for the onset of dynamical instability and chaos. It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Here we will show that the same underlying physical mechanism, the overlapping of secular resonances, responsible for the eventual destabilization of Mercury and recently proposed to explain the orbital architecture of extrasolar planetary systems (Lithwick Y., Wu Y., 2014, PNAS; Batygin K., Morbidelli A., Holman M.J., 2015, ApJ) is at the heart of the orbital instabilities of seemingly more mundane celestial bodies—the Earth’s navigation satellites. We will demonstrate that the occurrence and nature of the secular resonances driving these dynamics depend chiefly on one aspect of the Moon’s perturbed motion, the regression of the line of nodes. This talk will present analytical models that accurately reflect the true nature of the resonant interactions, and will show how chaotic diffusion is mediated by the web-like structure of secular resonances. We will also present an atlas of FLI stability maps, showing the extent of the chaotic regions of the phase space, computed through a hierarchy of more realistic, and more complicated, models, and compare the chaotic zones in these charts with the analytical estimation of the width of the chaotic layers from the heuristic Chirikov resonance overlap criterion. The obtained results have remarkable practical applications for space debris mitigation and for satellite technology, and are both of essential dynamical and theoretical importance, with broad implications for planetary science.

Notes

  • Motivation: space debris problem
    • Active debris removal is becoming necessary
    • New: exploit resonant orbits to obtain relatively stable graveyards or highly unstable disposal orbits
  • Resonance overlap & chaos
    • asteroid belt resonances: cf. DeMeo & Carry 2014 (Nature Rev)
    • What is resonant structure of cislunar space?
      • actually less well known than resonant structure of asteroid belt
    • Cislunar resonant phenomena:
      • tesseral resonances
      • MMRs
      • lunisolar semi-secular resonances (sun-synchronous, evection resonance)
      • secular resonances (crit. inclination, Kxxxx resonance)
    • Navsat orbits (European) are unstable!
      • Chao 2000, Jenkin & Gick 2002, Chao & Gick 2004
      • Also: interference from sats in disposal orbits
    • Ref: Mercury’s orbit and secular chaos
  • Harmonic analysis of Lunar perturbations
    • Tesseral and lunisolar semi-secular resonances cannot be the cause of orbital instabilities observed in numerical surveys
    • Role of secular resonances in producing chaos
      • simplifications:
        • 2nd order in ratio of semimajor axes
        • short periodic terms of disturbing function can be averaged out
      • resonance: $\dot{\psi} = (2-2p) \dot{\omega} + m \dot{\Omega} \pm s\dot{\Omega}_2 \approx 0$
    • chaotic diffusion (~250 yr)
      • Daquin et al. CMDA (in prep)
      • Chirikov res overlap criterion
      • chaotic web
        • plot: $e$ vs $i$
      • FLI stability maps
        • heat map: $e$ vs $i$
        • too many dimensions $\rightarrow$ far from understood

DDA 2015 – Increasing Space Situational Awareness for NEOs

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 J.G.J. Hestroffer (IMCCE/Paris observatory)

Abstract

Over the past years, Europe has strengthened its commitment to foster space situational awareness. Apart from the current efforts in tracking space weather, artificial satellites and space debris, Near Earth Asteroid threat assessment is a key task. NEOshield has been part of this European effort. We will give an overview over national projects and European programs with French participation such as PoDET, ESTERS, FRIPON, NEOShield, Gaia-FUN-SSO and Stardust. Future plans regarding Near Earth Object threat assessment and mitigation are described. The role of the IMCCE in this framework is discussed using the example of the post mitigation impact risk analysis of Gravity Tractor and Kinetic Impactor based asteroid deflection demonstration mission designs.

Notes

  • SSA:
    • debris
      • short-term & long-term stability
      • evolution of debris clouds
    • meteorites
      • fish-eye cameras
      • FRIPON network
      • triangulation, orbit/trajectory reconstruction
      • people don’t look up anymore
    • NEOs
  • NEOs
    • ~1500 detections/yr from various surveys
    • ~12,000 catalogued so far
      • still missing many — very incomplete
    • large NEOs: fairly complete census by now
    • GAIA: much ofunobservability cone overlaps NEO territory
      • ongoing ground-based surveys go fainter anyway
    • Need ~250 day arcs to get CEU $\le 1$ arcsec
    • http://neo.ssa.esa.edu
    • ESA appears to be starting to get serious about detection and mitigation schemes

DDA 2015 – The 2014 KCG meteor outburst: clues to a parent body

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 I

Althea V Moorhead (MSFC)

Abstract

The κ Cygnid (KCG) meteor shower exhibited unusually high activity in 2014, producing ten times the typical number of meteors. The shower was detected in both radar and optical systems and meteoroids associated with the outburst spanned at least five decades in mass. In total, the Canadian Meteor Orbit Radar, European Network, and NASA All Sky and Southern Ontario Meteor Network produced thousands of KCG meteor trajectories. Using these data, we have undertaken a new and improved characterization of the dynamics of this little-studied, variable meteor shower. The κ Cygnids have a diffuse radiant and a significant spread in orbital characteristics, with multiple resonances appearing to play a role in the shower dynamics. We conducted a new search for parent bodies and found that several known asteroids are orbitally similar to the KCGs. N-body simulations show that the two best parent body candidates readily transfer meteoroids to the Earth in recent centuries, but neither produces an exact match to the KCG radiant, velocity, and solar longitude. We nevertheless identify asteroid 2001 MG1 as a promising parent body candidate.

Notes

  • $\kappa$Cygnid shower:
    • Competes with the Perseids, so unfairly obscure.
    • Observations go back to 1869.
    • Orb elements unusually spread out.
    • Relatively slow: $v_g \sim 24$ km/s
    • Short trajectories
    • Multiple flares
    • Diffuse radiant
    • 2014:
      • Unusually active year
      • Canadian Meteor Orbit Radar
      • European network
  • Showermemberselection:
    • Use established shower orbits to select members (Drummond 1981)
    • Use observed (Sun-centered) radiantandvelocity to select members
      • drift over time — fit curve
      • peaks in stacked prob. distribution (in $a$) correspond to MMRs
  • Parent body search
    • Use D parameter to rank objects from JPL Small Body Database
    • 2002 LV,2008ED69,2001MG1
      • Not quite…
    • via location of descending node — intersection with Earth’s orbit (Jenniskens & Vaubaillon 2008)
    • Integrate and search for close encounters
    • Hard to get a parent body to match all search criteria

DDA 2015 – High precision comet trajectory estimates: the Mars flyby of C/2013 A1 (Siding Spring)

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

Davide Farnocchia (JPL, CalTech)

Abstract

The Mars flyby of C/2013 A1 (Siding Spring) represented a unique opportunity for imaging a long-period comet and resolving its nucleus and rotation period. Because of the small encounter distance and the high relative velocity, the goal of successfully observing C/2013 A1 from the Mars orbiting spacecrafts posed strict requirements on the accuracy of the comet ephemeris estimate. These requirements were hard to meet, as comets are known for being highly unpredictable: astrometric observations can be significantly biased and nongravitational perturbations significantly affect the trajectory. Therefore, we remeasured a couple of hundred astrometric positions from images provided by ground-based observers and also observed the comet with the Mars Reconnaissance Orbiter’s HiRISE camera on 2014 October 7. In particular, the HiRISE observations were decisive in securing the trajectory and revealed that nongravitational perturbations were larger than anticipated. The comet was successfully observed and the analysis of the science data is still ongoing. By adding some post-encounter data and using the Rotating Jet Model for nongravitational accelerations we constrain the rotation pole of C/2013 A1.

Notes

  • Observations
    • 140,000 km close approach
    • HiRISE FoV: 4×4 mrad $\rightarrow 280$ km
    • post-conjunction updates: positions kind of all over the place
      • center of light is not coincident with position
      • PSF is diffuse, not starlike
      • most astrometry was from amateurs
      • non-grav perturbations
        • previously unknown for this comet
      • use observations fromMRO!
        • provided good constraints on non-grav params
  • post-flyby
    • astrometric predictions sucked!
    • rotating jet model
      • spin pole
      • thrust angle between jet and pole
      • superposition of two jets
      • avg over rotation
      • $\rightarrow$ better trajectory
      • $\rightarrow$ spin pole direction

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 variations due to synodic “chopping” effects

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Katherine Deck (CalTech)

Abstract

Gravitational interactions between planets in transiting exoplanetary systems lead to variations in the times of transit (TTVs) that are diagnostic of the planetary masses and the dynamical state of the system. I will present analytic formulae for TTVs which can be applied to pairs of planets on nearly circular orbits which are not caught in a mean motion resonance. For a number of Kepler systems with TTVs, I will show that synodic “chopping” contributions to the TTVs can be used to uniquely measure the masses of planets without full dynamical analyses involving direct integration of the equations of motion. This demonstrates how mass measurements from TTVs may primarily arise from an observable chopping signal. I will also explain our extension of these formulae to first order in eccentricity, which allows us to apply the formulae to pairs of planets closer to mean motion resonances and with larger eccentricities.

Notes

  • Still don’t know much about formation and evolution of exoplanet systems
  • Use TTVs to measure planet masses?
  • e.g. Kepler 36
    • TTV amplitude ~2 hr p-p
    • mass constraints: Carter et al. 2012
    • composition constraints: Rogers et al. in prep
  • TTVs largest nearMMRs
    • Lithwick et al. 2012
    • $\dfrac{\delta t}{P} \propto \dfrac{M_{pert}}{M_{star}}$
    • short-period components and res components
  • Derive formula for synodicTTVs
    • sums of sinusoids, linear in mass ratios and periods [duh]
    • constrain masses
      • measure harmonic component period $\rightarrow$ mass ratio
    • Near first order MMR, degeneracy between mass and eccentricity breaks
    • Schmidt et al. 2015
    • Can use to set upper bounds, even in absence of TTVs
  • see also Algol et al. 2005, Nesvorny & Vokrouhlicky 2014

DDA 2015 – Dynamical Analysis of the 6:1 Resonance of the Brown Dwarfs Orbiting 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 Observaons 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 formation – Applicaon 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 Evolution 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 Observations 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 Relative 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
    • Add dissipation
    • $\rightarrow$ 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 – Instabilies 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.

Anne-Marie Madigan (UC Berkeley) (invited)

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
    • Madigan 2009
    • 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
    • Prograde precession case
      • torque from disk grav. reducesang. momentum, increasing eccentricity.
        • produces oscillations
        • but stable disks
      • Andromeda
    • Retrograde precession:
  • Inclination instability
    • Madigan 2015
    • 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 relativistic orbits and gravitational 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:
    • AdvLIGO/Virgo+ (~2015+)
      • 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
      • Orbital and radiation-reaction timescales
      • 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
    • radiative effects important
    • 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)