This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Eric Ford (Penn State)
Abstract
Observations of strongly interacting planetary systems in or near a mean motion resonance are unusually sensitive to planet masses and orbital properties, including dynamical properties that can help illuminate planet formation. Having developed a powerful toolbox for translating Doppler and/or transit timing observations into physics parameters, now we are able to characterize the resonant and secular behaviour of several strongly interacting planetary systems. I will present recent results for selected resonant and near-resonant planetary systems and discuss implications for planet formation. In particular, I will address implications for the nature and extent of orbital migration for giant and low-mass planets.
Notes
How didSTIPs form?
Three strawman models:
In situ formation: wrong
Large-scale disk formation: wrong
Nearly in situ formation plus modest early radial drift
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
William Newman (UCLA)
Abstract
Öpik (1976) proposed that close-range gravitational interactions between planetesimal material and planets could be approximated by a two-step integration scheme: (1) while the planetesimal was outside the gravitational sphere of influence of the planet, its orbit would be described by a heliocentric Keplerian orbit; and (2) once its orbit entered the sphere of influence of the planet, its trajectory would then become a planetocentric Keplerian orbit until it exited the sphere of influence and resumed a heliocentric path. This approximation, however, was also limited by the requirement noted by Öpik that the perihelion or aphelion distance of the planetesimal differ from the orbital distance of the planet from the sun. This approximation proved to be a useful tool during early solar system dynamical investigations but this process was often employed as a numerical integration method without checking Öpik’s requirements, as well as establishing whether the orbital passage through the sphere of influence was sufficiently accurate. Öpik’s scheme was used to establish many features of solar system evolution, including the commonly-held belief that the giant planets serve as a shield preventing substantial numbers of planetesimals from entering the inner solar system. Wetherill (1994) in a pioneering work that exploited the Öpik approximation as an integration scheme estimated that present-day Jupiter could prevent 99.9% of planetesimals from entering the inner solar system. Here, we employ high precision first principles calculations of the orbits of swarms of planetesimals emerging from the Jupiter-Saturn, Saturn-Uranus, and Uranus- Neptune zones and have shown (1) the conditions necessary for Öpik’s approximation to be valid fail for a substantial fraction of the planetesimal population during their lifetimes, and (2) approximately 44% of the planetesimal swarm originating in the Jupiter-Saturn zone alone are injected into the inner Solar System while 18% ultimately become Earth-crossers.
Notes
Does Jupiter shield the inner solar system?
Impact history
Öpik:
novel scheme for solar system integrations
but identified a useful criterion for valid (numerical) results
exploits near-Keplerian orbits of inner SS
Öpik’s method:
Keplerian time step
…
…
Criterion: aphelion or perihelion must be different from mean distance. [um…duh]
Öpik’s result: 99.95% of outer solar system planetesimals could not have entered inner solar system!
Newman: let’s check, using an extremelyaccurate numerical integrator.
counted fraction of particles in each planet-planet outer zone fail Öpik’s criterion
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Session: Planet Formation II
Kevin Graves (Purdue)
Abstract
We model the effects of tidal dissipation in the late stages of planetary accretion. We investigate the tidal dissipation during close encounters between embryos and nearly-formed planets using a modified version of the N-body integrator SyMBA. We calculate a total energy lost due to tides per close encounter and estimate the change in velocities of the bodies at each encounter. We measure the effects on the dynamics, evolution, and final outcome of the planets. Our initial results show a clear separation between the tidal and non-tidal case for a relatively strong tidal dissipation factor. We compare these results to traditional late stage simulations both with and without fragmentation.
Notes
Overview of late-stage terrestrial planet accretion
a few dozen embryos
a few thousand planetesimals
Morby 2012
giant plant migration?
increases AMD of inner solar system
must therefore start with a lower deficit
AMD: Jacobson & Morbidelli 2014
Tidal effects on planetary embryos
Lots of heat generation from various processes $\rightarrow$ magma oceans
Simple model for energy loss during a close encounter (Kaula & Harris 1973): tides
free parameters: tidal Love numbers, dissipation param
combine to a “tidal parameter”: $\frac{h_2 (k_2 + 1)}{Q}$
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Hal Levison (SwRI)
Abstract
Building the terrestrial planets has been a challenge for planeVormation models. In particular, classical theories have been unable to reproduce the small mass of Mars and instead predict that a planet near 1.5 AU should roughly be the same mass as the Earth (Chambers 2001, icarus 152,205). Recently, a new model, known as ‘slow pebble accretion’, has been developed that can explain the formation of the gas giants (Levison+ 2015, Nature submitted). This model envisions that the cores of the giant planets formed from 100 to 1000 km bodies that directly accreted a population of pebbles (Lambrechts & Johansen 2012, A&A 544, A32) – centimeter- to meter-sized objects that slowly grew in the protoplanetary disk. Here we apply this model to the terrestrial planet region and find that it can reproduce the basic structure of the inner Solar System, including a small Mars and a low-mass asteroid belt. In particular, our models show that for an initial population of planetesimals with sizes similar to those of the main belt asteroids, slow pebble accretion becomes inefficient beyond ~1.5 AU. As a result, Mars’s growth is stunted and nothing large in the asteroid belt can accumulate.
Notes
Standard view:
disk forms, dust settles to midplanet
dust accumulates, ~1-10 km
runaway growth
oligarchic growth
late-stage
violent endgame for terrestrial planets
main problem: Mars is way to small
possible solution: pebble accretion
dust
settling dust creates turbulence
~10 mm – 1 m pebbles
large planetesimals can accrete pebbles very effectively
strong gas drag $\rightarrow$ huge collision cross section (~Hill sphere)
Can this explain the low mass of Mars?
low-pebble-mass exponential cutoff
encounter time too short
A Ceres can grow if $r \lt \sim 1$ AU, but it can’t grow if $r \gt \sim 1$ AU.
$\rightarrow$ leaves ~20 planets inside of ~1 AU
subsequently very unstable and < 1 AU largely clears out
leaves behind essentially the Solar System architecture
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Brett Gladman (CITA)
Abstract
Continuing the established tradition in the field of speculative “fairy tales”, we postulate that our Solar System once had a set of several additional Earth-scale planets interior to the orbit of Venus. This would resolve a known issue that the energy and angular momentum of our inner-planet system is best explained by accreting the current terrestrial planets from a disk limited to 0.7-1.1 AU; in our picture the disk material closer to the Sun also formed planets, but they have since been destroyed. By studying the orbital stability of systems like the known Kepler systems, Volk and Gladman (companion abstract) demonstrate that orbital excitation and collisional destruction could be confined to just the inner parts of the system. In this scenario, our Mercury is the final remnant of the inner system’s destruction via a violent multi-collision (and/or hit-and-run disruption) process.This would provide a natural explanation for Mercury’s unusually high eccentricity and orbital inclination; it also fits into the general picture of long-timescale secular orbital instability, with Mercury’s current orbit being unstable on 5 Gyr time scales. The common decade spacing of instability time scales raises the intriguing possibility that this destruction occurred roughly 0.6 Gyr after the formation of our Solar System and that the lunar cataclysm is a preserved record of this apocalyptic event that began when slow secular chaos generated orbital instability in our former super-Earth system.
Notes
Motivation
inner edge of terrestrial planet zone
Mercury is weird.
Why don’t we have a STIP (system of tightly-packed inner planets)?
Mercury:
surfing the edge of secular chaos
not clear how it got to $e^2 + i^2 \sim (0.25)^2$
tough to strip mantle without it quickly falling right back
Ausphaug & Reiner (2014): Mercury is the end state of a sequence of collisions.
Why is there an inner edge?
Wetherill 1978 (Protostars & Planets): E and L of terrestrial planets requires an inner edge ~0.6 AU.
Historical way out: it’s too hot.
But modern studies indicate $T < 1500$K until much later.
If there is (collision) debris, where does it go?
radiation pressure: days
PR drag: kyr
meteoritic transfer: kyr-Myr
planetary interactions: ~10 Myr
$\rightarrow$ disappears quickly
if self-collisional, it will still disappear quickly
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Session: Planet Formation I
Doug Lin (UC Santa Cruz) (Brouwer award winner)
Abstract
Radial velocity and transit surveys indicate the presence of super Earth around half of the main sequence stars regardless of their mass and metallicity. In contrast, the frequency of gas giants is much lower and increases with stellar mass and metallicity. I will show how the emergence of super-Earth is a robust process whereas the formation of gas giant planets is a threshold phenomena. The topics to be discussed include physical barriers in the planet building process, the role of migration in their evolving natal disks, planets’ interaction with each other and with their host stars. I will also discuss some key observations which may provide quantitative tests for planet formation theories.
Notes
Observed properties of exoplanets: Howard 2013 (Science)
Showstoppers:
disk formation
grain growth: the “meter barrier”
Trapping of refractory grains beyond the magnetospheric cavity
Tends to pile up at boundary
grain growth: the “kilometer barrier”
collisional fragmentation vs. grav.
oligarchic barrier: isolation mass
typically very small
embryo retention barrier — Type I migration
planet-disk tidal interaction
get to high mass $\rightarrow$ migrate outward
resonant sweeping $\rightarrow 2^{nd}$ generation
core barrier: embryo resonant trapping
bypass the resonant barrier
inner scattered outward, outer scattered inward $\rightarrow$ collisions $\rightarrow$ impacts of super Earths
gas accretion barrier
Is there a threshold mass for gas accretion?
runaway accretion
Why didn’t this happen for observed super Earths?
plenty of material left over: why didn’t they turn into gas giants?
Measured disk accretion rate…?
metal rich stars: no observed dependence, despite theory
But metallicity of star and disk need not be the same.
Rapid growth of proto gas giants
grand design barrier: dynamical instability
How did gas giants acquire their eccentricities?
Type II migration
provides constraint on growth process
Why did hot Jupiters stop their inward migration?
Close in planets
e.g. Kepler-78
8-hour period
Star is magnetic
~15 g
analogous to Jupiter-Io
induced EMF (unipolar induction) $\rightarrow$ energy dissipation at expense of planet’s orbit
Planet surface cannot be iron; must be silicates.
Flux tube footprints on star should move at period of planetary orbit, not stellar rotation.
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
The Kepler Mission has detected dozens of compact planetary systems with more than four transiting planets. This sample provides a collection of close-packed planetary systems with relatively liRle spread in the inclination angles of the inferred orbits. We have explored the effectiveness of dynamical mechanisms in exciting orbital inclination in this class of solar systems. The two mechanisms we discuss are self-excitation of orbital inclination in initially (nearly) coplanar planetary systems and perturbations by additional unseen larger bodies in the outer regions of the solar systems. For both of these scenarios, we determine the regimes of parameter space for which orbital inclination can be effectively excited. For compact planetary systems with the observed architectures, we find that the orbital inclination angles are not spread out appreciably through self-excitation, resulting in a negligible scaRer in impact parameter and a subsequently stable transiting system. In contrast, companions in the outer solar system can be effective in driving variations of the inclination angles of the inner planetary orbits, leading to significant scatter in impact parameter and resultantly non-transiting systems. We present the results of our study, the regimes in which each excitation method – self-excitation of inclination and excitation by a perturbing secondary – are relevant, and the magnitude of the effects.
$2^{nd}$ order secular Laplace-Lagrange theory (Murray &Dermott)
inc. & ecc. decoupled
Inclination as function of time (analytical)
Use Kepler 4+ planets as model systems
Conclusions:
Self-excitation in compact solar system planets does not appear to be a significant mechanism
Current Kepler systems with non-transiting planets could have started out transiting but driven out of transit by self-excitation
Excitation by compact solar system planets themselves (smear their mass into a disk) does notappearto be a significant mechanism
It might be possible to see multi-transiting systems with Jovian masses (if they exist)
Dynamical transit duration variations due to secular interactions will be small ($10^{-4}$ to $10^{-7}$ sec) but potentially observable (via statistics on long time series)
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Christopher Spalding (CalTech) (Duncombe prize winner)
Abstract
A recent paradigm shift in exoplanetary astronomy has come with the detection of a substantial number of planets possessing orbits that are misaligned with respect to the spin axes of their host stars. Moreover, observations of misalignments now include coplanar, multi-transiting systems, suggesting that these planets inherited their orbital planes from a protoplanetary disk which was once itself inclined with respect to the star. It has been proposed that mutual star-disk inclination may arise as a consequence of turbulence within the collapsing molecular cloud core, out of which both the star and its disk form. Alternatively, misalignments may be aRained later on, through secular interactions between the disk and companion stars. In this work, we examine the secular dynamics of the stellar spin axis arising in response to the gravitational and accretional torques communicated between the star and its disk throughout the epoch of star and planet formation. Our analysis shows that even though the disk forms from turbulent material, and is thus expected to exhibit a stochastic variation in its orientation with time during the star formation process, gravitational disk-star coupling adiabatically suppresses the excitation of mutual star-disk inclination under all reasonable parameter regimes. As such, the excitation of mutual star-protoplanetary disk inclination must occur later on in the disk’s lifetime, by way of an encounter with a secular resonance between stellar precession and the gravitational perturbations arising from an external potential, such as a binary companion.
Notes
Motivation: our solar system, Laplace 1796
Ecliptic disk oriented approx perp to Sun’s spin axis
Goldreich & Tremaine 1980:
disk-driven migration
Jupiters eaten by stars
Why aren’t observed hot Jupiters eaten?
$\rightarrow$ hot Jupiters should be aligned with their disks
But significant fraction is seriously misaligned!
Tends to be more massive planets
How to getmisalignments?
Disk-driven migration doesn’t work
High-eccentricity + tidal?
Cannot explain multi-transiting misaligned systems (Huber et al. 2013)
$\rightarrow$ Are disks really aligned with their stars?
Hypothesis 1: misalignment during formation
Spalding et al. 2014 (ApJ)
Cores are turbulent
Spin dir varies randomly by $\approx30^{\circ}$ every ~0.01 pc
Shell infall time $\approx 10^4$ yr
Disk adopts plane of whatever shell falls last (Bate et al. 2010)
Star-disk system forms misaligned
BUT: disk-star coupling?
Young stars spin rapidly $\rightarrow$ oblate
Dynamically equivalent to massive wire around point mass
$\rightarrow$ disk precession
Use Laplace-Lagrange secular theory
Disk annuli act as outer perturbers upon stellar irientation
$\rightarrow$ precession period ~100 years(!)
Numerical simulation — will star spin axis follow motion of disk?
Star trails disk, even though motion stochastic
Hypothesis 2: binary companion in orbit around star+disk — disktorquing
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Konstantin Batygin (CalTech)
Abstract
The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this talk, we will show how an integrable Hamiltonian formalism for planetary resonances that allows both secondary bodies to have finite masses and eccentricities, can be used to construct a comprehensive theory for resonant capture. Employing the developed analytical model, we shall examine the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets, and demonstrate that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean motion commensurabilities on slightly eccentric (e ~ 0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary disks.
Notes
SeeMécaniqueCéleste, Laplace 1805!
But origins not really understood until Roy & Ovenden 1954, Goldreich 1964 (MNRAS)
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Kathryn Volk (U. British Columbia)
Abstract
Kepler revealed the common existence of tightly-packed planetary systems around solar-type stars, existing entirely on orbits with periods shorter than ~200 days. Those systems must have survived for the ages of their host stars (~5 Gyr), so their formation mechanism must provide inter-planet spacings that permit long-term stability. If one postulates that most planetary systems form with tightly-packed inner planets, their current absence in some systems could be explained by the collisional destruction of the inner system after a period of meta-stability. The signatures of such intense collisional environments may have been observed around stars in the form of rapidly varying debris disks; in these observed disks, collisional products are being disposed of via drag down onto the star or grinding to the nearly instantaneous dust blow-out limit. We use the orbital spacings and planet masses of the observed Kepler multi-planet systems to investigate the stability and long-term behavior of the systems. We find that many of our Kepler system analogs are unstable on 100 Myr timescales, even for initially small eccentricities (0-0.05); the instability timescales in these systems are distributed such that equal fractions of the systems experience planetary collisions in each decade in time. We discuss the likely outcomes of collisions in these systems based on the typical collision speeds from our numerical integrations and what implications this has for interpreting the observed Kepler multi-planet systems. The possible implications for our Solar System are discussed in a companion abstract (Gladman and Volk).
Notes
Architectures of close-in (closely packed) planetary systems (from Kepler)
Fabrycky 2014
~5-10% ofFGK field stars
These systems must be stable on Gyr timescales
Are all stars formed tightly packed?
Modeled 13 such Kepler systems
Preserved $a$ and masses, orbital angles randomized
Allowed $e_0$ to vary $0 < e_0 < 0.05$
Sudden onset of instability in 11 of these 13 after tens to ~100 Myr
[why is she surprised?]
These eccentricities are in range of observed values
Decay rates consistent with e.g. Holman & Wisdom (1992 AJ)
Why sudden onset?
History is very sensitive to ICs [duh]
Consolidation (low-speed collisions) vs. Destruction (high-speed collisions)
First collision is often near the accretion/erosion boundary — i.e., low-speed
Masses in 4-5 planet systems tend to be lower, while individual masses in ~3-planet systems are higher: mergers?
Tracked collision speeds during integrations.
Second collision often goes into erosion regime (i.e., high-speed)
Observing debris should be rare (but see Meng et al. 2012)
Ergodicity allows large variety of outcomes
$\Rightarrow$ tightly packed systems could be ubiquitous initially
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Session: Exoplanet Theory II
Maryame El Moutamid (Cornell)
Abstract
Many bodies in the Solar System and some exo-planets are close to or captured in Mean Motion Resonances (MMR). Capture into such resonances has been investigated by many authors. Indeed, the Hamiltonian equations of motion in presence of migration are given by Sicardy and Dubois Cel. Mech. & Dyn. Astron., 86, 321-350 (2003). Fleming and Hamilton, Icarus 148, 479-493 (2000), studied the problem in a less generic context. In these two papers, the authors studied the problem of 1:1 corotation (Lagrange points L4 and L5), rather than m+1:m corotations (El Moutamid et al, Cel. Mech. & Dyn. Astron, 118, 235-252 (2014)). We will present a generic way to analyze details of a successful (or not) capture in the case of an oblate (or not) central body in the context of Restricted Three Body Problem (RTBP) and a more General Three Body Problem in the context of known statistics for captured exoplanets (candidates) observed by Kepler.
Notes
Captures partial near MMR (Fabrycky et al. 2012)
No generic study on coupling between associated resonances (ERTB vs. general TB)
1) simple model,2DoF — $(m+1) n’ \approx m n$
splitting the corotation and Lindblad resonances (by $J_2 \neq 0$)
Lindblad: vary $e$
corotation: pendular motion (conserves $e$)
plot: $J_c – J_L$ vs. $\phi_C$
2) general case
can define a constant of motion: $J_{c,relat} = \frac{A^2 \xi}{m} – \frac{A’^2 e’}{m+1} – ?? = const.$
Add dissipationforMMR capture
ratio: potential barrier of one vs. other body
plot: potential energy vs. critical angle of corotation
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Russell Deitrick (U. Washington)
Abstract
In order to properly assess the potential for habitability and prioritize target selection for the characterization of exoplanets, we need to understand the limits of orbital and rotational dynamics. Large satellites may be rare and very difficult to detect. Consequently, it is necessary to quantify the likelihood of a planet’s having extreme obliquity cycles in the absence of a moon and to model the potential impact on the planet’s climate. We explore the obliquity evolution of (1) known exoplanet systems that could contain Earth-like planets in the habitable zone and (2) hypothetical planets in mutually inclined, chaotic resonant configurations that experience some of the most extreme orbital evolution possible. We use a secular obliquity model coupled to either an N-body models or a 4 order secular orbital model. We find that in some known systems, planets’ obliquity variations are small and unlikely to have a major effect on climate, unless undetected planets are present. Systems with three or more planets are significantly more dynamically rich, with planets that undergo obliquity changes of ~10° over 50,000 years and >30° over a few million years. In resonant configurations, Earth-like exoplanets can undergo dramatic and chaotic evolution in eccentricity and inclination while remaining stable for over 10 Gyr. In configurations in which eccentricities and inclinations stay below ~0.1 and~10°, respectively, obliquities oscillate quasi-periodically with amplitudes similar to the non-resonant, three-planet configurations. In more dynamically active configurations, in which eccentricities and inclinations evolve to e > 0.3 and i > 15°, obliquities can extend from ~0° to well past 90°. In extreme cases eccentricities can reach >0.9999 and inclinations >179.9 degrees, driving precession rates in excess of degrees per year. However, these planets can graze or impact the stellar surface and are probably not habitable.
Notes
$\upsilon$Andromedae c and d
obliquity oscillations
Model description
Barnes, Deitrick et al. 2015
Using the secular disturbing function (Murray & Dermott) and a secular obliquity model (Kinoshita 1975, 1977)
HD190360
obliquity varies w large amplitude in a “strip” in $\Delta i_0$ – $e_0$ plane — WTH?
two planets interacting (an Earth and a super-Jupiter) … somehow
Inside the “strip”, a commensurabilitylibrates
$(\varpi’ – \varpi) – (\Omega + p_A)$
outside the “strip”: no libration
Analogous to a compound pendulum
Summary
Non-coplanar systems in MMR exhibit long-lived chaos.
These systems can be formed by scattering.
Possible way to form misaligned hot Jupiters.
Earth-like planets in these systems can also have chaotic obliquity variations.
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Christa Van Laerhoven (CITA)
Abstract
Considering the secular dynamics of multi-planet systems provides substantial insight into the interactions between planets in those systems. Secular interactions are those that don’t involve knowing where a planet is along its orbit, and they dominate when planets are not involved in mean motion resonances. These interactions exchange angular momentum among the planets, evolving their eccentricities and inclinations. To second order in the planets’ eccentricities and inclinations, the eccentricity and inclination perturbations are decoupled. Given the right variable choice, the relevant differential equations are linear and thus the eccentricity and inclination behaviors can be described as a sum of eigenmodes. Since the underlying structure of the secular eigenmodes can be calculated using only the planets’ masses and semi-major axes, one can elucidate the eccentricity and inclination behavior of planets in exoplanet systems even without knowing the planets’ current eccentricities and inclinations. I have calculated both the eccentricity and inclination secular eigenmodes for the population of known multi-planet systems whose planets have well determined masses and periods. Using this catalog of secular character, I will discuss the prevalence of dynamically grouped planets (‘groupies’) versus dynamically uncoupled planets (‘loners’) and how this relates to the exoplanets ‘long-term eccentricity and inclination behavior. I will also touch on the distribution of the secular eigenfreqiencies.
Notes
Secular character of multi-planet system
planet-planet interactions
only need masses and semimajor axes (not eccentricity, not inclination) to set secular structure
two-planet system: two eccentricityeigenmodes
$h = e \sin \varpi$, $k = e \cos \varpi$ plot: $e$ is a vector
each $e$ vector is the sum of two eigenvectors
3-planet system: “groupie”-ness and loners
groupies:
$e$ highly variable
$\varpi$ precession not uniform
loners:
$e$ does not vary by much
$\varpi$ precesses steadily
Kepler-11
outer planet is a loner — does not interact with others
5 inner planets are groupies — interact strongly with each other
Summary: most planets are “groupies”, “loners” are rare.
This is one of a series of notes taken during the 2015 meeting of the AASDivision on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.
Session: Exoplanet Theory I
Ruth Murray-Clay (UC Santa Barbara) (invited)
Abstract
[none]
Notes
How do giant planets and brown dwarfs form?
Architecture of Solar System is atypical.
Lots of gas giants at large distances, small distances (“hot Jupiters”), but not much in between a la Solar System. Why?
This (see photo) is how I spent my afternoon and evening, today. I have a conference to attend next week and must present a poster paper on some recent research results. Because I know by now that both Old Man Murphy and Loki the Trickster always lie in wait, snickering — I hear you, you bastards — I go to check the large-format printer. It is a Beast, and it turns electrons into poster papers. I flip the power switch, and it makes a horrible noise, won’t boot up, freezes, then whines plaintively, “call HP … call HP … please, won’t you call HP ….” Not very encouraging. Screw you, Loki — thou art a Puck.
As with all things computer that misbehave, I keep trying the same thing over and over, hoping for a different result, though I know full well that no different result will … um … result. Indeed, no dice. Run around the building and check with everybody: nobody knows what’s wrong or what happened. Yeah, sure.
What to do? Go find some screwdrivers, of course. The horrible noise emanates from somewhere around the power supply. Sort of. It’s buried in the guts of the Beast, so it’s hard to tell from the outside. It is a place to start, anyway. I roll up the sleeves of my robe, pick up a Holy Implement of Torx, and get to work …
Several hours later, I finally have figured out, cuss word by cuss word (proper ordering is important), how to get past all the barriers cleverly designed by Evil HP Engineers to make rational disassembly near-impossible. (Ever disassemble a laptop computer, down to the bare metal? This is harder, I kid you not.) Sixty screws later (I count them, twice), I get to the power supply fan. The heart of the Beast is diseased, despoiled. It is not turning quite right, and the motor shaft wiggles a little. It is not supposed to wiggle. Even a little. Culprit apprehended at last? Perhaps. Fortunately, it’s just a cheap $8 cooling fan you can pick up at any Radio Shack.
But Radio Shack does not exist anymore. When did that happen?
We have come round to this place again: what to do? Rummage around in the junk spare parts room, of course. It is a glorious room, beloved of tinkerers on staff. Bingo: six salvaged computer power supplies, just lying there on a shelf, calling to me. No, seven! But I am wise to their siren song. One after another, a closer look reveals frightening ugliness — mostly in the form of caked-on dust and dirt and grime. Their hearts spin, but they are Unclean and Decrepit. Sigh … last one: yay, Cleanliness! The Blessed One, Savior of the Beast, is found.
It believes it has been bestowed a new chance at life. I wish I could be happy for it. Little does it know its fate. Surely it deserves to be told of its pending doom? Yet that would crush its new-found hopes. You are perverse and cruel, you Fates! I do not have the heart to tell it.
True to my calling as Lord High Tinkerer, I pick up the Holy Implement of Torx and sacrifice the Blessed One upon the Ancient Altar of Gorthung (a fifty-year-old, government-issue desk, solid and heavy as a tank, with an ice-cold slate top). I flay its body and cut out its heart. I know no mercy.
Fan in bloody hand (a blood blister acquired some time during printer pieces-parts separation has popped), I trundle down the hill to the electronics lab. There, a colleague — the Wizard of Wire, Lord of Circuit — performs minor surgery. Lo, and behold! Upon application of the Lightning of Zoltar (a 12-volt power supply), the heart of the Blessed One lives again, spinning round and round in a most pleasing whir. Back up the hill.
That dreaded niggle squatting in the back of my mind finds a crack and blossoms. It dawns on me: now I have to put it all back together. Sixty screws. I realize I am tired. I’ll never remember where they all go. Come back tomorrow with freshly caffeinated veins? Pffft. Such is for wusses, unbecoming of a Tinkerer. So, since the operation of my memory — even on a good day — resembles most closely that of a sieve, I have little choice but to re-figure out how to take apart the Beast but in reverse. I am reminded of Ginger Rogers. I miss Ann Richards and her rapier wit. Today is not a good day.
Another hour passes by. I wave hi. We do that a lot, Time and I. My finger leaks on the table; I wipe it. And also on the housing of the reassembled printer power supply. I look at the smear, and I do not wipe it. I have left my mark upon this Beast, I think to myself. I shall not remove it. It will be buried amidst your guts; only you and I will ever know it is there. This token of my toil is enough, I decide. I move on.
At last, it is back together, despite all the King’s men staying home, watching TV. I do not want to plug it in. I’m sure you understand. Don’t you? Even so, I still roll the Beast back to its lair. I reattach its stiff black tail. I notice it is dirty, the cord, this conduit of the Lightning of Zoltar.
We have arrived at the moment of truth: I flip the switch. And wait. As with a pot of water that has yet to boil, it is best not to stare at a booting computer, especially one as slow and dumb as the Beast’s. I stare anyway. I wave hi to passing Time again, then it whirs with a pleasing sound. And dies. And tells me to call HP.
Naturally, I turn it off, wait ten seconds (capacitors can be slow to bleed, you know), and then turn it on again. Maybe something different will happen this time.
