Institute for Advanced Study
1) The case for stochastic orbital migration 2) Open Exoplanet Catalogue
Hanno Rein @ UoT St. George Campus, March 2013 Extra-solar planet census All discovered extra-solar planets
869 confirmed extra- solar planets
• Super-Jupiters
• (Hot) Jupiters
• Neptunes
• Super-Earths
• Earth-like planets
Open Exoplanet Catalogue (Rein 2012b) All multi-planetary systems
327 confirmed planets in multi-planetary systems
• Multiple Jupiters
• Densely packed systems of Neptunes and (Super)-Earths
• 1 Solar System
• Some systems are deep in resonance
Open Exoplanet Catalogue (Rein 2012b) Radial velocity planets
80 RV planets 70 Cumulative period ] 60 ratio in multi- 10 jup
[M planetary systems 2 50 + m 1 • Periods of systems with 40 massive planets tend to pile up near integer number of pairs 30 ratios 1 20 total planet mass m • Most prominent 10 features at 4:1, 3:1, 2:1,
0 3:2 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio 300 0.1 KOI planets
250 ] jup
[M Rein, Payne, Veras & Ford (2012) 200 2 + m 1
150 number of pairs 100 total planet mass m 50
0 0.01 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio Kepler candidates
2740 planet candidates
• Probing a different regime
• Small mass planets
• A lot of planets
Open Exoplanet Catalogue (Rein 2012b) Kepler candidates with multiple planets
Kepler multi-planetary systems
• Small mass planets
• Hierarchical systems
• Densely packed
• Not many are in resonance
Open Exoplanet Catalogue (Rein 2012b) 80 RV planets
70 ] 60 10 jup [M 2 50 + m 1
40
number of pairs 30
1 20 total planet mass m
Kepler's 10 transiting planet candidates
0 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio 300 0.1 KOI planets
250 • Period ratio ]
jup distribution much [M 200 2 smoother for small
+ m mass planets 1
150 • Deficiencies near 4:3, number of pairs 3:2, 2:1 100
total planet mass m • Excess slightly outside 50 of the exact commensurability 0 0.01 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio
Rein, Payne, Veras & Ford (2012) Stochastic orbital migration Migration - Type I
• Low mass planets • No gap opening in disc • Migration rate is fast • Depends strongly on thermodynamics of the disc
2D hydro code Prometheus (Rein 2010) Migration - Type II
• Massive planets (typically bigger than Saturn) • Opens a (clear) gap • Migration rate is slow • Follows viscous evolution of the disc
2D hydro code Prometheus (Rein 2010) How does a real protoplanetary disk look like?
Image credit: NASA/JPL-Caltech Why think about stochastic migration?
• Angular momentum transport • Magnetorotational instability (MRI) • Density perturbations interact gravitationally with planets • Stochastic forces lead to random walk • Large uncertainties in strength of forces
Animation from Nelson & Papaloizou 2004 Random forces measured by Laughlin et al. 2004, Nelson 2005, Oischi et al. 2007 Random walk in all orbital parameters
pericenter
eccentricity
semi-major axis
time
Rein & Papaloizou 2009 Analytic growth rates for 1 planet
0.01 Dt 2 [AU] 0.001 ( a) =4 RMS a>
2 n < 0.0001
1
2 2.5 Dt 0.1 RMS ( $) = >
2 2 2 < e n a 0.01
0.001
0.01
Dt RMS 2 0.001 e> ( e) =2.5 < n2a2 0.0001 Simulations Analytic solution Analytic solution ( =1) 10 100 1000 10000 100000 time [years]
Rein & Papaloizou 2009, Adams et al 2009, Rein 2010 Analytic growth rates for 2 planets
2 ( ⇥1) 9 f 2 = 2 2 Dt (p + 1) a1⇤lf
2 5 s ( ( ⌅)) = 2 2 2 Dt 4a1n1e1
Rein & Papaloizou 2009 Multi-planetary systems in mean motion resonance
GJ876
Earth
• Stability of multi-planetary systems depends strongly on diffusion coefficient • Most planetary systems are stable for entire disc lifetime
Rein & Papaloizou 2009 The formation of Kepler-36 Kepler-36 c as seen from Kepler-36 b
• Would appear 2.5 times the size of the Moon • Very close orbits, near a 7:6 resonance • Very different densities Formation of Kepler-36
2.8 • Migration rate and mass 106 ratio determine the final 2.6 resonance stable resonances 2.4 105 8:7 7:6 6:5 5:4 4:3 3:2 Higher order resonances 2.2 • require faster migration 2.0 rates 104 1.8 time [yrs] period ratio • Higher mass planets end up crossing of the 2:1 resonance 1.6 in lower order resonances 103 1.4 • Once in resonance, planets
convergent migration 1.2 often stay there for the rest 102 of the disc lifetime 1.0 103 104 105 migration timescale [yrs]
Pardekooper, Rein & Kley (in prep) Problem with Kepler-36
<1000 years
Need extremely fast migration rate to capture into a high order resonance.
Unrealistically fast.
Planets are not large enough to migrate in Type III regime.
Pardekooper, Rein & Kley (in prep) τa = 2000 yrs τa = 10000 yrs low order resonances low order resonances high order resonances 2.8 2.8 104 7:6 2.6 2.6
2.4 5:4 2.4 104 2.2 2.2 3:2 2.0 2.0 103 1.8 1.8 time [yrs] time [yrs] period ratio 3 period ratio crossing of the 2:1 resonance 1.6 10 crossing of the 2:1 resonance 1.6
1.4 1.4
102 convergent migration 1.2 convergent migration 1.2 1.0 1.0 Solution: Stochastic10-3 migration10-2 10-1 10-3 10-2 10-1 amplitude of stochastic forces [F0] amplitude of stochastic forces [F0]
τa = 20000 yrs τa = 100000 yrs low order resonances high order resonances low order resonances high order resonances 2.8 2.8 105 2.6 2.6 4:3 4:3 2.4 2.4 105 2.2 2.2 3:2 3:2 104 2.0 2.0
1.8 1.8 time [yrs] time [yrs] period ratio 104 period ratio crossing of the 2:1 resonance 1.6 crossing of the 2:1 resonance 1.6
103 1.4 1.4
convergent migration 1.2 convergent migration 1.2
1.0 103 1.0 10-3 10-2 10-1 10-3 10-2 10-1
amplitude of stochastic forces [F0] amplitude of stochastic forces [F0]
Pardekooper, Rein & Kley (in prep) A statistical analysis 80 RV planets
70 ] 60 10 jup [M 2 50 + m 1
40
number of pairs 30
1 20 total planet mass m
Kepler's 10 transiting planet candidates
0 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio 300 0.1 KOI planets
250 • Period ratio ]
jup distribution much [M 200 2 smoother for small
+ m mass planets 1
150 • Deficiencies near 4:3, number of pairs 3:2, 2:1 100
total planet mass m • Excess slightly outside 50 of the exact commensurability 0 0.01 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio
Rein, Payne, Veras & Ford (2012) Testing stochastic migration: Method
Architecture and masses from observed KOIs
Placing planets in a MMSN, further out, further apart, randomizing all angles
N-body simulation with migration forces
Rein 2012 Testing stochastic migration: Advantages
Comparison of Comparison of statistical quantities individual systems • Period ratio distribution • Especially interesting for • Eccentricity distribution multi-planetary systems • TTVs • Can create multiple realizations of each system
No synthesis of a planet population required • Observed masses, architectures • Model independent Preliminary results
600 Observed KOI planets 4 -6 Migration τ =10 years, α=10 a 4 -5 Migration τ =10 years, α=10 500 a
400
300 300 number of pairs 200 250
100 200
1.9 2 2.1 0 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio
Rein 2012 Future expansions
Physical disk model Completeness • 1D hydrodynamic simulation • Include planets missed by Kepler • Coupled to N-body simulations GPU based integrators Other physical effects • Allows for much bigger samples • Tidal damping • Wider parameter space exploration • Evaporation Saturn’s Rings REBOUND
c ESO 2011 • Code description paper published by A&A, Rein & Liu 2012
manuscript no. paper
Astronomy & Astrophysics 2,3 Multi-purpose N-body code October 20, 2011 • collisional1 and dynamics Shang-Fei Liu Hanno Rein REBOUND: An open-source multi-purpose N-body code for
Only public N-body code that can 1 Institute [email protected] Advanced Study, 1 Einstein Drive, Princeton, NJ 08540 ABSTRACT • e-mail: 2 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, P. R. China 3 [email protected] of Astronomy, Peking University, Beijing 100871, P. R. China can use a Barnes-Hut tree to calculate both self- e-mail: REBOUND be used for granular dynamics Submitted: 13 September 2011 ↵erent problems in astrophysics and beyond. is a new multi-purpose N-body code which is freely available under an open-source, the license. philosophy It was behind designed the for code’s collisional structure as well as REBOUND REBOUND dynamics such as planetary rings but can also solve the classical N-body problem. It is highly modular and can be customized easily to work on acomes wide variety with three of di symplectic integrators: leap-frog, the symplectic epicycle integrator (SEI) and a Wisdom-Holman mapping REBOUND (WH). It supports open, periodic and shearing-sheet boundary conditions. erent algorithms↵erent modules. implemented We present in results of accuracy and scaling tests which show that the code gravity and collisions. These modules↵ are fully parallelized with MPI as well as OpenMP. The formercan makes also use be usedof a static with domain little modification in situa- decomposition and a distributed essential tree. Two new collision detection modules based on a plane-sweep algorithm are also implemented. The performance of the plane-sweep algorithm is superior to a tree codeREBOUND for simulations in which one dimension is Written in C99, open source, GPL much longer than the other two and in simulations which are quasi-two dimensionaltions with where less than only one a million statistical particles. measure of the collision frequency • In this work, we discuss the di ciently on both desktop machines and large computing clusters. is required such as in transitional and debris discs. In such sys- implementation specific details of the di can be used to simulate classical N- can run e Methods: numerical – Planets and satellites: rings – Proto-planetarytems, disks individual collisions between particles are not modeled ex- actly, but approximatedREBOUND by the use of super-particles (Stark & Key words. Kuchner 2009; Lithwick & Chiang 2007). Furthermore, body problems involving entirely collision-less systems. A sym- plectic and mixed variable integrator can be used to follow the↵erent mod- 1. Introduction trajectories of both test-particles and massive particles. is a new open-source collisional N-body code. This We describe the general structure of the code, how to ob- can be of great use for many REBOUND 1 tain, compile and run it in Sect. 2. The time-stepping scheme Freely available at code, and precursors of it, haveREBOUND already been used in wide variety . • and our implementation of symplectic integrators are described of publications (Rein & Papaloizou 2010; Crida et al. 2010; Rein in Sect. 3. The modules for gravityciency are of the described parallelization in Sect.with 4. The the help et al. 2010, Rein & Liu in preparation; Rein & Latter in prepa- algorithms for collision detection are discussed in Sect. 5. In ration). We believe that ↵erent problems and have a wide reach in astrophysics and Sect. 6, we present results of accuracy tests for di di ules. We discuss the e http://github.com/hannorein/rebound other disciplines. To our knowledge, there is currently no pub- of scaling tests in Sect. 7. We finally summarize in Sect. 8. licly available code for collisional dynamics capable of solving the problems described in this paper. This is why we decided to requires no external make it freely available under the open-source license GPLv3 2. Overviewis written of the entirely code structure in C and conformsREBOUND to the ISO C99 Collisional N-body simulations are extensively used in as- REBOUND trophysics. A classical application is a planetary ring (see standard. It compiles and runs on any modern computer platform 1 e.g. Wisdom & Tremaine 1988; Salo 1991; Richardson 1994; which supports the POSIX standard such as Linux, Unix and Lewis & Stewart 2009; Rein & Papaloizou 2010; Michikoshi & Mac OSX. In its simplest form, Kokubo 2011, and references therein) which have often a colli- libraries to compile. sion time-scale that is much shorter than or at least comparable Users are encouraged to install the OpenGL and GLUT li- to an orbital time-scale. Self-gravity plays an important role, es- braries which enable real-time and interactive 3D visualizations. pecially in the dense parts of Saturn’s rings (Schmidt et al. 2009). LIBPNG is required to automatically save screen-shots. The These simulations are usually done in the shearing sheet approx- . It can also. be imation (Hill 1878). Collisions are also important during planetesimalREBOUND formation (Johansen et al. 2007; Rein et al. 2010, Johansen et al. in prepa- ration). Collisions provide the dissipative mechanism to form a planetesimal out of a gravitationally bound swarm of boulders. http://www.gnu.org/licenses/gpl.html 1 The full license is distributed together with downloaded from Symplectic Epicycle Integrator
1 1 H = p2 + ⇥(p r)e + ⇥2 r2 3(r e )2 + (r) 2 ⇥ z 2 · x Epicycle ⇥ ⇤ Kick
1/2 Kick Epicycle 1/2 Kick
Rein & Tremaine 2011 Mixed variable symplectic (MVS) integrator
non symplectic 100 symplectic mixed variable symplectic 10-2
10-4
-6 phase error 10
10-8
10-10 10-5 10-4 10-3 10-2 10-1 100 -1 timestep [2πΩ ]
Rein & Tremaine 2011 Symplectic Epicycle Integrator: Rotation • Solving for the orbital motion involves a rotation. • Formally det( D )=1 , but due to floating point precision det( D ) 1 only. ⇠ • Trick: Use three shear operators instead of one rotation. cos sin 10 1 sin 10 = sin cos tan 1 1 · 01 · tan 1 1 ✓ ◆ ✓ 2 ◆ ✓ ◆ ✓ 2 ◆ 1e-06
• det( D )=1 exactly for 1e-07 each shear operator, even 1e-08 in floating point precision.
relative energy error 1e-09 • No long term trend linear Quinn SEI trend anymore! SEKI 1e-10 0 10 20 30 40 50 60 70 80 90 100 -1 time [2πΩ ] Saturn is a smaller version of the Solar System Propeller structures in A-ring
Porco et al. 2007, Sremcevic et al. 2007, Tiscareno et al. 2006, NASA/JPL-Caltech/Space Science Institute Stochastic Migration
REBOUND code, Rein & Papaloizou 2010, Crida et al 2010, Pan, Rein, Chiang & Evans 2012 Motion is consistent with a random walk
Diagonalization
Pan, Rein, Chiang, Evans 2012 Observations
• Observational evidence for small scale structures • Typical size ~100m
Rein & Latter (2013), Thomson 2010 Close-up view of the viscous over-stability
Rein & Latter (2013) Numerical simulations with REBOUND
Symplectic Epicycle Plane-sweep Integrator algorithm - Fast - Fast - High accuracy - O(N) for elongated boxes - No long term drifts (important)
Direct particle simulations of Saturn's Rings - Longest integration time ever done - Widest boxes ever done
Rein & Latter (2013) Non-linear evolution
wavetrain wavetrain sink source
Rein & Latter (2013) Long-term evolution
6000
5
5000
4
4000
3
time [orb] 3000 optical depth
2 2000
1 1000
0 0 -23000 0 23000
x [rp]
Rein & Latter (2013) Institute for Advanced Study
Open Exoplanet Catalogue Why do we need another exoplanet catalogue?
openexoplanetcatalogue.com Common drawbacks of astronomical catalogues
Centralized Slow and outdated • Impossible to correct typos, add • It can take days/weeks/months for new data without sending an e-mail to planets to be added the person in charge • Maintainer can be holiday or abandon the project • Closed ecosystem
Web-based Old-fashioned formats • Static tables are not adequate to • Website are badly written represent diverse dataset • Requires flash or java plugin • Almost impossible to include binary/ • Need a constant internet connection triple/quadruple systems • Restricted to a very limited, • Not flexible when adding new data predefined set of possible queries • Unintuitive to parse Open Exoplanet Catalogue
Open source philosophy Ready to go • Unrestrictive MIT license • 674 systems, 51 binary system, 870 • Community project exoplanets, 9 solar system objects, 2740 • Everyone can contribute and modify data KOI objects • Everyone can expand it • ~10 million users • Distributed, no need for a server/website • Private clones with confidential data Hierarchical data structure • Uses plain XML Based on git • Can represent arbitrary configurations • Distributed version control system in systems with stellar multiplicity >1 • Used by Linux kernel and most other • Extremely easy and intuitive to parse in open source projects almost any language • Every single value, every change ever • Compresses extremely well made is logged, verifiable • size ~ 100KB
OpenExoplanetCatalogue.com, arXiv:1211.7121 Example of a system file: 42 Dra b
Confirmed planets
!!!
import xml.etree.ElementTree as ET, glob for filename in glob.glob("*.xml"): !tree = ET.parse(open(filename, ’r’)) !planets = tree.findall(".//planet") ! for planet in planets: !!print planet.findtext("./name") !!print planet.findtext("./mass") Open Exoplanet Catalogue
OpenExoplanetCatalogue.com
arXiv:1211.7121 Summary
The case for stochastic orbital migration • Stochastic migration is directly observable in Saturn’s rings.
• Protoplanetary disks are turbulent due to the MRI.
• Stochastic migration plays an important role for small mass planets.
• Resonances can easily get destroyed.
• Tendency to form high order resonance.
• Very soon, we will understand how most planets in the Kepler sample formed.
Open Exoplanet Catalogue Use it! Contribute to it!
Paardekooper, Rein & Kley (in prep), Rein, Payne, Veras & Ford (2012), Rein (2012a), Rein & Papaloizou (2009)