the planetary migration in planetesimal disks: … planetary migration in planetesimal disks:...

The planetary migration in planetesimal disks: analytical and

numerical results

Federico Panichi

XCIX Congresso Nazionale di Fisica

Trieste 23 – 27 Settembre 2013 (SISSA)

Icy Jupiter planets

Earth like planets

Hot Jupiter planets

First part: astrophysical problem

• A huge variety of extrasolar planets (723 confirmed) with distinct orbital parameters and which can be grouped into three different classes:

1. Hot Jupiters: planets with masses equal to or greater than

Jupiter and near (𝑎𝑎 < 1 𝐴𝐴.𝑈𝑈. ) the host stars; 2. Icy Jupiters: planets with masses equal to or less than Jupiter and far (𝑎𝑎 > 1 𝐴𝐴.𝑈𝑈. ) from the host stars; 3. Earth-like planets: planets with masses equal the Earth mass and distances beetwen 0.01 A. U. and 1 𝐴𝐴.𝑈𝑈.


Extrasolar systems with single planet confirmed

𝑀𝑀 sin 𝑖𝑖 Histogram


From J. T. Wright et al. (2009)


• The number of planets with masses greater than 0.1 𝑀𝑀𝐽𝐽sin(𝑖𝑖) is very high. One possible reason for this asymmetry is the greater stability that these planets have respect the less massive ones. What is observed in the histogram may be the result dynamical evolution process rather than a photograph of the formation mechanism or a simple bias effects.

• The aim of this communication is to explain the first possibility by using a dynamical toy-model of interaction between the planet and the protoplanetary disk.

• We decided to analyze the problem using a disk of planetesimals. A similar study could be done using a gaseous disk.


Mass Histogram of single planet sample

𝟎𝟎.𝟓𝟓 𝑴𝑴𝑱𝑱 < 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴(𝑴𝑴) < 𝟏𝟏𝟎𝟎 𝑴𝑴𝑱𝑱

𝟎𝟎.𝟎𝟎𝟏𝟏 < 𝒂𝒂 < 𝟎𝟎.𝟏𝟏

Histogram of Semimajor axis of single planet sample


Icy Jupiter planets

Earth like planets

Hot Jupiter planets


???

Planet migration models: planetesimal-driven migration

• Fast migration: Bromley 2011 • Slow migration: Del Popolo 2004 • Chaotic migration: from Ida et al. 2000 to H. Rein 2012 • In/outward migration: Kirsh et al. 2009 • Also counter rotating migration: this comunication


Planet formation models: counter-rotating scenarios

Theories that allow the creation of counter-rotating planet:

• Kozai resonance combinations, planet planet scattering and orbital

circularization phenomena (Nagasawa, 2008);

• Kozai oscillations (Correia, 2011);

• Chaotic scenarios (secular chaos) (Wu and Lithwich, 2011);

• scattering planet-planet (Kaib et al., 2011);

• Capture (Varuoglis, 2011).

Planets with inclinations higher than 100°: • γ1 Leob 172°; • HD 106225b 166°; • 2M0746 + 20b 138°.


• The model studied in this presentation analyze the distribution in mass and semimajor axis of extrasolar planetary systems with a single planet whit mass greater than or equal to 𝑴𝑴𝑱𝑱𝑴𝑴𝑴𝑴𝑴𝑴 𝑴𝑴 .

Restricted three-body problem: resonances

General three-body problem: planet migration

Numerical simulations

Collective phenomena: dynamical friction

& density waves

Simulation with ALMA

Second part: presentation map

• The Jacobi’s integral allows to constrain the motion of the test particle in a

precise region of the phase space: �̇�𝑥 𝑡𝑡 2 + �̇�𝑦 𝑡𝑡 2 = 2𝑈𝑈 − 𝐶𝐶𝐽𝐽

Third part: the restricted three-body problem

The study of the three-body problem is essential to understand the effects that the

planet exerts on the disc.

• Jacobi’s surface of section Equipotential surfaces. By

varying the initial conditions

also vary the section of the

Jacobi’s surface and with it

the regions of permitted

motion. 𝐽𝐽 𝑥𝑥, 𝑦𝑦, 𝑎𝑎 = 2 × 𝑈𝑈(𝑥𝑥,𝑦𝑦, 𝑎𝑎)


Ergodic orbit (chaotic): Over time this trajectory will cover densely the (n-1)-dimensional surface determined by the condition of energy conservation.

The existence, for the three-body problem, of regular or ergodic orbits is the fondation of the different stability of

prograde and retrograde orbits.

prograde (resonante) orbit retrograde (resonant) orbit


• Lyapunov exponent

Ergodic prograde orbit

Resonant regular prograde orbit


stable

instable

• Resonant irregular orbit NEAR the planet (instable)


• The resonance overlap is the mechanism that produces chaos in the vicinity of a planet.

Overlap regions

Third part: the restricted three-body problem ec

cent

rici

ty

Semimajor axis [A.U.]

Observed asteroids in Main Belt

• Libration width of 1-th and 2-th order resonance

The study of resonances and their overlap is used to calculate the gap extension within the disk of

planetesimals. This is fundamental to understand the numerical results.


• Propriety of gap in the conter- and co-rotational cases.

Gap dimension 7 𝑹𝑹𝑯𝑯

planet

x [A

.U.]

y [A.U.]

The resonance overlap controls the width of the gap

Gap dimension 2 𝑹𝑹𝑯𝑯

x [A

.U.]

y [A.U.]

planet


Ergodic orbit (chaotic): Over time this trajectory will cover densely the (n-1)-dimensional surface determined by the condition of energy conservation.

The existence, for the three-body problem, of regular or ergodic orbits is the key of the different stability of

prograde and retrograde orbits.

prograde (resonante) orbit retrograde (resonant) orbit


∆𝒂𝒂 = ±𝟑𝟑.𝟓𝟓𝑹𝑹𝑯𝑯

∆𝒂𝒂 = ±𝟏𝟏 𝑹𝑹𝑯𝑯


The resonance overlap controls the width of the gap

Counter-rotating

Co-rotating

1. The General three-body problem (G3bp) is a natural extension of

the Restricted three-body problem (R3bp) and allow the study of

the interaction of the disk ON the planet.

2. It allows to calculate the angular momentum CHANGE that the

disk exerts on the planet and consequently can provide an

estimate of the rate of planetary migration comparable with that

obtained by numerical simulations.

Fourth part: the general three-body problem

Г = −𝟏𝟏𝟏𝟏𝟕𝟕𝜮𝜮𝜴𝜴𝒌𝒌

𝟐𝟐𝑹𝑹𝑯𝑯𝟒𝟒 from Crida et al. (2010)

Fifth part: numerical simulations

Jupiter like planets Too massive planets fail to migrate quickly: they loss little amount of kinetic energy IF interactions take place in a low massive disk.

ecce

ntri

city

Earth like planets Low massive planets migrate too fast and may fall onto the central star rapidly. In this way it is possible to explains the mass distribution of observing extrasolar systems with a single planet.

High ρ Low ρ

ecce

ntri

city


Planetary migration rate While it is inside the disk

While it is outside the disk

- Disk dimension:

20 𝐴𝐴.𝑈𝑈.

- Planet mass:

320𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ≅ 1𝑀𝑀𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐸𝐸𝐽𝐽𝐸𝐸

- Integration time:

105 yrs

(B.C. Bromley & S.J. Kenyon , 2011)

Higher mass


A migration compendium Fixed planet mass of 1 𝑴𝑴𝑱𝑱 (corotating)

ρ = 500 𝑔𝑔𝑔𝑔/𝑐𝑐𝑚𝑚2 ρ = 50 𝑔𝑔𝑔𝑔/𝑐𝑐𝑚𝑚2 ρ = 5 𝑔𝑔𝑔𝑔/𝑐𝑐𝑚𝑚2

Counter- rotating disk

Co-rotating disk

𝑀𝑀𝐽𝐽 = 10 𝑀𝑀𝐽𝐽 , 𝜌𝜌 = 40 𝑔𝑔𝑔𝑔/𝑐𝑐𝑚𝑚2 , Ecc. Dist. = Rayleigh 𝑀𝑀𝐽𝐽 = 100 𝑀𝑀𝐽𝐽 , 𝜌𝜌 = 40 𝑔𝑔𝑔𝑔/𝑐𝑐𝑚𝑚2 , Ecc. Dist. = 0.0

Co-rotating disk

Conter- rotating disk

Analytic Torque vs.

Numerical Torque

#𝒑𝒑𝒑𝒑𝒂𝒂𝑴𝑴𝒑𝒑𝒑𝒑𝒑𝒑𝑴𝑴𝑴𝑴𝒑𝒑𝒂𝒂𝒑𝒑𝑴𝑴

Low ρ High ρ


Fixed disk mass 40 𝑴𝑴⊕

A migration compendium

Analytic Torque Numerical Torque

#𝒑𝒑𝒑𝒑𝒂𝒂𝑴𝑴𝒑𝒑𝒑𝒑𝒑𝒑𝑴𝑴𝑴𝑴𝒑𝒑𝒂𝒂𝒑𝒑𝑴𝑴

Low ρ High ρ


Fixed disk mass 40 𝑀𝑀⊕

Fixed disk dimension 20 A.U.

Fixed planet mass 1 𝑀𝑀𝐽𝐽

Simulated observations (ALMA) of the simulated disks (only R3bp)

Sixth part: ALMA

Image credits: ALMA (ESO/NAOJ/NRAO) / B. Saxton

Image credits: A. Boley et al. (2012) / orange, ALMA observation (350 GHz → 850 μm; 𝐹𝐹𝐸𝐸𝑡𝑡𝐸𝐸 = 80 mJy)

• Interferometer locate in Chile (Atacama Large Millimeter Array)

• frequency range : 31 GHz – 950 GHz (1 cm – 300 μm) • Maximum angular resolution: 0.2” λ(mm)/D(km) >

0.004”; • Total number of antenna: 66 • Maximum baseline: 16 chilometri • Cycle: Cycle 0 closed, Cycle 1 begin.

Simulated observations (ALMA) of the simulated disks (only R3bp)

Co-rotating disk

Tadpole orbits

planet

Total flux = 1 Jy Observed frequency = 700 GHz Disk distance = 60 pc Disk dimension = 20 A.U. PSF = 0.01’’

Sixth part: ALMA

ALMA - simulation

Co-rotating case Counter-rotating case

Sixth part: ALMA

20 A.U. 30 pc

100 A.U. 70 pc

ALMA - Model

z

y x

60° planet

y

60°

planet planet

Co-rotating case Counter-rotating case

Sixth part: ALMA

planet

x

z

ALMA - simulations Co-rotating case Counter-rotating case

Surface brightness profile

Sixth part: ALMA

Surface brightness profile

2 𝑅𝑅𝐻𝐻 7 𝑅𝑅𝐻𝐻

20 A.U. 30 pc

100 A.U. 70 pc

Conclusions 1. The counter-rotating case is more stable than the co-rotating one and for this reason

it presents a gap of smaller extension. The gap extension, in the co-rotating disk , allows a first understanding of the migration rate :

Higher the mass = larger the gap Larger the gap = Higher migration rate

In the counter-rotating case instead of the large number of interactions NEAR the planet the migration is more rapid. (remember the plots in the migration compendium slide!!!)

2. Fixing the density of the planetesimal disk, too massive planets do not migrate

quickly and therefore are more easily observed (in the mass histogram there are more massive planets). Fixing the disk density to planet mass ratio: less massive planets migrate more slowly.

Seventh part: Conclusions

High 𝑀𝑀𝐽𝐽𝑝𝑝𝐸𝐸𝑝𝑝𝐽𝐽𝐸𝐸

Low 𝑀𝑀𝐽𝐽𝑝𝑝𝐸𝐸𝑝𝑝𝐽𝐽𝐸𝐸

High 𝑀𝑀𝐽𝐽𝑝𝑝𝐸𝐸𝑝𝑝𝐽𝐽𝐸𝐸

Low 𝑀𝑀𝐽𝐽𝑝𝑝𝐸𝐸𝑝𝑝𝐽𝐽𝐸𝐸

Torque da/dt

Conclusions 3. Dynamical Friction:

describes the dependence of the planet mass and the disk density observed in the simulations:

𝒅𝒅𝒂𝒂𝒅𝒅𝒑𝒑∝ 𝑀𝑀𝑷𝑷×ρ𝑑𝑑𝐽𝐽𝑑𝑑𝑑𝑑𝑡𝑡 ;

describes the dependence of the migration rate by an anisotropic velocity distribution:

Ψ 𝑒𝑒, 𝑖𝑖 ∝𝒑𝒑𝑴𝑴

𝒑𝒑𝟐𝟐 𝑴𝑴𝟐𝟐𝒑𝒑𝒆𝒆𝒑𝒑 −

𝒑𝒑𝟐𝟐

𝒑𝒑𝟐𝟐−

𝑴𝑴𝟐𝟐

𝑴𝑴𝟐𝟐 ;

Describes the difference between co-rotating and counter-rotating planetesimal disks;

DO NOT describes the dependence of the radial migration due to the orbital resonances.

4. Density waves (linear theroy):

describes the dependence of the radial migration caused by orbital resonances;

Describes the difference between co-rotating and counter-rotating disks;

DO NOT describes the dependence of radial migration by an anisotropic velocity distribution.


Conclusions

5. Is it possible, with ALMA, to study protoplanetary disks whit a better resolution of the

arcsec and to gain information about the evolution, structure and internal dynamics.

In fact we can:

Observe with a lot of detail disks up to 70 pc and with a dimension of 100 astronomical units,

being able to distinguish the case of a counter-rotating from a co-rotating one (resonance structures, gap extensions, …);

Observe with a lot of detail disks up to 30 pc and with a dimension of 20 astronomical units, begin able to distinguish the case of a counter-rotating from a co-rotating one (resonance structures, gap extensions, …);

Study the surface brightness profile when is it impossible to distinguish between co- and

counter-rotating disks (small dimension and/or high distance) and observe the presence of a planet inside the disks;

Using the models proposed in this work to compare them with the first protoplanetary disks observed (es. TW Hydrae).


Conclusions Seventh part: Conclusions

Our co-rotating simulation - ALMA TW Hydrae real disk – HST

100 A.U. at 70 pc 300 A.U. at 60 pc

Future works

Future works

• Extend the results obtained by relaxing the assumption of a single planet S. N. Raymond et al. (2010);

• Extend introducing the hydrodynamics C. C. Capobianco (2010);

• Study the evolution of a self-gravity three dimensional disk W. Kley & R. P. Nelson (2012);

• Derive a synthetic SED with a realistic (radiative) emission model and compare it with the observational data.

Running now!!!!

Running now!!!!

Una visione tridimensionale di un orbita ergodica, proiettata sulla superficie di Jacobi costante ricavata dalle condizioni iniziali di velocità e posizione.

• La superficie di Jacobi, la costante di Jacobi e l’orbita della particella di prova

ORBITA NELLA SUPERFICIE DI JACOBI

Una visione tridimensionale di un orbita ergodica, proiettata sulla superficie di Jacobi costante ricavata dalle condizioni iniziali di velocità e posizione.

Lyapunov exponent Si può dimostrare che una misura della divergenza (locale) di due traiettorie (vicine) consente di calcolare il massimo degli esponenti di Lyapounov (LCE). È possibile calcolare la separazione finale tra le due traiettorie in modo molto semplice:

𝑑𝑑𝐸𝐸 = 𝑑𝑑0𝑒𝑒γ(𝐸𝐸−𝐸𝐸0) .

Lyapunov exponent Affinché la distanza tra le due traiettorie converga nel tempo si deve avere γ > 0. Per stabilire il valore di γ come il risultato di un integrazione numerica si ricorrere alla seguente formula:

γ = lim𝑝𝑝→∞

�ln 𝑑𝑑𝐽𝐽 𝑑𝑑0⁄

𝑛𝑛𝛥𝛥𝑡𝑡

𝑝𝑝

𝐽𝐽=1

.

L’andamento di γ in un grafico log-norm o log-log consente di distinguere immediatamente orbite caotiche ed orbite non caotiche. • Orbite ordinate: l’esponente di Lyapounov deve avere una slope in un grafico log-norm o log-

log negativa. (es.: −1 affinché si abbia 𝑑𝑑𝐸𝐸 ≈ 𝑑𝑑0).

• Orbite caotiche: γ deve tendere ad un valore positivo e, comunque, deve aumentare nel tempo o, al più, rimanere costante.

Lyapunov exponent Affinché la distanza tra le due traiettorie converga nel tempo si deve avere γ > 0. Per stabilire il valore di γ come il risultato di un integrazione numerica si ricorrere alla seguente formula: L’andamento di γ in un grafico log-norm o log-log consente di distinguere immediatamente orbite caotiche ed orbite non caotiche. • Orbite ordinate: l’esponente di Lyapounov deve avere una slope in un grafico log-norm o log-

log negativa. (es.: −1 affinché si abbia 𝑑𝑑𝐸𝐸 ≈ 𝑑𝑑0).

• Orbite caotiche: γ deve tendere ad un valore positivo e, comunque, deve aumentare nel tempo o, al più, rimanere costante.

𝑑𝑑0 𝑑𝑑𝑝𝑝

𝑑𝑑1 𝑑𝑑2

𝑑𝑑3 𝑑𝑑…

Il criterio di Chirikov

Prima parte: il problema dei tre corpi ristretto

Chirikov criterium

The anisotropic dynamical friction • Il caso di disco infinitamente sottile senza spettro di

massa

Isotropic

Anisotropic

Quinta parte: la frizione dinamica

Anisotropic dynamical friction

• Conclusioni (1)

▫ La differenza nelle due funzioni di distribuzione; ▫ Nel caso di velocità paragonabili con la σ la differenza è

maggiormente evidente; ▫ L’utilizzo della formula di Chandrasekhar sottostima di

un fattore circa due il valore vero della frizione dinamica nel caso anisotropo;

▫ La frizione dinamica è direttamente proporzionale alla massa e al profilo di densità del disco !

Quinta parte: la frizione dinamica

Osservazioni simulate (ALMA) Wolf

& D’Angelo

(2005)

• ALMA • 950 GHz • embedded

protoplanet of 1 Jupiter Mass

Sesta parte: le simulazioni numeriche

Le capacità osservative con cui è stato costruito ALMA garantiscono l’osservazione di dischi protoplanetari fino a 150 pc.

Diffusion of gas 𝜕𝜕Σ(𝑥𝑥, 𝑡𝑡)𝜕𝜕𝑡𝑡

=3𝑅𝑅𝜕𝜕𝜕𝜕𝑅𝑅

𝑅𝑅1/2 𝜕𝜕𝜕𝜕𝑅𝑅

νΣ(𝑥𝑥, 𝑡𝑡)𝑅𝑅1/2

Σ 𝑅𝑅, 𝑡𝑡 = 0 =𝑚𝑚

2π𝑔𝑔0δ(𝑅𝑅 − 𝑔𝑔0)

Initial condition at 𝑔𝑔0 = 1

Σ 𝑅𝑅, 𝑡𝑡 ~𝑅𝑅−1/4

𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 −1 + 𝑅𝑅2

𝑡𝑡 𝐼𝐼14

𝑅𝑅𝑡𝑡

Final condition

ν = 𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡.

Lynden-Bell &

Pringle (1974)



𝑅𝑅1/2 𝜕𝜕𝜕𝜕𝑅𝑅

νΣ(𝑥𝑥, 𝑡𝑡)𝑅𝑅1/2 + 𝑆𝑆𝑐𝑐𝑆𝑆𝑔𝑔𝑐𝑐𝑒𝑒

Yoshi et al. (2010)

K. E. Haish &

E. A. Lada (2001)



𝑅𝑅1/2 𝜕𝜕𝜕𝜕𝑅𝑅

νΣ(𝑥𝑥, 𝑡𝑡)𝑅𝑅1/2 + 𝑆𝑆𝑐𝑐𝑆𝑆𝑔𝑔𝑐𝑐𝑒𝑒

E. E. Mamajek et al. (2009)

Formation theory

(Kokubo & Ida, 2000)

𝑂𝑂𝑂𝑂𝑖𝑖𝑔𝑔𝑎𝑎𝑔𝑔𝑐𝑐𝑂𝑖𝑖𝑐𝑐𝑂 𝑔𝑔𝑔𝑔𝑐𝑐𝑔𝑔𝑡𝑡𝑂: quando un protopianeta

diventa abbastanza grande esso domina il tasso

di accrescimento

Runaway 𝑔𝑔𝑔𝑔𝑐𝑐𝑔𝑔𝑡𝑡𝑂: oggetti con massa

maggiore aumentano la massa in modo

più rapido

Formation theory: disk dimension

Morbidelli (2005)

The hidrodynamical similarity

G. Lufkin &

T. Quinn (2004)

- Isotermal

- 3D

- Self gravity

- accretion

Number of planetesimals

Bias From Stefano Meschiari

Bias

A. W. Howard, 2013; Cumming, 2008

ALSO WITH KEPLER THE

SAME HISTOGRAM…

…Ok the radius but what

about the mass? See Sozzetti’s

talk

Number of planetesimals

𝒑𝒑𝒑𝒑𝒑𝒑𝒂𝒂𝑴𝑴𝒑𝒑𝒑𝒑

Kokubo E , and Ida S Prog. Theor. Exp. Phys. 2012;2012:01A308

© The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.

𝑴𝑴𝒑𝒑𝒑𝒑𝒂𝒂𝑴𝑴𝒑𝒑𝒑𝒑

Initial condition generator • Il pianeta

Initial condition generator

• Disk:

Dynamical Friction vs.

Viscous stirring =

Rayleigh Distribution

• RESONANT RELAXATION IN PROTOPLANETARY DISKS, S. Tremaine (1998); • Growth of planets from planetesimals, J. J. Lissauer e G. R. Stewart (1993); • The gas drag effect on the elliptical motion of a solid body in the primordial solar

nebula, I. Adachi et al. (1976).

Entropy

σ𝑑𝑑𝐽𝐽𝑝𝑝=∑ 1

𝑎𝑎𝑖𝑖2

1−𝑒𝑒𝑖𝑖−1

2𝑁𝑁𝑖𝑖=0

𝑁𝑁−1

Entropia metrica:

𝐼𝐼 𝑡𝑡, ε = �𝑒𝑒𝐽𝐽

𝑝𝑝

𝐽𝐽=0

𝑡𝑡, ε log2(𝑒𝑒𝐽𝐽(𝑡𝑡, ε))

Entropia di Kolmogorov-Sinai:

𝐾𝐾 = limε→0

lim𝐸𝐸→𝑇𝑇𝑠𝑠

𝐼𝐼 𝑡𝑡, ε𝑡𝑡

Caso corotante Caso controrotante

Torque numerica e Torque analitica • Torque numerica (Cionco & Brunini, 2002 ):

Г = 0.5𝑎𝑎𝑑𝑑𝑎𝑎𝑑𝑑𝑡𝑡𝑀𝑀𝑃𝑃Ω𝑘𝑘 → Г = 0.5 𝑎𝑎

𝑑𝑑𝑎𝑎𝑑𝑑𝑡𝑡

𝑀𝑀𝑃𝑃 Ω𝑘𝑘

• Torque analitica -> DENSITY WAVES (Ward, 1992):

Г = 98ΣΩ𝑘𝑘2𝑅𝑅𝐻𝐻4 → Г =

98Σ Ω𝑘𝑘2 𝑅𝑅𝐻𝐻4

• Torque analitica -> G3BP(A. Crida et al., 2010):

Г = −117ΣΩ𝑘𝑘2𝑅𝑅𝐻𝐻4 → Г = −

117Σ Ω𝑘𝑘2 𝑅𝑅𝐻𝐻4

Imaging

the planetary migration in planetesimal disks: … planetary migration in planetesimal disks:...

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