Una (mica tanto) breve introduzione ai pianeti extrasolari. · 2019. 6. 4. · Una (mica tanto) breve introduzione ai pianeti extrasolari. Alla ricerca di nuovi mondi (abitabili)

Una (mica tanto) breve introduzioneai pianeti extrasolari.

Alla ricerca di nuovi mondi (abitabili) nella nostra galassia?

M. Sansottera

Risultati ottenuti in collaborazione conA. Giorgilli, U. Locatelli, A.-S. Libert e M. Volpi

Meccanica Celeste — Milano, 07.06.2018

Cos’è un pianeta extrasolare?

Un pianeta che orbita attorno ad una stella diversa dal Sole.

Cos’è un pianeta?

La definizione è mutata con il progredire delle conoscenze scientifiche:

tutti gli astri che si muovevano (attorno alla Terra) rispetto allestelle fisse (comete escluse);

modello eliocentrico: la Terra diventa un pianeta, il Sole e la Lunacessano di esserlo;

telescopio: Urano, Nettuno e Plutone;

Plutone, un pianeta anomalo: orbita eccentrica ed inclinata rispettoal piano dell’eclittica, massa pari ad un quinto della Luna.

serve una definizione precisa: 24 agosto 2006 Assemblea Generale diPraga dell’Unione Astronomica Internazionale.

La legge di Titius-Bode (1766-1772)

I semiassi maggiori ai delle orbite di ciascun pianeta seguonoapprossimativamente la relazione:

ai = 0.4 + 0.3× 2i , i = −∞, 0, 1, 2, . . . ,

dove 1 u.a = distanza Terra-Sole.

Urano (1781) occupava esattamente la posizione prevista!

Bode: “c’è un pianeta mancante tra il quarto ed il quinto!”

Giuseppe Piazzi (1 Gennaio 1801) scopre Cerere.

Cerere, il pianeta mancante?

Piazzi non poté seguire il moto di Cerere abbastanza a lungo (24osservazioni) prima che diventasse invisibile dalla Terra;

non fu possibile determinarne l’orbita, e Cerere andò perduto;

Carl Friedrich Gauss, a 24 anni, sviluppa un nuovo metodo dideterminazione dell’orbita di un corpo celeste con tre soleosservazioni basato sui minimi quadrati, che Gauss sviluppòappositamente per l’astronomia;

Gauss predisse la traiettoria di Cerere, in base ai dati raccolti daPiazzi, e Cerere venne recupoerato!

Cerere: pianeta per 50 anni, ma troppo piccolo per essere degno ditale nome in quanto furono scoperti corpi celesti più grandi tra Martee Giove, ribattezzato “asteroide” (Herschel) o “planetoide” (Piazzi).

Dal 2006 Cerere è l’unico asteroide del sistema solare internoconsiderato un pianeta nano!

Le orbite dei pianeti interni

Le orbite dei pianeti hanno piccole eccentricità ed inclinazioni.

Le orbite dei pianeti esterni

Ad eccezione di Plutone, le orbite dei pianeti hanno piccole eccentricitàed inclinazioni.

IAU 2006Un pianeta è un corpo celeste che:

è in orbita intorno al Sole;

ha una massa sufficiente affinché la sua gravità possa vincere le forzedi corpo rigido, cosicché assume una forma di equilibrio idrostatico(quasi sferica);

ha ripulito le vicinanze intorno alla sua orbita.

Un pianeta nano è un corpo celeste che:

è in orbita intorno al Sole;

ha una massa sufficiente affinché la sua gravità possa vincere le forzedi corpo rigido, cosicché assume una forma di equilibrio idrostatico(quasi sferica);

non ha ripulito le vicinanze intorno alla sua orbita;

non è un satellite;

IAU 2006Tutti gli altri oggetti, eccetto i satelliti, che orbitano intorno al Soledevono essere considerati in maniera collettiva come “piccoli corpi delsistema solare”.

La IAU inoltre decide che: Plutone è un pianeta nano secondo ladefinizione prima citata ed è riconosciuto come prototipo di una nuovacategoria di oggetti transnettuniani.

Perchè un paragrafo tutto per Plutone?

Il dibattito all’assemblea IAU fu molto acceso: Plutone era l’unicopianeta scoperto dagli statunitensi!

Nettuno — una scoperta carta e penna!

Il primo pianeta ad essere stato trovato tramite calcoli matematici.Il comportamento di Urano portò ad ipotizzare l’esistenza di un pianetasconosciuto che ne perturbava l’orbita.Nettuno fu scoperto entro appena 1◦ dal punto previsto da Le Verrier.

Quale futuro per il nostro sistama solare?

Tante sono le domande aperte circa il nostro sistema solare, a cuicerchiamo di rispondere con esplorazioni spaziali sempre più sofisticate.

Il sistema solare è stabile?

Cosa intendiamo precisamente con la parola stabilità?

La stabilità del sistema solareLe orbite dei pianeti rimarranno essenzialmente le stesse perl’eternità, o almeno per un tempo paragonabile all’età stimatadell’universo?

In un futuro remoto, potranno avvenire drastici mutamenti delsistema solare cos̀ı come lo conosciamo noi oggi, dovuti per esempioalle collisioni tra due pianeti, alla caduta di un pianeta sul Sole o aduna espulsione di un pianeta dal nostro sistema solare?

Sulle spalle dei giganti:

due teoremi fondamentali.

Il teorema di Kolmogorov (1954)

Consideriamo una Hamiltoniana

H(p, q) = h(p) + εf(p, q) ,

dove p e q sono variabili di azione-angolo. Assumiamo che la componenteimperturbata h(p) sia non-degenere e che esista p∗ ∈ Rn tale che lecorrispondenti frequenze ω = ∂h∂p (p

∗) soddisfino la condizione diofantea∣∣k · ω∣∣ ≥ γ|k|−τ , ∀0 6= k ∈ Nn ,dove γ ≥ 0 e τ ≥ n− 1 . Allora, per ε abbastanza piccolo,l’Hamiltoniana possiede un toro invariante sul quale il moto èquasi-periodico con frequenze ω .

Possibile soluzione per il problema della stabilità del sistema solare.

Problema: quanto deve essere piccola la perturbazione?

Hénon: massa di Giove paragonabile a quella di un protone!

Il teorema di Nekhoroshev (1977/79)

Consideriamo una Hamiltoniana

H(p, q) = h(p) + εf(p, q) ,

dove p e q sono variabili di azione-angolo. Assumiamo che la componenteimperturbata h(p) sia non-degenere allora, senza addentrarci in ulterioridettagli tecnici dell’enunciato, il teorema di Nekhoroshev afferma che, perε abbastanza piccolo,

|p(t)− p(0)| < ε1/4 , per |t| < exp

(c

(1

ε

)1/(2a)),

dove c è una costante ed a = n2 + n .

Stabilità esponenziale: si rinuncia all’invarianza perpetua delle orbite eci si accontenta di mostrare che gli elementi orbitali critici dei pianeti(semiassi maggiori, eccentricità ed inclinazioni) restino vicini a quelliiniziali per un tempo esponenzialmente lungo.

Che cosa sappiamo?

Breve rassegna di alcuni risultati più o meno recenti.

Laskar, 1988

Laskar, 1990

Laskar, 1994

Locatelli & Giorgilli, 2000

Locatelli & Giorgilli, 2007

Hayes, 2007

Laskar & Gastineau, 2009

Giorgilli, Locatelli & Sansottera, 2009

Giorgilli, Locatelli & Sansottera, 2017

Paneta Nove, un “gigante” (10 masse terrestri)

ai confini del sistema solare?

Batygin & Brown, 2016

Batygin & Morbidelli, 2017

232nd Meeting of the American Astronomical Society

4 June 2018Madigan & ZdericCollective gravity, not PlanetNine, may explain the orbitsof ’detached objects’.

I sistemi extrasolari:

la nuova era della meccanica celeste!

Osservare o scoprire i pianeti estrasolari?

Velocità radiali (752/3786 p., 558/2834 s.p. / 135/629 s.p.m.);

Transito (2815/3786 p., 2106/2834 s.p. / 462/629 s.p.m.);

Astrometria (moto proprio di una stella, nessun pianeta confermato);

Microlente gravitazionale (allineamento stella-pianeta-osservatore);

Dischi circumstellari e protoplanetari;

Rilevamento diretto (pianeti molto massivi, lontani e caldi).

Osservazione diretta (telescopio Hubble)

Metodo delle velocità radiali

Metodo del Transito — un pianeta

Misurazione dalla Terra vs. la sonda Kepler

Metodo del Transito — più pianeti

Mayor & Queloz (1995) — Il primo successo!

La scoperta del primo pianeta extrasolare, scoperto con il metodo dellavelocità radiale.

I primi 5 pianeti scoperti dalla sonda Kepler (hot Jupiters)

Tatooine — Star Wars

Il pianeta desertico orbitante attorno ad una stella binaria dove ècresciuto Luke Skywalker!

Kepler-16

Tatooine — Star Wars

Il pianeta desertico orbitante attorno ad una stella binaria dove ècresciuto Luke Skywalker!Kepler-16

Extrasolar Systems vs. Solar System

The main difference between theextrasolar systems and the SolarSystem regards the shape of theorbits.

In the extrasolar systems, themajority of the orbits describe trueellipses (high eccentricities) and nomore almost circular like in the SolarSystem.

The classical approach uses the circular approximation as a reference.Dealing with systems with high eccentricities we need to compute theexpansion at high order to study the long-term evolution of theextrasolar planetary system.

Semiassi vs. Eccentricità

Superterre

Trappist-1 (sistema ultracompatto)

I sistemi extrasolari:

una breve panoramica di studi analitici.

Hamiltonian of a planetary system

F (r, r̃) = T (0)(r̃) + U (0)(r) + T (1)(r̃) + U (1)(r) ,

where r are the heliocentric coordinates and r̃ the conjugated momenta.

T (0)(r̃) =1

2

∑j>0

‖r̃j‖2(

1

m0+

1

mj

),

U (0)(r) = −G∑j>0

m0mj‖rj‖

,

T (1)(r̃) =∑i

The Poincaré variables in the plane

Λj =m0mjm0 +mj

√G(m0 +mj)aj λj = Mj + ωj︸︷︷︸fast variables

ξj =√

2Λj

√1−

√1− e2j cos(ωj) ηj = −

√2Λj

√1−

√1− e2j sin(ωj)︸︷︷︸

secular variables

where aj , ej , Mj and ωj are the semi-major axis, the eccentricity, the

mean anomaly and perihelion argument of the j-th planet, respectively.

Expansion of the Hamiltonian

In order to compute the Taylor expansion of the Hamiltonian around thefixed value Λ∗, we introduce the translated fast actions,

L = Λ−Λ∗ .

The Hamiltonian reads,

H = n∗ · L+∞∑j1=2

h(Kep)j1,0

(L) + µ

∞∑j1=0

∞∑j2=0

hj1,j2(L,λ, ξ,η) .

where h(Kep)j1,0

is a hom. pol. of degree j1 in L and

hj1,j2 is a

hom. pol. of degree j1 in L ,

hom. pol. of degree j2 in ξ,η ,

with coeff. that are trig. pol. in λ .

We will choose the lowest possible limits in order to include thefundamental features of the system.

Expansion of the Hamiltonian

In order to compute the Taylor expansion of the Hamiltonian around thefixed value Λ∗, we introduce the translated fast actions,

L = Λ−Λ∗ .

The computed Hamiltonian reads,

H = n∗ · L+2∑

j1=2

h(Kep)j1,0

(L) + µ

1∑j1=0

12∑j2=0

hj1,j2(L,λ, ξ,η) ,

where h(Kep)j1,0


hj1,j2 is a




We will choose the lowest possible limits in order to include thefundamental features of the system.

First order averaging

We consider the averaged Hamiltonian,

H(L,λ, ξ,η) = 〈H(L,λ, ξ,η)〉λ .

Namely, we get rid of the fast motion removing from the expansion ofthe Hamiltonian all the terms that depend on the fast angles λ .

This is the so called first order averaging.

We end up with the Hamiltonian,

H(sec) = µ

12∑j2=0

h0,j2(ξ,η) .

Secular dynamics

Doing the averaging over the fast angles (as we are interested in thesecular motions of the planets), the system pass from 4 to 2degrees of freedom,

H(sec) = H0(ξ,η) +H2(ξ,η) +H4(ξ,η) + . . . ,

where H2j is a hom. pol. of degree (2j + 2) in (ξ, η), for all j ∈ N .

ξ = η = 0 is an elliptic equilibrium point, thus we can introduceaction-angle variables via Birkhoff normal form.

Having the Hamiltonian in Birkhoff normal form, we can easilysolve the equations of motion and finally obtain the motion of theorbital parameters.

Secular dynamics

If the remainder, Rr , is small enough, we can neglect it!

The equations of motion are

Φ̇j(0) = 0 , ϕ̇j(0) =∂H(r)

∂Φj

∣∣∣∣(Φ(0),ϕ(0))

.

The solutions are

Φj(t) = Φj(0) , ϕj(t) = ϕ̇j(0) t+ ϕj(0) .

Analytical integration

(η(0), ξ(0)

) (Φ(r)(0), ϕ(r)(0)

)

(Φ(r)(t), ϕ(r)(t)

)(η(t), ξ(t)

)

Secular + NF(r)

Φ(r)(t)=Φ(r)(0)ϕ(r)(t)=ϕ̇(r)(0)t+ϕ(r)(0)

Numericalintegration

(NF(r))−1

First order approximation (HD 134987)

HD 134987: the system is secular.

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06

Eccentricities analy_HD_134987_ord1

"./ecc_0.dat""./ecc_1.dat"

"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4

But. . . near mean-motion resonance (HD 108874)

HD 108874: the system is “close” to the 4:1 mean-motion resonance.

First order averaged Hamiltonian failed.

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10000 20000 30000 40000 50000 60000 70000




Second order averaging

Coming back to the original Hamiltonian,

H = n∗ · L+∑j1≥2

hj1,0(L) + µ∑j1≥0

∑j2≥0

hj1,j2(L,λ, ξ,η) .

If we consider the point L(0) = 0, we have

L̇j = −µ∑j2≥0

∂h0,j2(λ, ξ,η)

∂λj.

In order to get rid of the fast motion, instead of simply erasing the termsdepending on fast angles λ, we perform a canonical transformation viaLie Series to kill the terms

∂h0,0(λ, ξ,η)

∂λ,∂h0,1(λ, ξ,η)

∂λ,∂h0,2(λ, ξ,η)

∂λ, . . . .

The scheme of the preliminary perturbation reduction

We perform a “Kolmogorov-like” step of normalization.

We determine the generating function, χ, by solving the equation

n∗∂χ

∂λ+

KS∑j2=0

⌈h0,j2

⌉λ:KF

= 0 .

where d·eλ:KF means that we keep only the terms depending on λ and atmost of degree KF . The parameters KS and KF are chosen so as toinclude in the secular model the main effects due to the possibleproximity to a mean-motion resonance.

The transformed Hamiltonian reads

H(O2) = expLµχH =∞∑j=0

1

j!LjµχH .

This is our Hamiltonian at order two in the masses.

Analytical integration

(η(0), ξ(0)

) (Φ(r)(0), ϕ(r)(0)

)

(Φ(r)(t), ϕ(r)(t)

)(η(t), ξ(t)

)

Secular + NF(r)

Φ(r)(t)=Φ(r)(0)ϕ(r)(t)=ϕ̇(r)(0)t+ϕ(r)(0)

Numericalintegration

(NF(r))−1


HD 11506: the system is “close” to the 7:1 MMR (weak MMR).

0.2

0.25

0.3

0.35

0.4

0.45

0 10000 20000 30000 40000 50000 60000




Second order approximation (HD 11506)

HD 11506: the system is “close” to the 7:1 MMR (weak MMR).

0.2

0.25

0.3

0.35

0.4

0.45

0 10000 20000 30000 40000 50000 60000





HD 177830: the system is “close” to the 3:1 and 4:1 MMR.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10000 20000 30000 40000 50000





HD 108874: the system is “close” to the 4:1 MMR (strong MMR).

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10000 20000 30000 40000 50000 60000 70000




Proximity to a mean-motion resonance

We now want to evaluate the proximity to a mean-motion resonance.

The idea is to rate the proximity by looking at the canonical change ofcoordinates induced by the approximation at order two in the masses.

ξ′j = ξj − µ∂ χ

∂ηj= ξj

(1− µ

ξj

∂ χ

∂ηj

),

η′j = ηj − µ∂ χ

∂ξj= ηj

(1− µ

ηj

∂ χ

∂ξj

).

In particular, we focus on the coefficients of the terms

δξj =µ

ξj

∂ χ

∂ηjand δηj =

µ

ηj

∂ χ

∂ξj.

δ-criterion

System a1/a2 δ

Sec

ula

rHD 11964 0.072 9.897× 10−4 sin(−λ1 + 2λ2)HD 74156 0.075 9.681× 10−4 cos( 4λ1 − λ2)HD 134987 0.140 9.897× 10−4 sin(−λ1 + 2λ2)HD 163607 0.149 1.376× 10−3 cos( 3λ1 − λ2)HD 12661 0.287 1.760× 10−3 sin(−λ1 + 2λ2)HD 147018 0.124 2.455× 10−3 sin(−2λ1 + λ2)

nea

ra

MM

R

HD 11506 0.263 2.943× 10−3 cos( λ1 − 7λ2)HD 177830 0.420 2.551× 10−3 cos( λ1 − 4λ2)HD 9446 0.289 2.328× 10−3 sin(−2λ1 + λ2)HD 169830 0.225 2.316× 10−2 cos( λ1 − 9λ2)υ Andromedae 0.329 1.009× 10−2 cos( λ1 − 5λ2)Sun-Jup-Sat 0.546 2.534× 10−2 cos(2λ1 − 5λ2)

MM

R HD 108874 0.380 4.314× 10−2 sin(−λ1 + 4λ2)

HD 128311 0.622 6.421× 10−1 sin(−λ1 + 2λ2)HD 183263 0.347 5.253× 10−2 cos( λ1 − 5λ2)

Results

1 Can we predict the long-term evolution of extrasolar systems?

If the system is not too close to a mean-motion resonance, providingan approximation of the motions of the secular variables up to ordertwo in the masses, the secular evolution is well approximated viaBirkhoff normal form.

2 How can we evaluate the influence of a mean-motion resonance?

The secular Hamiltonian at order two in the masses is explicitlyconstructed via Lie Series, so the generating function contains theinformation about the proximity to a mean-motion.

We introduce an heuristic and quite rough criterion that we think isuseful to discriminate between the different behaviors:

(i) δ ≤ 2.6× 10−3 : secular;(ii) 2.6× 10−3 < δ ≤ 2.6× 10−2 : near mean-motion resonance;(iii) δ > 2.6× 10−2 : in mean-motion resonance.

Laskar & Petit, 2018

Angular Momentum Deficit

The AMD is defined as the difference between the norm of the angularmomentum of a coplanar and circular system with the same semi-majoraxis values and the norm of the angular momentum (G)

G =

n∑k=1

Lk

√1− e2k cos ik , Lk = mk

√µak

C =

n∑k=1

Lk

(1−

√1− e2k cos ik

)

AMD and instabilities (collisions)

The instabilities of a planetary system often result in a modification of itsarchitecture. A planet can be ejected from the system or can fall into thestar; in both cases this results in a loss of AMD for the system. Theoutcome of the AMD after a planetary collisions is less trivial, but itturns out that in all circumstances we have a decrease of the angularmomentum deficit during the collision.

AMD-Stability

We say that a planetary system is AMD-stable if the angular momentumdeficit (AMD) amount in the system is not sufficient to allow forplanetary collisions.The AMD is conserved in the secular system at all orders, thus inabsence of short period resonances, the AMD-stability ensures thepractical long-term stability of the system.

The condition of AMD-stability is obtained when the orbits of twoplanets of semi-major axis a, a′ cannot intersect under the assumptionthat the total AMD C has been absorbed by these two planets.

In the planar case one has

a(1 + e) = a′(1− e′)

m√µa(1−

√1− e2) +m′

√µa′(1−

√1− e′2) = C,

where (m, a, e) are the parameters of the inner orbit and (m′, a′, e′)those of the outer orbit (a ≤ a′).

Collisional condition

Denoting α = a/a′, γ = m/m′, we have

D(e, e′) = αe+ e′ − 1 + α = 0

C(e, e′) = γ√α(1−

√1− e2) + (1−

√1− e′2) = C/Λ′

with Λ′ = m′√µa′, and where C(e, e′) = C/Λ′ is called the relative

AMD.

The set of collisional conditions can be solved using Lagrange multipliers.

Critical AMD Cc(α, γ)

For a given pair of ratios of semi-major axes and masses, (α, γ), there isalways a unique critical value Cc(α, γ) of the relative AMD C = C/Λ′which defines the AMD-stability. The system of two planets isAMD-stable if and only if

C = CΛ′

< Cc(α, γ) .

Our approach (in pills)

Focus on exoplanets detected by RV method =⇒ inclinations unknown.

“Reverse approach” in the framework of the Stability Problem:

The stability of the observed multi-planet system is prescribed.

One can determine the range of possible values that are compatiblewith the stability of the system.

In the past, this has been done by means of many numerical integrationsand using frequency analysis (Laskar & Correia, A&A, 496 (2009)).

What’s new?We prescribe KAM-stability for the secular dynamics.

We analyze a stability indicator, given by the construction of theKolmogorov’s normal form (L.U. & Giorgilli, CMDA (2000)).

Pros:“Synthetic Coverage”(via interval arithetics).

Pros & Cons:Understanding.

Cons:Optimality.

Construction scheme of KAM tori: the secular SJS system

1 Start from the classical expansions of the three–body Hamiltonian inPoincaré heliocentric coordinates and reduce the angular momentum.

2 Average over the mean motion angles (λ1, λ2) , then remove thecorresponding actions (L1, L2) , that now are constant of motions.

3 The form of the secular Hamiltonian is now the following:

H(ξ1, ξ2, η1, η2) = P2(ξ1, ξ2, η1, η2) + P4(ξ1, ξ2, η1, η2) + P6 . . . ,

where (ξ, η) = O(ecc.) and P2j is a homogeneous polynomial ofdegree 2j in the arguments. Therefore, it is useful to first“diagonalize” the quadratic part by a linear canonical transformation.

4 Perform a Birkhoff normalization up to a fixed degree (say, 6).

5 In action–angle coordinates such that (ξ, η) =√

2I(cosϕ, sinϕ) :

H(I, ϕ) = ν1I1+ν2I2+P4(I1, I2)+P6(I1, I2)+P8(I1, I2, ϕ1, ϕ2)+. . . ,

6 “Shift the actions origin” so that the integrable part is centeredabout the torus corresponding to the wanted secular frequencies.

7 Perform the usual “Kolmogorov’s normalization algorithm”.

The Hamiltonian in Poincaré variables

1 Introduce new actions Lj = Λj − Λ∗j , where Λ∗j is the value of Λjfor the observed semi-major axis aj .

2 Expand the Hamiltonian in power series of variables L, ξ, η,parameter D2 and in Fourier series of λ , so to obtain

H(TF ) =

∞∑j1=1

h(Kep)j1,0

(L) + µ

∞∑j1=0

∞∑j2=0

Ds2 h(TF )s;j1,j2

(L,λ, ξ,η)

where h(Kep)j1,0


h(TF )s;j1,j2

is a




Parameter D2 ' a normalized Angular Momentum DeficitThe parameter D2, appearing in the expansions, is defined as

D2 =(Λ1 + Λ2)

2 − C2

Λ1Λ2,

where C is the norm of the total angular momentum.

Remark:D2 = 0 for orbits that are circular and coplanar, becauseD2 = O(e2 + i2).

Remark:The istantaneous value of the mutual inclination can be calculatedstarting from the parameter D2 and the variables L, ξ, η:

1− cos(i1 + i2) =D2 − 2

(1

Λ1+ 1Λ2

)(I1 + I2) +

1Λ1Λ2

(I1 + I2)2

2(

1− I1Λ1)(

1− I2Λ2) ,

where I1 =(ξ21 + η

21

)/2 and I2 =

(ξ22 + η

22

)/2.

The secular part of the Hamiltonian

Introducing the secular system (better, up to order 2 in the masses):average over the fast angles λ, and put L = 0 ;hereafter, we are considering a system with two degrees of freedom.

From the D’Alembert rules, it follows that

H(sec) = H(sec)1,1 +

2∑j=1

D2−j2 H(sec)2;j +

3∑j=1

D3−j2 H(sec)3;j + . . . ,

where H(sec)s,j is a hom. pol. of degree 2j in ξ and η , ∀ 1 ≤ j ≤ s .

ξ = η = 0 is an elliptic equilibrium point.A linear canonical transformation D make the quadratic termdiagonal. The new Hamiltonian is given by H(D) = H(sec) ◦ D ,being

H(D) = H(D)1,1 +D2H

(D)2,1 +H

(D)2,2 +D

22H

(D)3,1 +D2H

(D)3,2 +H

(D)3,3 +. . .

its decomposition in even homogeneous polynomials and

H(D)1,1 =

ν12

(ξ21 + η

21

)+ν22

(ξ22 + η

22

).

Let A be the can. transf. introducing action–angle variables (I,ϕ)so that Ij =

(ξ2j + η

2j

)/2 for j = 1, 2 . We define H(I) = H(D) ◦A .

Partial Birkhoff normalization of the secular Hamiltonian

Let us focus on the Hamiltonian in action–angle coordinates:

H(I)(I,ϕ) = ν · I +∞∑s=2

s∑j=1

Ds−j2 P(I)s;j(I,ϕ) ,

where Ps;j is an hom. pol. of degree 2j in the square roots of actions Iand a trigonometric pol. of degree 2s in angles ϕ .The following way to expand the Hamiltonian highlights both the size ofthe perturbation (with respect to both eccenticities and inclinations, inthe horizontal direction) and the degree in action (vertically):

·· ·

+ P(I)4,4(I,ϕ) . . .

+ P(I)3,3(I,ϕ) +D2P(I)4,3(I,ϕ) . . .

+ P(I)2,2(I,ϕ) +D2P(I)3,2(I,ϕ) +D

22P

(I)4,2(I,ϕ) . . .

H(I)(I,ϕ) = ν · I +D2P(I)2,1(I,ϕ) +D22P(I)3,1(I,ϕ) +D

32P

(I)4,1(I,ϕ) . . .


Let us solve the equation for the generating function B(II):{B(II) , ν · I

}+D2P(I)2,1(I,ϕ) + P

(I)2,2(I,ϕ) = D2Z2,1(I) + Z2,2(I) ,

where {·, ·} is a Poisson bracket, Z2,1 and Z2,2 are angular averages ofP(I)2,1 and P

(I)2,2, resp. That equation can be solved if |k · ν| 6= 0 ∀ k ∈ Z2

such that 0 < |k| ≤ 4 . Thus, we calculate the new HamiltonianH(II) = expLB(II)H(I), the expansion of which can be written as follows(with new terms P(II)s,j sharing the same properties with P

(I)s,j

∀ 1 ≤ j ≤ s ):

·· ·

+ P(II)4,4 (I,ϕ) . . .

+ P(II)3,3 (I,ϕ) +D2P(II)4,3 (I,ϕ) . . .

+ Z2,2(I) +D2P(II)3,2 (I,ϕ) +D22P(II)4,2 (I,ϕ) . . .

H(II)(I,ϕ) = ν · I +D2Z2,1(I) +D22P(II)3,1 (I,ϕ) +D

32P

(II)4,1 (I,ϕ) . . .


Let us solve the equation for the generating function B(III):

{B(III) , ν · I

}+

3∑j=1

D3−j2 P(II)3,j (I,ϕ) =

3∑j=1

D3−j2 Z3,j(I) ,

where {·, ·} is a Poisson bracket, Z3,j is the angular average of P(II)3,j and∀ j = 1, 2, 3. That equation can be solved if |k · ν| 6= 0 ∀ k ∈ Z2 suchthat 0 < |k| ≤ 6 . Thus, we calculate H(III) = expLB(III)H(II), theexpansion of which can be written as follows (with new terms P(III)s,jsharing the same properties with P(II)s,j ∀ 1 ≤ j ≤ s ):

·· ·

+ P(III)4,4 (I,ϕ) . . .

+ Z3,3(I) +D2P(III)4,3 (I,ϕ) . . .

+ Z2,2(I) +D2Z3,2(I) +D22P

(III)4,2 (I,ϕ) . . .

H(III)(I,ϕ) = ν · I +D2Z2,1(I) +D22Z3,1(I) +D32P(III)4,1 (I,ϕ) . . .

Initial translation of the actions

Remark:

The action vector I is nearly constant, i.e. I(t) ' I(0) where I∗ = I(0)can be approximately computed, by using the initial (observed) values ofsemi-major axes and eccentricities.

We shift the origin of the actions so to center the actions about I∗

Let the canonical transformation T be so thatT (I,ϕ) = (p+ I∗, q) , introduce the new HamiltonianH(0) = H(III) ◦ T and expand it:

......

......

...

f(0,0)2 (p) f

(0,1)2 (p, q) . . . f

(0,s)2 (p, q) . . .

H(0)(p, q) =∑

ω(0) · p f (0,1)1 (p, q) . . . f(0,s)1 (p, q) . . .

0 f(0,1)0 (q) . . . f

(0,s)0 (q) . . .

Some remarks about the algorithm constructing theKolmogorov’s normal form

With respect to the method described in L.&G., Cel. Mech. Dyn.Astr. (2000), we do not fix the frequency vector ω associated to thewanted invariant tori.

The terms f(0,s)j appearing in the formula giving H

(0) are defined soto have particular functional properties:

f(0,s)j is a hom. pol. of degree j in actions p and a trigonometric

pol. of degree 2s in q . Thus, the expansion of each f(0,s)j is

representable on a computer because it is finite.

The Kolmogorov’s normalization algorithm requires to eliminate allthe terms having degree equal to 0 or 1 in the actions, except ω · p .Where is the small parameter? The size of the perturbation is ruledby the translation vector I∗. Moreover, going to the right in theexpansion of H(0), the terms get smaller and smaller.








The Kolmogorov’s normalization algorithm requires to eliminate allthe terms having degree equal to 0 or 1 in the actions, except ω · p .

Where is the small parameter? The size of the perturbation is ruledby the translation vector I∗. Moreover, going to the right in theexpansion of H(0), the terms get smaller and smaller.








The Kolmogorov’s normalization algorithm requires to eliminate allthe terms having degree equal to 0 or 1 in the actions, except ω · p .Where is the small parameter? The size of the perturbation is ruledby the translation vector I∗. Moreover, going to the right in theexpansion of H(0), the terms get smaller and smaller.

A half of the first Kolmogorov’s normalization step

We define a new Hamiltonian Ĥ(1) = expLχ(1)1H(0) where the

generating function χ(1)1 (q) is such that{χ

(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,

being χ(1)1 a trigonometric polynomial of deg. 2.

The expansion of the new Hamiltonian Ĥ(1) is calculated by studying thefunctional properties of all the terms of expL

χ(1)1H(0); e.g.,

{χ(1)1 ,ω(0) · p} shares the same properties with f(0,1)0 , then

f̂(1,1)0 = f

(0,1)0 + Lχ(1)1 ω

(0) · p .

......

......

......

f(0,0)2 (p) f

(0,1)2 (p, q) f

(0,2)2 (p, q) . . . f

(0,s)2 (p, q) . . .∑

ω(0) · p f (0,1)1 (p, q) f(0,2)1 (p, q) . . . f

(0,s)1 (p, q) . . .

0 f(0,1)0 (q) f

(0,2)0 (q) . . . f

(0,s)0 (q) . . .




(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,



χ(1)1H(0);

e.g.,


f̂(1,1)0 = f

(0,1)0 + Lχ(1)1 ω

(0) · p .

......

......

......

f(0,0)2 (p) f

(0,1)2 (p, q) f

(0,2)2 (p, q) . . . f

(0,s)2 (p, q) . . .∑

ω(0) · p f (0,1)1 (p, q) f(0,2)1 (p, q) . . . f

(0,s)1 (p, q) . . .

0 f(0,1)0 (q) f

(0,2)0 (q) . . . f

(0,s)0 (q) . . .




(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,



χ(1)1H(0); e.g.,


f̂(1,1)0 = f

(0,1)0 + Lχ(1)1 ω

(0) · p .

......

......

......

f(0,0)2 (p) f

(0,1)2 (p, q) f

(0,2)2 (p, q) . . . f

(0,s)2 (p, q) . . .∑

ω(0) · p f (0,1)1 (p, q) f(0,2)1 (p, q) . . . f

(0,s)1 (p, q) . . .

0 f(0,1)0 (q) f

(0,2)0 (q) . . . f

(0,s)0 (q) . . .




(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,



χ(1)1H(0); e.g.,


f̂(1,1)0 = f

(0,1)0 + Lχ(1)1 ω

(0) · p .

......

......

......

f(0,0)2 (p) f

(0,1)2 (p, q) f

(0,2)2 (p, q) . . . f

(0,s)2 (p, q) . . .∑

ω(0) · p f (0,1)1 (p, q) f(0,2)1 (p, q) . . . f

(0,s)1 (p, q) . . .

0 0 f(0,2)0 (q) . . . f

(0,s)0 (q) . . .




(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,


The expansion of the new Hamiltonian Ĥ(1) is calculated by studying the

functional properties of all the terms of expLχ(1)1H(0);

e.g., {χ(1)1 , f(0,0)2 }

is really like f(0,1)1 , then f̂

(1,1)1 = f

(0,1)1 + Lχ(1)1 f

(0,0)2 .

......

......

......

f(0,0)2 (p) f

(0,1)2 (p, q) f

(0,2)2 (p, q) . . . f

(0,s)2 (p, q) . . .∑

ω(0) · p f (0,1)1 (p, q) f(0,2)1 (p, q) . . . f

(0,s)1 (p, q) . . .

0 0 f(0,2)0 (q) . . . f

(0,s)0 (q) . . .




(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,



functional properties of all the terms of expLχ(1)1H(0); e.g., {χ(1)1 , f

(0,0)2 }


(1,1)1 = f

(0,1)1 + Lχ(1)1 f

(0,0)2 .

......

......

......

f(0,0)2 (p) f

(0,1)2 (p, q) f

(0,2)2 (p, q) . . . f

(0,s)2 (p, q) . . .∑

ω(0) · p f (0,1)1 (p, q) f(0,2)1 (p, q) . . . f

(0,s)1 (p, q) . . .

0 0 f(0,2)0 (q) . . . f

(0,s)0 (q) . . .




(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,



functional properties of all the terms of expLχ(1)1H(0); e.g., {χ(1)1 , f

(0,0)2 }


(1,1)1 = f

(0,1)1 + Lχ(1)1 f

(0,0)2 .

......

......

......

f(0,0)2 (p) f

(0,1)2 (p, q) f

(0,2)2 (p, q) . . . f

(0,s)2 (p, q) . . .∑

ω(0) · p f̂ (1,1)1 (p, q) f(0,2)1 (p, q) . . . f

(0,s)1 (p, q) . . .

0 0 f(0,2)0 (q) . . . f

(0,s)0 (q) . . .




(1)1 , ω

(0) · p}

+ f(0,1)0 = 0 ,



χ(1)1H(0). Therefore, we can

get the recursive expressions of all f̂(1,s)j in the following expansion:

......

......

......

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .

Ĥ(1)(p, q) =∑

ω(0) · p f̂ (1,1)1 (p, q) f̂(1,2)1 (p, q) . . . f̂

(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .

Completing the first Kolmogorov’s normalization step

We define a new Hamiltonian H(1) = expLχ(1)2Ĥ(1) where the

generating function χ(1)2 (p, q) is such that{

χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,

where χ(1)2 is linear in p and a trigonometric polynomial of deg. 2 in q .

The expansion of the new Hamiltonian H(1) is calculated by studying thefunctional properties of all the terms of expL

χ(1)2Ĥ(1); e.g.,

{χ(1)2 ,ω(0) · p} shares the same properties with f̂(1,1)1 , then

f(1,1)1 = f̂

(1,1)1 + Lχ(1)2 ω

(0) · p ....

......

......

...

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(0) · p f̂ (1,1)1 (p, q) f̂(1,2)1 (p, q) . . . f̂

(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,



χ(1)2Ĥ(1);

e.g.,


f(1,1)1 = f̂

(1,1)1 + Lχ(1)2 ω

(0) · p ....

......

......

...

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(0) · p f̂ (1,1)1 (p, q) f̂(1,2)1 (p, q) . . . f̂

(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,



χ(1)2Ĥ(1);

e.g.,


f(1,1)1 = f̂

(1,1)1 + Lχ(1)2 ω

(0) · p .

......

......

......

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(0) · p f̂ (1,1)1 (p, q) f̂(1,2)1 (p, q) . . . f̂

(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,



χ(1)2Ĥ(1); e.g.,


f(1,1)1 = f̂

(1,1)1 + Lχ(1)2 ω

(0) · p ....

......

......

...

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(0) · p f̂ (1,1)1 (p, q) f̂(1,2)1 (p, q) . . . f̂

(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,



χ(1)2Ĥ(1); e.g.,


f(1,1)1 = f̂

(1,1)1 + Lχ(1)2 ω

(0) · p ....

......

......

...

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(0) · p 〈f̂ (1,1)1 〉 f̂(1,2)1 (p, q) . . . f̂

(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,


The expansion of the new Hamiltonian H(1) is calculated by studying the

functional properties of all the terms of expLχ(1)2Ĥ(1);

e.g., {χ(1)2 , f̂(1,0)2 }

shares the same properties with f̂(1,1)2 , then f

(1,1)2 = f̂

(1,1)2 +Lχ(1)2 f̂

(1,0)2 .

......

......

......

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(1) · p 0 f̂ (1,2)1 (p, q) . . . f̂(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,



functional properties of all the terms of expLχ(1)2Ĥ(1); e.g., {χ(1)2 , f̂

(1,0)2 }


(1,1)2 = f̂

(1,1)2 +Lχ(1)2 f̂

(1,0)2 .

......

......

......

f̂(1,0)2 (p) f̂

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(1) · p 0 f̂ (1,2)1 (p, q) . . . f̂(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,



functional properties of all the terms of expLχ(1)2Ĥ(1); e.g., {χ(1)2 , f̂

(1,0)2 }


(1,1)2 = f̂

(1,1)2 + Lχ(1)2 f̂

(1,0)2 .

......

......

......

f̂(1,0)2 (p) f

(1,1)2 (p, q) f̂

(1,2)2 (p, q) . . . f̂

(1,s)2 (p, q) . . .∑

ω(1) · p 0 f̂ (1,2)1 (p, q) . . . f̂(1,s)1 (p, q) . . .

0 0 f̂(1,2)0 (q) . . . f̂

(1,s)0 (q) . . .




χ(1)2 , ω

(0) · p}

+f̂(1,1)1 = 〈f̂

(1,1)1 〉 with 〈f̂

(1,1)1 〉 =

∫f̂

(1,1)1 dq1 . . . dqn

(2π)n,



χ(1)2Ĥ(1). Therefore, we can

get the recursive expressions of all f̂(1,s)j in the following expansion:

......

......

......

f(1,0)2 (p) f

(1,1)2 (p, q) f

(1,2)2 (p, q) . . . f

(1,s)2 (p, q) . . .

H(1)(p, q) =∑

ω(1) · p 0 f (1,2)1 (p, q) . . . f(1,s)1 (p, q) . . .

0 0 f(1,2)0 (q) . . . f

(1,s)0 (q) . . .

Completing all the Kolmogorov’s normalization algorithm

The Kolmogorov’s normalization step can be “infinitely” iterated, if

the frequency vectors ω(r) are non-resonant enough (e.g.,diophantine, i.e. |k · ω(r)| > γ

/|k|τ for each normalization step r,

with some fixed γ > 0 and τ > n− 1);the perturbation (or, equivalently, ‖I∗‖) is small enough;the hessian of f

(0,0)2 (p) is non-degenerate, then frequencies ω

(0)

(for which the procedure works) form a set with positive measure.

Therefore, the sequence of H(r) is convergent to a Hamiltonian H(∞) inKolmogorov’s normal form, the expansions of which is written as

......

......

...

f(∞,0)2 (p) f

(∞,1)2 (p, q) . . . f

(∞,s)2 (p, q) . . .

H(∞)(p, q) =∑

ω(∞) · p 0 . . . 0 . . .0 0 . . . 0 . . .

Definition of the computational algorithm

Input: masses, semi-major axes,eccentricities and a grid oftemptative values for D2 .

Final operations: the behavior ofthe norms is automaticallyanalyzed (they should decreaseexponentially to allow theconvergence of the algorithm).Output: for each value of D2 , ifthe norms are regularly decreasing,the final result is 1, else it is 0.

=⇒

⇐=

Procedure: algebraic manipulations ona computer so to implement all theprescribed expansions & transformations.First (partial) output: for each value

of D2 we plot ‖χ(r)2 ‖ as a function ofthe normalization step r.

↓

1.0e-12

1.0e-10

1.0e-08

1.0e-06

1.0e-04

1.0e-02

1.0e+00

2 4 6 8 10 12 14 16 18 20

| chi

2[r]

|

Normalization step r

Behaviour of the norms of the generating function chi2

"norme_delle_chi.out" using 1:4

Definition of the computational algorithm

Input: masses, semi-major axes,eccentricities and a grid ofinterval of values for D2 .

Final operations: the behavior ofthe norms is automaticallyanalyzed (they should decreaseexponentially to allow theconvergence of the algorithm).Output: for each set of D2’s, ifthe norms are regularly decreasing,the final result is 1, else it is 0.Remark: not a computer-assistedproof, but interval arithmetics isused (nearly) everywhere.

=⇒

⇐=

Procedure: algebraic manipulations ona computer so to implement all theprescribed expansions & transformations.First (partial) output: for each set of

D2’s we plot ‖χ(r)2 ‖ as a function of thenormalization step r.

↓

1.0e-12

1.0e-10

1.0e-08

1.0e-06

1.0e-04

1.0e-02

1.0e+00

2 4 6 8 10 12 14 16 18 20

| chi

2[r]

|

Normalization step r

Behaviour of the norms of the generating function chi2

"norme_delle_chi.out" using 1:4

Values of the mutual inclinations compatible with KAMstability for the secular dynamics: the case of HD141399

HD141399 – ParametersMass of the Star: 1.14 M�Masses of the planets:mc = 1.33 MJupmd = 1.18 MJup

Semi-major axes:ac ' 0.704 AUad ' 2.14 AU

Eccentricities:ec = 0.048± 0.009ed = 0.074± 0.025

ConclusionMutual inclinations up to ∼ 18◦are compatible with KAMstability (in the sense of theconstruction of normal forms).

0

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Tru

e (

1)

/ F

als

e (

0)

D2

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

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incli

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D2


HD143761 – ParametersMass of the Star: 0.99 M�Masses of the planets:mb = 1.045 MJupmc = 0.079 MJup

Semi-major axes:ab ' 0.228 AUac ' 0.427 AU

Eccentricities:eb = 0.037± 0.004ec = 0.050± 0.004


-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

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D2


HD40307 – ParametersMass of the Star: 0.77 M�Masses of the planets:mc = 0.0202 MJupmd = 0.0275 MJup

Semi-major axes:ac ' 0.081 AUad ' 0.134 AU

Eccentricities:ec = 0.060± 0.005ed = 0.070± 0.005


-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Mutu

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D2

Perspectives

Possible extensionsThe initial expansions can be adapted, by using intervals, so to takeinto account the factor 1/ sin(i) modifying the mass of each planet[not so trivial, modifying the initial expansions is hard].

Range of values for the eccentricities (when they are unknown)could be determined by adapting this same approach to extrasolarsystems where the inclinations are known [it should be easy].

The same analysis of the stability could be done by considering thecomplete planetary problem instead of just the secular dynamics[much harder from a computational point of view].

In order to consider richer systems, it is mandatory to consider morethan two planets [well... This is both harder from a computationalpoint of view and it requires also to modify the initial expansions...].

It is important to modify the normal form approach so to coveralso the case of extrasolar systems with large eccentricities,where our method actually fails [This is the most conceptualtask, some new ideas are required!].

Documents

Una (mica tanto) breve introduzione ai pianeti extrasolari. · 2019. 6. 4. · Una (mica tanto) breve introduzione ai pianeti extrasolari. Alla ricerca di nuovi mondi (abitabili)