Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and...
Transcript of Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and...
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Non-linear oscillations and long-term evolutionapplication to planetary systems and spin-orbit problem
Marco Sansottera [1]
Based on research works in collaboration with
Giorgilli[2] & Locatelli[3] , Libert[1] and Lhotka[1] & Lemaıtre[1]
[1]NAXYS, University of Namur[2]University of Milan[3]Univerity of Rome “Tor Vergata”
Planetary Motions, Satellite Dynamicsand Spaceship OrbitsMontreal, 23.07.2013
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Aim of the work
Problems
Stability of the Solar System (secular dynamics).Secular dynamics of exoplanetary system.Long-time stability around a Cassini state.
Models considered
Sun-Jupiter-Saturn-Uranus (plane).Systems with two coplanar planets.Spin-orbit problem: Saturn-Titan.
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Aim of the work
Problems
Stability of the Solar System (secular dynamics).Secular dynamics of exoplanetary system.Long-time stability around a Cassini state.
Models considered
Sun-Jupiter-Saturn-Uranus (plane).Systems with two coplanar planets.Spin-orbit problem: Saturn-Titan.
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A common point
The analytical form of the Hamiltonian is similar to that of a Hamiltonianin the neighbourhood of an elliptic equilibrium, namely
H(x, y) =1
2
∑l
ωl(x2l + y2
l ) +H1(x, y) +H2(x, y) + . . . ,
where Hs is a homogeneous polynomial of degree s+ 2.
This is a perturbed system of harmonic oscillators.
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Birkhoff normal form
We look for a near the identity canonical change of coordinates such thatthe Hamiltonian is in Birkhoff normal form up to order r, namely
H(r)(x, y) = H0(I) + Z1(I) + . . . Zr(I) +R(r)r+1(x, y) + . . . ,
where Il = 12 (x2
l + y2l ) are the actions of the system, Zs is a
homogeneous polynomial of degree (s+ 2)/2 in I and the terms
R(r)s (x, y) are homogeneous polynomial of degree s+ 2 in (x, y).
At each step one has to solve the equation
χ(r+1), ω · I+R(r)r+1(x, y) = Zr+1(I) ,
provided the non-resonance condition
k · ω 6= 0 for 0 < |k| ≤ r + 3 .
Thus we can write the new Hamiltonian as H(r+1) = expLχ(r+1) H(r).
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Birkhoff normal form
We look for a near the identity canonical change of coordinates such thatthe Hamiltonian is in Birkhoff normal form up to order r, namely
H(r)(x, y) = H0(I) + Z1(I) + . . . Zr(I) +R(r)r+1(x, y) + . . . ,
where Il = 12 (x2
l + y2l ) are the actions of the system, Zs is a
homogeneous polynomial of degree (s+ 2)/2 in I and the terms
R(r)s (x, y) are homogeneous polynomial of degree s+ 2 in (x, y).
At each step one has to solve the equation
χ(r+1), ω · I+R(r)r+1(x, y) = Zr+1(I) ,
provided the non-resonance condition
k · ω 6= 0 for 0 < |k| ≤ r + 3 .
Thus we can write the new Hamiltonian as H(r+1) = expLχ(r+1) H(r).
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Birkhoff normal form
We look for a near the identity canonical change of coordinates such thatthe Hamiltonian is in Birkhoff normal form up to order r, namely
H(r)(x, y) = H0(I) + Z1(I) + . . . Zr(I) +R(r)r+1(x, y) + . . . ,
where Il = 12 (x2
l + y2l ) are the actions of the system, Zs is a
homogeneous polynomial of degree (s+ 2)/2 in I and the terms
R(r)s (x, y) are homogeneous polynomial of degree s+ 2 in (x, y).
At each step one has to solve the equation
χ(r+1), ω · I+R(r)r+1(x, y) = Zr+1(I) ,
provided the non-resonance condition
k · ω 6= 0 for 0 < |k| ≤ r + 3 .
Thus we can write the new Hamiltonian as H(r+1) = expLχ(r+1) H(r).
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Sketch of the procedure
H0
R(0)1
R(0)2
R(0)3
. . .
χ(1), ω · I+R(0)1 (x, y) = Z1(I)
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Sketch of the procedure
H0
Lχ(1)H0 R(0)1
1
2!L2χ(1)H0 Lχ(1)R(0)
1 R(0)2
1
3!L3χ(1)H0
1
2!L2χ(1)R(0)
1 Lχ(1)R(0)2 R(0)
3
. . .
H(1) = expLχ(1) H(0)
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Sketch of the procedure
H0
Z1
R(1)2
R(1)3
. . .
χ(2), ω · I+R(1)2 (x, y) = Z2(I)
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Sketch of the procedure
H0
Z1
Lχ(2)H0 R(1)2
Lχ(2)Z(1) R(1)3
. . .
H(2) = expLχ(2) H(1)
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Sketch of the procedure
H0
Z1
Z2
R(2)3
. . .
χ(2), ω · I+R(2)3 (x, y) = Z3(I)
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Sketch of the procedure
H0
Z1
Z2
Lχ(3)H0 R(2)3
. . .
H(3) = expLχ(3) H(2)
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Sketch of the procedure
H0
Z1
Z2
Z3
. . .
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Effective stability time
The Hamiltonian H(r) admits approximated first integrals of the form
Il =1
2(x2l + y2
l ) ,
indeedIl = Il, H(r) = Il,R(r) ∼ 2Il,R(r)
r+1 .
Consider a polydisk
∆ρR =
(x, y) : x2l + y2
l ≤ ρ2R2l
.
Let ρ0 = ρ/2 and (x(0), y(0)) ∈ ∆ρ0R, then
I(0) =x2j + y2
j
2≤ρ2
0R2j
2.
Thus, there is T (ρ0) > 0 such that for |t| ≤ T (ρ0) we have
I(t) ≤ρ2R2
j
2so that (x(t), y(t)) ∈ ∆ρR .
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Effective stability time
The Hamiltonian H(r) admits approximated first integrals of the form
Il =1
2(x2l + y2
l ) ,
indeedIl = Il, H(r) = Il,R(r) ∼ 2Il,R(r)
r+1 .
Consider a polydisk
∆ρR =
(x, y) : x2l + y2
l ≤ ρ2R2l
.
Let ρ0 = ρ/2 and (x(0), y(0)) ∈ ∆ρ0R, then
I(0) =x2j + y2
j
2≤ρ2
0R2j
2.
Thus, there is T (ρ0) > 0 such that for |t| ≤ T (ρ0) we have
I(t) ≤ρ2R2
j
2so that (x(t), y(t)) ∈ ∆ρR .
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Effective stability
Given a homogeneous polynomial f(x, y) of degree s as
f(x, y) =∑
|j|+|k|=s
fj,kxjyk ,
we define the quantity |f |R as
|f |R =∑
|j|+|k|=s
|fj,k|Rj+kΘj,k , Θj,k =
√jjkk
(j + k)j+k.
Thus, for ρ > 0, we have
sup(x,y)∈∆ρR
∣∣f(x, y)∣∣ < ρs|f |R .
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Effective stability
We can now estimate
sup(x,y)∈∆ρR
∣∣Ij(x, y)∣∣ ≤ 2ρr+3
∣∣Ij ,R(r)r+1
∣∣R,
and get a lower bound for the time stability T (ρ0) as
τ(ρ0, r) = minj
(1− 1
2r+1
)R2j
2(r + 1)|Ij ,R(r)r+1|R ρ
r+10
.
Finally we can setT (ρ0) = max
rτ(ρ0, r) .
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Stability of the secular problem for the
planar Sun – Jupiter – Saturn – Uranus system
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The Solar system. . .
QuestionsIs the Solar System stable?Can we apply the Kolmogorov and Nekhoroshev theorems to realisticmodels of planetary systems?
Models considered
The complete Sun-Jupiter-Saturn system (SJS).The planar Sun-Jupiter-Saturn-Uranus system (SJSU).
Answers
The KAM theorem was applied to the realistic SJS system (L.&G. 2007).We applied the Nekhoroshev’s like exponential estimates for the stabilityof the SJS system, in the neighborhood of a KAM torus (G.,L.&S. 2009).We studied the secular problem for the SJSU system (S.L.&G. 2013).
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The Solar system. . .
QuestionsIs the Solar System stable?Can we apply the Kolmogorov and Nekhoroshev theorems to realisticmodels of planetary systems?
Models considered
The complete Sun-Jupiter-Saturn system (SJS).The planar Sun-Jupiter-Saturn-Uranus system (SJSU).
Answers
The KAM theorem was applied to the realistic SJS system (L.&G. 2007).We applied the Nekhoroshev’s like exponential estimates for the stabilityof the SJS system, in the neighborhood of a KAM torus (G.,L.&S. 2009).We studied the secular problem for the SJSU system (S.L.&G. 2013).
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The Solar system. . .
QuestionsIs the Solar System stable?Can we apply the Kolmogorov and Nekhoroshev theorems to realisticmodels of planetary systems?
Models considered
The complete Sun-Jupiter-Saturn system (SJS).The planar Sun-Jupiter-Saturn-Uranus system (SJSU).
Answers
The KAM theorem was applied to the realistic SJS system (L.&G. 2007).We applied the Nekhoroshev’s like exponential estimates for the stabilityof the SJS system, in the neighborhood of a KAM torus (G.,L.&S. 2009).We studied the secular problem for the SJSU system (S.L.&G. 2013).
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Dynamics of the SJSU systems
In order to study the secular dynamics of the SJSU system we must takeinto account the triple (3,−5,−7) mean-motion resonance. IndeedSaturn is close to the celebrated 5 : 2 resonance with Jupiter, whileUranus is near to the 7 : 1 . Moreover, 2nJ − 5nS ' 7nU − nJ .
Remark:From the study of the Sun-Jupiter-Saturn system we know that a carefulhandling of the secular part of the Hamiltonian is crucial. Before startingto manipulate the secular part of the Hamiltonian, we need to reduce themain part of the perturbation depending on the fast angles.
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Dynamics of the SJSU systems
In order to study the secular dynamics of the SJSU system we must takeinto account the triple (3,−5,−7) mean-motion resonance. IndeedSaturn is close to the celebrated 5 : 2 resonance with Jupiter, whileUranus is near to the 7 : 1 . Moreover, 2nJ − 5nS ' 7nU − nJ .
Remark:From the study of the Sun-Jupiter-Saturn system we know that a carefulhandling of the secular part of the Hamiltonian is crucial. Before startingto manipulate the secular part of the Hamiltonian, we need to reduce themain part of the perturbation depending on the fast angles.
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The Hamiltonian of the planetary system
The Hamiltonian is
F (r, r) = T (0)(r) + U (0)(r) + T (1)(r) + U (1)(r) ,
where r are the heliocentric coordinates and r the conjugated momenta.
T (0)(r) =1
2
3∑j=1
‖rj‖2(
1
m0+
1
mj
),
U (0)(r) = −G3∑j=1
m0mj
‖rj‖,
T (1)(r) =r1 · r2
m0+
r1 · r3
m0+
r2 · r3
m0,
U (1)(r) = −G(
m1m2
‖r1 − r2‖+
m1m3
‖r1 − r3‖+
m2m3
‖r2 − r3‖
).
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The Poincare variables in the plane
Λj =m0mj
m0 +mj
√G(m0 +mj)aj λj = Mj + ωj︸ ︷︷ ︸fast variables
ξj =√
2Λj
√1−
√1− e2
j cos(ωj) ηj = −√
2Λj
√1−
√1− e2
j 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.
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How to expand the Hamiltonian
1 The development of the Hamiltonian is a quite standard matter.
2 Choose a Λ∗ such that
∂〈F 〉λ∂Λj
∣∣∣∣Λ=Λ∗
ξ=η=0
= n∗j , j = 1, 2, 3 .
〈.〉λ means the average over the fast angles ,n∗j are the fundamental frequencies of the mean motion .
3 Introduce new actions Lj = Λj − Λ∗j .
4 Perform the canonical transformation TF translating the fast actions.
5 Expand the Hamiltonian in power series of L, ξ, η and in Fourierseries of λ .
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The expansion of the Hamiltonian
The transformed Hamiltonian reads
H(TF ) = n∗ · L +
∞∑j1=2
h(Kep)j1,0
(L) + µ
∞∑j1=0
∞∑j2=0
h(TF )j1,j2
(L,λ, ξ,η)
where h(Kep)j1,0
is an homogeneous polynomial of degree j1 in L and
h(TF )j1,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 .
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Truncation limits of the expansion
This is the Hamiltonian,
H(TF ) = n∗ · L +
∞∑j1=2
h(Kep)j1,0
(L) + µ
∞∑j1=0
∞∑j2=0
h(TF )j1,j2
(L,λ, ξ,η)
where we also truncate all the coefficients with harmonics of degreegreater than 16.
These are the lowest limits to include the fundamental features of thesystem.
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Truncation limits of the expansion
This is the computed Hamiltonian,
H(TF ) = n∗ · L +
2∑j1=2
h(Kep)j1,0
(L) + µ
1∑j1=0
12∑j2=0
h(TF )j1,j2
(L,λ, ξ,η)
where we also truncate all the coefficients with harmonics of degreegreater than 16.
These are the lowest limits to include the fundamental features of thesystem.
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The scheme of the preliminary perturbation reduction
We now aim to kill the terms
⌈µh
(TF )0,0
⌉λ:8
(λ, ξ,η),⌈µh
(TF )0,1
⌉λ:8
(λ, ξ,η), . . . ,⌈µh
(TF )0,6
⌉λ:8
(λ, ξ,η) ,
and
⌈µh
(TF )1,0
⌉λ:8
(λ, ξ,η),⌈µh
(TF )1,1
⌉λ:8
(λ, ξ,η), . . . ,⌈µh
(TF )1,6
⌉λ:8
(λ, ξ,η) .
where d·eλ:K means the truncation of the harmonics of degree greaterthan K.
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The details of the transformation
This procedure is essentially a “Kolmogorov’s like” step of normalization.
In order to kill the term⌈h
(TF )j1,j2
⌉λ:K
, one has to solve the equation
χ, n∗ · L+⌈µh
(TF )j1,j2
⌉λ:K
= 0 .
and find the generating function χ .
The generating function χ has the same structure of h(TF )j1,j2
, is of orderO(µ) and must depends on the fast angles λ.
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The details of the transformation
We now perform a canonical transformation of the Hamiltonian
expLχH =
∞∑j=0
1
j!LjχH .
This transformation, by construction, kill the terms⌈µh
(TF )j1,j2
⌉λ:K
, but
the transformed Hamiltonian still has a term of the same type, but atleast of order O(µ2).
This effect is due to Lie series algorithm, for example take j1 = 0, j2 = 0and consider the Poisson bracket
χ , µ h(TF )1,0
→ µ2 h
(TF )0,0 .
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Partial preliminary reduction of the perturbation
Firststep
n∗ · ∂χ
(O2)1
∂λ+ µ
6∑j2=0
⌈h
(TF )0,j2
⌉λ:8
(λ, ξ,η) = 0
H = expLχ(O2)1
H =
∞∑j=0
1
j!Ljχ(O2)1
H .
Secondstep
n∗ · ∂χ
(O2)2
∂λ+ µ
6∑j2=0
⌈h
(TF )1,j2
⌉λ:8
(L,λ, ξ,η) = 0
H(O2) = expLχ(O2)2 expL
χ(O2)1
H .
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Why these limits?
The secular variables:
H χ(O2)2 −−−−→ H(O2)
y y ycos(1λ1 − 7λ3) sin(1λ1 − 7λ3) secular termsy(ξ,η)
y(ξ,η)
y(ξ,η)
6 6 12
The fast angles:
(3,−5,−7) harmonics of order 15
< 16 .
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Why these limits?
The secular variables:
H χ(O2)2 −−−−→ H(O2)y y y
cos(1λ1 − 7λ3) sin(1λ1 − 7λ3) secular terms
y(ξ,η)
y(ξ,η)
y(ξ,η)
6 6 12
The fast angles:
(3,−5,−7) harmonics of order 15
< 16 .
![Page 37: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/37.jpg)
Why these limits?
The secular variables:
H χ(O2)2 −−−−→ H(O2)y y y
cos(1λ1 − 7λ3) sin(1λ1 − 7λ3) secular termsy(ξ,η)
y(ξ,η)
y(ξ,η)
6 6 12
The fast angles:
(3,−5,−7) harmonics of order 15
< 16 .
![Page 38: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/38.jpg)
Why these limits?
The secular variables:
H χ(O2)2 −−−−→ H(O2)y y y
cos(1λ1 − 7λ3) sin(1λ1 − 7λ3) secular termsy(ξ,η)
y(ξ,η)
y(ξ,η)
6 6 12
The fast angles:
(3,−5,−7) harmonics of order 15 < 16 .
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The Hamiltonian up to order two in the masses
H(O2) is the Hamiltonian up to order two in the masses.
No terms corresponding to any triple resonances before the“Kolmogorov’s like” step.
The “Kolmogorov’s like” step introduce the triple resonances, inparticular the (3,−5,−7) resonance.
Small limits don’t mean small expansion!
After the “Kolmogorov’s like” step, we have 94 109 751 coefficients.
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The secular part up to order two in the masses
Reduction to the secular system:
average over the fast angles λ, and put L = 0 ;hereafter, we are considering a system with three degrees of freedom.
From the D’Alembert rules, it follows that
H(sec) = H0 +H2 +H4 + . . . ,
where H2j is a hom. pol. of degree (2j + 2) in ξ and η , ∀ j ∈ N .
ξ = η = 0 is an elliptic equilibrium point.
We diagonalize the quadratic term by a linear canonicaltransformation D :
H(D)2 =
3∑j=1
νj2
(ξ2j + η2
j
).
![Page 41: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/41.jpg)
The optimal normalization order
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
r opt(ρ
0)
ρ0
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The estimated “stability time” of the secular Hamiltonian
100
105
1010
1015
1020
1025
1030
1035
1040
1045
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
T(ρ
0)
ρ0
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Comments about our results
We considered a secular Hamiltonian model of the planarSun-Jupiter-Saturn-Uranus system, providing an approximation ofthe motions of the secular variables up to order two in the masses.Our results ensure that such a system is stable for a timecomparable to the age of the universe just in a domain with a radiusthat is about a half of the real distance of the initial secularvariables from the origin.
This exponential stability estimate around the equilibrium point is a“too lazy option”. Indeed, we show that a preliminary constructionof a KAM torus for the planar SJSU system allows much betterestimates.
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Secular evolution of
extrasolar systems
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Extrasolar systems vs. Solar System
The main difference between the extrasolarsystems and the Solar System regards theshape of the orbits.
In the extrasolar systems, the majority ofthe orbits describe true ellipses (higheccentricities) and no more almost circleslike in the Solar System.
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 systems.
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Aim of the work
1 Can we predict the long-term evolution of extrasolar systems?
Numerical integrations are really accurate, but have highcomputational cost. One has to compute a numerical integration foreach initial condition.
Normal forms provide non-linear approximations of the dynamics in aneighborhood of an invariant object. In addition, an accurateanalytic approximation is the starting point for the study of theeffective long-time stability.
2 How can we evaluate the influence of a mean-motion resonance?
The semi-major axis ratio gives a rough indication of the proximity tothe main mean-motion resonance.
However, the impact of the proximity to a mean-motion resonanceon the secular evolution of a planetary system depends on manyparameters. This is due to the non-linear character of the system.
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Aim of the work
1 Can we predict the long-term evolution of extrasolar systems?
Numerical integrations are really accurate, but have highcomputational cost. One has to compute a numerical integration foreach initial condition.
Normal forms provide non-linear approximations of the dynamics in aneighborhood of an invariant object. In addition, an accurateanalytic approximation is the starting point for the study of theeffective long-time stability.
2 How can we evaluate the influence of a mean-motion resonance?
The semi-major axis ratio gives a rough indication of the proximity tothe main mean-motion resonance.
However, the impact of the proximity to a mean-motion resonanceon the secular evolution of a planetary system depends on manyparameters. This is due to the non-linear character of the system.
![Page 48: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/48.jpg)
Aim of the work
1 Can we predict the long-term evolution of extrasolar systems?
Numerical integrations are really accurate, but have highcomputational cost. One has to compute a numerical integration foreach initial condition.
Normal forms provide non-linear approximations of the dynamics in aneighborhood of an invariant object. In addition, an accurateanalytic approximation is the starting point for the study of theeffective long-time stability.
2 How can we evaluate the influence of a mean-motion resonance?
The semi-major axis ratio gives a rough indication of the proximity tothe main mean-motion resonance.
However, the impact of the proximity to a mean-motion resonanceon the secular evolution of a planetary system depends on manyparameters. This is due to the non-linear character of the system.
![Page 49: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/49.jpg)
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.
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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
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.
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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) .
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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.
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Secular dynamics
If the remainder, Rr , is small enough, we can neglect it!
The equations of motion are
Φj(0) = 0 , ϕj(0) =∂H(r)
∂Φj
∣∣∣∣(Φ(0),ϕ(0))
.
The solutions are
Φj(t) = Φj(0) , ϕj(t) = ϕj(0) t+ ϕj(0) .
![Page 54: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/54.jpg)
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)=ϕ(r)(0)t+ϕ(r)(0)
Numericalintegration
(NF(r))−1
![Page 55: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/55.jpg)
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
![Page 56: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/56.jpg)
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
Eccentricities analy_HD_108874_ord1
"./ecc_0.dat""./ecc_1.dat"
"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4
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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
Lj = −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, . . . .
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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.
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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)=ϕ(r)(0)t+ϕ(r)(0)
Numericalintegration
(NF(r))−1
![Page 60: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/60.jpg)
First 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
Eccentricities analy_HD_11506_ord1
"./ecc_0.dat""./ecc_1.dat"
"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4
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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
Eccentricities analy_HD_11506_ord2
"./ecc_0.dat""./ecc_1.dat"
"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4
![Page 62: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/62.jpg)
First order approximation (HD 177830)
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
Eccentricities analy_HD_177830_ord1
"./ecc_0.dat""./ecc_1.dat"
"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4
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Second order approximation (HD 177830)
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
Eccentricities analy_HD_177830_ord2
"./ecc_0.dat""./ecc_1.dat"
"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4
![Page 64: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/64.jpg)
First order approximation (HD 108874)
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
Eccentricities analy_HD_108874_ord1
"./ecc_0.dat""./ecc_1.dat"
"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4
![Page 65: Non-linear oscillations and long-term evolution · 2013. 9. 6. · Non-linear oscillations and long-term evolution application to planetary systems and spin-orbit problem Marco Sansottera](https://reader035.fdocuments.net/reader035/viewer/2022071508/6128fd11fb46fd4276796d98/html5/thumbnails/65.jpg)
Second order approximation (HD 108874)
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
Eccentricities analy_HD_108874_ord2
"./ecc_0.dat""./ecc_1.dat"
"./orbitals_1.dat" u 1:4"./orbitals_2.dat" u 1:4
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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.
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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)
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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.
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Effective stability around the Cassini state
in the spin-orbit problem
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Hamiltonian formulation
We consider a rotating body (e.g., Titan) with mass m and equatorialradius Re , orbiting around a point body (e.g., Saturn) with mass M .
The rotating body is considered as a triaxial rigid body whose principalmoments of inertia are A, B and C, with A ≤ B < C.
We closely follow the Hamiltonian formulation that has already been usedin previous works, see, e.g., Henrard & Schwanen (2004) for a generaltreatment of synchronous satellites.
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Reference planes
In order to describe the spin-orbit motion we need four reference frames,centered at the center of mass of the rotating body,
(i) the inertial frame, (X0, Y0, Z0) , with X0 and Y0 in the eclipticplane;
(ii) the orbital frame, (X1, Y1, Z1) , with Z1 perpendicular to the orbitplane;
(iii) the spin frame, (X2, Y2, Z2) , with Z2 pointing to the spin axisdirection and X2 to the ascending node of the equatorial plane onthe ecliptic plane;
(iv) the body frame, (X3, Y3, Z3) , with Z3 in the direction of the axis ofgreatest inertia and X3 of the axis of smallest inertia.
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Reference planes
The four reference frames and the relevant angles related to the Andoyer(left) and Delaunay (right) canonical variables.
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Andoyer variables
For the rotational motion we adopt the Andoyer variables,
Ls = Gs cos J , ls ,
Gs , gs ,
Hs = Gs cosK , hs ,
where Gs is the norm of the angular momentum.
In order to remove the two virtual singularity (J = 0 and K = 0), weintroduce the modified Andoyer variables,
L1 =GsnoC
, l1 = ls + gs + hs ,
L2 =Gs − LsnoC
, l2 = −ls ,
L3 =Gs −Hs
noC, l3 = −hs ,
where no is the orbital mean-motion of the rotating body.
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Delaunay variables
For the orbital motion, we introduce the classical Delaunay variables,
Lo = mõa , lo ,
Go = Lo√
1− e2 , go = ω ,
Ho = Go cos i , ho = Ω ,
Again, to remove the singularity (e = 0 and i = 0), we introduce themodified Delaunay variables,
L4 = Lo , l4 = lo + go + ho ,
L5 = Lo −Go , l5 = −go − ho ,L6 = Go −Ho , l6 = −ho .
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Free rotation
The rotational kinetic energy, (Deprit, 1967), reads
T =L2s
2C+
1
2(G2
s − L2s)
(sin2 lsA
+cos2 lsB
).
Thus, in our set of variables, we get
T
noC=noL
21
2+n0L3(2L1 − L3)
2
(γ1 + γ2
1− γ1 − γ2sin2(l3)
+γ1 − γ2
1− γ1 + γ2cos2(l3)
),
where
γ1 =2C −B −A
2Cand γ2 =
B −A2C
.
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Perturbation by another body
The perturbation induced by the point body mass on the rotation of therigid body, can be expressed via a gravitational potential, V , in the form
V =3
2
GMa3
(ar
)3 (C0
2
(x2
3 + y23
)− 2C2
2
(x2
3 − y23
)),
where (x3, y3, z3) are the components (in the body frame) of the unitvector pointing to the perturbing body. The coefficients C0
2 and C22 , can
be written in terms of the moments of inertia and of the dimensionlessparameters J2 and C22, as
C02 =
A+B − 2C
2= −mR2
eJ2 ,
C22 =
B −A4
= mR2eC22 .
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Simplified model
We now consider a simplified spin-orbit model, making some strongassumptions on the system.
(i) We assume that the wobble, J , is equal to zero. This means thatthe spin axis is aligned with figure one.
(ii) We introduce the resonant variables
Σ1 = L1 , σ1 = l1 − l4 ,Σ3 = L3 , σ3 = l3 − l6 ,
and make an over the fast angle, l4, namely
〈V 〉l4 =1
2π
∫ 2π
0
V dl4 .
(iii) We neglect the influence of the rotation on the orbit of the bodyand we model the time dependence of the Hamiltonian via the twoangular variables,
l4(t) = n t+ l4(0) and l6(t) = Ω t+ l6(0) .
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Simplified model
Finally, we end up with a Hamiltonian that reads
H =noΣ
21
2− noΣ1 + ΩΣ3 + 〈V 〉l4 .
This Hamiltonian possesses an equilibrium, the Cassini state, defined by
σ1 = 0 ,∂H
∂Σ1= 0 ,
σ3 = 0 ,∂H
∂Σ3= 0 .
We denote by Σ∗1 and Σ∗3 the values at the equilibrium.
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Stability around the Cassini state
We now aim to study the dynamics in the neighborhood of the Cassinistate defined here above. We introduce the translated canonical variables
∆Σ1 = Σ1 − Σ∗1 , σ1 ,
∆Σ3 = Σ3 − Σ∗3 , σ3 ,
and, with a little abuse of notation, in the following we will denote again∆Σi by Σi, with i = 1, 3 .
In these new coordinates, the equilibrium is set at the origin, thus we canexpand the Hamiltonian in power series of (Σ, σ). Let us remark that thelinear terms disappear, as the origin is an equilibrium, thus the lowerorder terms in the expansion are quadratic in (Σ, σ) .
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Stability around the Cassini state
Precisely, we can write the Hamiltonian as
H(Σ, σ) = H0(Σ, σ) +∑j>0
Hj(Σ, σ) , (1)
where Hj is an homogeneous polynomial of degree j + 2 in (Σ, σ) .The Hamitonian is almost in the “right” form, we just need todiagonalize the quadratic part, H0, via the so-called ‘untanglingtransformation” (Henrard & Lemaıtre, 2005), perform a rescaling andintroduce the action-angle coordinates.Finally, the transformed Hamiltonian can be expanded in Taylor-Fourierseries and reads
H(0)(U, u) = ωu · U +∑j>0
H(0)j (U, u) ,
where the terms Hj are homogeneous polynomials of degree j/2 + 1 inU , whose coefficients are trigonometric polynomials in the angles u .
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Estimated effective stability time
0
10
20
30
40
50
60
0 1 2 3 4 5
log10T
(ρ0)
ρ0
Effective stability time
Estimated effective stability time. The time unit is the year. The threelines correspond (from down to top) to three different normalizationorders: r = 10 (blue), r = 20 (pink) and r = 30 (red).
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Estimated effective stability time
Effective stability time (Ω vs. i)
-0.016-0.014-0.012-0.010-0.008-0.006
Ω
0.004
0.008
0.012
0.016
i
12
14
16
18
20
22
24
26
28
30
log10T
Effective stability time as a function of the mean inclination, i, and themean precession of the ascending node of Titan orbit, Ω .
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Estimated effective stability time
Effective stability time (Ω vs. C/MR2e)
-0.016-0.014-0.012-0.010-0.008-0.006
Ω
0.30
0.32
0.34
0.36
0.38
0.40
C/MR
2 e
20
20.5
21
21.5
22
22.5
23
23.5
24
log10T
Effective sability time as a function of the normalized greatest moment ofinertia, C/MR2
e , and the mean precession of the ascending node of Titanorbit, Ω .
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Estimated effective stability time
Effective stability time (i vs. C/MR2e)
0.004 0.008 0.012 0.016
i
0.30
0.32
0.34
0.36
0.38
0.40
C/MR
2 e
14
16
18
20
22
24
26
28
30
log10T
Effective stability time as a function of the mean inclination, i, and thenormalized greatest moment of inertia, C/MR2
e .
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Thanks for your attention!
Questions?
Comments?