DETERMINATION OF ASTEROID PROPERELEMENTS: CONTRIBUTION OF PAOLO
FARINELLA AND THE CURRENTSTATE-OF-THE-ART
Zoran Knezevic
Astronomical Observatory, Belgrade
Pisa, June 15, 2010.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Beginnings
Zappala, V., P. Farinella, Z. Knezevic, and P. Paolicchi: 1984,Collisional origin of the asteroid families: mass and velocitydistributions. Icarus 59, 261–285.
Mass and velocity distributions of family members ⇒morphological classification of families: asymmetric, dispersed,intermediate.
Results that did not fit:
the degree of fragmentation in real families lower than forlaboratory targets
relative velocities asymmetry
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Beginnings
Zappala, V., P. Farinella, Z. Knezevic, and P. Paolicchi: 1984,Collisional origin of the asteroid families: mass and velocitydistributions. Icarus 59, 261–285.
Mass and velocity distributions of family members ⇒morphological classification of families: asymmetric, dispersed,intermediate.
Results that did not fit:
the degree of fragmentation in real families lower than forlaboratory targets
relative velocities asymmetry
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Beginnings
Zappala, V., P. Farinella, Z. Knezevic, and P. Paolicchi: 1984,Collisional origin of the asteroid families: mass and velocitydistributions. Icarus 59, 261–285.
Mass and velocity distributions of family members ⇒morphological classification of families: asymmetric, dispersed,intermediate.
Results that did not fit:
the degree of fragmentation in real families lower than forlaboratory targets
relative velocities asymmetry
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Beginnings
q =
√3∆vT
√
∆v2T + ∆v2
S + ∆v2W
Expected q ∼ 1; obtained q ∼ 0.2!!
Williams’ proper elements for∼ 1800 asteroids.
Knezevic, Z. 1984, in preparation.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Beginnings
q =
√3∆vT
√
∆v2T + ∆v2
S + ∆v2W
Expected q ∼ 1; obtained q ∼ 0.2!!
Williams’ proper elements for∼ 1800 asteroids.
Knezevic, Z. 1984, in preparation.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Development
Hori, 1966
⇓≫ Kozai, 1979 ⇒ Yuasa, 1973
⇓Knezevic (et al.), 1986, 1988, 1989, 1990, ...
⇓Milani and Knezevic, 1990, 1992, 1994, 1999, 2000, ...
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Common papers
Knezevic, Z., M. Carpino, P. Farinella, Ch. Froeschle, Cl.Froeschle, R. Gonczi, B. Jovanovic, P. Paolicchi, and V.Zappala: 1988, Astron. Astrophys. 192, 360–369.
Farinella, P., M. Carpino, Ch. Froeschle, Cl. Froeschle, R.Gonczi, Z. Knezevic, and V. Zappala: 1989, Astron. Astrophys.217, 298–306.
Zappala, V., A. Cellino, P. Farinella, and Z. Knezevic: 1990,Astron. J. 100, 2030–2046.
Knezevic, Z., A. Milani, P. Farinella, Ch. Froeschle, and Cl.Froeschle: 1991, Icarus 93, 316–330.
Knezevic Z., A. Milani, and P. Farinella: 1997. TPlanet. SpaceSci. 45, 1581–1585.
Vokrouhlicky D., M. Broz, P. Farinella and Z. Knezevic Z.: 2001.Icarus 150, 78–93.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Asteroid proper elements
Definition:
Proper elements are quasi-integrals of the full N-bodyequations of motion.
In practice:
Proper elements are true integrals of the simplified problem.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Asteroid proper elements
Definition:
Proper elements are quasi-integrals of the full N-bodyequations of motion.
In practice:
Proper elements are true integrals of the simplified problem.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Elements:⇓
Osculating → Mean
Elimination of the short-periodic perturbations
Mean → Proper
Elimination of the long-periodic perturbations⇓
Averaging
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Canonical elements
Delaunay’s variables:(ℓ, ω,Ω, L, G, J). Actions (L, G, J) define canonical system:
L = K√
a
G = K√
a(1 − e2)
J = K√
a(1 − e2) cos I
where K is Gauss’ constant.
Hamiltonian:
H =µ
2L2 − K + R .
R is the perturbing function and K is the moment conjugated totime t(= k). 4 degrees of freedom.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Canonical elements
Poincare’s variables:
(λ, x , u,Λ, y , v), are a canonical analogue of the coordinatetransformation to eliminate singularities e = 0 and I = 0:
x =√
2(L − G) cos(ω + Ω)
u =√
2(G − J) cos(Ω)
λ = ℓ + ω + Ω
y = −√
2(L − G) sin(ω + Ω)
v = −√
2(G − J) sin(Ω)
Λ = L
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Equations of motion
Hamilton function H(X,Y) of the vectorial coordinates X and moments Y:
dXdt
=∂H∂Y
dYdt
= −∂H∂X
Solving by canonical transformations keeps the same generalform of the equations and enables use of general rules forsubsequent transformations;transformed system in new variables (X ′, Y ′) simpler;the goal is to end up with an integrable system H ′ = H ′(Y ′).
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Hamiltonian of the asteroid problem
Hamiltonian expanded up to degree 4 in e, I in the first orderwith respect to the perturbing mass, and degree 2 in thesecond order + several resonant terms of degree 6.
Generic term for the direct part:
K2εjh1
h2· (h3)
(i)(−1)h4 ih5eh6eh7j sinh8 I sinh9 Ij sinh10
I2
sinh11Ij2·
· cos[(i + k1)λj − (i + k2)λ + k3j + k4 + k5Ωj + k6Ω] ,
where(h3)(i) are LeVerrier’s coefficients depending on a/a′. ∀i
189 terms up to degree 4 in e, I.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Lie series
Lie transform of the function H with determining function W isdefined by an expansion in formal power series:
H ′ = TW H = H − H, W + 12H, W, W + . . .
where ., . is Poisson bracket:
H, W =∂H∂X
∂W∂Y
− ∂H∂Y
∂W∂X
and W is given as an expansion in some small parameter ε:
W = εW1 + ε2W2 + . . .
so that transformation is close to identity.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Lie series
Expansion of Lie series in powers of ε:
H ′ = H − εH, W1 + ε2[−H, W2 + 12H, W1, W1] + . . .
Asteroid Haniltonian is given as sum of the keplerian term andthe perturbation:
H = H0 + εH1
Substituting and expressing again in powers of ε:
H ′ = TW H = H0 + ε[H1 − H0, W1] +
+ ε2[−H0, W2 − H1, W1 + 12H0, W1, W1] + . . .
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Method of canonical transformations
In asteroid problem H0 is integrable (depends only onmomenta):
H = H0(Y ) + εH1(X , Y )
Equaling terms of the transformed and initial Hamiltonian of thesame degree in ε:
H ′
0(X′, Y ′) = H0(Y
′)
H ′
1(X′, Y ′) = H1(X
′, Y ′) − H0, W1(X ′, Y ′)
H ′
2(X′, Y ′) = −H0, W2 − H1, W1 + 1
2H0, W1, W1
the problem reduces to finding W1 i W2 such that one getssimpler Hamiltonian.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Method of canonical transformations
We define the linear operator L acting on any function F asPoisson bracket with the zero order Hamiltonian:
LF = H0, F
It defines decomposition of the function space into a direct sumof the kernel (null space) and the image of the operator L:
F = F + F F ∈ ImL ; F ∈ Ker L
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Method of canonical transformations
Decomposition of Hamiltonian H1 = H1 + H1 :
H ′
1 = H1 + H1 − LW1
gives an obvious solution:
W1 ∈ ImL = H1
and thus defines the transformed Hamiltonian of the first order:
H ′
1 = H1
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Method of canonical transformations
The second order equation:
H ′
2 = −12H1 + H1, W1 − LW2
in the same way gives the definitin of H ′
2:
H ′
2 = −12H1, W1
and the equation for W2:
LW2 = −H1, W1 − 12H1, W1 + 1
2H1, W1.H ′ and W are thus defined to order 2:
W ∈ ImL ; H ′ ∈ Ker L
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Method of canonical transformations
To compute the second order H ′, it is enough to know W toorder 1;Computation of the map FW to order 2 requires knowledge ofW2. For the transformation of variables:
Y ′ = Y + ε∂W1
∂X+ ε2 ∂W2
∂X+ 1
2ε2−∂W1
∂X, W1 + . . .
There are 378 terms in H1 in the asteroid problem, thus also inW1, as the latter is obtained by term by term integration.Iterative procedure accounts for the ”wrong” direction of themap (from osculating to proper). Typical accuracy ∼ 10−4 inproper semimajor axis, 0.003 in proper eccentricity and 0.001in proper (sine of) inclination; based on selected test cases.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Synthetic theory
1 numerical integration of asteroid orbits in the framework ofa realistic dynamical model;
2 online digital filtering of the short periodic perturbations ⇒mean (filtered) elements (proper semimajor axis as asimple average of the filtered data);
3 Fourier analysis of the output to remove the main forcedterms and extract proper eccentricity, proper inclination,and the corresponding fundamental frequencies;
4 check of the accuracy of the results by means of runningbox tests.
Knezevic Z. and A. Milani: 2000. Synthetic proper elements forouter main belt asteroids. CMDA 78, 17–46.
More than 220.000 asteroids (MB,Trojan,TNO,Hungaria).Accuracy by a factor of 3 better than the analytical properelements.
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
158 Koronis: osculating, mean and proper elements
Eccentricity Inclination
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Stable vs. chaotic motion
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Resonances in the Trans-Neptunian region
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Identification of asteroid families
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Chaotic chronology: 490 Veritas
0.05
0.055
0.06
0.065
0.07
0.075
3.15 3.155 3.16 3.165 3.17 3.175 3.18 3.185 3.19
e p
ap [AU]
3 3 -2 5 -2 -2 7 -7 -2
0.152
0.154
0.156
0.158
0.16
0.162
0.164
0.166
0.168
3.15 3.155 3.16 3.165 3.17 3.175 3.18 3.185 3.19
sin I
pap [AU]
3 3 -2 5 -2 -2 7 -7 -2
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Coefficient of diffusion
0
2e-008
4e-008
6e-008
8e-008
1e-007
1.2e-007
1.4e-007
1.6e-007
0 2e+006 4e+006 6e+006 8e+006 1e+007
<(∆
J 1)2
>
t [yr]
5 -2 -2 ap = 3.174 AU
0
2e-008
4e-008
6e-008
8e-008
1e-007
1.2e-007
1.4e-007
1.6e-007
0 2e+006 4e+006 6e+006 8e+006 1e+007<
(∆J 2
)2>
t [yr]
5 -2 -2 ap = 3.174 AU
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Coefficients of diffusion as functions of the semimajoraxis
0
2e-015
4e-015
6e-015
8e-015
1e-014
1.2e-014
1.4e-014
3.165 3.17 3.175 3.18 3.185
D(J
1)
[yr-1
]
ap [AU]
0
5e-016
1e-015
3.167 3.168 3.169 3.17
3 3 -2
0
5e-016
1e-015
3.1795 3.18 3.1805
7 -7 -2
0
2e-015
4e-015
6e-015
8e-015
1e-014
1.2e-014
1.4e-014
3.165 3.17 3.175 3.18 3.185D
(J2)
[yr-1
]
ap [AU]
0
2e-017
4e-017
3.167 3.168 3.169 3.17
3 3 -2
0
2e-017
4e-017
3.1795 3.18 3.1805
7 -7 -2
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Monte-Carlo simulations: age 8.7 ± 1.2 million years
5
6
7
8
9
10
11
0 1000 2000 3000 4000 5000 6000
τ [M
yr]
dt [yr]
n=2000, δJ1(0)=1.25 x 10-4, δJ2(0)=5.6 x 10-4
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
Monte-Carlo simulations: age 8.7 ± 1.2 million years
5
6
7
8
9
10
11
0 1000 2000 3000 4000 5000 6000
τ [M
yr]
dt [yr]
n=2000, δJ1(0)=1.25 x 10-4, δJ2(0)=5.6 x 10-4
5
6
7
8
9
10
11
0 1000 2000 3000 4000 5000 6000
τ [M
yr]
n
dt=2000, δJ1(0)=1.25 x 10-4, δJ2(0)=5.6 x 10-4
5
6
7
8
9
10
11
2 4 6 8 10 12
τ [M
yr]
δJ2(0) x 104
n=2000, dt=2000 yr, δJ1(0)=2.30 x 10-4
5
6
7
8
9
10
11
0 1000 2000 3000 4000 5000 6000
τ [M
yr]
dt [yr]
n=2000, δJ1(0)=2.3 x 10-4, δJ2(0)=11.3 x 10-4
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
A few of Paolo’s valuable contributions:
Continuous friendly support and encouragement;
Highly competent assistance and advice in problems solving;
Suggestion to measure the accuracy of analytical elements byusing their deviation from constancy;
He put me in contact with Andrea Milani.
Paolo Farinella, thank you!
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
A few of Paolo’s valuable contributions:
Continuous friendly support and encouragement;
Highly competent assistance and advice in problems solving;
Suggestion to measure the accuracy of analytical elements byusing their deviation from constancy;
He put me in contact with Andrea Milani.
Paolo Farinella, thank you!
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
A few of Paolo’s valuable contributions:
Continuous friendly support and encouragement;
Highly competent assistance and advice in problems solving;
Suggestion to measure the accuracy of analytical elements byusing their deviation from constancy;
He put me in contact with Andrea Milani.
Paolo Farinella, thank you!
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
A few of Paolo’s valuable contributions:
Continuous friendly support and encouragement;
Highly competent assistance and advice in problems solving;
Suggestion to measure the accuracy of analytical elements byusing their deviation from constancy;
He put me in contact with Andrea Milani.
Paolo Farinella, thank you!
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
A few of Paolo’s valuable contributions:
Continuous friendly support and encouragement;
Highly competent assistance and advice in problems solving;
Suggestion to measure the accuracy of analytical elements byusing their deviation from constancy;
He put me in contact with Andrea Milani.
Paolo Farinella, thank you!
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA
A few of Paolo’s valuable contributions:
Continuous friendly support and encouragement;
Highly competent assistance and advice in problems solving;
Suggestion to measure the accuracy of analytical elements byusing their deviation from constancy;
He put me in contact with Andrea Milani.
Paolo Farinella, thank you!
Knezevi c ASTEROID PROPER ELEMENTS: PAOLO FARINELLA