N.24 knezevic determination-of-asteroid-proper-elements-con

DETERMINATION OF ASTEROID PROPER
ELEMENTS: CONTRIBUTION OF PAOLO
FARINELLA AND THE CURRENT
STATE-OF-THE-ART

Zoran Kneˇ evi´
z c

Astronomical Observatory, Belgrade

Pisa, June 15, 2010.

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Kneˇ evic ASTEROID PROPER ELEMENTS: PAOLO FARINELLA

Beginnings

´
Zappala, V., P. Farinella, Z. Kneˇ evi´ , and P. Paolicchi: 1984,
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Collisional origin of the asteroid families: mass and velocity
distributions. 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 for
laboratory targets
relative velocities asymmetry

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Beginnings

√
3∆vT
q=
2 2 2
∆vT + ∆vS + ∆vW

Expected q ∼ 1; obtained q ∼ 0.2!!

Williams’ proper elements for
∼ 1800 asteroids.

Kneˇ evi´ , Z. 1984, in preparation.
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Development

Hori, 1966
⇓
Kozai, 1979 ⇒ Yuasa, 1973
⇓
Kneˇ evi´ (et al.), 1986, 1988, 1989, 1990, ...
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⇓
Milani and Kneˇ evi´ , 1990, 1992, 1994, 1999, 2000, ...
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Common papers

`
Kneˇ evi´ , Z., M. Carpino, P. Farinella, Ch. Froeschle, Cl.
z c
`
Froeschle, R. Gonczi, B. Jovanovi´ , P. Paolicchi, and V.
c
´
Zappala: 1988, Astron. Astrophys. 192, 360–369.
` `
Farinella, P., M. Carpino, Ch. Froeschle, Cl. Froeschle, R.
´
Gonczi, Z. Kneˇ evi´ , and V. Zappala: 1989, Astron. Astrophys.
z c
217, 298–306.
´
Zappala, V., A. Cellino, P. Farinella, and Z. Kneˇ evi´ : 1990,
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Astron. J. 100, 2030–2046.
`
Kneˇ evi´ , Z., A. Milani, P. Farinella, Ch. Froeschle, and Cl.
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`
Froeschle: 1991, Icarus 93, 316–330.
Kneˇ evi´ Z., A. Milani, and P. Farinella: 1997. TPlanet. Space
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Sci. 45, 1581–1585.
Vokrouhlick´ D., M. Broˇ , P. Farinella and Z. Kneˇ evi´ Z.: 2001.
y z z c
Icarus 150, 78–93.

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Asteroid proper elements

Deﬁnition:
Proper elements are quasi-integrals of the full N-body
equations of motion.

In practice:
Proper elements are true integrals of the simpliﬁed problem.

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Elements:
⇓
Osculating → Mean

Elimination of the short-periodic perturbations

Mean → Proper

Elimination of the long-periodic perturbations
⇓
Averaging

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Canonical elements

Delaunay’s variables:
( , 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:
µ˜
H= −K +R.
2L2
R is the perturbing function and K is the moment conjugated to
time t(= k). 4 degrees of freedom.

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Canonical elements

´
Poincare’s variables:
(λ, x, u, Λ, y, v), are a canonical analogue of the coordinate
transformation to eliminate singularities e = 0 and I = 0:

x = 2(L − G) cos(ω + Ω) y = − 2(L − G) sin(ω + Ω)
u = 2(G − J) cos(Ω) v = − 2(G − J) sin(Ω)
λ = +ω+Ω Λ = L

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Equations of motion

Hamilton function H(X,Y) of the vectorial coordinates X and moments Y:

dX ∂H
=
dt ∂Y
dY ∂H
= −
dt ∂X
Solving by canonical transformations keeps the same general
form of the equations and enables use of general rules for
subsequent transformations;
transformed system in new variables (X , Y ) simpler;
the goal is to end up with an integrable system H = H (Y ).

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Hamiltonian of the asteroid problem

Hamiltonian expanded up to degree 4 in e, I in the ﬁrst order
with respect to the perturbing mass, and degree 2 in the
second order + several resonant terms of degree 6.
Generic term for the direct part:

h1 I Ij
K2 εj · (h3 )(i) (−1)h4 i h5 eh6 ejh7 sinh8 I sinh9 Ij sinh10 sinh11 ·
h2 2 2
· cos[(i + k1 )λj − (i + k2 )λ + k3 j + k4 + k5 Ωj + k6 Ω] ,

where(h3 )(i) are LeVerrier’s coefﬁcients depending on a/a . ∀i
189 ˇerms up to degree 4 in e, I.
t

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Lie series

Lie transform of the function H with determining function W is
deﬁned by an expansion in formal power series:

1
H = TW H = H − {H, W } + 2 {{H, W }, W } + . . .

where {., .} is Poisson bracket:

∂H ∂W ∂H ∂W
{H, W } = −
∂X ∂Y ∂Y ∂X
and W is given as an expansion in some small parameter ε:

W = εW1 + ε2 W2 + . . .
so that transformation is close to identity.

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Lie series
Expansion of Lie series in powers of ε:

H = H − ε{H, W1 } + ε2 [−{H, W2 } + 1 {{H, W1 }, W1 }] + . . .
2

Asteroid Haniltonian is given as sum of the keplerian term and
the 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 } + 1 {{H0 , W1 }, W1 }] + . . .
2

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Method of canonical transformations

In asteroid problem H0 is integrable (depends only on
momenta):

H = H0 (Y ) + εH1 (X , Y )
Equaling terms of the transformed and initial Hamiltonian of the
same degree in ε:

H0 (X , Y ) = H0 (Y )
H1 (X , Y ) = H1 (X , Y ) − {H0 , W1 }(X , Y )
H2 (X , Y ) = −{H0 , W2 } − {H1 , W1 } + 1 {{H0 , W1 }, W1 }
2

the problem reduces to ﬁnding W1 i W2 such that one gets
simpler Hamiltonian.

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We deﬁne the linear operator L acting on any function F as
Poisson bracket with the zero order Hamiltonian:

LF = {H0 , F }

It deﬁnes decomposition of the function space into a direct sum
of the kernel (null space) and the image of the operator L:
˜
F =F +F ˜
F ∈ Im L ; F ∈ Ker L

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˜
Decomposition of Hamiltonian H1 = H1 + H1 :

˜
H1 = H1 + H1 − LW1
gives an obvious solution:
˜
W1 ∈ Im L = H1

and thus deﬁnes the transformed Hamiltonian of the ﬁrst order:

H1 = H1

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The second order equation:

H2 = − 1 {H1 + H1 , W1 } − LW2
2

in the same way gives the deﬁnitin of H2 :

1 ˜
H2 = − 2 {H1 , W1 }

and the equation for W2 :

˜ ˜
LW2 = −{H1 , W1 } − 1 {H1 , W1 } + 1 {H1 , W1 }.
2 2

H and W are thus deﬁned to order 2:

W ∈ Im L ; H ∈ Ker L

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To compute the second order H , it is enough to know W to
order 1;
Computation of the map FW to order 2 requires knowledge of
W2 . For the transformation of variables:
∂W1 ∂W2 1 2 ∂W1
Y = Y +ε + ε2 + 2 ε {− , W1 } + . . .
∂X ∂X ∂X
˜
There are 378 terms in H1 in the asteroid problem, thus also in
W1 , as the latter is obtained by term by term integration.
Iterative procedure accounts for the ”wrong” direction of the
map (from osculating to proper). Typical accuracy ∼ 10−4 in
proper semimajor axis, 0.003 in proper eccentricity and 0.001
in proper (sine of) inclination; based on selected test cases.

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Synthetic theory
1 numerical integration of asteroid orbits in the framework of
a realistic dynamical model;
2 online digital filtering of the short periodic perturbations ⇒
mean (filtered) elements (proper semimajor axis as a
simple average of the filtered data);
3 Fourier analysis of the output to remove the main forced
terms and extract proper eccentricity, proper inclination,
and the corresponding fundamental frequencies;
4 check of the accuracy of the results by means of running
box tests.

Kneˇ evi´ Z. and A. Milani: 2000. Synthetic proper elements for
z c
outer 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 proper
elements.
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158 Koronis: osculating, mean and proper elements

Eccentricity Inclination

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Stable vs. chaotic motion

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Resonances in the Trans-Neptunian region

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Identiﬁcation of asteroid families

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Chaotic chronology: 490 Veritas

0.075 3 3 -2 5 -2 -2 7 -7 -2 0.168 3 3 -2 5 -2 -2 7 -7 -2
0.166
0.07
0.164

0.065 0.162

sin Ip
ep

0.16
0.06 0.158

0.156
0.055
0.154
0.05 0.152
3.15 3.155 3.16 3.165 3.17 3.175 3.18 3.185 3.19 3.15 3.155 3.16 3.165 3.17 3.175 3.18 3.185 3.19
ap [AU] ap [AU]

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Coefﬁcient of diffusion

1.6e-007 1.6e-007

1.4e-007
5 -2 -2 ap = 3.174 AU
1.4e-007
5 -2 -2 ap = 3.174 AU

1.2e-007 1.2e-007

1e-007 1e-007
<(∆J1)2>

<(∆J2)2>
8e-008 8e-008

6e-008 6e-008

4e-008 4e-008

2e-008 2e-008

0 0
0 2e+006 4e+006 6e+006 8e+006 1e+007 0 2e+006 4e+006 6e+006 8e+006 1e+007
t [yr] t [yr]

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Coefﬁcients of diffusion as functions of the semimajor
axis

1.4e-014 1.4e-014

3 3 -2 7 -7 -2 3 3 -2 7 -7 -2
1.2e-014 1.2e-014
1e-015 1e-015
4e-017 4e-017
1e-014 5e-016 5e-016 1e-014
2e-017 2e-017
D(J1) [yr-1]

D(J2) [yr-1]
8e-015 0 0 8e-015 0 0
3.167 3.168 3.169 3.17 3.1795 3.18 3.1805 3.167 3.168 3.169 3.17 3.1795 3.18 3.1805

6e-015 6e-015

4e-015 4e-015

2e-015 2e-015

0 0
3.165 3.17 3.175 3.18 3.185 3.165 3.17 3.175 3.18 3.185
ap [AU] ap [AU]

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Monte-Carlo simulations: age 8.7 ± 1.2 million years

-4 -4
n=2000, δJ1(0)=1.25 x 10 , δJ2(0)=5.6 x 10

11
10
9
τ [Myr]

8
7
6
5
0 1000 2000 3000 4000 5000 6000
dt [yr]

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Monte-Carlo simulations: age 8.7 ± 1.2 million years

n=2000, δJ1(0)=1.25 x 10-4, δJ2(0)=5.6 x 10-4 dt=2000, δJ1(0)=1.25 x 10-4, δJ2(0)=5.6 x 10-4

11 11
10 10
9 9
τ [Myr]

τ [Myr]
8 8
7 7
6 6
5 5
0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
dt [yr] n

n=2000, dt=2000 yr, δJ1(0)=2.30 x 10-4
n=2000, δJ1(0)=2.3 x 10-4, δJ2(0)=11.3 x 10-4
11 11
10
10
9
τ [Myr]

9

τ [Myr]
8 8
7 7
6 6
5 5
2 4 6 8 10 12 0 1000 2000 3000 4000 5000 6000
δJ2(0) x 104 dt [yr]

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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 by
using their deviation from constancy;
He put me in contact with Andrea Milani.

Paolo Farinella, thank you!

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