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

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


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.


Development

Hori, 1966

⇓≫ Kozai, 1979 ⇒ Yuasa, 1973

⇓Knezevic (et al.), 1986, 1988, 1989, 1990, ...

⇓Milani and Knezevic, 1990, 1992, 1994, 1999, 2000, ...


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.


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.


Elements:⇓

Osculating → Mean

Elimination of the short-periodic perturbations

Mean → Proper

Elimination of the long-periodic perturbations⇓

Averaging


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.


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


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 ′).


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.


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.


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] + . . .


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.



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



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



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



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.


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.


158 Koronis: osculating, mean and proper elements

Eccentricity Inclination


Stable vs. chaotic motion


Resonances in the Trans-Neptunian region


Identification of asteroid families


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


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


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


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


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


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!


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

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