Transcript
Page 1: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Stability of an asteroid satellite

Anne Lemaître, Audrey Compère, Nicolas Delsate

Department of Mathematics FUNDP Namur

14 June 2010

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

1 Introduction

2 Satellites of asteroidsSystem Ida-DactylPrevious results

3 Stability testsNumerical simulationsClassical calculation of the potentialMacMillan potentialChaos indicator : MEGNO

4 Chaos MapsGravitational resonancesFrequency analysis

5 Analytical developmentMacMillan potentialApproximated formulation1:1 resonanceEquatorial resonant orbitsPolar resonant orbits

6 Conclusions

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Motivation

Previous studies : rotation of the planets and natural satellitesand space debris on geostationary orbits

Collaboration Grasse - CNES : stability conditions for themotion of a probe around an asteroid

To test our methods on asteroid satellites (PhD - notpublished)

Stability : numerical tests and dynamical models

Several approaches of the potential of a non spherical body

Trace-free tensors in elliptical harmonicsGeometrical approachMacMillan potential : the only one presented here (Paolo)

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Vocabulary

Binary asteroid : system of two asteroids

Two categories :

1 The two bodies have the same size : double asteroidEx : Antiope - Dynamics intensively studied in particular by Scheeres andcollaborators

2 A body is much smaller than the other one : asteroid and its satelliteEx : Ida-Dactyl

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Natural system Ida-Dactyl

Ida : main belt asteroid (Koronis family), very irregular shape and fast spin

Ida Dactyl

Mass (4.2 ± 0.6) × 1016 kg ∼ 4.1012 kg

Diameter 59.8 × 25.4 × 18.6 km 1.6 × 1.4 × 1.2 km

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Dactyl :

Orbit data :

Semimajor axis (a) : 108 kmOrbital period (P) : 1.54 daysEccentricity (e) : ≥ 0.2?

Other data :

Mean radius : 0.7 kmPrincipal diameters : 1.6 × 1.4 × 1.2 kmShape : less irregular then IdaEllisoidal �t (radii) : 0,8 × 0,7 × 0,6 kmMass : ∼ 4.1012 kgSurface area : 6,3 km2

Volume : 1,4 km3

Spin period : > 8 hr

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Ida-Dactyl simulations

J-M Petit et al : 1998, Belton, 1996Context :

Ida mass is not known precisely.

Each value of the mass corresponds to a Keplerian orbit for Dactyl

To constraint the mass of Ida by Dactyl's orbit

Belton,1996

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Petit et al

1. Stability bounds on Ida mass

First model

Ida is represented by an ellipsoid.Gravitational potential : elliptic integralsIntegrator : Bulirsch and Stoer with a precision of 10−10

Masses : between 3.65× 1016 and 5.7× 1016 kg

Results :

Orbits with M > 4.93× 1016 kg (q < 63 km) are very unstable.→ crash or escape after several hours or days

The other orbits are stable for hundreds of years.

Second model

Approximation of Ida by a collection of 44 spheres of di�erent sizes.

⇒ more precise bounds.

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Resonant stable orbits

The Ida-Dactyl system should be stable for long time ⇒ search for resonancesbetween the rotation of Ida and the orbital frequency of Dactyl.

Simulations results :

Most probable resonances 5:1 and 9:2

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Numerical simulations

Model : a point mass orbiting an ellipsoid

Parameters : shape, mass and spin of the primary, initialconditions of the satellite

Purpose : search for stable or resonant systems

Technique : chaos maps (MEGNO)

Software : NIMASTEP (N. Delsate) written for numericalintegration of an arti�cial satellite around a telluric planet

Di�erences : irregular shape and fast rotation of the primary,large eccentricity of the satellite, relative importance of theperturbations

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

First calculation of the potential

Based on the spherical harmonics as for the telluric planetsSmall deformations of a sphere

V (r , θ, λ) =GM

r

[1 +

∞∑n=2

n∑m=0

(Re

r

)n

Pnm(sin θ) (Cnm cos mλ+ Snm sin mλ)

]

(r , θ, λ) are the spherical coordinates

Re is the equatorial radius

Pnm are the Legendre's polynomials

Cnm et Snm are the spherical harmonics coe�cients

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Check of the integrations

Paper of A. Rossi, F. Marzari and P. Farinella (1999) : Orbitalevolution around irregular bodies in Earth, Planets, Space.Four approaches of the potential :

Ivory's approach : direct calculation of the potential of anhomogeneous triaxial ellipsoid

Spherical harmonics approach (4th order)

Mascons approach : the body is approximated by a set ofpoint masses placed in a suitable place to reproduce the massdistribution

Polyhedral approach : the body is approximated by apolyhedron with a great number of faces

Axisymmetric ellipsoid (a = b = 10 km, c= 5 km) or triaxialellipsoid (a=30 km, b=10 km and c = 6.66 km).

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Tests of Rossi, Marzari and Farinella

Four cases :

Case 0 : Sphere (not considered here)

Case 1 : Axisymmetric ellipsoid with inclined circular orbits(i = 10◦) at a distance of 20km - 5835 mascons - 1521 faces -Mass = 2.0831015 kg - ρ = 1g cm3.

Case 2 : Axisymmetric ellipsoid with inclined elliptic orbits(e = 0.2)

Case 3 : Axisymmetric ellipsoid with distant inclined ellipticorbits at a distance of 40km

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Comparisons

First tests : Variation of the ascending node (in radians s−1) :

Secular Theory (J2) polygones mascons spherical harmonics

Case 1

circular inclined -7.7 10−6 -1.09 10−5 -1.11 10−5 -1.07 10−5

Case 2

elliptic inclined -8.37 10−6 -1.25 10−5 -1.33 10−5 -1.27 10−5

Case 3

elliptic, inclined and distant -7.10 10−7 -7.76 10−7 -7.92 10−7 -7.85 10−7

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

MacMillan potential

New potential : Potential for an ellipsoid : MacMillan (1958)

V (x , y , z) =3

2GM

Z +∞

λ1

„1− x2

s2− y2

s2 − h2− z2

s2 − k2

«ds√

s2 − h2√s2 − k2

where

h2 = a2 − b2 et k2 = a2 − c2 (a, b et c are the semi-majoraxes of the ellipsoid with a ≥ b ≥ c)

(x , y , z) are the cartesian coordinates of the point

λ1 is the �rst ellipsoidal coordinate of the point

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

For each (x , y , z) :

x2

s2+

y2

s2 − h2+

z2

s2 − k2= 1 Equation of degree 3 in s2

Roots : λ21, λ22 et λ23 with 0 ≤ λ23 ≤ h2 ≤ λ22 ≤ k2 ≤ λ21.

Geometrically (x , y , z) is the intersection between

an ellipsoid with axes (√λ21,√λ21 − h2,

√λ21 − k2)

an hyperboloid of one sheet with axes

(√λ22,√λ22 − h2,

√k2 − λ22)

an hyperboloid of two sheets with axes

(√λ23,√h2 − λ23,

√k2 − λ23)

Ellipsoidal coordinates : (λ1, λ2, λ3)

Page 17: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

New tests and comparisons with Rossi et al

Calculation of the force components explicitly (partialderivatives)

Gauss-Legendre quadrature for the integrals

Introduction in NIMASTEP

New tests : Variation of the ascending node (in radians s−1) :

Secular Theory (J2) polygones mascons spherical harmonics Mac Millan

Case 1 -7.7 10−6 -1.09 10−5 -1.11 10−5 -1.07 10−5 -1.11 10−5

Case 2 -8.37 10−6 -1.25 10−5 -1.33 10−5 -1.27 10−5 -1.33 10−5

Case 3 -7.10 10−7 -7.76 10−7 -7.92 10−7 -7.85 10−7 -7.86 10−7

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Orbits

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Tests on the system Ida-Dactyl

Test on eccentric Dactyl orbits :

Resultats :

Crash or escapes for M ≥ 5× 1016 kg

Regular orbits for M ≤ 5× 1016 kg

⇒ same results as Petit et al. (1998)

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Chaos indicator : MEGNO

MEGNO = Mean Exponential Growth factor of Nearby Orbits(Cincotta et Simo, 2000)

Dynamical system : : d

dtx(t) = f (x(t)), x ∈ IR2n.

φ(t) a solution function of time t

δφ(t) the tangent vector along φ(t) with δφ = ∂f∂x

(φ(t))δφ(t).

The MEGNO is :

Yφ(t) =2

t

Zt

0

δφ · δφδφ · δφ

s ds and Yφ =1

t

Zt

0

Yφ(s) ds

= measure of the divergence rate between two close orbits.

Periodic orbit : Yφ → 0

Quasi-periodic orbit : Yφ → 2

Chaotic orbit : Yφ is increasing with time

Page 21: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Chaos Maps

We set :

the mass and the rotation rate of the asteroid (ellipsoid)

the initial conditions of the satellite (a=148.8km, i = 3 rad)

a the largest semi-axis of the ellipsoid

Variations of the primary shape (through the semi-axes b and c).

Integrator : Runge-Kutta-Fehlberg with variable stepPrecision : 10−12

Results of the chaos indicator MEGNO are given in the plane (b/a,c/a)

Page 22: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

M=3.895551 106 kg, rotation rate = −3.76687× 10−4 rad/s

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

M=3.745722 106 kg, rotation rate = −3.76687× 10−4 rad/s

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

M=5.693498 106 kg, rotation rate = 3.76687× 10−4 rad/s

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

However let us remind that the mass is constant in these graphics, some ofthese cases correspond to impossible values of the densities (chosen between 1and 3 gr/cm3) - The mass M and the axis a are �xed.

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Evolution of the MEGNO with time

After 0.1 year after 1 year

after 5 years after 10 years

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Spin

v = −2.5 10−4 rad/s v = −4.0 10−4 rad/s

v = −3.76687 10−4 rad/s In�uence of the spin v

Page 28: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Semi major-axis

a=130 km a=170 km

a=148.8 km Evolution with semi-major axis

Page 29: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Reference case

M=3.895551 106 kg, initial orbit i ' 3 rad)

rotation rate v = −3.76687× 10−4 rad/s

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Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Gravitational resonance

A resonance between

the rotation of the primary (P = 4, 63 hours)

the orbital period of the satellite (speci�c to each point)

Tests on a few points

Q1 : b=18.6 km, c=8.9 km and Y → 2 - period of 2.50 daysQ2 : b=18.9 km, c=8.9 km and Y → +∞ - period of 2.48 daysQ3 : b=20.1 km, c=8.9 km and Y → 2 - period of 2.48 days

Gravitational resonance 1:13

Page 31: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Frequency analysis (J. Laskar)

c = 8.9 km is constant and b variesAnalysis (a ∗ cos(M), a ∗ sin(M)) :

Page 32: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Second case

M=3.745722 106 kg, i ' 2.99), v = −3.76687× 10−4 rad/s

Page 33: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Choosing again c = 8.9 km and b varies with time

Analysis of (a ∗ cos(M) , a ∗ sin(M))

Several high order resonances (1:21 or 1:20 or 2:43)

Page 34: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Analytical development

MacMillan Potential for an ellipsoid (1958) :

V (x , y , z) =3

2GM

Z +∞

λ1

„1− x2

s2− y2

s2 − h2− z2

s2 − k2

«ds√

s2 − h2√s2 − k2

with h2 = a2 − b2 and k2 = a2 − c2

a, b et c are the semi- axes of the ellipsoid with a ≥ b ≥ c.

Chauvineau, B., Farinella, P. and Mignard, F. (Icarus, 1993)Planar orbits about a triaxial body - Application to asteroidal

satellites

Scheeres, D. (Icarus, 1994) Dynamics about uniformly rotating

triaxial ellipsoids : applications to asteroids

Page 35: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Expansion of the potential

Expansion of MacMillan potential in powers of h/R and k/Rwhere R2 = x2 + y2 + z2

Keplerian orbit about a rotating body (about its vertical axis)perturbed by MacMillan potential

Delaunay's Hamiltonian momentum : L =õa

H = − µ

2L2− µ

10R3(h2 + k2) +

10R5(y2h2 + z2k2)

Page 36: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

1:1 resonance, circular and equatorial

The curve corresponds to an curve : k2 − 2h2 ' 0

Page 37: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

1:1 resonance model

Simpli�cations : z = 0 and y = L2

µ sin(M +$ − φ)

φ = v t

Resonant variable : σ = M +$ − φSame equilibria as Scheeres or others

H = − µ

2L2− v L− µ4

10L6(h2 + k2) +

3µ4

20L6h2(1− cos 2σ).

The exact 1:1 resonance : v = n : k2 − 2h2 = 0

Page 38: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Other resonances in the equatorial cases

z = 0 and y = L2

µ sin(f +$ − φ)

The eccentricity is used to develop f in multiples of M

Extraction of the resonant angle σ

σ is now conjugated to P = L− G .

Introduction of the pericentre motion (second degree offreedom) responsible for the multipliers of the exact resonance

Higher orders of resonances require higher powers of theeccentricity

Case Ida - Dactyl : potential 5:1 or 9:2 resonance (eccentricityof Dactyl high)

Page 39: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Non-equatorial cases : polar case

Map of the resonances between the rotation of Vesta and theorbital motion of a polar satellite : numerical work

300 400 500 600 700 800 900 1000Initial Radius [km]

300

400

500

600

700

800

900

1000

Dis

tance

Ran

ge

[km

]

1:1

3:2

4:3

1:2

2:3

HAMOLAMO

2

3

4

5

6

7

8

9

10

11

12

13

14

Orb

ital

Per

iod [

hour]

2

3

4

5

6

7

Figure 4: Distance range as a function of the initial radius of a circular orbit, computed over aperiod of 50 days. The central mark in each bar represents the median of the range. The rotationperiod used for Vesta is of 5.3421288 hours (Harris et al., 2008). Five spin-orbit resonances havebeen identified and marked in the plot. The 1:1 resonance a!ects the largest interval in initialradius, but the strongest perturbations come from the 2:3 resonance. The leftmost data point,with initial radius of 370 km, reaches a lowest distance just below 300 km, only a few km awayfrom Vesta’s surface. The orbital radius of HAMO and LAMO is also marked.

19

Paper of Tricarico and SykesThe dynamical environment of Vesta

submitted to Planetary and Space Science

Page 40: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Our results

Numerical integration with NIMASTEP (especially drawn forpolar orbits)

Resonance map : position and importance of each resonance

Complete agreement with Tricarico and Sykes

Discovery of smaller structures ignored by Tricarico and Sykes

Analysis of each resonance to compare their width and shape

Page 41: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Page 42: N4.Lemaitre - "Stability of an asteroid satellite"

Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions

Conclusions

MEGNO is very e�cient for the detection of gravitationalresonances

Use of the frequency map for the identi�cation of theresonances

E�ciency and precision of MacMillan potential for ellipsoidalbodies

Explicit approximated formulation in h and k

Speci�c i : j resonance models : strength, width, equilibria

Equatorial and polar cases (Ida and Vesta)

Paolo's contribution : pioneer and omnipresent in theliterature about asteroid dynamics


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