Stability of the Equilibrium Points of a
Rotating Ellipsoid
Thomas Reppert
[email protected]
University of Saragossa, Saragossa, Spain
October 8, 2008
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
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Table of Contents
1
Introduction
Objectives and applications
Asteroid 1999 KW4
Asteroid Itokawa
2
Existence of the equilibrium points
Ellipsoid model
Equations of motion
Equilibrium points
3
Stability of the equilibrium points
Stability of points P1
Stability of points P2
Stability of points P3
Summary of the stability conditions
Integrations without surface constraint
Integrations with surface constraint
4
Conclusions
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
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Modelling the motion of rubble about the surface
of an asteroid
Objectives
Present the dynamics that exist between a rotating ellipsoid
and a particle constrained to the surface of the ellipsoid
Model rubbles migrations about the surface of an asteroid
Applications
Knowledge about asteroid origins
Preparation for future missions that involve landing upon an
asteroid
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
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Asteroid 1999 KW4: a representative example of
binary asteroids
The asteroid consists of two principal bodies: the primary
(Alpha) and the secondary (Beta)
Many binary asteroids have near-Earth orbits
Eccentricity e (relative orbit): 0,0113
Period P (relative orbit): 17,4 hours
Separation distance of the two bodies: 2,54 km
Estimated age less than 106 years
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
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Asteroid Itokawa: How has the surface rubble
been dispersed?
Appears to consist mainly of rubble and conglomerate rocks
Contains three smooth surface regions
Principal axes: 2α = 535 m, 2β = 294 m y 2γ = 209 m
Mass: 3, 51 × 1010 ± 0, 105 × 1010 kg
Density: 1, 9 ± 0, 13 g cm−3
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
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Description of the ellipsoid model
Physical and geometric characteristics
Three semi-axes α, β y γ, with α > β > γ
Constant density ρ
Constant angular velocity ω about semi-axis γ
Gravitational parameter µ = mG = ρV G = ρ 34 παβγ G
Gravitational potential on the surface of the ellipsoid:
Z
3µ ∞
du
V (x, y, z) =
φ(x, y, z, u)
,
4 0
∆(u)
with
φ(x, y, z, u) =
x2
y2
z2
+
+
−1
α2 + u β 2 + u γ 2 + u
and
∆(u) =
T. Reppert (UniZar)
p
(α2 + u)(β 2 + u)(γ 2 + u).
Rotating Ellipsoid
October 2008
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Description of the ellipsoid model
Physical and geometric characteristics
Three semi-axes α, β y γ, with α > β > γ
Constant density ρ
Constant angular velocity ω about semi-axis γ
Gravitational parameter µ = mG = ρV G = ρ 34 παβγ G
Constraint which assures that the particle does not leave the surface:
x2
y2
z2
S(r) = 2 + 2 + 2 − 1 = 0,
α
β
γ
where r denotes the position vector with respect to the body system. Since the particle’s motion is tangent to the surface, we have:
Sr · ṙ = 0,
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Sr · r̈ = 0.
Rotating Ellipsoid
October 2008
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Parameter normalizations
The ellipsoid parameters are normalized in order to work with
relative values. The normalizations made are the following:
x̂ =
x
,
α
ŷ =
y
,
α
ẑ =
z
,
α
t̂ = ωt,
δ=
µ
ω 2 α3
.
Developing the integral of the potential function leads to:
3
V̂ (x̂, ŷ, ẑ) = δ(Iα̂ x̂2 + Iβ̂ ŷ 2 + Iγ̂ ẑ 2 − I),
4
where
Z ∞
du
du
, Iβ̂ =
,
Iα̂ =
2
(1 + u)∆(u)
(β̂ + u)∆(u)
0
0
Z ∞
Z ∞
du
du
Iγ̂ =
, I=
.
2
(γ̂ + u)∆(u)
∆(u)
0
0
Z
∞
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
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Development of the equations of motion
In the case of the rotating ellipsoid without constraints, the
Lagrangian is calculated as such:
1
L(r, ṙ) = T − V = (ṙ + ω ∧ r) · (ṙ + ω ∧ r) − V (r).
2
The application of the Lagrangian is extended to the constrained
case in the following manner:
L0 = L(r, ṙ) + λS(r).
The equations of motion are calculated as follows:
d ∂L0
∂L0
∂L0
=
,
= 0.
dt ∂ ṙ
∂r
∂λ
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
8 / 20
Development of the equations of motion
In the case of the rotating ellipsoid without constraints, the
Lagrangian is calculated as such:
1
L(r, ṙ) = T − V = (ṙ + ω ∧ r) · (ṙ + ω ∧ r) − V (r).
2
The application of the Lagrangian is extended to the constrained
case in the following manner:
L0 = L(r, ṙ) + λS(r).
Upon substituting the augmented Lagrangian, the equations of motion take the following form:
r̈ + 2ω ∧ ṙ + ω ∧ (ω ∧ r) = −Vr + λSr ,
S(r) = x2 +
T. Reppert (UniZar)
y2
z2
+
− 1 = 0.
β2 γ2
Rotating Ellipsoid
October 2008
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Three principal pairs of equilibrium points are
discovered
Upon substituting r̈ = ṙ = 0, the equations of motion which
define the surface equilibrium points are found:
3
y2
4
x 1 − 2 x2 + 2 − δIα +
|Sr |
β
2
2
y
y
4
3
β 2 − 2 x2 + 2 − δβ 2 Iβ +
β2
|Sr |
β
2
2
z
4
y
3
γ 2 − 2 x2 + 2 − δγ 2 Iγ +
γ2
|Sr |
β
2
z2
y2
I
+
I
β
γ
β2
γ2
y2
z2
I
+
I
β
γ
β2
γ2
y2
z2
Iβ + 2 Iγ
β2
γ
2
z2
y
x2 + 2 + 2 − 1
β
γ
6
δ
x2 Iα +
|Sr2 |
6
δ
x2 Iα +
|Sr2 |
6
δ x2 Iα +
|Sr2 |
=
0
=
0
=
0
=
0
Three pairs of equilibrium points stand out: P1 = (±1, 0, 0),
P2 = (0, ±β, 0) y P3 = (0, 0, ±γ).
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
9 / 20
Preparations for the stability analysis
The general equations of motion are rewritten here in scalar form:

3
ẍ
−
2
ẏ
=
x
1
−
δI
+
2λ
α

2



 ÿ + 2ẋ = y 1 − 32 δIβ + 2 βλ2
3
λ

z̈ = z − 2 δIγ + 2 γ 2



 2 y2 z2
x + β2 + γ2 − 1 = 0
The Lagrange multiplier has the following value:
λ=
−1
1
3
2y ẋ
y2
y2
z2
y2
z2
−2xẏ − x2 + δ x2 Iα + 2 Iβ + 2 Iγ + 2 − 2
x2 + 4 + 4
2
2
β
γ
β
β
β
γ
In order to analyze stability, a linealization of the equations of
motion is performed for small regions about the equilibrium
points.
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
10 / 20
Stability conditions for P1
For a small region about P1 , we have that x = 1 + ξ, y = η y
z = ζ, with |ξ| 1, |η| 1 y |ζ| 1. Upon performing the
linealization, the following is obtained:

ξ¨ = 0 h



i

 η̈ = η 1 − 3 δIβ + 12 −1 + 3 δIα
2
β
2
h
i
3
3
1

ζ̈ = ζ − 2 δIγ + γ 2 −1 + 2 δIα




ξ=0
In order for P1 to be stable, the two following conditions must be
obeyed:
3
1
3
1 − δIβ + 2 −1 + δIα < 0,
2
β
2
3
1
3
− δIγ + 2 −1 + δIα < 0.
2
γ
2
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
11 / 20
Stability conditions for P2
For a small region about P2 , we have that x = ξ, y = β + η y
z = ζ, with |ξ| 1, |η| 1 y |ζ| 1. Upon performing the
linealization, the following is obtained:

ξ¨ = ξ 1 − 23 δIα + β 2 −1 + 32 δIβ



 η̈ = 0
h
i
β2
3
3
−1
+
ζ̈
=
ζ
−
δI
+
δI

γ
β

2
γ2
2


η=0
In order for P2 to be stable, the two following conditions must be
obeyed:
3
3
2
1 − δIα + β −1 + δIβ < 0,
2
2
3
β2
3
− δIγ + 2 −1 + δIβ < 0.
2
γ
2
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
12 / 20
Stability conditions for P3
For a small region about P3 , we have that x = ξ, y = η y
z = γ + ζ, with |ξ| 1, |η| 1 y |ζ| 1. Upon performing the
linealization, the following is obtained:

ξ¨ − 2η̇ = ξ 1 − 23 δIα + 32 δγ 2 Iγ 


2

η̈ + 2ξ˙ = η 1 − 32 δIβ + 32 δ βγ 2 Iγ


ζ̈ = 0


ζ=0
Since the equations in ξ¨ y η̈ are coupled, they are transformed
into a system of four first-order equations. The characteristic
equation which results is the bi-quadratic equation
R(λ) = λ4 − λ2 (A + B − 4) + AB = 0, with
3
3
A = 1 − δIα + δγ 2 Iγ ,
2
2
3
3 γ2
B = 1 − δIβ + δ 2 Iγ .
2
2 β
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
13 / 20
Stability conditions for P3
For a small region about P3 , we have that x = ξ, y = η y
z = γ + ζ, with |ξ| 1, |η| 1 y |ζ| 1. Upon performing the
linealization, the following is obtained:

ξ¨ − 2η̇ = ξ 1 − 23 δIα + 32 δγ 2 Iγ 


2

η̈ + 2ξ˙ = η 1 − 32 δIβ + 32 δ βγ 2 Iγ


ζ̈ = 0


ζ=0
Upon performing the change of variable σ = λ2 , the characteristic
equation may be written as R(σ) = σ2 − σ(A + B − 4) + AB = 0. In
order for P3 to be stable, the roots σ1,2 must be real and negative,
and thus the stability conditions are obtained:
A + B − 4 < 0 −→
AB > 0 −→
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3
δ(Iα − γ 2 Iγ ) > 1,
2
3
δ(β 2 Iβ − γ 2 Iγ ) > β 2 .
2
Rotating Ellipsoid
October 2008
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Collection of calculated stability conditions
The six stability conditions that correspond to equilibrium points
P1 , P2 y P3 are the following:
 3
δ(Iα − β 2 Iβ ) < 1 − β 2

2



3

δ(Iα − γ 2 Iγ ) < 1

2



 3 δ(Iα − β 2 Iβ ) > 1 − β 2
2
3

δ(β 2 Iβ − γ 2 Iγ ) < β 2

2



3

δ(Iα − γ 2 Iγ ) > 1


2

 3
δ(β 2 Iβ − γ 2 Iγ ) > β 2
2
Note that it is impossible for two points to be stable at one time.
In fact, Condition (1) es the opposite of (3), (2) the opposite of
(5) and (4) the opposite of (6).
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
14 / 20
Numerical integrations without surface constraint
In order to analyze the unconstrained system, the Lagrange
multiplier is ignored: λ = 0. In this case, the equations of motion
are much simpler:

 ẍ − 2ẏ = x 1 − 23 δIα
ÿ + 2ẋ = y 1− 32 δIβ

z̈ = z − 32 δIγ
Since the equations are already linear, no simplifications are
applied. The three following stability conditions are obtained:
3
−2 − δ(Iα + Iβ ) < 0,
2
3
3
1 − δIα
1 − δIβ < 0,
2
2
3
− δIγ < 0.
2
It is important to note that these stability conditions do not
correpond to equilibrium points, but rather particle trajectories.
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
15 / 20
Numerical integrations without constraint
For ρ = 1, 9 kg m−3 , α = 100 m, β = 50 m y γ = 25 m, the critical value of
the angular velocity is ω = 4, 2313 × 10−4 rad s−1 .
ω = 2,5 × 10−4 rad s−1
y
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
1
1.5
x
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
16 / 20
Numerical integrations without constraint
For ρ = 1, 9 kg m−3 , α = 100 m, β = 50 m y γ = 25 m, the critical value of
the angular velocity is ω = 4, 2313 × 10−4 rad s−1 .
ω = 2,5 × 10−4 rad s−1
y
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
1
1.5
1
1.5
x
ω = 3,5 × 10−4 rad s−1
y
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
x
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
16 / 20
Numerical integrations without constraint
For ρ = 1, 9 kg m−3 , α = 100 m, β = 50 m y γ = 25 m, the critical value of
the angular velocity is ω = 4, 2313 × 10−4 rad s−1 .
ω = 2,5 × 10−4 rad s−1
y
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
1
1.5
1
1.5
1
1.5
x
ω = 3,5 × 10−4 rad s−1
y
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
x
ω = 4,5 × 10−4 rad s−1
y
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
x
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
16 / 20
Numerical integrations with constraint
The equations of motion of the constrained system are repeated
here:

3
ẍ
−
2
ẏ
=
x
1
−
δI
α + 2λ 
2



 ÿ + 2ẋ = y 1 − 32 δIβ + 2 βλ2
3
λ

z̈
=
z
−
δI
+
2
γ
2

2
γ


 2 y2 z2
x + β2 + γ2 − 1 = 0
We test out the stability conditions obtained for P1 :
3
1
3
1 − δIβ + 2 −1 + δIα < 0,
2
β
2
1
3
3
− δIγ + 2 −1 + δIα < 0.
2
γ
2
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
17 / 20
Numerical integrations with constraint
For ρ = 1, 9 kg m−3 , α = 100 m, β = 50 m y γ = 25 m, the critical value of
the angular velocity is ω = 3, 4495 × 10−4 rad s−1 .
ω = 4,5 × 10−4 rad s−1
z
0.5
0
−0.5
−0.5
0
0.5
y
T. Reppert (UniZar)
1
0
0.5
−0.5
−1
x
Rotating Ellipsoid
October 2008
18 / 20
Numerical integrations with constraint
For ρ = 1, 9 kg m−3 , α = 100 m, β = 50 m y γ = 25 m, the critical value of
the angular velocity is ω = 3, 4495 × 10−4 rad s−1 .
ω = 4,5 × 10−4 rad s−1
z
0.5
0
−0.5
−0.5
0
0.5
y
1
z
0.5
0
0.5
−0.5
−1
−0.5
−1
x
ω = 3,5 × 10−4 rad s−1
0
−0.5
−0.5
0
0.5
y
T. Reppert (UniZar)
1
0
0.5
x
Rotating Ellipsoid
October 2008
18 / 20
Numerical integrations with constraint
For ρ = 1, 9 kg m−3 , α = 100 m, β = 50 m y γ = 25 m, the critical value of
the angular velocity is ω = 3, 4495 × 10−4 rad s−1 .
ω = 4,5 × 10−4 rad s−1
z
0.5
0
−0.5
−0.5
0
0.5
y
1
z
0.5
−0.5
−1
0
−0.5
−1
0
−0.5
−1
0
0.5
x
ω = 3,5 × 10−4 rad s−1
0
−0.5
−0.5
0
0.5
1
0.5
y
z
0.5
x
ω = 2,5 × 10−4 rad s−1
0
−0.5
−0.5
0
0.5
y
T. Reppert (UniZar)
1
0.5
x
Rotating Ellipsoid
October 2008
18 / 20
Conclusions
It is found that, in general, the ellipsoid has six equilibrium
points that correspond to the two extremes of each principal
body axis.
The stability of each of these six points is analyzed as a
function of the angular velocity for fixed values of the
semi-axes and density.
A critical value of the angular velocity is established for the
equilibrium points, if such a value exists.
It is determined and verified that the equilibrium points of
two different principal ellipsoid axes cannot be stable at the
same time.
In order to offer a better model of the surface dynamics, it is
necessary to incorporate a rigid body capable of sliding and
rolling upon the ellipsoid surface, thus involving friction.
T. Reppert (UniZar)
Rotating Ellipsoid
October 2008
19 / 20
References
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The rubble-pile asteroid itokawa as observed by hayabusa.
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Dynamical configuration of binary near-earth asteroid (66391) 1999 kw4.
Science, 314:1280–1283, 2006.
Hideaki Miyamoto et al.
Regolith migration and sorting on asteroid itokawa.
Science, 316:1011–1014, 2007.
S.J. Ostro et al.
Radar imaging of binary near-earth asteroid (66391) 1999 kw4.
Science, 314:1276–1280, 2006.
Donald T. Greenwood.
Principles of Dynamics.
Prentice-Hall, 1965.
V. Guibout and D.J. Scheeres.
Stability of surface motion on a rotating ellipsoid.
Celestial Mechanics and Dynamical Astronomy, 87:263–290, 2003.
Jerrold E. Marsden and Anthony J. Tromba.
Vector Calculus.
W. H. Freeman and Company, 1988.
Daniel J. Scheeres.
Rotational fission of contact binary asteroids.
ICARUS, 189:370–385, 2007.
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Rotating Ellipsoid
October 2008
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