celestial mechanics of asteroid systems

D.J. Scheeres, A. Richard Seebass Chair, University of Colorado at Boulder

Celestial Mechanics of Asteroid Systems

D.J. ScheeresDepartment of Aerospace Engineering Sciences

The University of [email protected]

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Tuesday, January 15, 2013

mailto:[email protected]

mailto:[email protected]


Granular Mechanics and Asteroids

• Asteroid systems are best modeled as self-gravitating granular systems under self-attraction and cohesion

2



Motivated by Observations

Muses Sea

TD1 Site

All images courtesy JAXA/ISAS



Discrete Granular Mechanics and Celestial Mechanics

• The theoretical mechanics of few-body systems can shed light on more complex aggregations and their evolution

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Discrete Granular Mechanics and Celestial Mechanics

• The theoretical mechanics of few-body systems can shed light on more complex aggregations and their evolution

4Movies by S.A. Jacobson



Fundamental and Simple Question:What are the expected configurations for

collections of self-gravitating grains?

5



Fundamental Concepts:

• The N-body problem:

• Mass: – In the Newtonian N-Body Problem each particle has a total mass

mi modeled as a point mass of infinite density

6

rjk = rk � rj

rjk = |rk � rj |

miri = �@U

@ri

i = 1, 2, . . . N

U = �G2

NX

j=1

NX

k=1, 6=i

mjmk

rjk

0 =NX

j=1

mjrj



• Angular Momentum:

– Mechanical angular momentum is conserved for a closed system, independent of internal physical processes.

– The most fundamental conservation principle in Celestial Mechanics.

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H =NX

j=1

mjrj ⇥ rj

=1

2M

NX

j=1

NX

k=1

mjmkrjk ⇥ rjk M =NX

j=1

mj



• Energy:

– Not necessarily conserved for a closed system– Additional non-modeled physical effects internal to the system can

lead to dissipation of energy (e.g., tidal forces, surface friction)– Physically occurs whenever relative motion exists within a system

– motivates the study of relative equilibria8


E = T + U

T =12

NX

j=1

mj rj · rj

=1

4M

NX

j=1

NX

k=1

mjmkrjk · rjk



Leads to a more precise question ...

Q: What are the minimum energy configurations for the Newtonian N-body problem at a fixed Angular Momentum?

A: There are none for N ≥3.

A surprising and untenable result – all mechanical systems should have a minimum

energy state...

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I =NX

i=1

mir2i =

12M

NX

j,k=1

mjmkr2jk


Sundman’s Inequality

• To investigate this we start with Sundman’s Inequality– Apply Cauchy’s Inequality to the Angular Momentum

• Sundman’s Inequality is:

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H2 2IT

Polar Moment of Inertia

H2 =1

4M2

��

NX

j,k=1

mjmkrjk ⇥ rjk

��

2

14M2

0

@NX

j,k=1

mjmkr2jk

1

A

0

@NX

j,k=1

mjmkr2jk

1

A = 2IT



Minimum Energy Function and Relative Equilibrium

• Leads to a lower bound on the energy of an N-body system by defining the “minimum energy function” Em (also known as the Amended Potential).

– Em is only a function of the relative configuration Q of an N-body system

• Theorem: Stationary values of Em are relative equilibria of the N-body problem at a fixed value of angular momentum (Smale, Arnold)– Equality occurs at relative equilibrium– Can be used to find central configurations and determine energetic stability

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H2 2IT T = E � U

Em(Q) =H2

2I(Q)+ U(Q) E

Q = {rij : i, j = 1, 2, . . . , N}



Example: Point Mass 2-Body Minimum Energy Configurations

• Point Mass 2-Body Problem: Minimum is a circular orbit

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Em =h2

2d2� 1

d

@Em

@d= �h2

d3+

1d2

= 0

d⇤ = h2

E⇤m = � 1

2h2

@2Em

@d2

��⇤

=3h2

d4� 2

d3=

1h6

> 0

Separation

Ener

gyAngular Momentum



Point Mass N-Body Minimum Energy Configurations, N ≥ 3

• Point Mass 3-Body Problem:– Relative equilibria occur at the Lagrange and Euler Solutions– Euler solutions are always unstable ≠ minimum energy solutions– Lagrange solutions are never minimum energy solutions

• Point Mass N-Body Problem:– Central configurations are never minimum energy configurations,

c.f. proof by R. Moeckel. – For any Point Mass N ≥ 3 Problem, Em can always –> -∞ while

maintaining a constant level of angular momentum

For the Point Mass N ≥ 3 Problem there are no non-singular minimum energy configurations

... does our original question even make sense?

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Non-Definite Minimum of the Energy Function for N ≥ 3

• Consider the minimum energy function for N=3:

– Choose the distance and velocity between P1 and (P2 , P3) to maintain a constant value of H.

– Choose a zero-relative velocity between (P2 , P3) and let d23 –> 0, forcing Em –> -∞ while maintaining H.

– Under energy dissipation, there is no lower limit on the system-level energy until the limits of Newtonian physics are reached.

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Em =H2

m3 [d2

12 + d223 + d2

31]� Gm2

1

d12+

1d23

+1

d31

�

P1P2P3



Non-Definite Minimum of the Energy Function for N ≥ 3

• Consider the minimum energy function for N=3:

– Choose the distance and velocity between P1 and (P2 , P3) to maintain a constant value of H.

– Choose a zero-relative velocity between (P2 , P3) and let d23 –> 0, forcing Em –> -∞ while maintaining H.

– Under energy dissipation, there is no lower limit on the system-level energy until the limits of Newtonian physics are reached.

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Em =H2

m3 [d2

12 + d223 + d2

31]� Gm2

1

d12+

1d23

+1

d31

�

d12 ⇠ d13 P1P2P3

d23 ! 0



The Role of Density

• The lack of minimum energy configurations in the Point Mass N-body problem arises due to the infinite density of Point Masses– The resolution of this problem is simple and physically well motivated – allow

for finite density – but has profound consequences:

– Bodies with a given mass must now have finite size, when in contact we assume they exert surface normal forces and frictional forces

– Moments of inertia, rotational angular momentum, rotational kinetic energy and mass distribution must now be tracked in I, H, T and U, even for spheres.

– For low enough angular momentum the minimum energy configurations of an N-body problem has them resting on each other and spinning at a constant rate

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d! 0 d > 0⇢!1 ⇢ <1



Finite Density (Full-Body) Considerations

• Energy, angular momentum and polar moment of inertia all generalize to the case of finite density, along with the Sundman Inequality (Scheeres, CMDA 2002):

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H =1

2M

NX

j,k=1

mjmkrjk ⇥ rjk +NX

j=1

Ij · ⌦j

E = T + U +12

NX

j=1

⌦j · Ij · ⌦j

H2 2IT Em =H2

2I+ U E

I =1

2M

NX

j,k=1

mjmkr2jk +

12Trace

0

@NX

j=1

Ij

1

A



Modified Sundman Inequality

• A sharper version of the Sundman Inequality can be derived for finite body distributions (Scheeres, CMDA 2012):– Define the total Inertia Dyadic of the Finite Density N-Body Problem:

– Define the angular momentum unit vector

– The modified Sundman Inequality is sharper and defines an updated Minimum Energy Function

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H

IH = H · I · H

H2 2IHT 2IT Em Em =H2

2IH+ U E

I =NX

i=1

⇥mi

�r2i U� riri

�+ Ai · Ii · AT

i

⇤



Minimum Energy Configurations

• Theorem: For finite density distributions, all N-body problems have minimum energy configurations.

• Proof (Scheeres, CMDA 2012): – Stationary values of are relative equilibria, and include (for

finite densities) resting configurations. – For a finite value of angular momentum H, the function is

compact and bounded. – By the Extreme Value Theorem, the minimum energy function

has a Global Minimum.

• Resolves the problem associated with minimum energy configurations of the Newtonian (Point Mass) N-Body Problem.

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Em

Em

Em



... back to the original question

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• Question: What is the Minimum Energy configuration of a finite density N-Body System at a specified value of Angular Momentum?

• Answer: The Minimum Value of across all stationary configurations, both resting and orbital.

QF = {rij |rij � (di + dj)/2, i, j = 1, 2, . . . , N}

Em(QF ) =H2

2I(QF )+ U(QF ) E

Em

�2E = �Q · @2E@Q2

· �Q > 0�E =@E@Q

· �Q � 0

8 Admissible �Q

Relative Equilibrium Stability

For Simple Spheres...


IH =110

NX

i=1

mid2i +

NX

i=1

mir2i


Minimum Energy Configurations of the Spherical Full Body Problem

• For definiteness, consider the simplest change from point mass to finite spheres (then U is unchanged)– For a collection of N spheres of diameter di the only change in is to IH

• But this dramatically changes the structure of the minimum energy configurations... take the 2-body problem for example with equal size spheres, normalized to unity radius

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Em =h2

2d2� 1

dversus Em =

h2

2(0.4 + d2)� 1

d

Em



2-Body Problem

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Separation

Separation

Ener

gy

Ener

gy

Angular Momentum

Angular MomentumOnly one stationary orbit energy configuration

Zero or Two stationary orbit energy configurations per AM

Em =h2

2d2� 1

d

Separation

Em =h2

2(0.4 + d2)� 1

d

Point Mass Case Finite Density Case



Reconfiguration and Fission

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Reconfiguration: Occurs once the relative resting configuration becomes unstable. For the 3BP occurs at a rotation rate beyond the Lagrange solution

Multiple resting configurations can exist at one angular momentum.Resting and orbital stable configurations can exist at one angular momentum.

• As a system’s AM is increased, there are two possible types of transitions between minimum energy states:– Reconfigurations, dynamically change the resting locations – Fissions, resting configurations split and enter orbit about each

other




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Fission: Occurs once the relative resting configuration becomes unstable. For the 3BP occurs at a rotation rate beyond the Euler solution




Internal Degrees of Freedom for Spherical Grains

• 2-Body Results: – Contact case has 0 degrees of freedom– Orbit case has 1 degree of freedom

• 3-Body Results– Contact case has 1 degree of freedom– Contact + Orbit case has 2 degrees of freedom– Know all of the orbit configurations

• 4-Body Results – Contact case has 2 degrees of freedom, multiple topologies– Many more possible Orbit + Contact configurations– 3-dimensional configurations– Don’t even know precisely how many orbit configurations exist...

but they are all energetically unstable (Moeckel)!23


Lagrange Resting

Static & Variable Resting Configurations

Euler Resting

V Resting

Lagrange Orbiting

Orbiting Configurations

Euler Orbiting

Mixed Configurations

Aligned Mixed

Transverse Mixed


All minimum energy states can be uniquely identified in the finite density 3 Body Problem


1.98375 4.002... 5.07

6.6125 ∞5.32417

0

5.07

H2

Bifurcation Chart


-3

-2.5

-2

-1.5

-1

0 1 2 3 4 5 6 7 8H2

E

∆E

∆E

∆E

∆E = Excess Energy Increasing AM Decreasing AM

∆E


Static Resting Equilibrium Configurations

Mixed Equilibrium Configurations

0 1 2

43 5

Variable Resting Equilibrium Configurations

A

B

C

D

E

G

F

Finite Density 4 Body Problem


= 70.53°

= 180°

Static Rest Configurations 0 and 1

Static Rest Configurations 1, 2, 5

1

Static Rest Configurations 1, 3, 4

2

1

2


-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

0 5 10 15 20 25

Configuration ABCD

E

H2


-4

-3.5

-3

-2.5

-2

6 8 10 12 14 16 18 20 22 24

Configuration ABCD

E

H2

∆E2,3

∆Ea

∆Eb

∆Ec,d

Detail


D.J. Scheeres, A. Richard Seebass Chair, University of Colorado at Boulder 31

Generalizations to Non-Spherical Bodies



Migration of Surface Material

• If an asteroid’s rotation rate changes the minimum energy resting points of particles will change (Guibout & Scheeres, Celestial Mechanics 2003)





Slow rotation





Intermediate rotation





Fast rotation





For an ellipsoid, transition rotation rates are related to the Jacobi Sequence



From Miyamoto et al., Science, 2007




As spin rate increases or decreases, an aggregate can be placed into a non minimum energy state.ω0

(K0,E0)

A perturbation can trigger a shape change, conserving AM, decreasing energy, and dissipating excess energy via friction and seismic waves.

ω1

(K1 = K0, E1 < E0)



ω ω

The theory of minimum energy configurations can be extended to arbitrary finite density shapes, e.g. an equal density ellipsoid/ellipsoid system

Minimum energy configuration for small Angular Momentum

Minimum energy configuration for large Angular Momentum

Generalization to Non-Spherical Bodies



ω

ω

ω

(Icarus 189: 370-385)Tuesday, January 15, 2013


Rubble pile partitions

As spin rate increases, the transition limit for each “mutual” set should be computed and compared with each other.



Fission Conditions

• For fission of an arbitrary rubble pile split into two collections I and J the general condition becomes:

– For mass distributions only a weak form of Euler’s Theorem of Homogenous functions applies which allows us to reduce this inequality to:

– where

• This is equivalent to “the two components with the largest separation between their centers of mass will fission first at the lowest spin rate”

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TIJ + �UIJ � 0

RIJ · ! · ! · RIJ � �MI + MJ

MIMJ

@UIJ

@RIJ· RIJ

α >1



BODYHEAD

ω ∗∗

Asteroid Itokawa’s peculiar mass distribution will “fission” when its rotation period < 6 hours – spin period can change due to the “YORP Effect”, slowly changes total angular momentum...Body = 490 x 310 x 260 meters Head = 230 x 200 x 180 meters


-3

-2.5

-2

-1.5

-1

0 1 2 3 4 5 6 7 8H2

E

∆E

∆E

∆E

∆E = Excess Energy Increasing AM Decreasing AM

∆E



Reconfigurations and Fission Events• When a Local Minimum reaches its reconfiguration or fission

state it cannot directly enter a different minimum energy state– Excess energy ensures a period of dynamics where dissipation may occur

Movie by S.A. Jacobson



Fission

• Fission can be a smooth transition for a rubble pile• Energy and AM are ideally conserved, but are

decomposed:– Kinetic Energy

– Potential Energy

– The mutual potential energy is “liberated” and serves as a conduit to transfer rotational and translational KE



Fission

Fission spin rate is the minimum rate for two partitions of the asteroid to enter orbit.

Corresponds to the maximum center of mass separation across all possible partitions



Orbital Evolution



Post-Fission Dynamics

• The problem changes to the dynamical motion of two arbitrary mass distributions orbiting about each other– Strong coupling between translational and rotational motion– Generalized version of a classical celestial mechanics problem



Orbit Mechanics after Fission

• The relevant energy for orbital motion is the “free energy,” which is conserved under dynamical evolution:

• Energy transfer between orbit and rotation happen rapidly– If EFree > 0, system can “catastrophically disrupt”

– If EFree < 0, system cannot “catastrophically disrupt”

• Orbits with EFree > 0 are highly unstable and usually will send the components away on hyperbolic orbits



E > 0

Unstable

“Ground State” Equilibrium

for initially touching bodies

Ideal Stability of a Sphere-Sphere System

– Fissioned binaries with a mass ratio < ~0.2 have sufficient energy to escape in a timescale of days to months (Scheeres, Celestial Mechanics & Dynamical Astronomy 2009).

Stability of a fissioned system


Fission Dynamics of a proto 1999 KW4 System

Initial conditions are chosen at the precise “fission limit” when the two components enter orbit about each other. Total movie duration is several days.

Movie by S. Jacobson


0.0 0.2 0.4 0.6 0.8 1.01

10

100

1000

104

105

106

Mass Ratio

SeparationDistance

�PrimaryRadius⇥ Zero Free

Energy for Two Spheres

Jacobson & Scheeres, Icarus 2011

Fissioned Asteroids with mass ratios < 0.2 eventually escape, the closer to the cutoff, the more energy is drawn from the

primary spin.Tuesday, January 15, 2013

Pravec et al., Nature 2010

Observation:The mass ratios and primary spin periods of Main Belt asteroid pairs match with our Asteroid Fission Theory

Prediction:The mass ratio between asteroid pairs should be < 0.2The primary spin period should grow long near the cut-off

Prim

ary

Spin

Per

iod

Asteroid Pair Primary Spin vs Mass Ratio

Comment:The theory matches two independent outcomes, mass ratio cut-off and primary spin period lengthening



Energetics of a fissioned system

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• When a body fissions, it’s components enter a chaotic dynamical phase:– Fissioned binaries with a mass ratio < ~0.2 have sufficient energy

to evolve to escape in a timescale of days to months (Scheeres, Celestial Mechanics & Dynamical Astronomy 2002).

– The theory makes specific predictions that are consistent with observations of Asteroid Pairs

– Prior to escape, a sizable fraction of the secondaries are spun fast enough to undergo fission again

• Inner satellite usually impacts the primary, should cause reshaping• Outer satellite is stabilized

– This sequence can repeat and, along with gravity, friction, and sunshine, can create the observed class of NEA asteroid systems (Jacobson & Scheeres, Icarus 2011).


Tidal Process⇥ 106 � 107 yrs



Doubly SynchronousBinary

Asteroid

Chaotic Binary

Chaotic Binary

Chaotic Ternary

BYORP Process⇥ 105 � 106 yrs

Asteroid Pair

BYORP Process⇥ 105 � 106 yrs

Contact Binary

Asteroid Pair

YORP Process⇥ 105 � 106 yrs

YORP Process⇥ 105 � 106 yrs

q � 0.2

q � 0.2

Dynamic Processes� 1 yrs

DynamicProcesses� 1 yrs

Re-Shaped Asteroid

Stable Ternary1.1± 1.1%

Synchronous Binary8.6± 4.0%

Asteroid Pair64.2± 7.1%

Re-Shaped Asteroid28.3± 6.6%

Jacobson & Scheeres, Icarus 2011



Precise Modeling of the Mechanics of an Asteroid System

54

M1M2

M1 + M2r =

@U12

@rr = r2 � r1

H1 = H1 ⇥⌦1 + M1

H2 = H2 ⇥⌦1 + M2

⌦1 = I�11 · H1

⌦2 = I�12 · A · H2

A = A⇥⌦1 �⌦2 ⇥A

U12(r,A) = GZ

�1

Z

�2

dm1dm2

|r + ⇢1 �A · ⇢2|A1 = A1 ⇥⌦1

High dimensional system with coupled rotational and translational motions.Interest in both short-time dynamics and evolutionary dynamics.



Summary

• Study of asteroids leads directly to study of minimum energy configurations of self-gravitating grains– Only possible for bodies with finite density

• For finite density bodies, minimum energy and stable configurations are defined as a function of angular momentum by studying the minimum energy function:

– only a function of the system configuration– Globally minimum energy configurations are denumerable

• Simple few body systems can be fully explored– Need theories for polydisperse grains and N >> 1

55

Em =H2

2IH+ U


celestial mechanics of asteroid systems

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