Transcript Celestial Mechanics V

Celestial Mechanics V
Circular restricted three-body problem
Jacobi integral, Hill surfaces
Stationary solutions
Tisserand criterion
Second part of the course
N-BODY
PROBLEM
Le Problème
Restreint
Numerical
Simulations
Perturbation
Theory
Modern
Science
Circular restricted 3-body
problem
• Two point masses with finite mass move in
a circular orbit around each other
• A third, massless body moves in the
combined gravity field of these two
• We study the dynamics of the third body
• Relevant approximation for much of smallbody dynamics in the Solar System
Inertial frame, massive bodies
Origo at CM of the two massive bodies:
Mutual distance = a; mean motion = n
Use Gaussian units:
Equation of motion of third body
This is independent of the mass m3
But we consider m3 infinitesimally small, so that
m1 and m2 are not perturbed
Co-rotating frame, massive
bodies
Transformation matrix:
The two bodies are at rest on the x-axis
Equation of motion of third body
Transform the position vector:
Transform the velocity and acceleration vectors:
Insert into the ‘inertial’ equation of motion:
Coriolis
Centrifugal
Gravitationa
l
The Jacobi Integral
Scalar multiplication by the velocity vector:
(the Coriolis term disappears)
This yields an energy integral:
kinetic
centrifugal
(Jacobi Integral)
potential
CJ is a constant of integration that corresponds to minus
the total energy of motion in the co-rotating frame
The Hill Surface
The physically accessible space domain is that
where
v2  0
3
The Hill surface of zero velocity is the locus of v3=0 in
(dx,dy,CJ) space, assuming dz=0

Motion is possible only below or on the Hill surface
Cuts of the Hill surface
‘zero-velocity
curves’
• For the smallest values of CJ, the whole (dx,dy) plane is
available
• For the largest values, only small disjoint regions around
the Sun and Jupiter plus another disjoint region of infinite
extent outside are available for motion of the third body
Constraints on the motion
• The zero-velocity curves are cuts of
surfaces in (dx,dy,dz) space with the
dz=0 plane
• The small regions around m1 and m2
indicate 3D lobes to which the motion is
constrained for large CJ
• The third object is unable to pass from
one disjoint region to another
• A separate lobe around the planet is a
region of stable satellite motion
Stationary solutions
Acceleration  0; velocity  0; insert into equation of motion:
This means the object stays in co-rotation, forming a
rigid
configuration
withcomponents:
m1 and m2
The three scalar
 dz  0
Stationary solutions, ctd
We know that the 3rd body has to stay in the orbital
plane of m1 and m2
From the second equation we get:
either
Collinear
solutions
or
Triangular solutions
Now, search for solutions of the first scalar equation
satisfying either of these conditions
Euler’s collinear solutions
Left of body #1:
Parametrise the position by :
(>1)
Insert into the first equation:
Unique
solution
for  > 1
Euler’s collinear solutions, ctd
Right of body #2:
Between the bodies:
Unique solutions in both cases
(>0)
(0<<1)
Limiting case: the Hill sphere
If m2 << m1 (as is the case for all the planets of the
Solar System), then  << 1 in the latter two cases
We get:
and:

LHS  m1  m1  m1 (1 2 ) 
m2
LHS  m1  m1  m1 (1 2 ) 
m2
2
2
1 m2 
    
3 m1 
1/ 3
In both cases this reduces to:

the radius of the “Hill sphere”
(largest region of stable
satellite motion)
Temporary satellite captures
• Jupiter’s orbit is slightly
eccentric
• An object approaching the
Hill sphere with a nearcritical value of CJ may enter
through an opening that then
closes for some time
• Temporary satellite captures
(TSC) are found for some
short-period comets
TSC for comet 111P/HelinRoman-Crockett predicted
for the 2070’s
Lagrange’s triangular solutions
Rearrange the first scalar equation:

But this expression is
zero according to the
condition from the
second equation!

Hence this must be
zero too!
r31  r32  a
Equilateral triangles with respect to m1 and m2
The Lagrange points
L1, L2, L3 are the
Collinear points
L4, L5 are the
Triangular points
Trojan asteroids
Stability of Lagrangian points
• This means that a slight push away
from the L point leads to an oscillatory
motion staying in its vicinity
• In this sense the collinear points are
unstable
• The triangular points are stable, if (m1–
m2)/(m1+m2) > 0.9229582
• This holds for the Sun-Jupiter case (and
for any other planet too)
Trojans have been detected for Mars and Neptune too
The Tisserand criterion
(after F.F. Tisserand 1889)
Start from the Jacobi integral:
Assume that r32 is not very small:
Transform the velocity squared to non-rotating axes:
Approximate by putting the Sun at origo!
The Tisserand criterion, ctd
Use the vis-viva law and the expression for
angular momentum:
Approximate by putting:
and multiply by
We get:
aJk-2

Tisserand parameter
/2
n  ka3
J
The Tisserand parameter
• TJ is a quasi-integral in the 3-body
problem comet-Sun-Jupiter in the
presence of close encounters
• It is used to classify cometary orbits
• T relates to the speed of the encounter
• TP may be defined for other planets too,
but they are less stable in case the
orbits cross that of Jupiter