1. Celestial Mechanics V Circular restricted three-body problem
Jacobi integral, Hill surfaces
Stationary solutions
Tisserand criterion

2. Second part of the course

3. 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 small-body dynamics in the Solar System

4. Inertial frame, massive bodies
Origo at CM of the two massive bodies:
Mutual distance = a; mean motion = n
Use Gaussian units:

5. 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

6. Co-rotating frame, massive bodies
Transformation matrix:
The two bodies are at rest on the x-axis

7. 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
Gravitational

8. The Jacobi Integral (Jacobi Integral)
Scalar multiplication by the velocity vector:
(the Coriolis term disappears)
This yields an energy integral:
(Jacobi Integral)
kinetic
centrifugal
potential
CJ is a constant of integration that corresponds to minus
the total energy of motion in the co-rotating frame

9. The Hill Surface The physically accessible space domain is that where
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

10. 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

11. 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

12. Stationary solutions
Acceleration  0; velocity  0; insert into equation of motion:
This means the object stays in co-rotation, forming a rigid
configuration with m1 and m2
The three scalar components:
 dz  0

13. 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

14. Euler’s collinear solutions
Left of body #1:
Parametrise the position by :
(>1)
Insert into the first equation:
Unique solution
for  > 1

15. Euler’s collinear solutions, ctd
Right of body #2:
(>0)
Between the bodies:
(0<<1)
Unique solutions in both cases

16. 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:
In both cases this reduces to:
(largest region of stable
satellite motion)
the radius of the “Hill sphere”

17. Temporary satellite captures
Jupiter’s orbit is slightly eccentric
An object approaching the Hill sphere with a near-critical 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/Helin-
Roman-Crockett predicted
for the 2070’s

18. 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!
Equilateral triangles with respect to m1 and m2

19. The Lagrange points L1, L2, L3 are the Collinear points L4, L5 are the
Triangular points
Trojan asteroids

20. 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) >
This holds for the Sun-Jupiter case (and for any other planet too)
Trojans have been detected for Mars and Neptune too

21. 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!

22. The Tisserand criterion, ctd
Use the vis-viva law and the expression for
angular momentum:
Approximate by putting:
and multiply by
aJk-2
We get:
Tisserand parameter

23. 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