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

Second part of the course

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

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 Gravitational

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

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

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 with m1 and m2 The three scalar components:  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: (>0) Between the bodies: (0<<1) Unique solutions in both cases

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”

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

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

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 aJk-2 We get: Tisserand parameter

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