These are notes I wrote about some elementary topics in celestial
mechanics. They grew out of assignments and projects done during the
fall of 2006 in Dr. Cesar Ocampo's Celestial Mechanics course at UT Austin.
They focus primarily on numerical methods
for studying n-body problems (especially the
circular restricted three body problem), but aim to include enough
background material and references so that they are readable
outside the context of that course.
Description: These notes contain introductory material for the
n-body problem, including the equations of motion and some important
invariance properties. The ten classical integrals are derived. The
solutions of the Kepler, and two body problems are found, and design of
conic trajectories is discussed . Several other configurations are
studied numerically, including the Sitnikov problem.
Introduction to the N-Body Problem
MatLab Code for Introduction Notes
Description: This set of notes describes the method of
differential corrections, and some applications.
The variational equations and the state transition matrix for
the n-body problem are developed. Examples are given which show
how to use the method to solve boundary value or `targeting'
problems in celestial mechanics. In particular a scheme for
computing choreography orbits is presented.
Differential Corrections (pdf)
Differential Corrections (ps)
MatLab Code for Differential Corrections
Description: These notes deal with the general three body problem.
The special triangular solutions of Lagrange and collinear solutions
of Euler are treated numerically, with emphasis on numerical and
dynamical stability. In addition the equations of motion for the
three body problem in Jacobi coordinates are developed. Finally some
kinematic results for dynamics in rotating coordinate frames are derived.
Special Solutions in the Three Body Problem (pdf)
Special Solutions in the Three Body Problem (ps)
Description: These notes are an introduction to the circular restricted three
body problem. In this problem two massive bodies orbit one another in
circular orbits, and a smaller body moves in their field. The
equations of motion and the Jacobi integral are derived. After
the five libration points (equilibria), the linearized equations about
these points are found and studied. The notes also contain numerical
studies of the resulting dynamics including the occurrence of ballistic capture.
Circular Restricted Three Body Problem (pdf)
MatLab Code for Note Set 4
Description: In this set of notes, procedures for computing
and analyzing symmetric periodic orbits in the circular restricted
three body problem are developed. In addition, the stable and
unstable manifolds of such orbits are defined and a method is
presented for computing these objects.
These methods are applied to halo orbits in the Earth/Sun system.
Halo orbits are out-of-plane periodic orbits near the collinear
libration points. These orbits have received much attention in recent
years as they have properties that make them ideal for astronomical
purposes such as solar observatories, or communication posts (say for
uninterrupted communication with a mission on the dark side of the
We analyze a halo orbit about L1 given by approximate physical data, and use
this to compute a one parameter family of halo orbits parameterized by
the Jacobi integral. From this we develop a method that allows one
to compute such a family of halo orbits near any of the collinear
libration points at any value of mu. We apply this method at L2 and
find the halo family there.
These tools are also applied to the Lyapunov orbits near L1, where we
compute the stable and unstable manifolds of these inplane periodic
Symmetric Periodic Orbits in the CRTBP and
their Stable and Unstable Manifolds (pdf)
MatLab Code for Note Set 5
Description: The frame on the left above shows a halo orbit in the
Earth/Moon system, with the branch of it's stable manifold that passes
near the Earth. The halo orbit is the thick black ring, and it's
stable manifold is the blue tube. Orbits on this tube approach the
halo orbit in forward time. The zero velocity surface at the energy
level of the halo orbit is shown in red. L1 is the black star in the
center of the halo orbit. An orbit which begins near the Sun, passes
through the L1 neck, performs a flyby of the Earth, and then returns
to the Sun through the L1 neck is shown as well.
The picture in this frame is a close up of the picture above. Here
we can see right through the L1-L2 `neck' in the energy surface. L2
is seen as a second black star on the far side of the neck. The
stable manifold is wrapped around the earth, obscuring it from
view. Its location is however roughly half way between L1 and L2.
Description: Another view.
Description: The picture shows a Poincare section for the
Earth/Moon CRTBP in an integral manifold. The energy (Jacobi
constant) of the manifold is between the energy of L1 and L2,
so that the L1 ``neck'' is open while the L2 ``neck'' is closed.
The coordinates of the section are x and xdot. The section is
at y=0 (ydot is then determined by the fixed value of the energy).
In this section the dominant features are the secondary KAM tori
due to a resonance near x=0.5, and the L1 neck near x=0.8. The
``static'' around the KAM tori is the Birkhoff instability zone.
Trajectories in this zone exhibit chaotic motion.
The section was constructed by taking a grid of 100 orbits with
initial conditions on the x axis (so xdot_0 = 0). For each such
orbit, 1000 crossings of the Poincare section are computed. This
is repeated three more times with varying initial xdot. The the
resulting image (a large part of which is shown here) shows
400,000 crossings. This computation was done in MatLab, and
took roughly 100hrs (in FORTRAN or C++ this would run much faster).
The data is available in .mat form on my MatLab page.
Description: This section illustrates the rich structure of the
problem. The picture
shows x between about 0.1 and 0.28 and xdot between about 0.0 and 1.5.
Then we are much closer to the earth than to L1, and the phase space
is packed quite tightly with secondary tori (secondary when we think
of the CRTBP as a perturbation of the Kepler problem).
Even though the region is rich with tori there are Birkhoff zones
between them, and it is possible for orbits to escape or diffuse
from the region via ``resonance hopping''. They can wind out from
between the tori chaoticly, and move into regions where the influence
of the moon is stronger.
Description: The black dots in this picture are taken from
Poincare section data near the moon, similar to the pictures
described above. The red and blue dots (or curves) are data obtained
by computing the intersection of the unstable, and stable manifolds of
a Lyapunov orbit with the Poincare section. Then a blue dot
approaches the Lyapunov orbit in forward time, and a red dot
approaches it in backward time.
It can be seen from the picture that the stable and unstable manifolds
intersect transversely. This implies much. In the first place such an
intersection point would approach the Lyapunov orbit in both forward
and backward time, so orbits homoclinic to the Lyapunov orbit exist.
On the other hand, the transverse crossing of the stable and unstable manifolds
also implies the existence of a topological horse shoe near the
Lyapunov orbit, so the dynamics there must be very rich.
In particular, there exist infinitely many periodic orbits, of arbitrarily
long period which ``shadow'' the homoclinic orbit. In other words,
they begin near the Lyapunov orbit, move out
to near an intersection of the stable and unstable manifold, and
return to the vicinity of the Lyapunov orbit. Since the stable and
unstable manifolds intersect on the far side of the moon, very far
from the Lyapunov orbit, this is global behavior.
Description: Knowing that there must be periodic orbits near the
Lyapunov orbit and having the intersections of the
stable and unstable manifolds allows us to actually compute them (here
the low dimension of the problem is essential as we are basically
doing graphical analysis). This picture shows a periodic orbit (in
configuration space) that begins near the Lyapunov orbit, flies by the
moon several times, passes near the intersection, returns, and closes
itself off. The outward trip is blue and the return is green. A
homoclinic orbit (which can be read off the above Poincare section) is
used as the initial guess for a differential corrections procedure
which then finds the initial conditions of a nearby periodic orbit.
(The material from the last several slides will eventually be added to
Note set five).
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