This was practically my only communication with Roger Broucke, my advisor in celestial mechanics. I had been having trouble integrating my Mars trajectory close to Earth, and he decided to write a formal paper on the problem. Nothing quite like your own professor stealing your ideas. He totally ignored me for the following 3 years. Totally.
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Page 1
Date: Wed, 3 Jun 1998 17:04:05 -0500 (CDT)
From: “R. Broucke” <broucke@uts.cc.utexas.edu>
To: whcii@ix.netcom.com
Subject: thesis.
hello Bill, do you have v.S.’s book? If you look up regularization, it would be the Thiele-Burrau thatwe need. page 103. Because this regularizes two close approaches at the same time. : Earth + Mars. But we would need fancy translation + rotation to get the Earth and Mars to (-1/2,0) and (+1/2,0), as in figure 3.10, page 102. I actually plan on trying for a serious contribution on the 3-body problem in the next 2 years. In the past 22 years, my access to this problem was limited, but now the circumstances are different. I attach my abstract submitted to the Belgium meeting of July. This is one of the fronts I will attack…
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<center> MOTION NEAR THE UNIT CIRCLE IN THE THREE-BODY PROBLEM.<br> ROGER A. BROUCKE<br> Department of Aerospace Engineering<br> University of Texas<br> Austin, TX, 78712 </center>
Abstract.
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Many of the important applications of the circular planar restricted problem of three bodies involve motion in the vicinity of the unit circle, (as defined in canonical units). It is then of interest to develop simplified models which are valid in this region. These models preserve the gross characteristics of the original system but they possess simpler equations of motion.
We will also show that several simplified models can be seen as a perturbation of a very well known simple linear system: the Clohessy – Wiltshire equations used by NASA in all their rendez-vous operations. These are actually very close to the well-known Hill problem. We will thus consider the Restricted problem as a perturbed Hill or Clohessy – Wiltshire problem. We also introduce the Clohessy – Wiltshire Lagrangian in Polar coordinates.
Actually we use polar coordinates throughout, in a rotating frame with canonical units of lenght and time. In this frame, the larger primary is at the origin, while the smaller primary is at (1,0). We are especially interested in the motion near the unit circle, (r=1). This type of situation has numerous applications, such as for instance the Trojan Asteroids as well as the co-orbital satellites of Saturn, as well as the Earth-orbit crossing asteroids. Another application is in the theoretical study of the existence and stability of the Lagrangian points L3, L4, L5.
Finally, a generalization is also being studied where several interacting particles are present and all moving near the unit circle, although this is not the classical restricted three-body problem any more. One of the most interesting features that is seen here is the effect of the “Repulsive Gravity”, (which is really a consequence of the Coriolis acceleration). The numerical integrations show that the particles have a tendency to avoid each other and consequently, that collisions are not as frequent as one would first expect. This fact may have some importance in relation to the physics of planetary rings.
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Date: Wed, 3 Jun 1998 10:34:38 -0500 (CDT)
From: “R. Broucke” <broucke@uts.cc.utexas.edu>
To: whcii@ix.netcom.com
Subject: Re: Thesis
Thanks for the new text. It looks beatiful to me. But I have many questions and remarks. So, we need to get together.
For instance, do you know about regularization. That allows you to do very good numerical integration. even when you go very close to the planets, when the denominators go to zero. You know those r**3 in the denominators.
Please look up regularization in V.S.’s theory of orbits.
I will also give you my notes. R.B.
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R. Broucke, 06:01 PM 6/28/98 , No Subject
Delivered-To: whcii@mail.utexas.edu
Date: Sun, 28 Jun 1998 18:01:16 -0500 (CDT)
From: “R. Broucke” <broucke@uts.cc.utexas.edu>
To: whcii@mail.utexas.edu
<br>
Suggested for presentation at the 1999 Cortina meeting.
Poincare’s characteristic exponents and bifurcations
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<center>
in Dynamical Systems.<br>
———————<br>
<br>
ROGER A. BROUCKE.
</center>
Poincare’s Characteristic Exponents are one of the most important tools that we have available for the analysis of the intrinsic structure of Dynamical Systems. This is because they allow us to define manifolds of stable solutions as well as manifolds of chaotic solutions. The boundaries between the two manifolds of solutions correspond to the zero characteristic exponents.
We will first review the applications to the restricted problem that were obtained in the sixties. The first and best known results relating to the Copenhagen problem ( mu = 0.5) are due mainly to M. Henon who found the boundary orbits with zero Characteristic exponents. We also review some important results corresponding to small values of mu, especially some bifurcations of periodic orbits near the Lagrange points L4 and L5.
Next, we will review the applications to the Artificial Satellite problem. We limit ourselves to the first few harmonics, J2, J3, J4 and J22. We believe that the most important result is the fact that the well-known critical inclination (63.4 degrees) is in fact a double pitchfork bifurcation. We have also been able to show that this critical inclination is really a rather generic phenomenon, because it occurs in a large class of potentials with axial symmetry.
Finally, we show that the Poincare characteristic exponents may be useful in the study of the stability of mass-spring systems, which shows that the applications of this tool may be as far-reaching as the field of Mechanical Engineering.
<br>
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Printed for “Bill H. Clark” <whcii@mail.utexas.edu>
