# Periodic Orbits in the Isosceles Three-Body Problem

• Cristina Chiralt Monleón
• José Martínez Alfaro
Part of the NATO ASI Series book series (NSSB, volume 272)

## Abstract

The Saturn’s satellites Janus and Epimetheus are the first known bodies in the Solar System that has horseshoe orbits in a frame that rotates with uniform angular velocity. Both satellites have similar masses and orbital elements when they are far from one another. Moreover, their orbits are nearly symmetric. In fact, in the past, they have been identify as a unique satellite and afterwards, some mathematical theories about their orbits has been necessaries to understand why they do not collide. In particular, the interest in planar three-body problem with two small masses has increased6. We assume that the two small masses have similar symmetric initial conditions. The aim of this paper is to find what kind of motion, can present these bodies. We find an infinity of different types. To prove that we analyze the limit case, when both small masses are equal and in symmetric opposite positions, i.e, the isosceles three- body problem.
$$\frac{{\text{d}^2 \text{x}}}{{\text{dt}^2 }} = - \frac{{\text{Gm}_0 \text{x}}}{{(\text{x}^2 + \text{y}^2 )^{3/2} }} - \frac{{\text{Gm}_1 }}{{4 \times ^2 }}$$
$$\frac{{\text{d}^2 \text{y}}}{{\text{dt}^2 }} = - \text{GM}\frac{\text{y}}{{(\text{x}^2 + \text{y}^2 )^{3/2} }}$$
where M is the total mass m0 + m1 + m2.

## Keywords

Periodic Orbit Triple Collision Symmetric Periodic Orbit Collision Orbit Hyperbolic Periodic Orbit

## References

1. 1.
Birman, J. and Williams, R.F. Knotted Periodic Orbits in Dynamical Systems I: Lorenz’s Equations. Topology 22:47 (1983)
2. 2.
Birman, J. and Williams, R.F. Knotted Periodic Orbits in Dynamical Systems II: Knot Holders for Fibered Knots. Contemporary Mathematics 20:1 (1983)
3. 3.
Broucke, R. On the Isosceles Triangle Configuration in the Planar General Three -Body Problem. Astronomy and Astrophysics. 73:303 (1979)
4. 4.
Franks, J. and Williams R.F. Entropy and Knots. Transactions of the American Mathematical Society. 291, No. 1:241 (1985)
5. 5.
Simó, C. and Martinez, R. Qualitative Study of the Planar Isosceles Three- Body Problem. Celestial Mechanics. 41:179 (1988)
6. 6.
Spirig, F. and Waldvogel, J.. The Three - Body Problem with Two Small Masses: A Singular- Perturbation Approach to the Problem of Saturn’s Coorbiting Satellites, in “Stability of the Solar System and Its Minor Natural and Artificial Bodies”. Nato Asi Series. (1984)Google Scholar