Sitnikov problem

Figure 1: Configuration of the Sitnikov Problem

Sitnikov problem is a problem about restricted three-body problem.

In 1911, United States Scientist William Duncan MacMillan found one special solution about a restricted three-body problem. In 1961, Russian mathematician Sitnikov improve this problem.

Defenition

The system consists of two primary bodies with the same mass $\left(m_{1}=m_{2}={\tfrac {m}{2}}\right)$, which move in circular or elliptical Kepler orbits around their center of mass. The third body, which is substantially smaller than the primary bodies and whose mass can be set to zero $(m_{3}=0)$, moves under the influence of the primary bodies in a plane that is perpendicular to the orbital plane of the primary bodies (see Figure 1). The origin of the system is at the focus of the primary bodies. The combined mass of the primary bodies $(m=1)$, the orbital period of the bodies $(2\pi )$, and the radius of the orbit of the bodies $(a=1)$ are used for this system. In addition, the gravitational constant is 1. In such a system that the third body only moves in one dimension - it moves only along the z-axis.

Equation of Motion

In order to derive the equation of motion (which for circular orbits are the primary bodies) the total energy $\,E$ has to be determined first:

$E={\frac {1}{2}}\left({\frac {dz}{dt}}\right)^{2}-{\frac {1}{r}}$

After differentiating with respect to time, the equation becomes:

${\frac {d^{2}z}{dt^{2}}}=-{\frac {z}{r^{3}}}$

This, according to Figure 1, is also true:

$r^{2}=a^{2}+z^{2}=1+z^{2}$

Thus, the equation of motion is as follows:

${\frac {d^{2}z}{dt^{2}}}=-{\frac {z}{\left({\sqrt {1+z^{2}}}\right)^{3}}}$

Significance

Although it is nearly impossible in the real world to find or arrange three celestial bodies exactly as in the Sitnikov Problem, the Sitnikov Problem is still widely and intensively studied for decades: althought it is a simple case of the more general 3-body problem, all the characteristics of a chaotic system can nevertheless be found within the problem, making the Sitnikov Problem ideal for general studies on effects in chaotic dynamical systems.

See Also

• Celestial mechanics
• Two body problem
• Chaos theory

External links

• scholarpedia
• Fault Analysis of the Sitnikov Problem for High Orders using automated derivation Methods in Mathmatics
• Florian Freistetter, Seltsame Welten: Sitnikov Planets

Literature

• K. A. Sitnikov: The existence of oscillatory motions in the three-body problems. In: Doklady Akademii Nauk SSSR, 133/1960, S. 303–306, ISSN 0002-3264 (English Translation in Soviet Physics. Doklady., 5/1960, S. 647–650)
• K. Wodnar: The original Sitnikov article - new insights., In: Celestial Mechanics and Dynamical Astronomy, 56/1993, S. 99–101, ISSN 09232958, pdf
• D. Hevia, F. Rañada: Chaos in the three-body problem: the Sitnikov case. IN: European Journal of Physics, 17/1996, S. 295–302, ISSN 0143-0807, pdf (255 KB)
• Rudolf Dvorak, Florian Freistetter, J. Kurths, Chaos and Stability in Planetary Systems., Springer, 2005, ISBN 3540282084