Sitnikov problem

The Sitnikov problem is a special case of the restricted three-body problem named after the Russian mathematician Kirill Alexandrowitsch Sitnikow (* 1926) and describes the movement of three celestial bodies under their mutual gravitational attraction.

definition

Fig. 1: Configuration of the Sitnikov problem

The system consists of 2 primary bodies (e.g. stars ) with the same mass , which move on circular or elliptical Kepler orbits around their common center of gravity . The third body, which is much smaller than the two primary bodies and whose mass can therefore be set equal to zero , moves under the influence of the primary bodies in a plane that is perpendicular to the plane of the primary body (see Figure 1). The origin of coordinates is in the center of gravity of the two primary bodies. The unit of mass is the total mass of the primary bodies , the unit of time is their period around the center of gravity and the unit of length is the radius of the path ; in addition, the gravitational constant is set equal to 1. In such a system the movement of the third body is one-dimensional - it only moves along the z-axis. ${\ displaystyle \ left (m_ {1} = m_ {2} = {\ tfrac {m} {2}} \ right)}$ ${\ displaystyle (m_ {3} = 0)}$ ${\ displaystyle (m = 1)}$ ${\ displaystyle (2 \ pi)}$ ${\ displaystyle (a = 1)}$

Equation of motion

To derive the equation of motion (for the case of circular orbits of the primary bodies) one first determines the total energy of the system: ${\ displaystyle \, E}$

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

Derived from time , this gives:

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

The following applies (see Figure 1):

{\ displaystyle r ^ {2} = a ^ {2} + z ^ {2} = 1 + z ^ {2}}

And therefore the equation of motion follows:

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

meaning

Although it is almost impossible for three celestial bodies to arrange or form in a Sitnikov configuration, the Sitnikov problem has been studied intensively for decades: although it is a very simple case of the three-body problem, all the properties of one can be found in the elliptical Sitnikov problem chaotic system , which is why it is ideal for general investigations into chaotic effects in dynamic systems.

Web links

literature

KA Sitnikov: The existence of oscillatory motions in the three-body problems. In: Doklady Akademii Nauk SSSR , 133/1960, pp. 303-306, ISSN 0002-3264 (English translation in Soviet Physics. Doklady. , 5/1960, pp. 647-650)
K. Wodnar: The original Sitnikov article - new insights. , In: Celestial Mechanics and Dynamical Astronomy , 56/1993, pp. 99-101, ISSN 0923-2958 , bibcode : 1993CeMDA..56 ... 99W
D. Hevia, F. Rañada: Chaos in the three-body problem: the Sitnikov case . In: European Journal of Physics . tape 17 , 1996, ISSN 0143-0807 , p. 295-302 , doi : 10.1088 / 0143-0807 / 17/5/009 .
Rudolf Dvorak, Florian Freistetter, J. Kurths, Chaos and Stability in Planetary Systems. , Springer, 2005, ISBN 3540282084