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# Orbits in Isotropic Collections of Stars

Stars group together as star clusters and galaxies. The orbit of a star within one of these star collections is determined by how the mass within the system is distributed. At one extreme, the mass is concentrated at the center of the system, so that the mass appears as a singe point-mass; stars outside of this mass concentration have Keplerian orbits. More generally, however, most of the mass of a system is far from the center of the system; this causes the the magnitude of the gravitational field to either falls less rapidly than the inverse-square law, or it rises with radius. From a practical standpoint, the mass density of a massive star system is uniform at the core of the system, and it falls with distance from the center outside of the core.

When the mass distribution of a stellar system is isotropic and constant in time, the orbit of a star within the collection is bounded between a maximum and a minimum distance from the center. This bound is a consequence of the conservation of energy and of angular momentum. A star's orbit in such a stellar system is confined to a plane that passes through the center of the system. The conservation of angular momentum means that the area swept out by a star per unit time is the same over every part of the star's orbit. These properties are also properties of Keplerian orbits.

Two properties of Keplerian orbits that are not possessed by orbits in more general isotropic systems is that the orbits are not usually closed, and the square of the orbital period is not proportional to the cube of the average of the maximum and minimum orbital radius.

An important point about isotropic mass distributions is that the gravitational field a star sees at a given distance from the center is identical to the gravitational field produced by a point source with a mass equal to the mass inside a sphere with a radius equal to the distance from the center. An implication is that the stars at larger distances from the center than a given star does not exert a gravitational force on that star.

## Orbits in a Homogeneous Medium

The core of a galaxy is never truly uniform, because there is usually a compact massive object, generally thought to be a black hole, at its center. But the effect of the compact object on the gravitational field is limited to the central several parsecs of the system, and for most stars in the core, the gravitational field appears to be that produced by a uniform mass density.

In a homogeneous medium, the magnitude of the force on an object towards the center of the star system is proportional to the distance r from the center. The equations of motion for this force are easily solved. One finds that the orbits are ovals, with the center of the oval lying on the center of the stellar system. The maximum and minimum distances are separated by 90° in angle. The orbital period, which is independent of either the value of the semimajor axis or eccentricity of the orbit is determined solely by the mass density; in this time, every star in the core returns to its starting position.

## General Orbits

From a practical standpoint, one does not find gravitational fields with magnitudes that rise faster than that produced by a homogeneous distribution, because this would required the density to increase with radius, which is a situation that does not arise in nature. One can therefore consider the Keplerian orbit and the orbit in a constant-density core as bounds on the possible orbital behavior in a simple isotropic mass distribution. For a gravitational force with strength proportional to rn - 2, where r is the distance from the galactic center and the index n is constrained by 0 ≤ n ≤ 3, with the lower limit corresponding to the gravitational field around a point mass, and the upper limit corresponding to a homogeneous mass distribution. Orbits in this general power law gravitational field are not closed. The angle between the maximum and minimum values of r range between 90° and 180°, with the angle close to 180° for n close to 0 and close to 90° for n close to 3.

## Orbits in the Plane of a Spiral Galaxy

While the mass distribution in a spiral galaxy is not isotropic, it is nearly axisymmetric within the plane of the galaxy. The conclusions about spherically-symmetric stellar systems can be applied to stellar orbits confined to the galactic plane.

The stars with circular orbits in the galactic plane are found to have velocities that are nearly independent of orbital radius when the radius is greater than about 5 kpc. This requires the gravitational force to be inversely proportional to the distance (n = 1). The stars in the galactic plane therefore move in open orbits. The maximum distance from the center is separated from the minimum distance by approximately 250°, although this value is weakly dependent on the eccentricity of the orbit, decreasing as the orbit becomes more eccentric. Orbits with the same average value of the minimum and maximum orbital radii have nearly the same orbital period, with the period increasing weakly as the orbit becomes more eccentric. These characteristics of orbits within the galactic disk of a spiral galaxy are crucial to the development of a stable spiral structure.