10 Central-Field Motion I

Ashok Goyal

1. Introduction

A particle in a central–force field may execute bounded or open noncircular motion depending on the nature of the force. If the particle in executing one complete revolution does not return to its original position it signals a deviation from the inverse – square law force however slight. An irregularity in the motion of the planet mercury was observed. It was observed that the perihelion of mercury that the semi-major axis advances at the rate of roughly 574 sec. of arc per century. Calculation of the influence of other planets predicted an advance of approximately 531 sec of arc per century leaving a deficit of about 40 sec of arc per century. Einstein’s General Theory of Relativity in post – Newtonian approximation was able to account for this difference of 43 sec of arc per century and became one of the greatest triumphs of General Relativity. In this unit we will discuss the conditions for closed orbits, stability of the orbits and an estimate of the advance of the perihelion of mercury in the presence of deviations from the inverse-square law.

2. Open and Closed Orbits:

We saw earlier that the radial velocity of a particle in a central field is given by

The motion is therefore confined in the region  and these points are the turning points of the motion. For certain value of the potential , there may be only one root in which case the motion is circular. If the motion of the particle is such that the orbit is closed after finite number of round travel between  and , the central potential satisfies certain conditions. If the orbit after finite number of oscillations does not come back to the initial orbit, the orbit is said to be open as shown below:

To find the condition for closed orbit, we need to find the change in angle  after the particle has made one complete oscillation from  and  and back; thus

Where and  are integers. Thus after  oscillations  will change by  times an integer and would return to its original position i.e. after  periods the particle would have made  complete oscillations.

Example: Show that if the potential  varies as some integer power of radial distance a closed non-circular path will result only if  or +1.

In terms of the variable , equation (11.4) can be written as

The integral (11.5) can be expressed in terms of circular functions provided the radical in the integral (11.5) can be expressed in the form

 implies constant force and is not of interest,  corresponds to the inverse square for law and  corresponds to inverse-cube force law. The case of  is the familiar case of planetary motion in gravitational field which we considered in detail in the previous unit. The equation of the orbit was found to be  and in this obviously the orbit is closed because as, the radial is  vector returns to its original position.

The case of inverse-cubic force law  is  will be treated when we evaluate the advance in the perihelion of mercury. We will find that in this case the orbit does not close on itself. The case of harmonic force  corresponds to  and can be solved in terms of circular functions and has a closed orbit.

3. Orbit in a Linear Force Field

We have the force  which can be derived from the potential function . The equation of the orbit expressed in the variable  is

Substituting back  and  .

For the stability of the orbit, it is required that  is imaginary  will increase exponentially making the  orbit to be unstable. Therefore the stability of the orbit is ensured provided

5. Perihelion of Mercury

We mentioned in the introduction that the perihelion of Mercury that is semi-major axis was found to advance at the rate of roughly 574 sec. of arc per century. This would happen if the law of force deviated to the inverse square law.

We will assume the presence of a small  force in addition to the inverse square law to understand the advance in the perihelion. This small  force could for example arise due to the pressure of other gravitating bodies i.e. other planets, comets, etc. apart from the sun which provides the major inverse square force. We take the potential  to be

Now for , the eccentricity  , the orbit is an ellipse, but it is not a closed ellipse because as  the particle does not come back to its original position. In the presence of a small respective inverse-cubic force.

6. Summary

•  A particle moving in a force field where the force is either inverse-square or a linear harmonic force has a closed orbit.
• The inverse cubic force law results in an open precessing elliptical orbit.
• The circular orbit of a particle moving in a general central force field is stable provided where ρ is the radius of the circular orbit.
• The perihelion of mercury arises because of the presence of a small repulsive inverse cubic force.