Next: Exercises
Up: Planetary Motion
Previous: Motion in a General
In principle, a circular orbit is a possible orbit for any attractive central force.
However, not all forces result in stable circular orbits.
Let us now consider the stability of circular orbits in a general central forcefield. Equation (250) generalizes to

(305) 
where is the radial force per unit mass. For a circular orbit,
, and the above equation reduces to

(306) 
where is the radius of the orbit.
Let us now consider small departures from circularity. Let

(307) 
Equation (305) can be written

(308) 
Expanding the two terms involving as power series in ,
and keeping all terms up to first order, we obtain

(309) 
where denotes a derivative. Making use of Equation (306),
the above equation reduces to

(310) 
If the term in square brackets is positive then we obtain a simple harmonic
equation, which we already know has bounded solutionsi.e., the orbit is stable to small perturbations. On the other hand,
if the term is square brackets is negative then we obtain an equation
whose solutions grow exponentially in timei.e., the orbit
is unstable to small oscillations. Thus, the stability criterion for a circular
orbit of radius in a central forcefield characterized by a radial force
(per unit mass) function is

(311) 
For example, consider an attractive powerlaw force function of the form

(312) 
where . Substituting into the above stability criterion, we obtain

(313) 
or

(314) 
We conclude that circular orbits in attractive central forcefields which decay
faster than are unstable. The case is special, since the firstorder terms in the expansion of Equation (308) cancel out exactly, and it
is necessary to retain the secondorder terms. Doing this, it
is easily demonstrated that circular orbits are also unstable for
inversecube () forces.
An apsis is a point in an orbit at which the radial distance, , assumes either a
maximum or a minimum value. Thus, the perihelion and aphelion points
are the apsides of planetary orbits. The angle through which the radius vector
rotates in going between two consecutive apsides is called the apsidal
angle. Thus, the apsidal angle for elliptical orbits in an inversesquare
forcefield is .
For the case of stable nearly circular orbits, we have seen that oscillates sinusoidally
about its mean value, . Indeed, it is clear from Equation (310) that
the period of the oscillation is

(315) 
The apsidal angle is the amount by which increases in going
between a maximum and a minimum of . The time taken
to achieve this is clearly . Now,
, where
is a constant of the motion, and is almost constant. Thus,
is approximately constant. In fact,

(316) 
where use has been made of Equation (306). Thus, the apsidal angle,
,
is given by

(317) 
For the case of attractive powerlaw central forces of the form
, where
, the apsidal angle becomes

(318) 
Now, it should be clear that if an orbit is going to close on itself then the apsidal angle needs to be a rational fraction of . There are, in fact,
only two small integer values of the powerlaw index, , for which this
is the case. As we have seen, for an inversesquare force law (i.e., ), the
apsidal angle is . For a linear force law (i.e., ),
the apsidal angle is see Section 4.2. However, for quadratic (i.e., ) or cubic (i.e., ) force laws, the apsidal angle is an irrational
fraction of , which means that noncircular orbits in such forcefields
never close on themselves.
Let us, finally, calculate the apsidal angle for a nearly circular orbit of radius in a slightly modified (attractive) inversesquare force law of the
form

(319) 
where
is small. Substitution into Equation (317) yields

(320) 
Expanding to firstorder in
, we obtain

(321) 
We conclude that if then the perihelion (or aphelion) of the orbit advances by an angle
every rotation period.
It turns out that the general relativistic corrections to
Newtonian gravity give rise to a small modification (with ) to
the Sun's gravitational field. Hence, these corrections generate
a small precession in the perihelion of each planet orbiting the Sun. This
effect is particularly large for Mercurysee Section 12.13.
Next: Exercises
Up: Planetary Motion
Previous: Motion in a General
Richard Fitzpatrick
20110331