Encyclopedia of the Solar System 2nd ed

Solar System Dynamics: Regular and Chaotic Motion 789

Z

R

X N

O P

f

i

M



Ω

FIGURE 2 Geometry of an orbit in three dimensions. The Sun
is at one focus of the ellipse (O) and the planet is instantaneously
at locationR. The location of the perihelion of the orbit isP.
The intersection of the orbital plane (X−Y) and the reference
plane is along the lineON(whereNis the ascending node). The
various angles shown are described in the text. The mean
anomalyMis an angle proportional to the areaOPRswept out
by the radius vectorOR(Kepler’s second law).

three quantities (a,e, andi) are often referred to as the
principal orbital elements, as they describe the orbit’s size,
shape, and tilt, respectively.

2. The Two-body Problem

In this section the general solution to the problem of the
motion of two otherwise isolated objects in which the only
force acting on each body is the mutual gravitational inter-
action is discussed.

2.1 Newton’s Laws of Motion and the Universal Law
of Gravitation

Although Kepler’s laws were originally found from careful
observation of planetary motion, they were subsequently
shown to be derivable from Newton’s laws of motion to-
gether with his universal law of gravity. Consider a body
of massm 1 at instantaneous locationr 1 with instantaneous
velocityv 1 ≡dr 1 /dtand hence momentump 1 ≡m 1 v 1. The
accelerationdv 1 /dtproduced by a net forceF 1 is given by
Newton’s second law of motion:

F 1 =

d(m 1 v 1 )

dt

. (4)

Newton’s universal law of gravity states that a second body
of massm 2 at positionr 2 exerts an attractive force on the

first body given by

F 1 =−

Gm 1 m 2

r^312

r 12 =−

Gm 1 m 2

r^212

rˆ 12 , (5)

wherer 12 ≡r 1 – r 2 is the location of particle 1 with respect

to particle 2, ˆr 12 is the unit vector in the direction ofr 12 ,

andGis the gravitational constant. Newton’s third law states

that for every action there is an equal and opposite reaction;

thus, the force on each object of a pair is equal in magnitude

but opposite in direction. These facts are used to reduce

the two-body problem to an equivalent one-body case in

the next subsection.

2.2 Reduction to the One-body Case

From the foregoing discussion of Newton’s laws, and the

two-body problem the force exerted by body 1onbody 2 is

d(m 2 v 2 )

dt

=F 2 =−F 1 =

Gm 1 m 2

r^312

r 12 =

Gm 1 m 2

r^212

rˆ 12

(6)

Thus, from Eqs. (4) and (6)

d(m 1 v 1 +m 2 v 2 )

dt

=F 1 +F 2 = 0. (7)

This is of course a statement that the total linear momentum

of the system is conserved, which means that the center of

mass of the system moves with constant velocity.

Multiplying Eq. (6) bym 1 and Eq. (5) bym 2 and sub-

tracting, the equation for the relative motion of the bodies

can be cast in the form

μr

d^2 r 12

dt^2

=μr

d^2 (r 1 −r 2 )

dt^2

=−

GμrM

r^312

r 12 , (8)

whereμr≡m 1 m 2 /(ml+m 2 ) is called the reduced mass and

M≡ml+m 2 is the total mass. Thus, the relative motion is

completely equivalent to that of a particle of reduced mass

μrorbiting afixedcentral massM. For known masses, spec-

ifying the elements of the relative orbit and the positions

and velocities of the center of mass is completely equivalent

to specifying the positions and velocities of both bodies. A

detailed solution of the equation of motion (8) is discussed

in any elementary text on orbital mechanics and in most

general classical mechanics books. In the remainder of Sec-

tion II, a few key results are given.

2.3 Energy, Circular Velocity, and Escape Velocity

The centripetal force necessary to keep an object of massμr

in a circular orbit of radiusrwith speedvcisμrvc^2 /r. Equat-

ing this to the gravitational force exerted by the central body