Begin2.DVI

where Ais the area of the ellipse and τ is the period of one orbit. The area of an

ellipse is given by the formula A=πab, where ais the semi-major axis and b=a

√ 1 − 2

is the semi-minor axis. Equation (9.118) can therefore be expressed in the form

A=πa^2

√ 1 − 2 =

h 2

τ

from which the period of the orbit is

τ=^2 πa

2 h

√ 1 − 2.

With the substitutions

1 − 2 =

p

a and p=

h^2 GM ,

the period of the orbit can be expressed

τ=^2 πa

3 / 2 √ GM

or τ^2 =^4 π

(^2) a 3
GM

. (9 .119)

This result is known as Kepler’s third law and depicts the fact that the square of

the period of one revolution is proportional to the cube of the semi-major axis of

the elliptical orbit.

Planets, comets, and asteroids have either elliptic, parabolic or hyperbolic orbits

about the sun.

Vector Differential Equations

A homogeneous vector differential equation, such as

d^2 y dt^2 +α

dy dt +βy=

0 (9 .135)

where αand βare scalar constants is solved by first solving the homogeneous scalar

differential equation

d^2 y dt^2

+αdy dt

+βy = 0 (9 .137)

The solution of the homogeneous differential equation is called the complementary

solution and is expressed using the notation yc. By assuming an exponential so-

lution y =eλt and substituting it into the homogeneous equation one obtains the

characteristic equation

λ^2 +αλ +β= 0 (9 .136)