A Classical Approach of Newtonian Mechanics

12 ORBITAL MOTION 12.6 Planetary orbits

e

Planet

r



e

r

Sun

Figure 105: A planetary orbit.

These expressions are more complicated that the corresponding cartesian expres-

sions because the unit vectors er and eθ change direction as the planet changes

position.

Now, the planet is subject to a single force: i.e., the force of gravitational

attraction exerted by the Sun. In polar coordinates, this force takes a particularly

simple form (which is why we are using polar coordinates):

f = −

G MⓈ m

e. (12.25)

r^2 r^

The minus sign indicates that the force is directed towards, rather than away

from, the Sun.

According to Newton’s second law, the planet’s equation of motion is written

m a = f. (12.26)

The above four equations yield

̈r − r θ ̇^2 = −

G MⓈ

, (12.27)

r^2

r θ ̈^ + 2 ̇r θ ̇ = 0. (12.28)