A Classical Approach of Newtonian Mechanics

12 ORBITAL MOTION 12.6 Planetary orbits

This equation possesses the fairly obvious general solution

u = A cos(θ − θ 0

where A and θ 0 are arbitrary constants.

) +

G MⓈ

, (12.40)

h^2

The above formula can be inverted to give the following simple orbit equation

for our planet:
1
r
A cos(θ − θ 0 ) + G MⓈ/h^2

. (12.41)

The constant θ 0 merely determines the orientation of the orbit. Since we are only
interested in the orbit’s shape, we can set this quantity to zero without loss of
generality. Hence, our orbit equation reduces to

1 + e

where

r = r 0

1 + e cos θ

A h^2

, (12.42)

and

e =

r 0 =

G MⓈ

h^2

, (12.43)

. (12.44)
G MⓈ (1 + e)

Formula (12.42) is the standard equation of an ellipse (assuming e < 1 ), with

the origin at a focus. Hence, we have now proved Kepler’s first law of planetary

motion. It is clear that r 0 is the radial distance at θ = 0. The radial distance at

θ = π is written

r 1 = r 0

1 + e

1 − e.^ (12.45)^

Here, r 0 is termed the perihelion distance (i.e., the closest distance to the Sun)

and r 1 is termed the aphelion distance (i.e., the furthest distance from the Sun).
The quantity

e =

r 1 − r 0

r 1 + r 0

(12.46)

iscircularity. termed the Thus, eccentricity e = 0 correspondsof the orbit, toand a purelyis a measure circular of orbit, its departure whereas frome
→

=