As we have already seen, gravity is a conservative force. Hence, the gravitational force in Equation (4.1) can be written (see Section 2.4)
 |
(4.3) |
where the potential energy,
, of our planet in the Sun's gravitational field takes the form
 |
(4.4) |
(See Section 3.5.)
It follows that the total energy of our planet is a conserved quantity. (See Section 2.4.) In other words,
 |
(4.5) |
is constant in time. Here,
is actually the planet's total energy per unit
mass, and
.
Gravity is also a central force. Hence, the angular momentum
of our planet is a conserved quantity. (See Section 2.5.) In other
words,
 |
(4.6) |
which is actually the planet's angular momentum per unit mass, is constant
in time. Assuming that
, and taking the scalar product of the preceding equation with
, we
obtain
 |
(4.7) |
This is the equation of a plane that passes through the origin, and
whose normal is parallel to
. Because
is a constant vector,
it always points in the same direction. We, therefore, conclude that
the orbit of our planet is two-dimensional; that is, it is confined to some fixed plane that passes through the origin. Without loss of generality, we can let this plane coincide with the
-
plane.