(4.3) |

where the potential energy, , of our planet in the Sun's gravitational field takes the form

(See Section 3.5.) It follows that the total energy of our planet is a conserved quantity. (See Section 2.4.) In other words,

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,

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.