Planetary Conservation Laws

As we have seen, gravity is a conservative force. Hence, the gravitational force (1.275) can be written

$${\bf f} = -\nabla U,$$ (1.278)

where the potential energy, $U({\bf r})$, of our planet in the Sun's gravitational field takes the form

$$U({\bf r})= - \frac{G\,M\,m}{r}.$$ (1.279)

[See Equation (1.264).] It follows that the total energy of our planet is a conserved quantity. (See Section 1.3.5.) In other words,

$${\cal E} = \frac{v^{2}}{2} - \frac{G\,M}{r}$$ (1.280)

is constant in time. Here, ${\cal E}$ is actually the planet's total energy per unit mass, and ${\bf v}=d{\bf r}/dt$.

Gravity is also a central force. This means that the gravitational force exerted on our planet is always directed toward the origin of our coordinate system (i.e., the Sun), which implies that the force exerts zero torque about the origin. Hence, the angular momentum of our planet (about the origin) is a conserved quantity. (See Section 1.4.5.) In other words,

$${\bf h} = {\bf r}\times {\bf v},$$ (1.281)

which is actually the planet's angular momentum per unit mass, is constant in time. Taking the scalar product of the previous equation with ${\bf r}$, we obtain

$${\bf h}\cdot{\bf r} = 0.$$ (1.282)

This is the equation of a plane that passes through the origin, and whose normal is parallel to ${\bf h}$. Because ${\bf h}$ is a constant vector, it always points in the same direction. We, therefore, conclude that the motion of our planet is two-dimensional in nature; 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 $x$-$y$ plane.

Figure 1.12: Plane polar coordinates.