Animated Solution for Physics - Gravitation: The angular momentum of a planet of mass $M$ moving around the Sun in an elliptical orbit is $L$. The magnitude of the areal velocity of the planet is

Imagine standing far above the solar system, watching a planet glide silently along its elliptical path around the Sun. As it moves, the imaginary line connecting the Sun to the planet sweeps across the vastness of space.

If we look at a tiny snapshot of time, $dt$, the planet moves from a point $P$ to a new point $P^{\prime }$. The radius vector $r$ sweeps out a small, almost triangular sliver of area, which we call $dA$. From the geometry of polar coordinates, the area of this infinitesimally small sector is given by:

To find how fast this area is being swept—the areal velocity—we simply divide by our tiny time interval $dt$:

Since the rate of change of the angle $\frac{d\theta }{dt}$ is just the angular velocity $\omega$, our areal velocity equation beautifully simplifies to:

The Rotational Drive

Now, let's shift our perspective from pure geometry to the physical forces driving this motion. The planet has a mass $M$ and is moving with some velocity $\vec{v}$. Because it's orbiting a central body (the Sun), it possesses angular momentum, denoted by $L$.

The angular momentum of a particle about a central point is the product of its mass, its distance from the center, and its perpendicular velocity:

In circular or instantaneous rotational motion, the perpendicular velocity is intimately tied to the angular velocity by the relation $v_{\perp }=r\omega$. Substituting this into our angular momentum equation gives:

We now hold two powerful equations in our hands. One describes the geometric sweeping of space, and the other describes the physical conservation of rotational motion. Let's bring them together.

From our angular momentum equation, we can isolate the angular velocity $\omega$:

Now, we take this expression for $\omega$ and substitute it back into our areal velocity equation:

Look at what happens next. The $r^2$ in the numerator and the $r^2$ in the denominator perfectly cancel each other out. The varying distance of the planet from the Sun completely vanishes from the equation! We are left with a stunningly simple and profound result:

Because the angular momentum $L$ is conserved (due to the absence of external torques) and the mass $M$ is constant, the areal velocity $\frac{L}{2M}$ must also be a constant. This elegant mathematical dance is the exact proof of Kepler's Second Law of Planetary Motion.