How does Kepler's Second Law show that a planet further from the sun will move slower?

Question

This is probably a very stupid question. We are told that due to Kepler's Second Law, which according to this very straightforward explanation:

"Kepler's second law of planetary motion describes the speed of a planet traveling in an elliptical orbit around the Sun. It states that a line between the Sun and the planet sweeps equal areas in equal times. Thus, the speed of the planet increases as it nears the Sun and decreases as it recedes from the Sun."

But it seems to me that a line between the Sun and the planet would always sweep out equal areas, regardless of its distance from the sun or how fast the planet is moving. What am I missing?

Answer 1

Kepler's Second Law is a precursor to the conservation of angular momentum. In a time $\delta t$, the area swept out by the orbit of a planet is (to first order in $\delta t$) $\frac{1}{2}\left|\mathbf{r}\times\mathbf{v}\delta t\right|=\frac{1}{2}rv\sin(\theta)\delta t$, where $\mathbf{r}$ is the vector pointing from the Sun to the planet, $\mathbf{v}$ is the velocity of the planet, and $\sin(\theta)$ is the angle between $\mathbf{r}$ and $\mathbf{v}$.

So, we see that, when $r$ or $v_\perp$ are increased, the area swept out in a certain time $\delta t$ increases as well. This allows us to conclude that, if the area swept out per unit time is constant, then as $r$ increases, $v_\perp$ must decrease.

Answer 2

The heading that you wrote How does Kepler's Second Law show that the planets further from the sun move slower? is a question about the whole assembly of planet$\color{red}s$ [plural] orbiting the Sun, whereas Kepler's second law is related to an individual planet.

From the text that you quoted, Thus, the speed of the planet [singular] increases as it nears the Sun and decreases as it recedes from the Sun.

Here is a pictorial representation of the law.

Distance 2 > Distance 1 hence Speed 2 > speed 1.