Kepler's second law — equal areas in equal times, from angular momentum conservation: Foundations

Before you can feel why "equal areas in equal times" is true, you need to own every symbol the parent note throws at you. This page builds each one from nothing — plain words, a picture, and the reason the topic needs it. Read top to bottom; each block uses only what came before.

1 — A vector, and what an arrow means

The little hat of an arrow on top, $r$ , is our shorthand for "this is an arrow, not just a number." A plain letter like $r$ (no arrow) means only the length of that arrow — a single positive number, the distance from Sun to planet.

Figure — Kepler's second law — equal areas in equal times, from angular momentum conservation

Why the topic needs it: Kepler's law is about the line "joining planet to Sun." That line is $r$ . Everything — area, angle, speed — is measured off this one arrow.

2 — Velocity $v$ : the arrow of "where it's heading"

Picture the planet a tiny instant later: it has shifted by a small arrow we call $d r$ ("a tiny piece of $r$ 's change"). The little $d$ in front of anything means "a tiny sliver of." Velocity is exactly that tiny shift divided by the tiny time it took:

$v = \frac{d r}{d t}$

Why the fraction? $d r$ is how far it moved; $d t$ is how long that took; distance over time is speed — and keeping them as arrows keeps the direction too.

3 — The angle $ϕ$ between two arrows, and $sin ϕ$

Two arrows starting from the same corner open up an angle between them. We name the angle between $r$ and $v$ with the Greek letter $ϕ$ ("phi").

Why the topic needs it: Only the sideways part of the velocity actually sweeps out new area — the part running straight along $r$ just lengthens or shortens the arrow without painting a slice. So the "how much area per second" always carries a hidden $sin ϕ$ .

4 — The cross product $a \times b$ : an area machine

Here is the one genuinely new tool. Suppose two arrows $a$ and $b$ start from the same point and open a parallelogram (a slanted rectangle). The cross product measures that parallelogram's area.

Read it as a recipe: base length $\times$ height. One arrow is the base; $∣ b ∣ sin θ$ is how tall the parallelogram stands (the sideways part again!). Cut the parallelogram in half along a diagonal and you get a triangle of area $\frac{1}{2} ∣ a \times b ∣$ .

Why the topic needs this exact tool and not multiplication: We want the area swept by two arrows, and area depends on how slanted they are — two parallel arrows sweep zero area, two perpendicular arrows sweep the most. Ordinary multiplication ( $∣ a ∣∣ b ∣$ ) can't see the slant; the cross product's built-in $sin θ$ can. That is precisely why the parent's very first step writes $d A = \frac{1}{2} ∣ r \times d r ∣$ . See Cross Product and Area.

5 — Area $A$ and areal velocity $d A / d t$

Kepler's Second Law, in these symbols, is simply: $d A / d t$ never changes. Fat short slices near the Sun, long thin slices far away — same area per second.

Why the topic needs it: this single quantity is the law. Everything else on the parent page exists to prove this one number is constant.

6 — Angle $θ$ , and its rate $\dot{θ}$

We also track the planet by the angle $θ$ ("theta") that $r$ has swung through, measured from some starting line. As the planet orbits, $θ$ grows.

In polar (angle) form the tiny slice is a thin wedge of a circle: $d A = \frac{1}{2} r^{2} d θ$ . Divide by $d t$ :

$\frac{d A}{d t} = \frac{1}{2} r^{2} \dot{θ}$

Why a wedge of $r^{2}$ ? A pie slice of radius $r$ and tiny angle $d θ$ is almost a triangle with base $r d θ$ (arc length) and height $r$ , so area $\approx \frac{1}{2} (r d θ) (r) = \frac{1}{2} r^{2} d θ$ .

7 — Angular momentum $L$ and mass $m$

Its length works out to $L = m r^{2} \dot{θ} = m r v sin ϕ$ . Comparing with step 6:

$\frac{d A}{d t} = \frac{1}{2} r^{2} \dot{θ} = \frac{L}{2 m}$

So the geometric area-rate is the physical angular momentum divided by $2 m$ . Since Angular Momentum Conservation keeps $L$ fixed for a central force, $d A / d t$ is fixed. See Central Forces and Torque for why a radial force keeps $L$ constant (zero torque).

Why the topic needs it: $L$ is the bridge. It turns a picture (area) into a conserved physics quantity, and conserved quantities are the ones that let you predict without solving the whole orbit.

How it all feeds the law

Equipment checklist

Recall Can you answer each before moving on?

What does the arrow $r$ represent, and what does plain $r$ mean? ::: $r$ is the arrow from Sun to planet (length + direction); $r$ is only its length (the distance). What is $v$ , and how does it relate to $d r$ and $d t$ ? ::: Velocity, the arrow of motion; $v = d r / d t$ , the tiny displacement over the tiny time. What does $sin ϕ$ measure between $r$ and $v$ ? ::: The fraction of the velocity that points sideways to the radius — the part that actually sweeps area. What does the cross product $∣ a \times b ∣$ compute, and what's its formula? ::: The area of the parallelogram the two arrows span; $∣ a ∣∣ b ∣ sin θ$ . Why is the cross product the right tool for swept area? ::: Because area depends on the slant between the arrows, and the built-in $sin θ$ captures that (parallel = zero, perpendicular = max). What does a dot over a symbol mean, e.g. $\dot{θ}$ ? ::: The rate of change per second; $\dot{θ}$ is how fast the radius sweeps angle. What is areal velocity $d A / d t$ in words? ::: How much area the radius vector paints each second. Write $L$ two equivalent ways and connect it to $d A / d t$ . ::: $L = m r^{2} \dot{θ} = m r v sin ϕ$ ; and $d A / d t = L / (2 m)$ . When is $L = m v r$ valid (no $sin ϕ$ )? ::: Only when $v ⊥ r$ — at perihelion, aphelion, or on a circle.

Connections

Parent: the full topic note
Cross Product and Area — the area machine defined here
Angular Momentum Conservation — why $L$ (and so $d A / d t$ ) is fixed
Central Forces and Torque — why gravity gives zero torque
Kepler's First Law (Ellipse) · Kepler's Third Law (T² ∝ a³) · Vis-viva Equation