Kepler's first law — orbits are conic sections: Visual walkthrough

Step 0 — The cast of characters (draw the scene first)

Before any formula, let us name what is on the stage. Everything lives in a flat plane (the orbit is flat — gravity never pushes sideways out of the plane, so the body can never leave it).

Step 1 — The one arrow: Newton's gravity

WHAT. Write down the acceleration of $m$ . Acceleration means "how the velocity arrow is changing" — we write it $\ddot{r}$ , where each dot means "rate of change in time." Two dots = rate of the rate = acceleration.

WHY. Every trajectory in all of physics is dictated by one rule: force sets acceleration ( $F = m a$ ). If we know the acceleration at every point, the path is fixed. Gravity is the only force here, so this single arrow is the whole physics.

PICTURE. The arrow always points from the planet back toward the Sun (attraction), and it grows longer as the planet swings in close.

Step 2 — The sweep rate never changes: angular momentum

WHAT. Introduce $h = r^{2} \dot{θ}$ and claim it is a fixed number for the whole orbit. Here $\dot{θ}$ is "how fast the angle turns" (the spin rate of the direction line).

WHY. We want the shape $r (θ)$ — distance as a function of direction — not a minute-by-minute timetable. To get shape we must erase time $t$ . The quantity $h$ is the bridge: it ties $t$ and $θ$ together permanently, so we can trade one for the other. It is conserved because gravity points straight along $\overset{r}{^}$ and therefore can never twist the body about the Sun (zero torque). Full argument: Conservation of angular momentum in central forces.

PICTURE. In equal time slices the planet sweeps equal areas — a fat short wedge near the Sun, a thin long wedge far away. That equal-area fact is $h$ = constant, and it is exactly Kepler's second law — equal areas in equal times.

Step 3 — The magic swap $u = 1/ r$

WHAT. Define a brand-new variable $u = 1/ r$ ("inverse distance"). Big $r$ (far away) ↔ small $u$ ; small $r$ (close in) ↔ big $u$ .

WHY. The equation of motion has an ugly $1/ r^{2}$ in it — non-linear, unsolvable by hand as written. Flipping to $u$ is a change of ruler: it straightens the ugly curve into a clean one. This is the single trick that makes the whole derivation doable with paper.

PICTURE. Picture the same orbit measured on an "inverse ruler." The bunched-up far region stretches out; the equation smooths into something we already know.

Using the chain rule and $h = r^{2} \dot{θ}$ (so $\dot{θ} = h u^{2}$ ), the two time-derivatives of $r$ transform into clean $θ$ -derivatives of $u$ :

Step 4 — The disguise drops: it's a spring

WHAT. Plug the swapped pieces into the radial law $\overset{r}{¨} - r \dot{θ}^{2} = - μ / r^{2}$ and simplify.

WHY. The left side, $\overset{r}{¨} - r \dot{θ}^{2}$ , is the true inward acceleration in polar coordinates: the first term is speeding-up/slowing-down along the radius, the second $- r \dot{θ}^{2}$ is the centrifugal-looking term from going in a circle. Setting it equal to gravity closes the physics.

PICTURE. After the dust settles, the equation is the world's most famous oscillator — a mass on a spring, nudged off-center by a constant. That constant nudge is what pins the orbit to one focus.

Substituting and dividing through by $- h^{2} u^{2}$ collapses everything to:

Step 5 — Solve it (a shifted cosine)

WHAT. Write the solution: a constant plus a cosine.

WHY. A spring pushed by a steady force settles at a new center and wobbles around it. The steady center is $μ / h^{2}$ ; the wobble is a cosine with some amplitude $A$ and starting phase. We slide our angle-zero to sit at the wobble's peak, killing the phase.

PICTURE. The graph of $u$ against $θ$ is a cosine floating above zero. Its peak is where $u$ is largest → where $r$ is smallest → perihelion, the closest approach.

Flip back with $r = 1/ u$ :

$r (θ) = \frac{1}{\frac{μ}{h ^{2}} + A c o s θ} = \frac{h ^{2} / μ}{1 + \frac{A h ^{2}}{μ} c o s θ}$

Step 6 — Read off the conic

WHAT. Compare our result with the standard polar conic $r = \frac{p}{1 + e cos θ}$ .

WHY. If two formulas have the same form, their pieces must match. Matching term-for-term names our constants and proves the shape is a conic — which is Kepler's first law.

PICTURE. Sun at the origin (a focus), the closest point on the right, the whole curve leaning toward that side. That lean is the eccentricity.

The geometry of these shapes lives in Conic sections — geometry of ellipse, parabola, hyperbola.

Step 7 — Which conic? Every case, by energy

WHAT. The amplitude $A$ (hence $e$ ) is set by the launch conditions, and it packages into one clean energy formula. Let $E$ = total energy per unit mass = $\frac{1}{2} v^{2} - μ / r$ (kinetic minus depth in the well).

WHY. We must cover all outcomes — not just the tame planet ellipse. Energy is the switch: negative = trapped, zero = barely free, positive = free with speed left over. This connects to Escape velocity and orbital energy and the Vis-viva equation.

PICTURE. One focus, four launches, four fates — circle, ellipse, parabola, hyperbola — nested outward as energy climbs from most-negative to positive.

Energy $E$	$e$	Shape	Fate
most negative ( $E = - μ^{2} /2 h^{2}$ )	$0$	circle	steady loop
$E < 0$	$0 < e < 1$	ellipse	bound, returns
$E = 0$	$e = 1$	parabola	just escapes, arrives at $\infty$ with zero speed
$E > 0$	$e > 1$	hyperbola	escapes with leftover speed (flyby)

The one-picture summary

One breath: an inward $1/ r^{2}$ arrow → constant sweep $h$ lets us swap time for angle → $u = 1/ r$ turns the mess into a spring $u^{''} + u = μ / h^{2}$ → its shifted cosine flips back to $r = p / (1 + e cos θ)$ → a conic with the Sun at a focus, and energy picks which conic.

Recall Feynman retelling — say it to a friend with no math

Picture the Sun holding an invisible rubber band on a planet, tugging harder the closer the planet swings in. That tug never sideways-twists the planet, so the planet always sweeps the same area per second — fast and fat near the Sun, slow and thin far out. That "same area" rule is our secret handshake: it lets us forget about time and just ask, "how far is the planet in each direction?" When we measure distance in an upside-down way (call it $u = 1/ r$ ), the tangled tug straightens out into the exact math of a mass bouncing on a spring — the most familiar wiggle in physics. A spring's answer is a cosine wave, a smooth up-and-down. Flip that cosine back to real distance and — surprise — it is precisely the equation of a squashed circle with the Sun sitting off to one side, at a focus. How squashed depends on how much energy the planet was born with: a little energy → a gentle ellipse that comes home; exactly enough → a one-time parabolic goodbye; more than enough → a hyperbolic slingshot that leaves forever. That whole chain is Kepler's first law, and we drew every link.

Recall Rebuild the chain from memory

Start-of-story arrow ::: $\ddot{r} = - (μ / r^{2}) \overset{r}{^}$ — inward $1/ r^{2}$ gravity. What lets us drop time $t$ ? ::: Conserved $h = r^{2} \dot{θ}$ (central force, zero torque). The magic substitution ::: $u = 1/ r$ . What the ODE becomes ::: $u^{''} + u = μ / h^{2}$ — driven simple harmonic motion. Solution shape ::: $u = μ / h^{2} + A cos θ$ , a shifted cosine. Final orbit ::: $r (θ) = \frac{h ^{2} / μ}{1 + e cos θ}$ , with $p = h^{2} / μ$ , $e = A h^{2} / μ$ . Which conic? ::: Set by energy: $e = 1 + 2 E h^{2} / μ^{2}$ .