Kepler's Laws from Newtonian Gravity

Source

Learning outcome: L37 — Kepler's three laws follow from Newton's law of gravity.
Simulation: Single orbital core with three law-modes (K1, K2, K3).
Concept: K1 orbits are ellipses with the Sun at a focus; K2 equal areas in equal times; K3 T² ∝ a³.
Misconception: Planets move at constant speed and heavier planets orbit slower. In fact, speed varies (K2), and period depends only on the semi-major axis a, not on planet mass (K3).
Inquiry: Orient to the ellipse → predict speed → equal areas → predict period vs mass → build T² vs a³.

Pedagogy principles applied

ID	Principle	How it shows up here
GI	Guided inquiry	5 stepper cards: orient, predict speed, equal areas, predict period, build T² vs a³.
H1	Structural misconceptions	Wrong intuitions live inside predict choices — never as standing text.
A2	Numbers earn their place	Only the readouts each step reads: swept areas A₁, A₂, and T², a³, T²/a³.
E1	Animation honesty	Symplectic velocity-Verlet integrator; K2 areas accumulated by the moving planet; K3 periods measured by perihelion crossings.
A3	Formal tools off by default	Newton→Kepler derivations live behind ⚛︎ The Physics — a pop-up revision sheet.

Kepler's Laws

Teach me Hide Text

Speed

K1 · ellipse with Sun at focus

Guided Inquiry

1 · Orient to the ellipse

Let’s start with Kepler’s first law. K1 says a planet’s orbit is an ellipse with the Sun at one focus. The empty focus is marked with ◆, and the Sun sits at the other focus. Try varying the eccentricity and watch the ellipse deform.

2 · Where is the planet fastest?

Now watch the planet trace its ellipse for a little while. Where do you think it moves fastest?

3 · Equal areas in equal times?

Here’s where the second law comes in. As the planet moves, it sweeps out area continuously. Two windows add up the area actually swept over the same Δt: one across perihelion (orange), one across aphelion (cyan). What’s your prediction — do they enclose equal area?

4 · Does a heavier planet orbit slower?

Now for a question that trips a lot of people up. The canvas shows two planets on the same orbit — a light one (blue) and one 10× heavier (magenta). Watch closely: do they stay together, or does one lag behind?

5 · Build T² ∝ a³

One last law to build. Add at least two planets from the chip row. Each one is integrated by Newton’s law, and its period is measured from perihelion crossings and plotted. Watch what shape the points make on the log–log plot.

Now make a prediction. Jupiter orbits at a = 5.2 AU — roughly what period do you expect it to have?

Listen

Law mode

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K1 — ellipse with the Sun at one focus.

Orbit

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Eccentricity e0.60

0 (circle)0.85

Swept areas

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A₁ (perihelion window)—

A₂ (aphelion window)—

A₁ / A₂—

Each window accumulates the area the planet actually sweeps over the same ΔT_win = 0.10·T_orbit. K2 predicts A₁ = A₂.

Add planets

▾

Click a chip to add. Each planet's period is measured from perihelion crossings and dropped on the T²–a³ plot.

added / measured0 / 0

T² / a³ yr² / AU³

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Unit system: AU–yr–M☉

From Newton to Kepler

The Physics

Kepler's laws · what this sim shows, the physics behind it, and every control

At a glance

Kepler found three rules in Tycho Brahe's careful records of planet positions: the orbit's shape, how the speed changes along it, and how the period grows with orbit size. This sim moves its planets by Newton's inverse-square pull alone, so you can watch all three rules come out of that one force law.

What you'll learn

What an ellipse with the Sun at a focus looks like, and how the eccentricity e shapes it.
How the planet's speed and its swept area depend on where it is on the orbit.
Whether a planet's own mass plays any part in its motion.
How the period T depends on the orbit size a, across the Solar System.

The Sun sits at one focus of the ellipse, not at its centre. a is the semi-major axis.

What's on screen

Sun (amber) at one focus; ◆ empty focus (grey) at the other.
Teal ellipse: the exact Kepler orbit for your e, drawn from its formula.
Blue planet and trail: moved step by step by Newton's law, with a faint line to the Sun.
K2 wedges: orange near perihelion, cyan near aphelion, with their areas A₁, A₂.
K3 view: planets round the Sun (left) and a log–log plot of T² against a³ (right).

See itRun the orbit at the opening setting, e = 0.60. Watch for: the Sun sitting well to the right of the ellipse's centre, and the planet lapping it once every 2 seconds.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Law mode	Which law the stage shows: K1 the orbit, K2 the swept areas, K3 the whole system.	K1 / K2 / K3 opens at K1
Eccentricity e (K1, K2)	The orbit's shape, from a circle to a long ellipse. Each change restarts the planet at perihelion.	0–0.85 opens at 0.60
Swept areas (K2)	A₁ and A₂: the areas swept in two windows of equal duration, 0.10 T each, and their ratio.	readout
Planet chips (K3)	Adds a real planet, Mercury to Saturn. Its period is measured, then plotted.	a = 0.39–9.54 AU
T² / a³ (K3)	The average of T²/a³ over the measured planets, in yr²/AU³, with the count added / measured.	readout
▶ Play / ⏸ Pause	Runs the planets, or freezes them where they are.	top bar
Speed	Plays the motion faster or slower. At 1×, one year takes 2 s. The physics stays the same.	0.25×–4× opens at 1×
↻ Reset	Puts e back to 0.60, clears added planets, and restarts the mode you are in.	top bar

Fixed: in K1 and K2, a = 1 AU, so T = 1 yr. The Sun is held still. Units are AU, years and solar masses, in which GM_☉ = 4π² AU³/yr².

One force law, one ellipse

The planet is pulled straight toward the Sun, with a strength that falls off as 1/r². That is all the sim uses to move it:

Newton's law of gravity, with the Sun at r = 0.

Every bound orbit under this force is an ellipse with the Sun at one focus. That is Kepler's first law (K1). In polar form, with r measured from the Sun:

The orbit, with θ measured from perihelion. The sim's slider covers 0 ≤ e ≤ 0.85.

a sets the size, and e sets the shape. The Sun is a distance ae from the centre.
Closest approach is r = a(1 − e) (perihelion); farthest is r = a(1 + e) (aphelion). At e = 0.60 these are 0.40 AU and 1.60 AU.
The other focus is empty. Nothing sits there; it is pure geometry.

See itStretch the orbit to e = 0.85. Watch for: the Sun almost touching the orbit at perihelion, and the ◆ far out on the other side.

See itSet e = 0. Watch for: the ◆ disappearing into the Sun as the two foci merge and the orbit becomes a circle.

Equal areas in equal times

The pull points straight at the Sun, so it exerts no torque about the Sun, and the planet's angular momentum L stays constant. The rate at which the Sun–planet line sweeps out area is L/2m, so it is constant too:

This is Kepler's second law (K2): equal areas in equal times.
Near the Sun the wedge is short, so it must be wide: the planet covers more of its orbit. Far out, the wedge is long and thin.
K2 needs only a central force. The 1/r² form is what makes the orbit a closed ellipse.

Computed at e = 0.60, the opening setting.

See itSwitch to K2 at e = 0.60 and let both windows close. Watch for: a short, fat orange wedge and a long, thin cyan one, with A₁ and A₂ both reading about 0.251 AU².

Fastest near the Sun

At perihelion and aphelion the velocity is at right angles to the Sun–planet line, so constant L gives r_pv_p = r_av_a. The speed ratio then depends on e alone:

Left: the speed the sim gives the planet at perihelion. Right: the ratio from K2.

This slider moves the sim too.

At e = 0.60 the planet is 4 times faster at perihelion (59.6 km/s) than at aphelion. At e = 0 the speed is constant, 2π AU/yr ≈ 29.8 km/s, which is close to Earth's.

See itColour the trail by speed at e = 0.60. Watch for: red (fast) on the perihelion side, blue (slow) on the aphelion side.

The planet's mass drops out

The m that gravity pulls on is the same m that resists acceleration.

Gravity pulls a heavier planet harder, in exact proportion to its mass. Its inertia grows in the same proportion, so its acceleration is unchanged.
At 1 AU, every planet accelerates at GM_☉/r² = 4π² ≈ 39.5 AU/yr², whatever its mass.
The same start then gives the same orbit and the same period. No planet mass appears in K1, K2 or K3.
One idealisation: the sim holds the Sun still. Strictly, T depends on M_☉ + m; for Jupiter, the heaviest planet, that shifts T² by about 0.1%.

See itLaunch a planet 10× heavier alongside the blue one, from the same point with the same velocity. Watch for: the violet ring staying wrapped around the blue planet, lap after lap.

T² = a³ across the Solar System

Newton's law gives Kepler's third law (K3): T² = 4π²a³/GM_☉. In AU, years and solar masses, GM_☉ = 4π², so T² = a³.
T grows as a^3/2, faster than a. Jupiter, at a = 5.20 AU, takes √(5.20³) ≈ 11.9 yr.
In K3 mode each planet is moved by Newton's law, and its period is timed between perihelion passages. A hollow point is still being timed; a solid point is measured.
On the log–log plot, T² = a³ is a straight line of slope 1. Around a heavier star the line shifts down, with the same slope.

See itAdd Mercury, Earth, Mars and Jupiter in K3. Watch for: the points turning solid on the slope-1 line, and T²/a³ reading close to 1.000.

Where it matters

Weighing the Sun. Rearranged, K3 gives the central mass: M = 4π²a³/GT². Earth's orbit (1 AU, 1 yr) gives M_☉ ≈ 2.0 × 10³⁰ kg.
The black hole at the centre of our galaxy. The star S2 orbits it every 16 years at about 1000 AU. K3 then gives about 4 million solar masses.
Geostationary satellites. The same law around Earth: an orbit that takes one sidereal day must have a radius of about 42 000 km from Earth's centre, which is why they all sit at the same height.

Go deeper: why only 1/r² gives a closed ellipse

Any central force sweeps equal areas, but most bound orbits never close: each lap the perihelion shifts, and the path traces a rosette.
Bertrand's theorem says only two force laws close every bound orbit: the inverse square, and the spring force F ∝ r. Of these, only the inverse square has the Sun at a focus.
Mercury's perihelion does shift slowly, partly because of the other planets and partly because of general relativity, which adds a small correction to 1/r².

Summary

In one line. Kepler's three laws all follow from one central, inverse-square pull: equal areas because the pull is central, and the closed ellipse and T² ∝ a³ because it falls off as 1/r².

Key points

K1: the orbit is an ellipse with the Sun at one focus; e sets its shape.
K2: no torque about the Sun keeps L constant, so equal areas take equal times and v_p/v_a = (1+e)/(1−e).
The planet's mass cancels, so it plays no part in the orbit.
K3: T² = a³ in AU and years, for every planet of the Sun.

Common mix-ups

"The Sun is at the centre of the ellipse." It is at a focus, a distance ae from the centre: 0.60 AU at the opening setting.
"Planets move at a steady speed." Only on a circle. On an ellipse they are fastest at perihelion.
"Equal areas only works for 1/r² gravity." Any central force gives equal areas. The 1/r² form is what gives the closed ellipse and T² = a³.

Things to try on your own

In K2, take e from 0 to 0.85 and compare the shapes of the two wedges with the A₁/A₂ readout.
In K3, add all seven planets and check that T²/a³ stays close to 1.000 as Saturn joins.
Use T = a^3/2 to predict Saturn's period from a = 9.54 AU, then compare with its point on the plot.