IB Physics SLTopic 4 — Force FieldsPaper 1 & 2T² ∝ r³~8 min read
Kepler’s Laws
Long before anyone knew why, Johannes Kepler stared at years of planetary data and pulled out three astonishingly simple rules for how the planets move. Newton’s gravity later explained every one of them. Together they turn the messy night sky into three lines you can state — and one you can calculate with.
📘 What you need to know
Kepler’s three laws describe how planets and satellites orbit — spotted from observations first, then explained by Newton’s law of gravitation
First law: every orbit is an ellipse (a squashed circle) with the central mass at one of the two foci — not at the centre
Second law: the line from the Sun to the planet sweeps out equal areas in equal times — so a planet moves fastest when closest and slowest when farthest
Third law: for orbits around the same central mass, T² ∝ r³ — bigger orbits take disproportionately longer
The full form is T² = 4π²r³ ÷ GM, which drops straight out of setting gravity equal to the centripetal force
Only a graph of log T against log r is a straight line (gradient 3/2); plain T against r is a curve
For SL you must be able to state all three laws in words, and use the third in calculations and ratios
Three Rules for the Sky
Kepler’s laws work for anything orbiting a much larger central mass: planets round the Sun, moons round a planet, satellites round the Earth. He found the patterns; a generation later Newton showed they’re exactly what his single law of gravitation predicts. Let’s take them one at a time.
First Law — Orbits Are Ellipses
Kepler’s first law fixes the shape of an orbit:
The orbit of a planet is an ellipse, with the Sun at one of the two foci.
An ellipse is just a stretched circle. It has two special inside points called foci; the Sun sits at one of them, and the other is simply empty space. The key exam trap lives right here: the Sun is not at the centre — it’s off to one side, at a focus. Some orbits are very stretched (comets, Pluto); others, like Earth’s, are so close to circular you’d struggle to tell by eye.
An orbit is an ellipse with the Sun at one focus (the other focus is empty). What makes it an ellipse: the two focus-distances r1 + r2 add up to the same total everywhere on the curve.
Pin a loop of string around two tacks, pull it taut with a pencil, and trace — you’ve just drawn an ellipse. Each tack is a focus, and one of them is where the Sun would sit.
Second Law — Equal Areas in Equal Times
Kepler’s second law is about speed:
A line joining the Sun to a planet sweeps out equal areas in equal time intervals.
Picture an imaginary elastic band stretching from the Sun to the planet. As the planet moves, that band sweeps out area like a windscreen wiper. Kepler’s rule: in any fixed chunk of time — say one month — the band always paints the same amount of area, wherever the planet is in its orbit.
That has a beautiful consequence. Near the Sun the planet is close, so to sweep a fat wedge of area it has to race along a long arc — it moves fast. Far from the Sun the band is long, so even a short, slow shuffle sweeps the same area — it moves slow.
Equal times sweep equal areas. Close to the Sun the wedge is short and fat (planet racing along a long arc); far away it’s long and thin (planet barely creeping) — yet the two areas match. Fast when close, slow when far.
planet near the Sun
short “elastic band” → must cover a long arc for the same area
moves faster
Third Law — Period vs Orbital Radius
Kepler’s third law links an orbit’s size to how long it takes:
For bodies orbiting the same central mass, the square of the orbital period is proportional to the cube of the orbital radius.
Kepler’s third lawT² ∝ r³
So a planet twice as far out doesn’t just take twice as long — because of the cube-versus-square, it takes 23/2 ≈ 2.8 times as long. Push out to four times the radius and the year stretches by 43/2 = 8. Distant planets crawl through very long years.
Where the law comes from
For a (near-)circular orbit, gravity is the centripetal force that keeps the planet turning. Set the two equal, and bring in the orbital speed v = 2πr ÷ T (once round the circle in one period):
gravity = centripetal
GMm ÷ r² = mv² ÷ r
v² = GM ÷ r
v = 2πr ÷ T
substitute & rearrange
T² = 4π²r³ ÷ GM
Period from orbital radiusT² = 4π²r³ ÷ GM
The test mass m cancels again, so the orbit doesn’t care about the planet’s own mass — only about M, the thing being orbited. And since 4π² ÷ GM is just a constant for a given central body, this isT² ∝ r³.
The straight-line test
Plot T against r and you get a curve — not much use. But take logs of both sides of T² ∝ r³ and it becomes 2 log T = 3 log r + constant, so a graph of log T against log r is a straight line with gradient 3/2. That’s the classic data-question payoff.
Take logs and the third law becomes a straight line: log T against log r has gradient 3/2. A plain T-against-r plot would curve, so the log–log version is the one to reach for in data questions.
🧭 Tackling a Kepler question
Name the law — shape or foci → first; speeds or areas → second; period vs radius → third
Pick the third-law form — need an actual number or the central mass? use T² = 4π²r³ ÷ GM. Comparing two orbits? use T² ∝ r³
For a ratio, cancel the constants — (TA ÷ TB)² = (rA ÷ rB)³, then square-root
Use the right r — orbital radius from the centre, in metres; T in seconds
Shortcut the powers — scale r by k and T scales by k3/2 (scale T by k and r scales by k2/3)
Quick recap: First law — orbits are ellipses with the Sun at a focus. Second law — equal areas in equal times, so fast when close and slow when far. Third law — T² ∝ r³, or in full T² = 4π²r³ ÷ GM.
WE 1
A comet follows a very elongated orbit around the Sun. (a) State Kepler’s second law. (b) Use it to explain why the comet travels fastest as it passes closest to the Sun.
Part (a) — the law
The line from the Sun to the comet sweeps out equal areas in equal time intervalsPart (b) — the explanation
In a fixed time the swept area is always the same.
Close to the Sun the Sun–comet line is short, so to sweep that fixed area the comet must travel along a long arclong arc in the same time → high speedFar out the line is long, so the same area is covered by only a short, slow shuffle — hence slowest at its farthest point.
WE 2
A planet orbits a star of mass 2.5 × 10³⁰ kg with a mean orbital radius of 3.0 × 10¹¹ m. (a) Calculate its orbital period. (b) A second planet orbits the same star at four times this radius. State its period as a multiple of the first planet’s.
Part (a) — use T² = 4π²r³ ÷ GMT² = 4π² × (3.0 × 10¹¹)³ ÷ (6.67 × 10⁻¹¹ × 2.5 × 10³⁰)T² = (39.5 × 2.7 × 10³⁴) ÷ (1.67 × 10²⁰) = 6.4 × 10¹⁵T ≈ 8.0 × 10⁷ sThat’s about 2.5 Earth-years.Part (b) — use T² ∝ r³ (the star’s mass cancels)
Four times the radius → period × 43/243/2 = √(4³) = √64 = 88 × the first planet’s period
💡 Top tips
Marks are for the words: “ellipse, Sun at a focus” / “equal areas in equal times” / “T² proportional to r³”. Learn to state each law cleanly, not just the equation
Square and cube: it’s T² ∝ r³. A classic slip is T ∝ r or muddling the powers — cube the radius, square the period
Ratios rarely need G or M — they cancel. Only reach for T² = 4π²r³ ÷ GM when you need a real number or the central mass
Straight-line test: log T vs log r is straight (gradient 3/2); plain T vs r curves — know which is which for data questions
⚠ Common mistakes
Putting the Sun at the centre of the ellipse — it sits at a focus, off to one side, and the other focus is empty
Forgetting the speed changes around the orbit — fastest at the closest point, slowest at the farthest
Muddling the powers in T² ∝ r³ (writing T ∝ r³ or T² ∝ r²)
Using a diameter, or a height above the surface, instead of the orbital radius from the centre, for r
And that closes out Gravitational Fields — from Newton’s law of gravitation, through field strength and field maps, to Kepler’s three laws that tie the whole sky together. You can now describe an orbit, read a field, and calculate a period. Try the worked examples again from a blank page — that’s where these marks are won.
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