Kepler's Laws of Planetary Motion

Eight minutes of arc#

When Tycho Brahe died in 1601, he left behind the most precise naked-eye observations ever made — decades of planetary positions accurate to about two arcminutes, a fifteenth of the Moon's width. Johannes Kepler inherited them, along with the specific assignment of explaining the orbit of Mars.

He expected it to take eight days. It took nearly six years.

Kepler tried circle after circle. He tried circles offset from the Sun, circles with the planet sweeping uniformly about some other point, nested combinations of circles — the full inherited toolkit of two thousand years of astronomy, in which heavenly motion had to be built from perfect circles at constant speed. He eventually found a circular model that fit Tycho's Mars data to within eight minutes of arc.

By the standards of every astronomer before him, that was a triumph. Eight arcminutes is a quarter of the Moon's diameter; Copernicus would have been delighted. But Kepler knew Tycho's instruments were better than that. Those eight minutes could not be observational error, and he refused to hide them.

"Because these eight minutes could not be ignored, they alone have led the way toward a complete reformation of astronomy."

He abandoned the circle. What replaced it was an ellipse, and with it came three laws that turned astronomy from geometry into physics.

The first law: orbits are ellipses#

Kepler's first law: every planet moves on an ellipse with the Sun at one focus.

An ellipse is the set of points whose distances to two fixed foci sum to a constant. Its shape is fixed by two numbers: the semi-major axis $a$ (half the long diameter) and the eccentricity $e$, a dimensionless number between 0 and 1 measuring how squashed it is. The semi-minor axis follows:

$$b = a\sqrt{1 - e^{2}}$$

and each focus sits a distance $c = ae$ from the centre. Setting $e = 0$ puts both foci on top of each other and recovers a circle — so a circle is not the wrong answer, just the special case where the eccentricity happens to vanish.

The orbit is easiest to write in polar coordinates centred on the Sun, with $\theta$ measured from perihelion:

$$r(\theta) = \frac{a\left(1 - e^{2}\right)}{1 + e\cos\theta}$$

At $\theta = 0$ this gives the closest approach, perihelion, at $r = a(1-e)$; at $\theta = 180°$ it gives the farthest point, aphelion, at $r = a(1+e)$.

There is a trap hiding here, and almost every textbook diagram falls into it. Real planetary eccentricities are tiny. Earth's is $e \approx 0.0167$, which makes $b/a = 0.99986$ — draw Earth's orbit accurately at page size and you cannot distinguish it from a circle with a ruler. Even Mars, the orbit that broke the circle, has only $e \approx 0.0934$, giving $b/a \approx 0.9956$. What Kepler detected was not a visibly oval path. It was the fact that the Sun sits noticeably off-centre: the offset is $c = ae$, which for Mars is about 4.4% of the orbit's size, twenty times more visible than the flattening itself.

The second law: equal areas in equal times#

Kepler's second law: the line joining a planet to the Sun sweeps out equal areas in equal intervals of time.

This is the law that killed uniform circular motion outright. A planet does not travel at constant speed. Near perihelion the radius vector is short, so the planet must cover a long arc to sweep a given area; near aphelion the radius vector is long, and a stubby little arc suffices. The planet races when it is close and crawls when it is far.

Formally, the areal velocity is constant:

$$\frac{dA}{dt} = \frac{1}{2}r^{2}\dot{\theta} = \text{constant}$$

Drag the eccentricity slider and watch the shaded wedges. Each wedge is swept in exactly the same amount of time. At $e = 0.02$ — roughly Earth — the wedges look almost identical and the planet's speed barely varies; this is why the ancients got away with circles for so long. Push the slider to $e = 0.6$ and the wedges become wildly different shapes: fat and stubby at aphelion, long and thin at perihelion, all with the same area. Notice also that the star stays fixed at the focus, not the centre, and drifts further off-centre as you increase $e$. Try $e = 0.2$, close to Mercury's real value of $0.206$, and watch how much the speed readout swings.

The third law: the harmonic law#

Kepler published the first two laws in 1609. The third took him another decade, and he found it — by his own account, on 15 May 1618 — after years of hunting for a numerical relationship between a planet's distance and its period.

Kepler's third law: the square of the orbital period is proportional to the cube of the semi-major axis.

$$T^{2} \propto a^{3}$$

If you measure $T$ in years and $a$ in astronomical units, the constant of proportionality becomes exactly 1 for the Sun's planets:

$$T^{2} = a^{3} \qquad \text{(} T \text{ in years, } a \text{ in AU)}$$

Check it on Mars: $a = 1.524$ AU, so $a^{3} = 3.54$, and $\sqrt{3.54} = 1.88$ years — the observed period. Check it on Neptune: $a = 30.07$, $a^{3} = 27{,}190$, $\sqrt{27{,}190} = 164.9$ years. Observed: 164.8.

Note what the law does not contain. It says nothing about eccentricity. Two objects with the same semi-major axis have the same period even if one traces a near-circle and the other a violently elongated cigar — the semi-major axis alone sets the clock.

A relationship of the form $T^2 = a^3$ becomes a straight line the moment you take logarithms:

$$2\log T = 3\log a \quad\Longrightarrow\quad \log T = \tfrac{3}{2}\log a$$

which is a line of slope exactly $3/2$. That is the cleanest way to see the law, because it turns a claim about eight scattered planets into a single visual test:

Every planet in the solar system, spanning a factor of 78 in distance and 684 in period, lands on one line. Step the slider through the planets and watch the $T^2/a^3$ readout: it stays pinned near 1.000 from Mercury to Neptune. Try to find a planet that misses the line — you cannot, and that is the point. Kepler had no theory for why the exponent was $3/2$ rather than 1 or 2. He simply found it in the numbers and trusted them.

Newton: three laws from one force#

Kepler's laws were empirical. They described the solar system beautifully and explained nothing. Newton, some seventy years later, showed in the Principia that all three follow from a single assumption: a central force falling off as the inverse square of distance,

$$F = \frac{GMm}{r^{2}}$$

The chain of reasoning is remarkably economical.

The second law needs almost nothing. For any central force — one directed always along the line to the Sun, whatever its strength — the torque about the Sun is zero, so angular momentum $L = mr^{2}\dot{\theta}$ is conserved. Since $dA/dt = \tfrac{1}{2}r^{2}\dot{\theta} = L/2m$, equal areas in equal times is just conservation of angular momentum in disguise. The equal-area law does not test the inverse square at all; it only tests that gravity points at the Sun.

The first law is where $1/r^{2}$ earns its keep. Solving the equation of motion for an inverse-square force gives orbits that are conic sections — ellipses when the total energy is negative, parabolas at exactly zero, hyperbolas when positive. Bound orbits must be ellipses with the centre of force at a focus. Change the exponent, and closed orbits generally stop closing: the orbit precesses into a rosette instead. Ellipses are a fingerprint of the exponent 2.

The third law falls out of the circular case in one line. For a circular orbit of radius $r$, gravity supplies the centripetal acceleration:

$$\frac{GMm}{r^{2}} = \frac{mv^{2}}{r}, \qquad v = \frac{2\pi r}{T}$$

Substituting and rearranging gives

$$T^{2} = \frac{4\pi^{2}}{GM}\,r^{3}$$

which for general elliptical orbits becomes $T^{2} = 4\pi^{2}a^{3}/G(M+m)$. Kepler's mysterious constant of proportionality is now something meaningful: it is $4\pi^2/GM$, a measurement of the Sun's mass. Watch any moon or satellite orbit anything, time one revolution, measure the semi-major axis, and you have weighed the central body. Every planetary mass in the textbooks was determined this way.

Newton's version also contains a correction Kepler could not have seen: the $(M + m)$ term. Because the Sun is over a thousand times more massive than Jupiter, the correction is well under 0.1% for the planets — inside Tycho's error bars. For binary stars of comparable mass it is essential.

Where it shows up — and one thing it does not explain#

The laws long outlived their solar-system origin. Every artificial satellite is placed using $T^{2} = 4\pi^{2}a^{3}/GM_{\oplus}$: a geostationary satellite needs a period of one sidereal day, which pins its orbital radius at 42,164 km and nothing else. Hohmann transfers between planets are half-ellipses whose travel time is read straight off the third law. Exoplanet hunters invert it — a transit gives the period, the third law gives the orbital distance, and distance gives the temperature that decides whether a planet is habitable. Astronomers weigh the Milky Way's central black hole by tracking the star S2 on its 16-year ellipse.

They also have limits. The laws describe two bodies; add a third and no closed-form solution exists. And they are not exactly right even for two bodies: Mercury's perihelion precesses 43 arcseconds per century faster than Newtonian gravity predicts, a discrepancy that stood unexplained until general relativity. Kepler's eight arcminutes broke the circle; Mercury's 43 arcseconds per century broke Newton.

The seasons misconception#

Here is the most common wrong application of the first law. Earth's orbit is an ellipse, so Earth's distance from the Sun changes over the year — therefore, the reasoning goes, summer is when we are close and winter is when we are far.

Every step of that is testable, and it fails. Earth reaches perihelion around 3 January at about 147.1 million km and aphelion around 4 July at about 152.1 million km. Those dates are precisely backwards for the Northern Hemisphere: we are closest to the Sun in the depth of northern winter.

Worse, the mechanism is the wrong shape. Distance affects the whole planet at once, so a distance-driven season would make both hemispheres cold together. Instead they are always opposite: it is July in Sydney and it is winter there. Any correct explanation must be something that can point one hemisphere at the Sun and the other away — and that is Earth's 23.4° axial tilt, which changes both the angle at which sunlight strikes the ground and the number of daylight hours.

The distance effect is real but small. With $e \approx 0.0167$, the perihelion-to-aphelion distance ratio is $(1+e)/(1-e) \approx 1.034$, and since intensity falls as $1/r^{2}$, Earth receives about 7% more sunlight in January than in July. That is a genuine effect — it slightly moderates northern winters and slightly intensifies southern summers — but it is a footnote riding on top of the tilt, not the cause. Kepler's ellipse tells you the shape of the year, not the reason for it.