Astronomy & Celestial Mechanics
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Orbits

Equal areas in equal times, which is angular momentum in disguise

Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.
12 min read 6 figures The sky as a clockTwo bodies, and no more

Of Kepler’s three laws, the second is the strangest to have found. The first is a shape and the third is a proportionality; both are the kind of thing a person looking for patterns might look for. The second is about the area swept out by an imaginary line, which is not an obvious quantity to think of measuring, and which nobody had any reason to expect would be constant.

It is constant. And eighty years later it turned out to be conservation of angular momentum, which is one of the deepest statements in mechanics, discovered by someone who had no concept of momentum at all.

Equal areas in equal times, at eccentricity 0.65Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach.fast hereslow here16 equal intervals of time, one full orbit
Fig. 1 Positions at sixteen equal intervals of time around one orbit, obtained by solving Kepler’s equation rather than by spacing points along the curve. The two shaded sectors cover the same interval of time and enclose the same area.

What the law says

Draw the line from the primary to the orbiting body — the radius vector — and watch it sweep as the body moves. In any given hour it sweeps out some area. The law is that this area is the same for every hour, wherever in the orbit the body happens to be.

Near the close approach the body is moving fast and the radius is short, so the sector swept is stubby and wide. At the far end the body crawls and the radius is long, so the sector is a long thin sliver. The two look nothing alike, and the figure above shows them side by side because the claim that they have equal area is not something the eye will grant without being shown.

The immediate consequence is a speed law. Area swept per unit time is roughly 12rv⊥\tfrac{1}{2} r v_\perp, so if that is constant then

r v⊥=constant,r\,v_\perp = \text{constant},

and the transverse speed varies inversely with distance. A body at periapsis on the orbit above, four times closer than at apoapsis, is moving four times faster across the line of sight.

For the Earth the effect is small and real. The planet is about 3.4% closer to the Sun in early January than in early July, and its orbital speed is about 3.4% higher — 30.3 km/s against 29.3. That difference is the reason the seasons are not of equal length: northern winter, which contains the close approach, is about five days shorter than northern summer. A calendar has an unequal number of days between equinoxes because of Kepler’s second law, which is a pleasingly concrete consequence of a statement about swept area.

Equal areas in equal times, at eccentricity 0.25Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach.fast hereslow here12 equal intervals of time, one full orbit
Fig. 2 The same construction at a much lower eccentricity. The crowding is still there and is now subtle, which is why the law was extracted from Mars — the most eccentric orbit then measurable — rather than from anything nearer.

Why area, of all things

The reason the law is about area is that area is what a central force cannot change.

A central force is one that always points along the line joining the two bodies — toward the primary, never off to the side. Gravity is central. That is a much weaker assumption than the inverse square: it says nothing about how the force varies with distance, only about its direction.

A force with no sideways component exerts no torque about the primary, and with no torque, the angular momentum L=mrv⊥L = m r v_\perp cannot change. The rate at which the radius vector sweeps area is exactly L/2mL/2m. So constant areal rate and constant angular momentum are the same statement, written in two vocabularies separated by eighty years.

This makes the second law far more general than the other two. The first and third laws require the inverse square specifically. The second holds for any central force whatsoever — an inverse cube, a linear spring, a force that switches sign at some radius. Anything that always points at the centre sweeps equal areas.

That generality is worth dwelling on because it inverts the usual reading. The second law is not a fact about gravity. It is a fact about direction, and it would survive the discovery that the force law was completely different, which is exactly the property that makes it useful for orbits nobody understands. When a star’s motion sweeps unequal areas about a companion, the conclusion is not that gravity is odd there — it is that something else is pulling.

The timing problem

Knowing the areal rate is not the same as knowing where a body will be next Tuesday, and the gap between them is one of the most stubborn small problems in mathematics.

The chain has three steps. The mean anomaly MM is the fictitious angle a body on a circular orbit of the same period would have swept — it is just time, rescaled, and it runs at a perfectly constant rate. The eccentric anomaly EE is a geometric intermediary defined on the circle that circumscribes the ellipse. The true anomaly ν\nu is the real angle from periapsis, which is what a telescope sees.

The second law connects them through Kepler’s equation,

M=E−esin⁡E,M = E - e\sin E,

and this is where the trouble is. Given EE, finding MM is arithmetic. Given MM, finding EE is not: the equation is transcendental, and Kepler said explicitly that he could not solve it and did not expect anyone to. Nobody has. Every position in every figure on this page was obtained by iterating — guessing EE, computing the residual, correcting, repeating — which converges in three or four steps and is what almanacs have done for four centuries.

There is a mild irony in it. The law that fixes the timing exactly is the one that makes the timing hardest to compute.

An orbit at eccentricity 0.65An orbit of eccentricity 0.65. The primary sits at a focus, offset from the centre by 0.65 of the semi-major axis, and the closest and furthest points differ by a factor of 4.71.empty focusrperiapsisapoapsis
Fig. 3 The same orbit as a shape alone. Nothing in this picture says when the body is anywhere, which is precisely the information the second law adds and the first law cannot.

Speed, read off the orbit

The second law gives the transverse speed. The full speed at any point needs one more ingredient, and the combination is worth having because it answers the practical question directly.

Speed against distance, for orbits of the same periodOrbital speed against distance from the primary. Every orbit with the same semi-major axis follows the same curve; the eccentricity decides only which stretch of it the body uses.00.511.50123distance from the primary (a = 1)e = 0e = 0.4e = 0.8circular speed at r = a
Fig. 4 Speed against distance for orbits sharing a semi-major axis. Every one of them lies on the same curve; the eccentricity decides only which stretch of it the body uses, from a short arc near the middle for a circle to a long sweep for an eccentric orbit.

That curve is the vis-viva relation,

v2=GM(2r−1a),v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right),

and the striking thing about it is what is missing: the eccentricity. Speed depends only on where the body is and how big the orbit is. Two bodies at the same distance from the Sun, one on a nearly circular orbit and one on a comet’s path, are moving at completely different speeds — but only because their semi-major axes differ, not because their shapes do.

The relation is the working tool of orbital manoeuvring, because a burn changes the speed at a known distance and the equation converts that immediately into a new orbit. It is also, read the other way, an energy statement: the term in 1/a1/a is the total energy per unit mass, and the fact that it depends on aa alone is the reason the boundary between returning and not returning is a statement about energy rather than about shape.

The same law without an orbit

Because the second law is about central forces rather than about gravity, it turns up in places with no astronomy in them.

A skater pulling in their arms spins faster, by exactly the factor that keeps rv⊥r v_\perp fixed — the same bookkeeping that makes a tidally receding moon slow down as it gains angular momentum. A collapsing gas cloud spins up as it shrinks, which is why almost everything in astronomy that forms by collapse ends up rotating fast and flattened — accretion discs, protoplanetary discs, and the reason the solar system is a plane rather than a swarm.

The same conservation sets a hard limit on what collapse can achieve. A cloud a light-year across, rotating imperceptibly, has enough angular momentum that by the time it has shrunk to the size of a star it would be spinning far too fast to hold together. Star formation is therefore mostly a problem of getting rid of angular momentum, not of gravity — and the leftover is what the planets are.

The wobble of a star, companion at eccentricity 0.4The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.00.511.52-101orbitstoward the observeraway from the observere = 0.4: skewed, and the skew is the measurement
Fig. 5 A star’s velocity along the line of sight, over two orbits of an unseen companion. The curve is skewed rather than sinusoidal, and the skew is the second law showing up in a measurement: the star swings through its close approach quickly and lingers at the far end.

That last figure is the second law used as an instrument. A star with a planet traces its own small ellipse about the common centre of mass, and the line-of-sight component of that motion is measurable as a Doppler shift. A circular companion gives a pure sine wave. An eccentric one gives the skewed curve above, and fitting the skew recovers the eccentricity of an orbit nobody can see, of a body nobody has imaged, from a wavelength shift of a few metres per second.

Timing as a measurement

Because the areal rate is exactly constant, any departure from it is information.

Period against size for the planets, around the SunOrbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.0.321.03.210320.100.321.03.21032100316semi-major axis (AU)MercuryVenusEarthMarsJupiterSaturnUranusNeptuneslope 3/2 — P² ∝ a³period (years)
Fig. 6 Period against semi-major axis for the planets. The second law fixes how a body moves within its orbit; this one fixes how orbits of different sizes relate to each other, and together they make a position predictable years ahead.

The pair is what makes an ephemeris possible. The third law gives the period from the size of the orbit; the second gives the position within the period; the first gives the shape being traversed. Three statements, and a table of positions for centuries.

Departures from the predicted timing are then the residuals that find things. Uranus’s arrival at the wrong place by two arcminutes is what located Neptune, and a transit arriving early or late by minutes is how additional planets are found in systems where only one transits — timing as a detection method, which needs no new instrument at all.

Where the model stops

Two bodies. The angular momentum of a pair is conserved; that of one planet about the Sun is not, quite, because the other planets exert torques. The effect is tiny and cumulative, and it is what makes orbital elements slowly drift.

A central force. Anything that pulls off-axis breaks the law. A non-spherical primary does exactly that, which is how a satellite’s orbital plane is made to precess on purpose.

No drag. A low satellite meets atmosphere, loses angular momentum continuously, and spirals in — and does so, counterintuitively, while speeding up, because a smaller orbit is a faster one.

The picture is a plane. Angular momentum is a vector, and the fact that it points in a fixed direction is a second and separate consequence of a central force: the orbit stays in one plane. No figure here shows that, because they are all drawn in the plane the law confines the orbit to.

There is also something no version of this figure can show, which is that the law says nothing about where in the orbit a body is at any given moment. It fixes the rate of sweeping, not the phase. Supplying the phase takes one more number — the time of periapsis passage — and that number is the sixth orbital element and the one that has to be measured rather than derived.

The ladder from here

Later rungs: Kepler’s equation and the numerical methods that solve it, from Newton’s iteration to series in the eccentricity. The three anomalies and the geometry connecting them. The vis-viva relation derived properly from energy. Angular momentum as a vector, and the orbital plane. Torque-free precession. The specific angular momentum as an orbital element. Areal velocity in the restricted three-body problem, where it stops being constant and the Jacobi integral takes over. And the accretion disc, which is the second law’s most consequential astronomical child.

Kepler published the second law before the first, in the same book, and thought less of it. It is the one that generalises.

Where this goes next

empty focusabae = 0.6arperiapsisapoapsis The orbit is an ellipse, and the Sun is not in the middle of it Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty. 0.321.03.210320.100.321.03.21032100316semi-major axis (AU)MercuryVenusEarthMarsJupiterSaturnUranusNeptuneslope 3/2 — P² ∝ a³period (years) The law that links period to size, and weighs everything Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since. barycentrethe line joining them always passes through it Neither body is still, and the wobble is how planets are found A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.