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

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.

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 $\tfrac{1}{2} r v_\perp$, so if that is constant then

$$r\,v_\perp = \text{constant},$$

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.

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 $L = m r v_\perp$ cannot change. The rate at which the radius vector sweeps area is exactly $L/2m$. 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 $M$ 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 $E$ is a geometric intermediary defined on the circle that circumscribes the ellipse. The true anomaly $\nu$ is the real angle from periapsis, which is what a telescope sees.

The second law connects them through Kepler’s equation,

$$M = E - e\sin E,$$

and this is where the trouble is. Given $E$, finding $M$ is arithmetic. Given $M$, finding $E$ 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 $E$, 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.

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. That curve is the vis-viva relation,

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

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 $1/a$ is the total energy per unit mass, and the fact that it depends on $a$ 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 $r v_\perp$ 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. 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.

What Kepler actually had, and the two errors that cancelled #

The area law is usually presented as though Kepler had noticed a constant and reported it. The real route is stranger, and it is worth knowing because it is a case of a correct result reached by a method that does not support it.

His data were Tycho Brahe’s positions of Mars: directions against the fixed stars, good to about two arcminutes, over roughly twenty years. There were no distances and no speeds. Turning that into a statement about how fast Mars moves at each point of its orbit required first constructing the orbit, which Kepler did by the triangulation described in the essay on the first law, and then extracting a timing rule from it.

What he was actually looking for was a distance law — the idea, inherited from a physical picture of the Sun sweeping the planets round, that a planet’s speed should be inversely proportional to its distance. That is not true. It holds exactly only at periapsis and apoapsis, where the motion is purely transverse; everywhere else the true relation involves the transverse component alone, and the radial component is unaccounted for.

Summing the distances round the orbit to get the elapsed time was, in his own words, a procedure he was not comfortable with — he replaced a sum over a great many radii with the area they sweep, which is a step he justified by analogy rather than by proof. The result was the area law.

So the derivation used a false speed law and an unjustified integration, and produced a statement that is exactly correct. The reason is that the two departures compensate: the error in assuming speed goes as $1/r$ is second order in the eccentricity, and the error in replacing the sum of radii by an area is second order the other way. For Mars, at $e = 0.093$, the residual was well below Brahe’s two arcminutes.

That is a piece of luck rather than a piece of insight, and it is fair to say so — while also noting that Kepler tested the result against the data rather than resting on the argument, which is why it survived.

The plane, which is the law’s other half #

Angular momentum is a vector, and everything above has used only its magnitude. Its direction is conserved too, and that is a separate physical statement with consequences the figures cannot show.

A central force produces no torque at all, not merely no torque about one axis. So the angular momentum vector is fixed in direction as well as length, and since the position and velocity are always perpendicular to it, the orbit is confined to a single fixed plane forever. Two of the six orbital elements exist only to say which plane.

For the solar system this has a striking observational form. Every planet’s orbital plane lies within about 7° of every other’s, and most within 2°. That flatness is the fossil of the disc the system formed from, and it is angular momentum conservation that made the disc flat: a collapsing cloud can shed energy by radiating, but it cannot shed angular momentum nearly as easily, so it collapses along the rotation axis and stalls in the perpendicular direction.

The invariable plane — the plane perpendicular to the total angular momentum of the whole system — is the natural reference, and it is fixed no matter what the planets do to each other, because their mutual torques cancel in the sum. It is dominated by Jupiter, which carries about 60% of the solar system’s orbital angular momentum despite being a thousandth of its mass.

And there is a residual nobody has fully explained. The Sun’s own equator is tilted about 6° to that plane. If the Sun and the planets formed from one rotating cloud, the two should coincide. They do not, and the proposed explanations — a passing star early on, a torque from an undiscovered distant planet, an asymmetry in the collapse itself — are all still live. A conservation law this clean makes a 6° discrepancy into a question rather than a rounding error.

Timing as a measurement #

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

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.

The sixth element, and why it is the awkward one #

Five of the six numbers describe the orbit as a curve in space. The sixth describes where along it the body is at a chosen moment, and it is the one this law is about — and the one that is hardest to state well.

The natural choice is the time of periapsis passage: the epoch at which the body was last at its closest approach. It is a clean definition and it fails in exactly the case that is most common.

For a nearly circular orbit there is no periapsis worth speaking of. The eccentricity is tiny, the long axis is barely distinguishable from any other diameter, and the moment of closest approach is determined by a difference of quantities that are almost equal. A small error in the eccentricity moves the computed periapsis time by a large fraction of the period, and for a perfectly circular orbit the quantity does not exist at all.

The standard remedy is to replace it with a mean longitude at a stated epoch — an angle measured from a fixed reference direction rather than from a feature of the orbit itself. That is well defined for every eccentricity including zero, it changes at a uniform rate by the second law’s own arithmetic, and it carries exactly the same information.

The pattern is the one that recurs whenever an orbit approaches a symmetric limit: a parameter defined relative to a feature of the orbit fails when the feature does, and the repair is to define it relative to something external instead. It is the same repair as replacing the eccentricity and the argument of periapsis by their two combinations, and the same repair as measuring an arrival by a point in a plane rather than by a closest approach.

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 law is stated for any eccentricity and it is only at the extreme that its content becomes obvious rather than merely true.

That picture is the reason the law was worth discovering rather than merely worth stating. A uniform angular rate would put the marks evenly around the ellipse and would be wrong; a uniform linear speed would put them evenly along the path and would also be wrong. The correct rule is neither, and it is not a rule anybody would guess from watching a planet, because the planets whose motion was well measured in Kepler’s time all have eccentricities under a tenth and their marks are nearly evenly spaced.

Mars is the exception, at 0.093, and that is why it was the planet the law came out of. A tenth of an eccentricity is enough to make the discrepancy from a circle several times the observational error of the sixteenth century, and small enough that the orbit still looks like a circle drawn slightly off centre. Kepler had the one dataset in which the effect was measurable and the shape was not yet obvious, and he spent eight years on it.

The extreme case also shows why the law’s other half — the constancy of the plane — is the half nobody remembers. At eccentricity 0.95 the body’s distance from the focus changes by a factor of forty and its speed by a factor of six, and through all of that the areal rate does not change at all, nor does the plane the sweeping happens in. Both are statements about the same conserved vector, and the one about the plane is invisible precisely because nothing about it ever happens.

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.