The velocity that always lies on a circle

On 8 March 1964 Richard Feynman gave a guest lecture to the first-year physics students at Caltech, and set himself a challenge: to prove that a planet moving under an inverse-square force travels on an ellipse using nothing beyond the geometry of the Greeks — no calculus, no coordinates, nothing a schoolchild could not follow. The lecture was recorded but not included in the published Lectures on Physics, and the notes and tape were lost in the archive until 1992; David and Judith Goodstein reconstructed it as Feynman’s Lost Lecture in 1996. The proof turned on a single object, an old one: the circle on which a planet’s velocities lie.

The circle had been named half a century before Feynman’s birth. William Rowan Hamilton noticed in 1846 that if all the velocity vectors of a planet on its orbit are drawn from one point, their tips do not trace out an ellipse, or some egg-shaped curve, but an exact circle. He called the curve the planet’s hodograph, from the Greek for a path-drawing. It is the most direct way of seeing what is special about the inverse square, and it turns out to be the same fact, seen differently, as the arrow that the arrow that says which way the orbit points found fixed in every Kepler orbit.

An orbit and its velocities #

Take a planet on an orbit of eccentricity 0.6 round a star. It moves fastest at its closest approach and slowest at its furthest, and its velocity is always tangent to the orbit, swinging round once per revolution. Now take each velocity vector, at many points round the orbit, and draw them all starting from the same point.

The tips lie on a circle. The orbit was not drawn from a formula; it was integrated step by step from Newton’s law of gravitation, and the velocities were read off the integration, and they stay on the circle to within the accuracy of the arithmetic. The circle’s radius is $GM/h$, where $h$ is the planet’s angular momentum per unit mass, and its centre is displaced from the drawing point by exactly the eccentricity times the radius. The planet’s fastest velocity, at closest approach, is the top of the circle, $(1+e)\,GM/h$; its slowest, at the furthest point, the bottom, $(1-e)\,GM/h$.

The velocity tips are bunched together at the slow end of the circle and spread out at the fast end. The arrows on the orbit were drawn at equal intervals of time, and the planet spends most of its time far from the star, where its velocity changes slowly, so most of the arrows land near the bottom of the circle. The circle is a curve of velocities, not of times.

Why it is a circle #

The reason is short enough to state completely. Under gravity alone, the change in a planet’s velocity is the pull of the star times the time over which it acts. Cut the orbit into short pieces, each subtending the same small angle at the star. Near the star the pieces are short and the planet crosses them quickly; far away they are long and it crosses them slowly. By Kepler’s second law — conservation of angular momentum, which the quantity that survives a change of shape traced to the absence of any sideways force — the time spent crossing a piece of fixed angle is proportional to the square of the distance, $r^2$. The pull on the planet is proportional to $1/r^2$. So the change of velocity in each piece, the pull times the time, is the same size for every piece, wherever it is on the orbit.

And it points towards the star, which, from one piece to the next, has turned by the same angle. Successive velocity changes are therefore equal in length and turn by equal angles: they are the sides of a regular polygon. In the limit of small pieces, a regular polygon is a circle.

That is the whole of it, and it uses the inverse square exactly once — to cancel the $r^2$ of the time against the $1/r^2$ of the force. Any other force law leaves the two unbalanced: the steps of velocity are then longer where the force is relatively stronger, and the polygon is no longer regular. The figure checks it numerically: twenty-four pieces of equal angle, read off an integrated orbit, give twenty-four velocity steps equal to nine decimal places.

Feynman’s lecture ran the argument in reverse. Given that the velocities lie on a circle, he showed by a construction with a ruler and compass that the orbit itself must be an ellipse with the star at a focus: the direction of the orbit at each point is perpendicular to the line from the circle’s centre to the velocity tip, and the curve whose tangents are fixed in that way is an ellipse. It is Newton’s result, reached by geometry Newton would have recognised, and James Clerk Maxwell had used the same circle in his textbook Matter and Motion in 1877.

Closed orbits and open ones #

The argument never assumed the orbit was closed, so the hodograph is a circle for every orbit under an inverse square, including the ones that escape.

For a circular orbit the centre of the velocity circle sits on the drawing point: the speed is constant and the velocity simply turns. For an ellipse the circle is raised but still encloses the drawing point, so the velocity turns all the way round and never falls to zero. For a parabola, the boundary between bound and unbound orbits, the circle passes exactly through the drawing point: the body arrives from infinity and returns there with zero speed. For a hyperbola, the circle is raised so far that the drawing point lies outside it, and only an arc is used: the body comes in along one direction, swings round, and leaves along another, still moving at $\sqrt{e^2-1}\,GM/h$ — 1.12 times $GM/h$ for an eccentricity of 1.5. The missing part of the circle is the set of velocities the body would need to be bound.

One family of curves for every orbit, distinguished only by how far the circle is raised. The raising is the eccentricity, and it is a vector: the direction in which the circle is displaced, turned through a right angle, points to the perihelion. That vector is the Laplace–Runge–Lenz vector, the conserved arrow of the arrow that says which way the orbit points. The orbit’s axis cannot turn because the velocity circle cannot move: it is fixed by the energy and the angular momentum, both conserved, and its offset is a third conserved quantity that the inverse square, and only the inverse square, provides.

Every speed from one picture #

The circle also gives every speed on the orbit without solving anything. A point on a circle of radius $GM/h$ whose centre is raised by $e\,GM/h$ is at distance

$$v = \frac{GM}{h}\sqrt{1 + e^2 + 2e\cos\theta}$$

from the drawing point, where $\theta$ is the angle round the circle — the same angle the planet has moved round the star since perihelion. Squared and combined with the equation of the ellipse, that is the vis-viva equation, $v^2 = GM(2/r - 1/a)$, which gives the speed at any distance from the orbit’s size alone. The figure needs no algebra to show its two extremes: the top of the circle is perihelion, the bottom aphelion, and their ratio, $(1+e)/(1-e)$, is the ratio of the two distances inverted, as Kepler’s second law requires — 4 for an eccentricity of 0.6.

The circle’s radius, $GM/h$, carries the angular momentum, and its offset carries the eccentricity; the orbit’s size and period follow from those, and the third law that falls out of a change of scale found the period growing as the size to the power three halves by a scaling argument the hodograph respects: enlarge an orbit fourfold and every velocity circle shrinks by a half.

The curve that does not close #

Change the force law slightly and the circle falls apart.

With an extra inverse-cube term added to the attraction — the kind of correction the Sun’s flattening or general relativity introduces — the velocity’s tip no longer runs round a fixed circle. Its distance from the old centre wanders, and each revolution’s loop is rotated against the last, because the orbit itself no longer closes: its long axis turns a little every revolution. The orbit that does not come back to itself found Bertrand’s theorem, that among all attractions falling with distance only the inverse square gives closed orbits for every starting condition; the hodograph is that theorem made visible. A circle of velocities is a closed orbit’s signature; a rosette of velocities is a precessing one’s, the pattern the orbit special relativity cannot close found in Mercury’s motion.

The same circle hides inside a projectile’s flight. The parabola that is the top of an ellipse found that a thrown ball’s parabola is really the far end of a very elongated Kepler ellipse round the Earth’s centre; its hodograph is correspondingly an arc of a very large circle, which over the few seconds of a throw is indistinguishable from the straight vertical line that the textbook’s uniform gravity gives — the velocity’s horizontal part constant, its vertical part changing steadily. The circle curves only when the flight is long enough for the direction to the Earth’s centre to change.

A transfer between orbits, read off two circles #

The hodograph is not only a proof device. It is the natural picture for planning a change of orbit, because a rocket burn changes a spacecraft’s velocity almost instantly, without changing its position, so a burn is a jump in velocity space from one orbit’s circle to another’s.

To go from the Earth’s orbit to a circular orbit at 1.524 astronomical units, Mars’s distance, the cheapest two-burn route is the Hohmann transfer: an ellipse touching both circles. On the hodograph the circular orbits are circles centred on the drawing point — 29.78 km/s for the Earth, 24.12 for the outer orbit — and the transfer ellipse is a raised circle. At departure the spacecraft must jump from the Earth’s circle to the transfer circle, from 29.78 to 32.73 km/s, a burn of 2.95 km/s along its direction of motion; half an orbit later, arriving at the outer distance at only 21.47 km/s, it must jump to the outer circle, 2.65 km/s more. Each burn is a straight jump between two circles, and since the circles touch the vertical axis at their tops and bottoms, the shortest jump is along the direction of motion — the reason tangential burns are the efficient ones.

The total, 5.6 km/s, is the minimum for this transfer, though real missions to Mars need different numbers because Mars’s orbit is elliptical and tilted and a spacecraft must climb out of the Earth’s own gravity first. Mission planners do not draw hodographs; they compute. But the picture explains why some manoeuvres are cheap and others expensive: a burn that changes the direction of a fast velocity, crossing the velocity circle sideways, costs far more than one that lengthens it.

A slingshot as a turn round a circle #

The hodograph also explains how a spacecraft gains speed by flying past a planet. In the planet’s own frame the flyby is a hyperbola, and its velocities lie on an arc of a circle: the craft arrives at some speed, swings round, and leaves at exactly the same speed in a new direction — nothing has been gained. But the planet is moving round the Sun. In the Sun’s frame the planet’s velocity is added to every point of that arc, shifting the whole circle, and a velocity that has merely been turned in the planet’s frame has, in the Sun’s frame, been lengthened or shortened depending on which way it was turned. The wall that moves while the ball is in flight found a ball bouncing elastically off a moving wall gaining twice the wall’s speed; a gravity assist is the same bounce, with a planet’s gravity for the wall and an arc of a velocity circle for the bounce.

Voyager 2 used four such turns to visit Jupiter, Saturn, Uranus and Neptune. Each flyby rotated its velocity round a circle centred on the planet’s own velocity, and the turn, seen from the Sun, was worth several kilometres a second. The energy came from the planets’ orbital motion, which was slowed by an immeasurably small amount, and the direction of each turn was chosen by how close the craft passed: a closer pass bends the hyperbola more and uses more of the circle.

The circle in the hydrogen atom #

The inverse-square force between an electron and a proton is the same law, and the hidden symmetry that keeps a planet’s velocity circle fixed has a quantum version. In 1935 the Soviet physicist Vladimir Fock showed that the hydrogen atom’s wavefunctions, written in terms of the electron’s momentum rather than its position, are the vibrations of a sphere in four dimensions, projected down to three — the quantum descendant of the hodograph’s circle. That four-dimensional symmetry, generated by angular momentum and the Laplace–Runge–Lenz vector together, is why hydrogen’s levels with the same principal quantum number but different angular momentum have exactly the same energy, the degeneracy that the order the shells fill found broken in every atom with more than one electron. A planet’s velocity circle and hydrogen’s energy levels are the same symmetry, at two scales twenty-five orders of magnitude apart.

Two bodies, a fixed centre and a flat plane #

The orbits are integrated in two dimensions round a fixed point mass, with the planet’s own mass neglected; for two bodies of comparable mass the same circle holds for their relative velocity, but each body’s own velocity circle is scaled by its share of the mass. The inverse-cube perturbation in the precessing figure is chosen large enough to see in three revolutions; the perturbations of real planetary orbits are thousands of times smaller, and Mercury’s velocity circle turns by a fraction of a second of arc per century.

The transfer figure treats both orbits as circular and coplanar and the burns as instantaneous, and it measures speeds in the Sun’s frame, so it says nothing about leaving or entering a planet’s gravity. And the hodograph is a statement about a two-body problem: any third body — Jupiter tugging at an asteroid, the Moon at a satellite — perturbs the circle, and the long-term behaviour of those perturbations, the subject of the hilltop that holds the Trojans and of planetary dynamics generally, is where the simple circle stops being the whole story.

Still open: what the hodograph says about three bodies #

For two bodies the velocity circle is a complete description: energy, angular momentum and the Runge–Lenz vector between them fix the orbit for ever. Add a third body and the circle drifts, as its centre’s offset — the eccentricity vector — is pushed around by the third body’s pull. The secular theory of planetary motion, begun by Lagrange and Laplace, follows those drifts as slow cycles of eccentricity and orientation lasting tens of thousands of years, and they underlie the Milankovitch cycles of the Earth’s climate. Whether those cycles continue indefinitely or eventually become chaotic — whether, over billions of years, Mercury’s velocity circle could be pushed far enough off centre for the planet to collide with Venus or the Sun — is known only statistically: simulations find it happening in about one per cent of possible futures over the Sun’s remaining lifetime.

The circle itself is the clean part. Under an inverse-square force, equal angles of an orbit bring equal steps of velocity, so the velocities lie on a circle whose offset is the orbit’s eccentricity; and the circle survives only as long as the inverse square does. Hamilton drew it, Maxwell taught it and Feynman used it to rebuild the ellipse from scratch. A planet’s speed changes all the way round its orbit, but its velocity, drawn from one point, has nowhere to go but round a circle.