The parabola that is the top of an ellipse

The angle that throws furthest found 45° by the most familiar calculation in mechanics: gravity pulls straight down with a constant strength, the horizontal velocity carries the stone forward, the vertical velocity decides how long it stays up, and the product of the two is largest when they are equal. The best throw is a tangency recast the same result as geometry in the plane of launch velocities, where it survived moving platforms and sloping ground. Both arguments rest on two assumptions so natural they are rarely stated: that the ground is flat, and that gravity points the same way, with the same strength, everywhere the stone goes.

Neither is true, and over a cricket pitch the errors are a part in a hundred thousand. Over a few thousand kilometres they are not small at all. Isaac Newton drew the consequence in the Principia, as a cannon on a mountain firing faster and faster until its ball falls round the Earth without ever reaching it. What the drawing hides, and what this essay computes, is that every throw on the way there is already an orbit, and that the textbook parabola is an orbit seen from too close to tell.

Every throw is an orbit #

Gravity outside a spherical Earth falls off as the inverse square of the distance from the centre, as though all the mass were concentrated there. A stone thrown from the surface, once it leaves the hand, moves in that field and nothing else, and the motion of a body in an inverse-square field is a conic section with the centre of attraction at a focus. For a throw that comes back down it is an ellipse. The flight is the part of the ellipse above the ground; the rest of the ellipse runs underground, and would be followed if the Earth’s mass were shrunk to a point at its centre.

That one fact replaces the textbook calculation. From the launch speed and angle come the orbit’s angular momentum and energy, from them its size and eccentricity, and from those the point where it re-crosses the Earth’s surface. The range, measured along the ground, is the angle between launch and landing as seen from the centre, times the Earth’s radius.

Drawn to scale, the throws look like what they are. A three-kilometre-a-second throw is a small arch, nearly a parabola, landing a thousand kilometres away. A five-kilometre-a-second one rises to a height of several hundred kilometres and comes down three thousand kilometres off. A seven-and-a-half lands twelve thousand kilometres away, and the arch has become a long, low curve that follows the Earth round. The stone is falling the whole time; at the higher speeds the ground falls away nearly as fast.

The best angle falls #

On a flat Earth the best angle is 45° whatever the speed, because the range is proportional to $\sin 2\theta$ and nothing else depends on the angle. On a round Earth it is not.

The best angle stays close to 45° until the throw becomes a noticeable fraction of the Earth’s size, and then falls steadily, reaching zero at the circular-orbit speed of 7.91 kilometres a second. The reason is visible in the paths. A steep launch spends its energy climbing, and on a round Earth the climb is wasted twice over, because the ground the stone comes down on is curving away underneath it and lower than it would be on a flat Earth. A flatter launch that spends more of its energy going sideways keeps up with the curving ground and gains by it.

There is a neat closed form for the optimum. If the range, as an angle round the Earth’s centre, is $\phi$, the best launch angle is $(\pi - \phi)/4$: a quarter of the angle that remains when the range is subtracted from half a turn. For a short throw $\phi$ is nearly zero and the angle is $\pi/4$, 45°. For a throw to the far side of the planet $\phi$ is $\pi$ and the angle is zero. The figure finds the angle by searching over launches, independently, and the search lands on the formula to a hundredth of a degree.

This is the same statement the best throw is a tangency made in the plane of launch velocities, now on a sphere. The set of reachable launches is a circle of fixed speed; the curves of equal range are no longer the hyperbolae of a flat Earth but curves that bend towards the horizontal as the range grows; and the tangency slides down the circle towards a horizontal launch.

The range runs ahead #

It is natural to guess that a long throw on the real Earth goes less far than the flat formula says, since gravity weakens with height and that sounds like the main correction. The figure shows the opposite. The round-Earth range is longer, from the first per cent at a kilometre a second to three times longer near orbital speed. Two corrections compete, and the curvature of the ground wins: a stone that is still falling towards a surface that keeps falling away from it stays up much longer than one falling towards a flat floor. The weakening of gravity helps too, because it lets the stone rise higher, but by less.

At the orbital speed the competition ends. The round-Earth range reaches half the circumference — a throw that lands on the far side of the world — while the flat formula, which knows nothing of the far side, says about six and a half thousand kilometres. Any faster and the best throw does not land.

The parabola, found inside the ellipse #

The textbook parabola has not gone anywhere. It is still the right answer for a slow throw, and the ellipse has to contain it. It does, in a way that is worth seeing.

The ellipse of a throw has two foci. One is the Earth’s centre. The other, for a slow throw, sits just below the top of the flight: the distance from the top of an ellipse to its far focus is the same as the distance from its bottom to its near focus, and for a throw that barely leaves the ground the bottom of the orbit, deep inside the Earth, passes very close to the centre. That distance works out to $u^2/2g$, where $u$ is the speed at the top of the flight and $g$ the gravity there — which is exactly the focal distance of the parabola a flat-Earth calculation draws through the top of the same flight.

So the parabola is the top of the ellipse, with the same top, the same curvature there and the same nearby focus. What the parabola lacks is the other focus, which for a parabola is infinitely far away. Moving the Earth’s centre infinitely far down — making the Earth infinitely large, its surface infinitely flat and its gravity everywhere parallel — turns the ellipse into the parabola. For the five-kilometre-a-second throw in the figure the far focus is 896 kilometres below the top and the parabola’s focus 795, a difference of eleven per cent; for a cricket ball the difference is beyond any measurement.

The underground half of the ellipse is not a path anything follows. Inside the Earth gravity is not inverse-square — it grows on the way down for a while and then falls to zero at the centre — so a stone in a tunnel would follow a different curve. The dashed half is drawn to show which conic the flight belongs to, and the conic is fixed by the field outside.

The argument about the underground half #

The dashed half of that ellipse has a history older than the Principia. In November 1679 Robert Hooke wrote to Newton, then largely withdrawn from the Royal Society’s affairs, asking what he made of the idea that planetary motion was a straight-line motion bent by an attraction towards the Sun. Newton replied with a different question: a stone dropped from a high tower would land slightly east of the foot, because the top of the tower moves faster than its base, and if the Earth did not get in the way the stone would go on falling — along, he drew, a spiral winding into the centre.

Hooke answered that the spiral was wrong. With no resistance and an attraction falling off as the inverse square of the distance, the stone would not spiral in but swing round the centre and back up, on a path he called an “elleptueid”. Newton conceded the point with some irritation, and within a few years had proved that an inverse-square attraction produces exactly an ellipse with the centre of attraction at a focus. The underground continuation of a falling stone’s path, an argument conducted entirely in thought because nobody could watch it, was one of the threads that led to the law of gravitation. The flight above ground and the imagined path below are, as Hooke saw, one curve.

Turning the question round #

The best-angle problem has a twin: instead of asking how far a given speed can throw, ask how little speed will reach a given distance. On a flat Earth the answers coincide at 45°. On a round Earth they coincide too, at the same launch angle $(\pi - \phi)/4$, because the launch that throws furthest at a given speed is also the launch that reaches that distance with the least speed. The required speed, in units of the circular-orbit speed, is $\sqrt{2\sin(\phi/2)/(1 + \sin(\phi/2))}$.

For a target ten thousand kilometres away, a quarter of the way round the Earth, that is 7.20 kilometres a second at 22.5° above the horizontal, with no air and no rotation. The flat-Earth formula would ask for $\sqrt{gd}$, 9.9 kilometres a second, faster than the speed of orbit — an answer that is not merely inaccurate but impossible, since anything that fast would not come down. The round Earth makes long throws cheaper than the flat one predicts, by exactly the curvature that carries the ground away from the falling stone.

How wrong the parabola is #

The error grows as the square of the launch speed while the throw is small compared with the Earth, because what sets it is the ratio of the throw’s own size — the range or height, both proportional to $v^2/g$ — to the Earth’s radius. For a cricket ball it is a thousandth of a per cent. For a rifle bullet fired in a vacuum at its best angle it would be two thirds of a per cent. For a missile with an intercontinental range it is most of the answer.

None of this is the dominant correction for anything thrown through air. Air resistance changes a cricket ball’s range by tens of per cent, moves its best angle well below 45°, and swamps the curvature correction by a factor of tens of thousands. The round-Earth calculation matters only where the throw leaves most of the atmosphere behind — ballistic missiles, sounding rockets, suborbital spaceflight — and there it is the whole calculation.

Two other corrections are left out of every figure here. The Earth turns, so a throw to the east starts with an extra few hundred metres a second and one to the west with less, and on the ground the landing point is displaced as a ball thrown upward lands to the west. And the Earth is not a sphere: its equatorial bulge makes its gravity slightly stronger at the poles and slightly non-central, which turns the ellipse slowly over many orbits.

The edge of everywhere a throw can reach #

Everywhere a throw can reach drew all the parabolas a fixed launch speed can make, at every angle, and found their edge — the boundary of the region the stone can reach at all — to be another parabola, with its focus at the launch point. The round-Earth version has an equally clean answer, and it shows the same family resemblance. With the attraction coming from the Earth’s centre, the edge of the reachable region, for a fixed launch speed, is an ellipse whose two foci are the Earth’s centre and the launch point.

The two results are one result. Push the Earth’s centre infinitely far down and an ellipse with one focus at the launch point and the other at infinity becomes a parabola with its focus at the launch point, which is the flat-Earth envelope. The point on the ground where the safety ellipse meets the Earth’s surface is the maximum range; the launch that reaches it touches the envelope there; and the tangency is the same condition, transplanted to a sphere, that picked out 45° on the plane.

The same ellipse, the same law #

Once a throw is seen as an orbit, everything known about orbits applies to it. Its period, if the Earth’s mass were at the centre and the stone could pass through, is given by Kepler’s third law, which falls out of a change of scale: the ellipse of a short throw has a semi-major axis of a little over half the Earth’s radius, so its full period would be about a third of the period of a satellite skimming the surface, some thirty minutes. The flight is the slow part at the top of that orbit, and the stone spends a few seconds on it because the arc it covers is tiny.

The ellipse also has a fixed orientation, set by a conserved vector that points from the centre towards the bottom of the orbit, and it closes on itself because the force is exactly inverse-square. For a throw, the vector points nearly straight down, beneath the top of the flight, which is the symmetry that makes the ascent and descent mirror images: launch and landing at the same height means the same speed and the same angle to the ground, whatever the speed.

What the pictures cannot show #

The drawings are to scale, and so the slow throws are almost invisible: a three-kilometre-a-second throw rises to a few hundred kilometres, which on a picture of the whole Earth is a few pixels. The ellipse figure is drawn for a fast throw so that the two foci can be seen apart; for any throw a hand can make, the far focus and the parabola’s focus coincide to well within the line’s width, and the near focus is six thousand kilometres away.

No figure includes air. For speeds above about a kilometre a second at the surface, air would destroy the throw long before the curvature mattered; the round-Earth results are for a body launched above the atmosphere or with the atmosphere ignored. Nor do they include the Earth’s rotation or its oblateness.

The domain of the argument is a body moving freely outside a spherical Earth with no air, launched from the surface and landing on it. Inside that domain the flight is an arc of an ellipse focused on the Earth’s centre, the best angle is $(\pi - \phi)/4$, and the parabola is the flat-Earth limit of its top.

Still open: the cheapest throw between two points #

For a given range there is a launch that needs the least speed, and for long ranges it is not the same as the launch that, at a given speed, throws furthest — though on a sphere the two questions turn out to have the same answer, a geometric fact that makes the minimum-energy trajectory easy to find. What is not settled by geometry alone is the cheapest path when the throw is not a single impulse: when thrust can be spread along the flight, the atmosphere has to be crossed twice, and the Earth turns underneath. That is an optimisation over continuous controls with constraints, solved numerically for every rocket that flies, and its answers depend on the vehicle.

The textbook throw is a limit, not a separate law. On a round Earth a throw is an arc of an ellipse with the Earth’s centre at one focus; its best launch angle is (π − φ)/4, falling from 45° to zero as the speed rises to 7.91 km/s; its range runs ahead of $v^2/g$, to half the circumference at orbital speed; and the parabola is the top of that ellipse with its far focus — $u^2/2g$ below the top — kept and the Earth’s centre sent to infinity. Newton’s cannon on the mountain is the textbook throw, thrown hard enough to see what it always was.