A fall is half an orbit with no width

A single inverse-square force permits exactly one family of curves — circle, ellipse, parabola, hyperbola — and one number picks between them. The family has a member that no drawing of conics shows, because it has no width. Take away a body’s sideways motion entirely and the orbit equation’s single parameter, which is proportional to the square of the angular momentum, goes to zero. What is left is motion along a straight line through the centre: the body falls in, or is thrown straight out.

It is natural to treat that case as a curiosity, a degenerate end of the family with nothing to say. It is the opposite. The straight fall is the simplest motion gravity produces, it is the motion of every collapse that begins from rest, and its timing is one of the most-used numbers in astrophysics. It also behaves, in almost every respect, like an ordinary ellipse, which is the fact that makes it computable.

The line at the end of the ellipses #

The way into the radial orbit is through the family of ellipses that share a semi-major axis.

The semi-major axis fixes an orbit’s energy and, through the harmonic law, its period. The eccentricity fixes its shape, and at a given semi-major axis the shape is controlled entirely by the angular momentum: the width of the ellipse is proportional to $\sqrt{1-e^2}$, which is proportional to the angular momentum. Holding the semi-major axis fixed and taking the angular momentum away therefore narrows the ellipse without shortening it. Its far end stays two semi-major axes from the focus, its near end approaches the focus, and at zero angular momentum the ellipse is a line segment from the attracting mass to a point at twice the semi-major axis.

That is the radial orbit, and it is an ellipse in every sense that matters for timing. It has the semi-major axis of the family, so it has the family’s energy and period. A body on it starts at rest at the far end, falls to the centre, and — if the centre is a point that it can pass through — climbs back to the far end on the other side of its approach, in exactly one period. Real bodies collide with what they fall onto, so a real radial orbit is half an orbit, from rest at the far end to the centre.

It is not a parabola, though both have an eccentricity of one. A parabola has eccentricity one at a fixed closest approach and an infinite semi-major axis, and it is the boundary between bound and unbound orbits. A radial orbit has eccentricity one at a fixed semi-major axis and zero closest approach, and it is bound. The eccentricity reaches one by two different routes, and the two endpoints have nothing in common but the number.

A fifth of a year #

The timing follows at once. A body released at rest at distance $r_0$ is at the far end of a radial orbit whose length is $r_0$, so its semi-major axis is $r_0/2$. Its period is the period of any orbit with that semi-major axis, and it reaches the centre in half of it:

$$t_{\rm ff} = \frac{1}{2}\,2\pi\sqrt{\frac{(r_0/2)^3}{GM}} = \pi\sqrt{\frac{r_0^3}{8GM}} = \frac{P(r_0)}{4\sqrt2},$$

where $P(r_0)$ is the period of a circular orbit at the starting distance. The factor $1/(4\sqrt2)$ is 0.177. A body at the Earth’s distance, stopped dead, would fall into the Sun in 0.177 of a year: 64.6 days.

Between the start and the end, the motion has a parametric solution that the ellipse supplies too. The eccentric anomaly of a radial orbit — the angle that parametrises position and time along an ellipse — becomes a parameter $\eta$ in which the distance is $r = r_0(1 + \cos\eta)/2$ and the time is $\sqrt{r_0^3/8GM}\,(\eta + \sin\eta)$. That is the equation of a cycloid, the curve traced by a point on the rim of a rolling wheel, and it is the solid curve in the opening figure.

The dashed curve is what the fall would be if the pull stayed at its starting value. The two begin identically, since the pull has not yet changed, but the real pull grows as the inverse square of the shrinking distance and the real fall ends far sooner, at $\pi/4$ — 79 per cent — of the time the constant pull would take. Most of that difference is made in the last fifth of the fall, where the real pull is many times stronger.

A fall lingers where it starts #

The cycloid hides an asymmetry that is worth making explicit, because it is the opposite of what the end of the fall suggests.

A body released from rest moves slowly at first, because it starts with no speed and the pull is at its weakest. It reaches half its starting distance only after 81.8 per cent of the total time has passed. The second half of the distance takes the remaining 18.2 per cent, and the last tenth takes 1.4 per cent. A fall is a long hesitation followed by a short plunge.

That asymmetry has two practical consequences. The first is that the duration of a long fall depends almost entirely on where it started and hardly at all on where it ended: falling to the surface of the Sun rather than to its centre shortens the fall from the Earth’s distance by about a quarter of an hour in 64.6 days. The second is that a falling body is almost always found near where it started. A cloud of bodies all falling inward from rest at the same moment would, if photographed at a random instant, show most of them still near their starting radius — which is why a collapsing cloud looks, for most of its collapse, like a cloud that has barely begun.

How long anything takes to fall into the Sun #

Since the fall time is a fixed fraction of the circular period at the starting distance, it inherits Kepler’s third law: it grows as the starting distance to the power three-halves.

From Mercury’s distance the fall takes sixteen days, from Mars’s 121, from Jupiter’s 2.1 years and from Neptune’s twenty-nine. From the aphelion of Sedna, nearly a thousand astronomical units out, it would take five thousand years, and from the Oort cloud, where long-period comets spend most of their lives, two million.

The same arithmetic applies to any pair. If the Moon were stopped in its orbit it would fall to the Earth in four days and twenty hours — again a fifth of its circular period, since the Moon’s 27.3-day orbit is nearly circular. A satellite in low orbit, stopped, would hit the ground in about five minutes. A fifth of its ninety-minute orbit, eighteen minutes, is the time to fall all the way to the centre of a point mass; the ground is only four hundred kilometres down, a sixteenth of the way, and that part of the fall is nearly the constant-pull one.

Nothing in the solar system falls this way, because nothing in it has zero angular momentum. But some things come close. A comet perturbed into an orbit that grazes the Sun has an angular momentum tiny compared with that of a circular orbit at its aphelion, and its fall from aphelion to perihelion takes very nearly the radial fall time from the same distance. The comets arriving from the Oort cloud take the corresponding millions of years to fall inward, and the ones that dive closest — the sungrazers — are on orbits that are radial to a fraction of a per cent.

Near-radial orbits keep the same clock #

The radial orbit is the limit of the ellipses at fixed semi-major axis, and the approach to the limit can be followed in time as well as in shape.

Every orbit in the family has the same period, so every curve in the figure returns to its starting point at the same moment. They differ only near the closest approach, where the near-radial orbits swing around the mass at a distance of a tenth, a hundredth and a thousandth of the semi-major axis, and the radial orbit passes through it. Away from that moment the curves are indistinguishable; the angular momentum, however small, matters only at the turn.

That is the property that makes near-radial orbits hard to compute and their limit easy. Close to the mass, a near-radial orbit’s speed is enormous and its direction changes by nearly 360 degrees in a sliver of time, so a numerical integration with a fixed step fails exactly there — which is why the equations are rewritten in a regularised time that slows down near the centre. The limit has no such difficulty, because a radial orbit’s turn is not a turn at all: it is a reversal at the centre, and the cycloid describes it without any special care.

For comets the turn is where the physics is. A sungrazer’s perihelion passage takes hours, during which it is heated, outgassed and tidally stretched by the Sun, while its fall and its climb take centuries. The Kreutz family of sungrazers — fragments of one parent that broke up near perihelion — shares orbits with periods of several centuries and perihelia less than one solar radius above the Sun’s surface, and every fragment’s orbital period was set by the small change in velocity it received during a passage that lasted an afternoon.

A cloud whose shells all arrive at once #

The radial fall is most useful not for single bodies but for clouds, and there it produces a result that seems at first to be a coincidence.

A spherical cloud with no pressure, released from rest, collapses as a set of nested shells, each falling on a radial orbit under the mass inside it. As long as no shell overtakes another, the mass inside each one stays constant, and each shell’s fall time is $\pi\sqrt{r^3/8GM(r)}$, with $M(r)$ the mass it encloses.

In a uniform cloud the enclosed mass grows as the cube of the radius, $M = 4\pi\rho r^3/3$, and the cube in the numerator cancels the cube in the denominator. Every shell takes the same time, $\sqrt{3\pi/32G\rho}$, and the whole cloud collapses homologously to a point in that time, the free-fall time. It depends only on the density: the free-fall time of a molecular cloud with a hundred hydrogen molecules per cubic centimetre is about three million years, and of one with ten thousand, a third of a million.

The free-fall time is the basic clock of star formation. It is what a cloud that cannot support itself against its own gravity collapses on, it sets how fast a galaxy could turn its gas into stars if nothing slowed the process, and the fact that real galaxies turn their gas into stars about a hundred times more slowly than the free-fall time permits is one of the central puzzles of the subject.

A cloud that is denser towards its centre does not collapse together. Its inner shells enclose relatively more mass for their size, fall faster, and arrive first; the outer shells follow later. For a cloud whose density falls as the inverse square of the radius — the density of an isothermal sphere in equilibrium, and a good description of many cloud cores before collapse — the fall time grows in proportion to radius, and the collapse proceeds from the inside out, with a growing central object fed by successively more distant shells. That is the standard picture of how a single star assembles, and it is the radial orbit applied shell by shell.

A fall through a body instead of onto one #

The radial orbit assumes all the mass is at the centre. A fall through the inside of a body is a different motion, and the comparison is instructive.

Inside a uniform sphere, the pull at any radius comes only from the mass inside that radius, which grows as the cube of the radius, so the pull grows in proportion to the distance from the centre. A body falling through a tunnel drilled along a diameter therefore moves as a harmonic oscillator: its fall from the surface to the centre takes a quarter of a period that does not depend on where it started. For a uniform Earth that is twenty-one minutes, from the surface or from halfway down, as a tunnel through any chord of a uniform planet takes the same time.

A body falling onto the same mass concentrated at a point, from the same starting distance, takes about fifteen minutes. It falls faster because the pull grows as it falls, instead of weakening; the point mass’s pull at the end of the fall is unbounded, the uniform sphere’s is zero.

And the whole uniform Earth, released from rest with its pressure switched off, would collapse in the same fifteen minutes, every shell arriving at the centre together. Each shell falls under the mass inside it, which is the same throughout the collapse, so each shell’s fall is a radial orbit about a point mass equal to what it encloses. The free-fall time of a uniform body is the point-mass fall time evaluated at any of its shells, and it is shorter than the time a single stone would take to fall down a tunnel through it while it stood still — because in the collapse, the mass the stone falls towards is falling inward too.

Those three numbers — twenty-one minutes through a standing body, fifteen onto a point, fifteen for the whole body to collapse — are the same order because they are all set by the density. Any fall under a body’s own gravity takes of order one over the square root of $G\rho$, and the differences are geometry.

What was actually measured #

The radial orbit is never observed exactly; what is measured are things whose timing it sets.

Sungrazing comets. Coronagraphs on solar observatories have recorded several thousand small sungrazing comets since 1996, nearly all members of the Kreutz group, most of which do not survive their perihelion passage. Their orbits, computed from a few hours of observation near the Sun, are near-radial to a part in a few hundred.

Collapse times. The ages of young stars in star-forming regions, and the number of cloud cores in different stages of collapse, are compared with free-fall times to infer how long collapse actually takes. Cores without stars are common enough that their lifetimes must be several free-fall times, which says that something — magnetic fields, turbulence, rotation — holds them up for a while before the radial fall begins.

The Earth’s own fall. The Earth does not fall into the Sun, but its angular momentum is a conserved quantity that only a torque could remove, and the 64.6 days is a measure of how much of its orbital motion is sideways: all of it, to within the eccentricity of 1.7 per cent.

Where the line stops #

Point masses. The fall times are computed for a point attracting mass and ignore the radius of the body fallen onto. For the Sun that changes the fall time from the Earth’s distance by about a hundredth of a per cent, because the last few solar radii are crossed in about an hour.

No pressure, no rotation. The cloud figure treats the cloud as pressureless and non-rotating. Real clouds have both, and the angular momentum that must be shed is the main obstacle to their collapse; a cloud with any rotation does not fall radially at its edge, because its outer shells meet the rotational barrier long before they reach the centre.

No shell crossing. The shell argument assumes that shells keep their order. In a uniform cloud they arrive at once and cross at the centre, after which the description fails; in a centrally concentrated one, inner shells reach the centre and form a dense object before outer shells cross them, which is consistent with the assumption up to that point.

The degenerate member is the most used #

The general lesson is about limiting cases. A family of solutions often has a member that is excluded by the conditions under which the family was derived — here, a non-zero angular momentum, without which the orbit equation’s parameter vanishes. Excluded members are easy to dismiss as degenerate. This one is the motion of every collapse that starts from rest, and it inherits the family’s properties — its energy, its period, its parametrisation — because it is a limit of members that have them.

A limit of a family usually keeps the properties that depend only on the quantity held fixed, and discards the ones that depended on the quantity sent to zero. The radial orbit keeps its period because the period depends only on the semi-major axis, which was held fixed; it loses its turn, because the turn depended on the angular momentum, which was not. Choosing which quantity to hold fixed is choosing which properties the limit will keep — which is why the radial orbit and the parabola, both of eccentricity one, are nothing alike.

Still open: the visitors that arrive with energy to spare #

The radial orbit is the boundary of the bound family in one direction, with zero angular momentum. Its counterpart in the other is the hyperbola, with positive energy and an orbit that never returns, and among hyperbolic orbits the most informative are those of bodies from outside the solar system altogether. Two have been seen passing through in six years. What two detections in six years say about how many such bodies there are — and how strongly the Sun’s gravity has concentrated them near the Sun, where they can be seen — is a question about orbits with no period and about the volume of space a survey can see them in.