Orbits

Orbits

falling, and missing

Newton's plate, engraved for the 1728 Treatise of the System of the World. The stone is thrown from the mountain at V, harder each time: it comes down at D, then E, then F, then G, which is most of the way round. The three circles outside the globe are what it does once it stops coming down at all. The plate carries no title; it is captioned only Page 6.

What an orbit is	A fall that keeps missing
Requires	Sideways speed
Does not require	The absence of gravity
First stated as such	Newton, about 1685[1]
At the space station, 420 km up
Gravity there	88 per cent of the surface value[2]
Falls, each second	4.32 metres
Travels, each second	7.66 kilometres
Its orbit drops, over that run	4.32 metres
One circuit	92.8 minutes
To catch something ahead of you
Thrust towards it	You rise, and fall behind
Thrust away from it	You drop, and catch up
Established in flight	June 1965[5]

An orbit is a fall. A body given enough sideways speed still drops towards the ground at the ordinary rate, but the ground curves away beneath it at the same rate, so that it never arrives; and since nothing has been done to slow it, it goes on not arriving indefinitely. The Moon and a dropped apple are the same problem, which is the first great unification in physics and the thing Isaac Newton is chiefly remembered for.^[1]

Almost everything commonly said about orbits is a description of some other arrangement. They are not a balance between gravity and a second force; nothing hovers; and the people inside are not weightless, in the sense of having no weight. They are falling, and so is the floor, which is why they do not meet it.

The mountain

Newton set the whole thing out once, in plain language, for readers who were not going to manage the Principia.

The passage is in De mundi systemate, written in about 1685 and published posthumously in 1728 as A Treatise of the System of the World. A stone is thrown horizontally from the top of a high mountain, and thrown harder each time. It falls a mile away, then two, then a hundred; the air is set aside as a complication; and then:^[1]

…if the velocity was still more and more augmented, it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected.
– Newton, A Treatise of the System of the World, 1728

That is an orbit, defined as a throw that comes back to the thrower without stopping anywhere in between, and it is the only definition that has ever been needed. The plate bound with the passage shows the arcs getting longer, and three closed curves drawn outside the globe for the throws that stop coming down at all. It is reproduced above.

The closed curves on the plate are drawn as circles, and the circle is the special case. Thrown at any other speed the stone traces an ellipse with the Earth's centre at one focus, which is the shape Kepler had got out of Tycho Brahe's observations of Mars in 1609, without knowing that what he had found was the shape of a fall. Newton's result, and the reason a plate like this can be drawn at all, is that under an inverse-square attraction the path is always a conic section, and which one it is depends on nothing but the speed.

Escape speed from the surface is 11.19 kilometres a second, and the parabola is the row that has to be hit exactly. It is the boundary between coming back and not coming back, it has no width, and nothing in nature has ever been observed on it.^[7]

Falling and missing

The claim is quantitative, and it is worth doing the arithmetic rather than admiring it.

Take the space station at 420 kilometres. Gravity there is $GM/r^2$ with $r$ the distance from the Earth's centre, which comes to about 88 per cent of the surface value: the station has not escaped gravity and is nowhere near doing so. A body in free fall under 8.64 m/s² drops 4.32 metres in the first second. In that same second the station travels 7.66 kilometres along its path.

Now ask how far the circle of the orbit itself falls away beneath a straight line over a 7.66-kilometre run. The answer is $r - \sqrt{r^2 - d^2}$, with $d$ the 7.66 kilometres travelled, and it is 4.32 metres.^[2]

The two numbers are equal because that is what an orbit is, and they agree to the last figure quoted because the arithmetic is a definition wearing the clothes of a coincidence. The station falls four and a third metres every second of its existence and has been doing so since 1998.

Nothing is balanced

The usual gloss is that gravity is balanced by centrifugal force. This is not exactly wrong, and that is the difficulty with it.

In a frame of reference that rotates along with the station, a centrifugal term appears and it does cancel gravity. That frame is legitimate and engineers work in it daily. What the cancellation does not do is explain anything, because it is true by construction of any circular motion whatever, including a conker on a string; it is a bookkeeping entry for the fact that the station is not accelerating relative to itself.

In an inertial frame there is one force acting and it is balanced by nothing at all. If it were balanced, the station would obey the first law and go in a straight line, which is precisely what an orbit is not. The force is unopposed, the motion is accelerated the whole way round, and the acceleration points at the Earth every second of every circuit.

There is a third account in which the force is not there either. In general relativity the station is not being pulled off a straight line at all: it is going as straight as anything can through a region the Earth's mass has made not flat, and the person on the ground, held up by it, is the one who is accelerating. For the path of an orbit this low the two agree to more decimal places than anyone needs; for the rate of a clock carried round it they do not, which is a difference GPS pays for continuously. Where they part company visibly is Mercury, whose orbit closes a little later each century than Newton allows, and Einstein computed the difference in November 1915 with nothing adjusted to make it fit.

The GPS and geostationary rows are worth reading twice. A GPS satellite circles in half a sidereal day and a geostationary one in a whole one, which is why the first sweeps overhead and the second appears nailed in place; and the nail is an arithmetical accident of the Earth's rotation rate rather than a property of the height.

Weightless is a sensation

Nothing aboard the station weighs nothing. Everything aboard falls at the same rate, which is a different statement and produces the same view.

A scale reads the force the floor exerts upwards. In free fall the floor exerts nothing, because it is going down as fast as the person standing on it, and the scale reads zero while gravity continues at 88 per cent throughout. The correct term is free fall, or microgravity for the residual accelerations of a large structure;^[4] zero gravity names what it is like and not what it is.

The proof does not require leaving the ground by any great distance. An aircraft flown on a free-fall arc gives its passengers the identical sensation for twenty seconds at a time, at an altitude where nobody would suggest gravity had gone anywhere. Height is not what produces the effect. Falling is, and orbiting is a way of arranging to fall for a long time.

The wrong way round

The consequence that catches people out is not weightlessness. It is that the pedal marked faster is on the wrong side.

Two things in the same orbit, one a little ahead of the other. The natural move is to point at the target and accelerate. Doing so raises the orbit; a higher orbit has a longer period; and the pursuer, now moving on a slower schedule, watches the target draw steadily away. To close the distance the pursuer must brake, drop into a lower and quicker orbit, gain ground from underneath, and rise again at the right moment.

This was established the hard way. On 3 June 1965, in the first hours of Gemini 4, James McDivitt attempted to close on the spent Titan upper stage that had launched him. He flew at it, and it receded and sank; he was expending propellant to make the gap larger. The attempt was abandoned once it was clear what it was costing in propellant, and NASA's own summary is that nobody involved, aboard or in Mission Control, had yet reasoned the mechanics out.^[5] The rendezvous procedures that Apollo later relied on were written afterwards, by people who had watched a spacecraft chase something away from itself.

The air has not finished

There is no altitude at which the atmosphere stops. There is only an altitude at which people stop mentioning it.

Skylab on 8 February 1974, photographed by the last crew to leave it, during their final fly-around. It flew with one of its two main solar wings, the other having been torn away at launch, and under an improvised sunshade. The bright band along the limb is the air the article is about; the station is inside it, and came down five years later.

At 420 kilometres the residual gas is thin past any ordinary meaning of the word, and it is still there, and it still drags. Further out the forces that matter get stranger still: the Pioneer spacecraft were measurably pushed about by the heat leaving their own generators. The space station loses height continuously and is pushed back up several times a year;^[3] left alone it would come down inside about two years. The rate is not constant, because the upper atmosphere is heated and inflated by the solar cycle, so the drag on everything in low orbit rises and falls on an eleven-year period that has nothing to do with the spacecraft.

Skylab is the standing example. Its orbit decayed faster than the planning had allowed for, because the cycle that began in 1976 was stronger than forecast and the atmosphere swelled to meet it; the station re-entered on 11 July 1979, scattering debris across the Indian Ocean and Western Australia. The Shire of Esperance issued NASA a fine of four hundred dollars for littering, which went unpaid for thirty years.^[6]

Checking it yourself

An orbit is among the few things in this encyclopedia that a reader can verify from a garden, without instruments, on a stated evening.

The station's passes are computed from its orbital elements and published to the minute, months ahead. It is bright, it crosses in a few minutes, it does not blink, and it can go out in the middle of the sky rather than setting, when it passes into the Earth's shadow. Anyone who steps outside at the printed time and looks in the printed direction is performing the test, and the test has a result. Amateurs with handheld radios work the station directly and hear its transmissions rise and fall in pitch as it approaches and recedes, a Doppler shift of some kilohertz on the two-metre band, which is not otherwise easy to arrange.

The prediction, the object and the verification are all available at once, which is unusual. Most of what is argued about in these pages was settled by somebody else's instrument a long way off: see the Bedford Level experiment for what happens when the check is available and the checking is done badly anyway.

See also

• Isaac Newton – who put the apple and the Moon under one rule, and drew the mountain
• The Moon – the oldest of these, and the one the whole idea was worked out on
• Johannes Kepler – the three laws the fall obeys, found before anyone knew it was a fall
• The three-body problem – where the exact solutions stop, which is at three
• GPS – thirty-odd of these, each falling for eleven hours fifty-eight minutes a lap
• General relativity – the same motion described as going straight through a shape, with no force in it anywhere
• Earth – the thing being missed
• Gravity – the force this article takes as given, and the one nobody can measure properly
• The Pioneer anomaly – a force on a trajectory that took thirty years to identify, and came from inside

References

1. ^ I. Newton, A Treatise of the System of the World, London, 1728, pp. 5–7, being the English translation of De mundi systemate, drafted in about 1685 and published posthumously. The quotation is from p. 6; the plate faces it and is captioned Page 6. and nothing else. Newton asks the reader to suppose "either that there is no air about the Earth, or at least that it is endowed with little or no power of resisting", which is a larger assumption than it looks: see the section on drag above.
2. ^ Worked from $GM = 3.986004418 \times 10^{14}$ m³/s² and a mean Earth radius of 6,371 km, so that $r = 6{,}791$ km. Then $g = GM/r^2 = 8.643$ m/s², the drop in the first second is $\tfrac{1}{2}gt^2 = 4.32$ m, the speed is $\sqrt{GM/r} = 7.661$ km/s, and the sagitta of the orbital circle over that arc, $r - \sqrt{r^2 - d^2}$, is 4.32 m. The percentages in the table are against $GM/R^2$ for the same mean radius; the station's height varies by some tens of kilometres and the figures move in the third digit accordingly.
3. ^ The station is reboosted using the engines of a docked vehicle, several times a year, and its height is chosen as a compromise between the fuel that costs and the shielding the remaining atmosphere does not provide.
4. ^ Free fall is the general term; microgravity is preferred in the literature because a structure the size of the station is not a point, and its ends are in very slightly different orbits, which produces small residual accelerations that matter to some experiments and to nobody else.
5. ^ Gemini 4, launched 3 June 1965. The rendezvous with the Titan second stage was attempted on the first revolutions and abandoned; the stage was also venting residual propellant and moving unpredictably, and carried only two lights, so its attitude was hard to read. The crew's own account, and NASA's, is that the orbital mechanics of station-keeping had not yet been worked through by anybody.
6. ^ Skylab re-entered on 11 July 1979. Debris reached the ground near Esperance, Western Australia; the Shire issued the fine, which NASA did not pay, and which was settled in 2009 by listeners of an American radio programme. Guinness records it as the first fine for littering from space, which is a category with one member.
7. ^ The parabola requires the speed to be exactly escape speed, a condition of measure zero: any real trajectory is either a very long ellipse or a very slightly hyperbolic one. Long-period comets are routinely called parabolic, which means only that their eccentricity cannot be told from 1 over the arc anybody watched.
8. ^ The Moon's familiar 384,400 km is measured from the Earth's centre, and every other height in the table is measured from the surface, so the numbers are not comparable as printed; 378,000 km is the same distance stated the same way as its neighbours. The Moon is also the one row that is not a test particle. Its own mass shortens the period: the two-body formula with the Earth's $GM$ alone gives 27.45 days, and the sidereal month is 27.32.

External links

• A Treatise of the System of the World (London, 1728) – the full scan; the mountain is on p. 6 and the plate faces it
• Gemini IV – NASA's mission page, including the abandoned rendezvous
• Spot the Station – NASA's pass predictions, by which any reader can check the above