The balloon that floats at a density

The weight of the water that is not there found Archimedes’ principle: a body in a fluid is pushed up by the weight of the fluid it displaces. Why a ship comes back upright found what keeps a floating body the right way up, and the essays after it found the places the principle is misapplied: the block the water does not lift, the depth past which it must sink, where a compressible body below a certain depth has no way back up, the balloon that leans the wrong way in an accelerating car, and the square log that floats on its corner. Every one was set in a liquid, or in air near the ground, where the fluid’s density is the same at the top of the body as at the bottom, and the same an hour later.

The atmosphere is not like that. Its density falls by a factor of about three in every eight kilometres, a hundredfold by thirty, and a body that rises into it finds itself in a thinner fluid with every metre. A helium balloon let go therefore does not simply float or sink; it climbs into air that supports it less, until something stops it. What stops it, and where, depends on the balloon’s construction, and the three common kinds of balloon end their climbs in three different ways. In each case the answer is a density, not a height.

Air that thins as it rises #

The atmosphere’s density follows from hydrostatics: each layer of air carries the weight of all the air above it, so pressure falls with height by the weight of each layer, and the gas law turns pressure and temperature into density. The figure integrates the International Standard Atmosphere, the agreed average used to calibrate altimeters: 1.225 kilograms per cubic metre at sea level, 0.364 at eleven kilometres where airliners cruise, 0.088 at twenty, 0.018 at thirty and 0.004 at forty. The density falls by a factor of $e$ every six to eight kilometres, a little more slowly where the air is warmer, since warm air of a given pressure is less dense and its layers are thicker. Why the air thins with height derived that exponential; here it is the scale a balloon climbs.

A balloon’s lift is the mass of air it displaces minus the mass of its gas, times $g$. Helium at the air’s pressure and temperature is lighter than air in the ratio of their molecular masses, four to twenty-nine, so each cubic metre of helium in air lifts 86 per cent of the air’s density. At sea level that is just over a kilogram per cubic metre; at forty kilometres, three grams.

The lift that does not change #

A zero-pressure balloon — the kind that carries scientific instruments to the edge of space — is a huge, thin plastic envelope, open at the bottom through ducts, so that its gas is always at the pressure of the air around it. At launch it holds only a small fraction of its volume of helium, and the envelope hangs limp above the gas bubble at its top.

As it rises the surprising thing is that its lift does not change. The gas inside is at the air’s pressure, and as the pressure falls the gas expands, in exactly the proportion the air thins: half the pressure, twice the volume. The mass of air displaced is the density of the air times the volume of the gas, and since one halves as the other doubles, the displaced mass is the same at every height. The lift of an underfilled zero-pressure balloon is fixed at launch by the amount of helium and stays fixed all the way up. The figure shows it flat at 3.69 tonnes for the smaller fill and 5.28 for the larger, against three tonnes of weight.

The flat line ends when the envelope is full. From then on the gas cannot expand further; any excess spills out through the open base, the volume stays at its maximum, and the lift falls as the air thins. The balloon floats where the lift has fallen to its weight: where the air’s density times the full volume times 86 per cent equals the mass carried. For this balloon that is 41.3 kilometres, and the striking feature of the figure is that both fills reach it. More helium makes the balloon climb faster and fill lower, and then the surplus is simply vented. The float height is set by the size of the envelope and the mass it carries, and by nothing else.

Three ways to stop climbing #

The three kinds of balloon differ only in what happens when the gas inside wants more room. All three start the same way: the gas expands as the pressure falls, three times by ten kilometres, fourteen times by twenty. The zero-pressure balloon’s envelope fills and it vents. A superpressure balloon is sealed and nearly unstretchable, launched partly full; when it reaches its fixed volume it stops growing, and the gas inside, unable to expand further, ends up at a higher pressure than the air outside — hence the name. A latex weather balloon is sealed and elastic, launched fully inflated, and it just keeps stretching, at nearly the air’s pressure, until the rubber gives way. One that can stretch to seventy-five times its launch volume, four times its launch diameter, bursts at about thirty kilometres, and its instruments come down on a parachute.

The weather balloon never floats; it rises at a steady few metres a second, because its lift, like the open balloon’s, does not change on the way up while the drag on its growing surface roughly balances it, and it bursts at the density where its rubber runs out. The other two float, and they float differently.

Floating at a density #

A superpressure balloon has a fixed volume and a fixed mass — its envelope, its payload and its sealed-in gas. Its average density is fixed, and Archimedes’ principle says it floats where the air has that density. The sealed balloon of the first figure, 1,500 cubic metres weighing 95 kilograms with its gas, has an average density of 0.063 kilograms per cubic metre and floats at twenty-two kilometres in the standard atmosphere. On a day when the stratosphere is warmer and its air thinner at that height, it floats lower; on a colder day, higher. What it finds is the density, and the height is wherever the atmosphere has put that density that day.

The same is true of the open balloon at float, with the difference that its mass is not fixed: it carries ballast to drop, and its gas can vent. It floats where the air’s density makes the lift of its full volume equal its mass, and dropping ten per cent of its mass as ballast sends it up to where the air is ten per cent thinner, about seven hundred metres higher. Balloon flights are controlled entirely this way, by dropping weight to climb and venting gas to descend.

The night that makes an open balloon fall #

The difference between the two shows itself at sunset.

By day, sunlight warms a balloon’s gas several kelvin above the air around it; at night the gas cools to the air’s temperature or below, because it radiates to the cold sky. An open balloon’s gas is at the air’s pressure, so cooling shrinks it, by one per cent for every two or three kelvin at stratospheric temperatures, and a smaller volume displaces less air. The balloon, no longer full, loses lift — five per cent for ten kelvin, ten for twenty — and starts to sink. Worse, a slack open balloon’s lift does not change with height, as the flat lines of the second figure showed: descending into denser air does not help it. It keeps sinking until it drops ballast or the sun rises. Long-duration zero-pressure flights carry a tenth or more of their mass as ballast for each night, and run out within days unless they fly in the continuous sunshine of a polar summer.

The superpressure balloon does not care. Its volume is fixed; the night’s cooling lowers its gas’s pressure, which only reduces how much higher than the outside pressure it is, and its displaced air and its mass are both unchanged. As long as the gas stays above the outside pressure — which is what the design’s margin of superpressure is for — the balloon stays at its density level through the night. That is why superpressure balloons can fly for months; NASA’s science superpressure balloons have circled the Southern Hemisphere for more than a month, and the balloons used to carry internet relays over remote regions stayed up for several months.

The comparison that decides stability #

Why one design holds its level and the other does not is the same question the depth past which it must sink asked of a Cartesian diver, and it has the same answer: it depends on how compressible the body is compared with the fluid it floats in.

A body that is less compressible than its fluid is stable. Push it up, into thinner fluid, and it expands less than the fluid around it has thinned, so it is now denser than its surroundings and sinks back; push it down, and it is lighter than its surroundings and rises back. A sealed superpressure balloon, whose volume cannot grow, is far less compressible than the air, and it is held at its level as by a spring. A full open balloon at float, venting gas when pushed up and gaining none back when pushed down, is also held, more loosely.

A body that is more compressible than its fluid is unstable: push it down and it is squeezed more than the fluid, becomes denser than it, and sinks further. That is the diver, a pocket of air in water, and the reason a scuba diver who sinks past a certain depth must swim or inflate to come back. And a body exactly as compressible as its fluid is neutral: wherever it is put it stays, and nothing pushes it back. An underfilled open balloon, a bubble of gas at the air’s own pressure and temperature, is exactly that. It neither returns nor flees; it simply has whatever lift its gas gave it, at every height, which is why it climbs steadily by day and sinks steadily on a cold night.

Oceanographers use the same comparison deliberately. Seawater is slightly compressible — the sea that stands lower because water gives found it 4.7 per cent denser at the bottom of the ocean than at the top — and an instrument float with a rigid hull, less compressible than the water, settles at the depth where the water’s density matches its own and returns there if displaced, which is how the thousands of profiling floats that measure the world’s oceans hold their parking depths between dives. A float built instead to be exactly as compressible as seawater has no preferred depth and is carried up and down with the water around it, following a parcel of the ocean as it moves; such floats are made precisely to track the water’s own vertical motion. The open balloon at night is the accidental version of that design.

Bobbing about the level #

A stable balloon displaced from its level oscillates about it. For a sealed balloon the restoring force per metre of displacement is its volume times $g$ times the rate at which the air’s density changes with height, which is the density divided by its scale height $H$, about six kilometres in the lower stratosphere. The mass that has to be moved is the balloon’s own, equal to the air it displaces, plus about half as much again of air that moves with it. The period that results, $2\pi\sqrt{1.5H/g}$, is about three minutes and contains neither the balloon’s size nor its mass: every sealed balloon in that part of the sky bobs at the same rate, as every pendulum of a given length swings at the same rate whatever its bob.

The atmosphere has a bobbing period of its own, the buoyancy period of a parcel of air displaced vertically, which the layer a parcel cannot leave found from the air’s stratification. It is longer, around five minutes in the stratosphere, because a parcel of air expands as it rises and so becomes less dense than a sealed balloon would, which weakens its restoring force. Waves in the stratosphere at periods near these shake real superpressure balloons up and down by tens to hundreds of metres, and the balloons’ records of that motion are used to measure the waves.

A weather balloon’s arithmetic #

The weather balloon deserves its own sum, because it is launched twice a day from about nine hundred stations round the world and is the most common balloon there is. Filled with a few cubic metres of helium or hydrogen, enough to lift its instruments with a kilogram or two to spare, it rises at about five metres a second, reaching thirty kilometres in under two hours. Its lift does not change on the way up, for the same reason the open balloon’s does not: a sealed, slack, stretchy balloon at nearly the air’s pressure is a fixed amount of gas displacing a fixed mass of air. Its rubber stretches as the gas expands, and the burst height is set by how far the latex can stretch before it fails — a better balloon, or one launched less inflated, bursts higher, since it has more stretching left. The thirty kilometres in the figure correspond to a balloon that can grow to seventy-five times its launch volume.

The instruments measure temperature, humidity and pressure on the way up, and the drift of the balloon, tracked by satellite navigation, gives the wind at every level. Those soundings are among the most important inputs to every weather forecast, and the whole technique rests on the flat line: a balloon whose lift does not depend on height rises at a predictable rate, so its height at any moment can be checked against the pressure it reports.

What the pictures cannot show #

The figures use the standard atmosphere, an annual average for middle latitudes; the real atmosphere varies by tens of per cent in density at a given height between seasons, latitudes and days, which moves every float height by kilometres. They ignore the balloon’s temperature difference from the air in daytime, the weight and stretchiness of real envelopes, the small superpressure inside a latex balloon, and the drag that sets a rising balloon’s speed. The night figure assumes the gas cools while the air does not and that the open balloon was exactly full at sunset; real flights also lose lift through slow leaks and gain it from the warming of a sunlit envelope. The bobbing figure uses an added mass of half the displaced air, a value for a sphere, and an illustrative damping.

Still open: steering a balloon by choosing its density #

A balloon cannot steer, but the winds at different heights blow in different directions, and a balloon that can change its float density can choose which wind to ride. Superpressure balloons fitted with a second, internal bladder that pumps air in or out to change their total mass have been flown for exactly this, holding position over a region for weeks by moving between wind layers a few kilometres apart. How well such navigation can be done depends on how well the winds are forecast at every height, and the balloons themselves are now among the instruments measuring those winds; whether stratospheric balloons can hold station well enough to replace satellites for some tasks is being tried.

The habit worth carrying away is to ask what a floating body keeps fixed as its surroundings change. An open balloon keeps its lift fixed while it fills, because its gas thins exactly as the air does, and then floats where its full volume’s lift equals its weight; a sealed one keeps its density fixed and floats where the air matches it; a stretchy one keeps stretching and bursts — 41.3, 22.0 and 30.6 km for the three balloons drawn here. Which of them holds its level is decided by whether it is less compressible than the air, the same comparison that sinks a diver.