Gas pressure needs a container

Gas pressure needs a container

an objection, and its arithmetic

Both bodies in one exposure, from the International Space Station. The Moon has an edge. The Earth's atmosphere has none: it grades from white through blue into black, and the reader is invited to say at which pixel it stops.

The claim	A gas cannot lie against a vacuum unheld
The premise	Correct
The conclusion	Does not follow
The container	A gravitational well
The numbers
Scale height	8.4 km[1]
Pressure at that height	37 per cent of sea level
At 50 km	0.27 per cent
Escape velocity	11.2 km/s
Speed of a nitrogen molecule	0.51 km/s
Shortfall	A factor of twenty-two
The leak
Rate	About 90 tonnes a day[4]
Chiefly	Hydrogen, and some helium
Original hydrogen remaining	Essentially none
Upper boundary	There is not one
Figure usually given	Set by a sporting body

Gas pressure needs a container is an objection raised against the ordinary account of the atmosphere: a gas under pressure will expand to fill whatever is available to it, so an atmosphere at one bar cannot simply lie against the vacuum of space with nothing between them. Something must be holding it in, and that something is taken to be a solid sky: the firmament, which the flat-Earth cosmographies require for other reasons and are glad to have a physical argument for.^[2]

The premise is entirely correct and is worth granting without qualification. A gas does expand into a vacuum; the Earth's atmosphere is expanding into one at this moment, at something like ninety tonnes a day; and essentially all of the hydrogen the planet started with has already left by exactly the route the objection describes. What does not follow is the lid. The container is a gravitational well, the arithmetic runs to two lines, and it can be checked against other worlds.

What holds it

Périer reading the Torricelli tube on the Puy de Dôme in 1648, engraved by Yan Dargent two centuries later. Pascal sent his brother-in-law up a mountain to see whether the mercury would fall. It fell by about three inches, which is the first measurement in this article and settled that the sea of air has a top, or at any rate a thinning.

Take a column of air in equilibrium. The pressure at any height must support the weight of everything above it, which gives a differential equation with an exponential solution: pressure falls by a constant factor for each fixed increase in height. That fixed distance is the scale height,

with $k$ Boltzmann's constant, $T$ the temperature, $m$ the mean mass of an air molecule and $g$ the surface gravity. For air at 288 K this comes to 8.4 kilometres.^[1]

The consequence is the answer to the objection. At one scale height the pressure is 37 per cent of sea level; at 50 km, a quarter of one per cent; at 100 km, near enough nothing. The atmosphere is not held in at a boundary. It thins away, and it never quite stops. There is no surface for a container to be pressed against, because there is no surface.

The intuition behind the objection is a good one and comes from the laboratory, where a gas is kept in a vessel because over the size of a vessel gravity does nothing worth measuring. Over eight and a half kilometres it does a great deal. The mistake is one of scale, not of physics, and it is the same shape as the one made with eight inches per mile, squared: a true statement, correctly recalled, applied where it does not reach.

The Moon and Titan

The claim can be tested, because the solar system has run the experiment several times with different settings.

A body keeps a gas if its escape velocity is comfortably greater than the speed the gas molecules are moving at, which for a molecule of mass $m$ at temperature $T$ goes as $\sqrt{3kT/m}$. The usual rule of thumb is that the escape velocity must exceed the molecular speed by a factor of about six for the gas to survive over geological time.

Titan, backlit. The violet ring is the atmosphere, and it goes all the way round.

The last two rows are the argument. Titan and the Moon have almost the same escape velocity – 2.64 kilometres per second against 2.38, a difference of about a tenth. Titan carries a nitrogen atmosphere at 1.45 bar which, being cold, is four times as dense at the surface as sea-level air on Earth.^[3] The Moon carries about a hundred molecules per cubic centimetre, against Earth's twenty-seven billion billion: seventeen orders of magnitude, which is to say nothing at all.

Sharper still: Titan pulls less hard than the Moon does. Its surface gravity is 1.35 m/s² against the Moon's 1.62. It has the deeper well only because it is the larger body, so the escape velocity is higher while the pull underfoot is weaker.

Neither has a container. The variable is temperature: Titan sits at 94 K and the Moon's day side reaches 390 K, and the molecular speeds follow. If a gas required a wall, the two worlds would not differ, and the one you would weigh less on would not be the one with the weather.

Confinement without a wall

It is worth saying plainly that walls are the least of the ways a gas is held.

The physics of hot confined gas recognises three methods, and they are named. Gravitational confinement is what stars do and what this article is about; magnetic confinement is what a tokamak does, holding a plasma at a hundred million degrees in a field so that it never reaches the vessel; and inertial confinement is what a laser implosion does, holding a fuel pellet together for a few billionths of a second by the sheer inertia of matter that has not yet had time to move apart.^[6]

A wall is not on the list. In a tokamak there is a vacuum vessel, and it is emphatically not the container: its function is to be the thing the plasma must be kept off, since anything the plasma touches it ruins and anything that touches the plasma cools it. The confinement is done by the field.

And the extreme case of gravitational confinement is overhead in the daytime. The core of the Sun sits at something like 250 billion atmospheres, held by nothing whatever but the weight of the Sun, with no surface, no vessel and no lid. Whatever difficulty there may be in believing that gravity can hold one bar of nitrogen against a vacuum, it is a difficulty of the same kind, eleven orders of magnitude smaller.

It does leak

Having granted the premise, the article owes the reader the rate.

The Earth loses about ninety tonnes of atmosphere a day, rather more than a kilogram every second.^[4] Almost all of it is hydrogen, at some three kilograms a second, with helium at about fifty grams. The mechanism is the one the objection imagines: at the top of the atmosphere the gas is thin enough that a molecule travelling upward is unlikely to hit anything, and if it is moving fast enough it simply leaves.

This is why the numbers above matter rather than the principle. Run the same ratio for hydrogen and the Earth comes out at 5.9, which is below the retention threshold, and the record agrees: the planet's original hydrogen is gone. Helium comes out at 8.3, marginal, and helium is likewise escaping and is replaced from below by radioactive decay. Nitrogen comes out at 22 and is going nowhere.

So the atmosphere is not sealed in. It is sorted, by molecular mass, and what is left is what is too heavy to get out. Four and a half billion years of the objection being correct have produced the air currently in the room.

Where the atmosphere ends

It does not.

Above about 500 kilometres the gas is so sparse that molecules follow ballistic arcs between collisions rather than behaving as a fluid; this region is the exosphere, and it grades into the solar wind without a boundary anywhere. Traces of the atmosphere are detectable far beyond the Moon's orbit.

The line usually quoted for the edge of space, 100 kilometres, is not a physical feature. It was adopted by the Fédération Aéronautique Internationale, a sporting body, and whether it follows von Kármán's calculation or is simply a round number in a metric unit is disputed.^[5] The United States has never used it, preferring 80 kilometres; in 2018 an analysis of the orbits actually achieved by some fifty satellites concluded that 80 was the better figure, several having held orbit that low and none below about 70.

The encyclopedia notes the shape of this without comment. Asked where the sky stops, the answer is that it does not stop, and that the number in general use for where it stops was chosen by a committee that awards records for ballooning.

See also

• Firmament – the lid this objection is raised in support of
• Flat Earth – the doctrine that requires something to contain its sky
• Eight inches per mile, squared – another talking point that turns out to be real arithmetic, correctly stated and wrongly applied
• Moon – the control experiment, and the reason it has no air
• Sun – gravitational confinement at 250 billion atmospheres, with no vessel at all
• Earth – the atmosphere in question, and what is left of it

References

1. ^ $H = kT/mg$. Taking $T = 288$ K, the mean molecular mass of dry air as 28.96 u and $g = 9.807$ m/s², this gives 8.43 km. The figure varies with temperature and so with latitude and season; 8.5 km is the usual round value. The pressure at height $h$ is then $P_0 e^{-h/H}$, from which the percentages in this article are calculated directly.
2. ^ The objection circulates in several forms, of which "gas pressure needs a container" is the most compact. It is generally offered as a decisive point rather than a question, which is a pity, because as a question it is a good one and the answer is instructive.
3. ^ Titan's surface pressure is about 147 kPa and its surface temperature about 94 K, giving an atmospheric density near 5.3 kg/m³ against 1.2 for sea-level air on Earth. It is chiefly nitrogen, with a few per cent methane. The comparison with the Moon is not a rhetorical flourish: the two escape velocities differ by about a tenth, and everything else about the outcome differs by seventeen orders of magnitude.
4. ^ About one kilogram a second in total, of which roughly three kilograms a second is hydrogen and fifty grams helium; the totals are not straightforwardly additive because the mechanisms and the accounting differ between sources. The order of magnitude is not in dispute.
5. ^ The FAI adopted 100 km in the early 1960s. Von Kármán's own reasoning concerned the altitude at which an aircraft would need to fly at orbital speed to generate lift, which lands in the region of 80 km rather than 100. J. McDowell, Acta Astronautica (2018), argues for 80 ± 10 from the orbits satellites have actually held. The line has therefore been in the wrong place, by its own inventor's reasoning, for the whole of the space age, and nothing whatever has depended on it except who is called an astronaut.
6. ^ The three-way division is the standard one in fusion research, where gravitational confinement is listed and then set aside as unavailable to the engineer, there being no way to arrange a star in a laboratory. Magnetic and inertial confinement are the two that are built. Inertial confinement is the odd name of the three, because nothing confines anything: a fuel pellet a couple of millimetres across is struck from every side at once, its outer layer blows off, the recoil drives the rest inward to densities beyond that of lead, and the fusion is got over with in about a nanosecond. The fuel is held together only by not yet having had time to leave. It is confinement in the sense that a thrown ball is confined.

External links

• Statement about the Kármán line – the Fédération Aéronautique Internationale on its own boundary
• The curious case of Earth's leaking atmosphere – the European Space Agency on the escape rate
• Atmosphere of Titan – the pressure, temperature and composition used above
• The Moon, Earth's limb and the thin blue atmosphere – the photograph reproduced above, taken from the International Space Station by an Expedition 51 crew member; NASA, via Wikimedia Commons
• Titan's halo – the Cassini image reproduced above, PIA07774; NASA/JPL, via Wikimedia Commons
• Périer on the Puy de Dôme – the Yan Dargent engraving reproduced above, from the Library of Congress via Wikimedia Commons