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I don't remember the exact value, but because of the solar wind, a huge mass of air just disappears into space everyday.
If it's replaced by gas from inside the Earth, it wouldn't contain any oxygen. But ignoring gas from the Earth, if we lose air at that rate and it doesn't get replaced, how long before sea level air pressure gets too low for humans to breathe? A billion years? A trillion?
Would evaporation from water in the ocean make the air pressure last longer?
Most of the Earth's atmosphere is quite close to the surface. The outer layers are very tenuous, and are primarily composed of hydrogen and helium. The lower regions of the atmosphere are well mixed, but stratification occurs in the upper layers.
There isn't much free hydrogen in the lower atmosphere, but in the upper atmosphere UV light splits water molecules into hydrogen and oxygen.
The Earth contains helium produced by the alpha decay of various radioactive isotopes, it also contains some primordial helium. The helium that leaks out of the crust rises into the upper atmosphere.
From Wikipedia Atmosphere of Earth
Air pressure actually decreases exponentially with altitude, dropping by half every 5.6 km (18,000 ft) or by a factor of 1/e (0.368) every 7.64 km (25,100 ft), (this is called the scale height) - for altitudes out to around 70 km (43 mi; 230,000 ft). However, the atmosphere is more accurately modeled with a customized equation for each layer that takes gradients of temperature, molecular composition, solar radiation and gravity into account. At heights over 100 km, an atmosphere may no longer be well mixed. Then each chemical species has its own scale height.
In summary, the mass of Earth's atmosphere is distributed approximately as follows:
- 50% is below 5.6 km (18,000 ft).
- 90% is below 16 km (52,000 ft).
- 99.99997% is below 100 km (62 mi; 330,000 ft), the Kármán line.
The outermost layer of the atmosphere is known as the exosphere
The exosphere, observable from space as the geocorona, is seen to extend to at least 100,000 kilometres (62,000 mi) from Earth's surface
The exosphere is protected (to an extent) from the effects of solar radiation pressure and the solar wind by the Earth's magnetosphere. The base of the exosphere is at an altitude of ~700 km, well above the orbit of the ISS.
Here's a diagram of the exosphere / geocorona (from NASA / GSFC, via Wikipedia):
According to Wikipedia Atmospheric escape, the exosphere is currently losing about 3 kg/s of hydrogen and approximately 50 g/s of helium.
Sources linked in https://physics.stackexchange.com/q/548056/123208 claim that the Earth's atmosphere loses helium equivalent to its current total helium content in roughly one million years.
According to the American National Center for Atmospheric Research (via Wikipedia), the total dry mass of the atmosphere is ~$5.132 \pm 0.0003 × 10^{18}$ kg, and its mean water content is ~$1.27 × 10^{16}$ kg. The actual water content varies with the weather.
Using those numbers to make a crude estimate of the time scale to lose all the atmosphere, we get ~53.5 billion years.
Of course, the actual atmosphere isn't composed of just hydrogen and helium. :) But we would need to lose most of our hydrogen before the heavier molecules like nitrogen, oxygen, and carbon dioxide become significant components of the exosphere and get exposed to the solar wind and radiation pressure.
Hydrogen lost from the upper exosphere is replaced by fresh hydrogen produced by UV photodissociation, as I mentioned earlier. So we won't start losing appreciable quantities of oxygen etc until we've lost the ocean.
From Wikipedia Origin of water on Earth, the mass of the water on the Earth is ~$1.87 × 10^{21}$ kg. There's probably 3× that quantity of water locked up in the mantle (mostly as various hydrated minerals), and ~4-5× that quantity in the core.
The hydrogen content of water (by mass) is 1/9. So the mass of hydrogen in the surface water is ~$2.08 × 10^{20}$ kg, and at the current rate it would take ~2.16 trillion years to lose all our hydrogen.
However, the hydrogen loss rate will not remain constant. The Sun is gradually becoming more luminous. According to Wikipedia Timeline of the far future, in ~1.1 billion years,
The Sun's luminosity will have increased by 10%, causing Earth's surface temperatures to reach an average of around 320 K (47 °C; 116 °F). The atmosphere will become a "moist greenhouse", resulting in a runaway evaporation of the oceans.
The oceans will be almost completely evaporated in ~2 billion years, and by 2.8 billion years,
Earth's surface temperature will reach around 420 K (147 °C; 296 °F), even at the poles.
By 3.5 - 4.5 billion years,
The greenhouse effect caused by the massive, water-rich atmosphere will result in Earth's surface temperature rising to 1,400 K (1,130 °C; 2,060 °F)—hot enough to melt some surface rock.
So the Earth will become extremely unliveable long before it loses its atmosphere.
Short Answer:
According to my rough calculations at the present rate that Earth is losing its atmosphere into space, it should take over a trillion years - 1,000,000,000,000 years - for Earth to lose all its atmosphere.
Long Answer:
Planetary retention of atmosphere is discussed in Habitable planets for Man Stephen H. Dole, 1964.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
Pages 34 to 35 discuss a rough formula for estimating how long a planet can retain a particular gas in its atmosphere. The amount of time it takes for a planet or other world to retain a gas in its atmosphere depends on the ratio of the world's escape velocity (NOT its surface gravity) divided by the root-mean-square velocity of atoms of that gas in the escape layer of the atmosphere.
Gases escaped from the atmosphere in the exosphere of the atmosphere, the outermost layer where the gases are almost a rarified as in interplanetary space.
The root-mean-square velocity of a gas in the exosphere depends on the mass of an atom and its temperature. The lighter an atom is, the faster its velocity will be at a specific temperature. The temperature of gases in an exosphere doesn't depend on the total amount of stellar radiation, but apparently only on the amount of high ultraviolet radiation. So the cooler a star is, and the less ultraviolet it emits, the lower the escape velocity necessary to retain a gas for a certain amount of time.
Table 5 on page 35 gives the amount of time it will take for the amount of gas in an atmosphere to fall to only 1/e, or 0.3678 of the original amount.
If the ratio of the escape velocity divided by the root-mean-square velocity of that gas in the exosphere is one, the amount of gas will fall to 0.3678 of the original amount instantly.
If the ratio of the escape velocity divided by the root-mean-square velocity of that gas in the exosphere is two, the amount of gas will fall to 0.3678 of the original amount instantly.
If the ratio of the escape velocity divided by the root-mean-square velocity of that gas in the exosphere is three, the amount of gas will fall to 0.3678 of the original amount in a few weeks.
If the ratio of the escape velocity divided by the root-mean-square velocity of that gas in the exosphere is four, the amount of gas will fall to 0.3678 of the original amount in several thousand years.
If the ratio of the escape velocity divided by the root-mean-square velocity of that gas in the exosphere is five, the amount of gas will fall to 0.3678 of the original amount in about a hundred million years.
If the ratio of the escape velocity divided by the root-mean-square velocity of that gas in the exosphere is six, the amount of gas will fall to 0.3678 of the original amount in about an infinite length of time.
Page 15 says that humans need a partial pressure of oxygen of 60 to 400 millimeters of mercury (mmHg) to survive. At sea level on Earth the pressure of oxygen is 160 mmHg. If it fell to 0.3678 of that it would fall to about 58 mmHg.
On page 54 Dole says that the temperature in Earth's exosphere varies between 1000 degrees K and 2000 degrees K. Dole says that at 1000 K the velocity of atomic oxygen would be 1.25 kilometers per second. And Dole says that if a world has an escape velocity of five time that, or 6.25 kilometers per second, it will retain 0.3678 of its original oxygen for about a hundred million years.
The escape velocity of Earth is about 11.186 kilometers per second. Six times 1.25 kilometers per second is 7.5 kilometers per second. So Earth should be able to retain atomic oxygen with a temperature of 1000 K for an infinite period.
But atomic oxygen in the exosphere can reach temperatures as high as 2000k. As long as the speed at 2000 K is slower than 1.8643 kilometers per second Earth can retain the hottest exosphere oxygen for an infinite period. As long of the speed at 2000 K is is lower than 2.2372 kilometers per second, Earth can retain 0.3678 of even the hottest atomic oxygen for about one hundred million years.
There are other effects which can cause atmospheric loss.
Earth looses several hundred tons of atmosphere into space each day.
https://earthobservatory.nasa.gov/images/144386/toward-mapping-the-atmospheres-escape-from-earth
If that is 100 to 1,000 tons per day, the loss rate will be about 365.25 to 3,625 tons per year, 3,625 to 36,525 tons per decade, 36,525 to 365,525 tons per century, 365,250 to 3,652,500 tons per millennium, 3,652,500 to 36,525,000 tons per 10,000 years, 36,525,000 to 365,250,000 tons per 100,000 years, 365,250,000 to 3,652,500,000 tons per million years, 3,652,500,000 to 36,525,000,000 tons per ten million years, 36,525,000,000 to tons per 365,250,000,000 hundred million years, and 365,250,000,000 to 3,652,500,000,000 tons per billion years.
365,250,000,000 tons would be 3.6525 times 10 the 11th power tons, and 3,652,500,000,000 tons would be 3.6525 times 10 to the 12th power tons, lost in a billion years.
Earth's atmosphere has a mass of about 5.15 times 10 to the 18th power kilograms. A kilogram is about 0.00110231 tons. So the mass of Earth's atmosphere is about 0.005676896 times 10th to the 18th power tons, or 5.676896 times 10 to the 15th power tons.
So at the current rate of atmospheric loss it should take about 1,554 to 15,542 times a billion years for the Earth to lose its atmosphere.
