The escape of atmospheres

Builds on: Sections 3.2 and S4
Reading time: 25 min
Exercises: 4
Olympiad problems: 2

The Moon has no air, while Titan, a moon of Saturn only 1.5 times the Moon's size, has an atmosphere thicker than Earth's. The difference is temperature. Titan is so cold, at about 94 K, that its nitrogen molecules move too slowly to escape its weak gravity, while on the Moon's sunlit side, near 390 K, they would be lost. Earth, in turn, keeps its nitrogen and oxygen but has lost almost all of its hydrogen and helium. In this section we compare the speeds of gas molecules with the escape speed of a planet to see which gases it can keep.

3.10.1Thermal speed against escape speed

Before you readMake a guess first

A planet holds a gas only if its molecules are slow enough. Which gas does Earth lose most easily?

The molecules of a gas at temperature $T$ move at a spread of speeds, with a root-mean-square speed $v_{rms} = \sqrt{3 k T / m}$ ((S.26)). A few move much faster than this. A molecule high in the atmosphere that moves outwards faster than the escape speed ((S.21)), and does not hit another molecule on the way, is lost. Even if the average speed is well below the escape speed, the fastest few keep leaking away, and over billions of years they can empty the atmosphere of that gas. Detailed calculations of this leak give a working rule.

ResultKeeping an atmosphere(3.22)

\begin{aligned} v_{rms} & < \frac{v_{esc}}{6} \\ m & > \frac{54 k T R}{G M} \end{aligned}

$v_{rms}$	the root-mean-square speed of the gas's molecules, $\sqrt{3 k T / m}$
$v_{esc}$	the escape speed of the planet, $\sqrt{2 G M / R}$
$m$	the mass of one molecule
$T$	the temperature of the gas high in the atmosphere, where molecules escape
$M, R$	the mass and radius of the planet

A planet keeps a gas for the age of the Solar System if the first line holds, and loses most of it if not. The second line is the first turned round, the smallest mass of molecule that the planet can keep. The factor 6 is a rule of thumb, and some problems use 10 in its place. The problem's own value should always be used.

Quick checkOxygen on Mercury

Mercury's escape speed is 4.25 km/s and its sunlit side reaches about 700 K. What is six times the root-mean-square speed of oxygen molecules, of mass 32.00 u, at 700 K, and can Mercury keep them?

km/s

The temperature that matters is the temperature where molecules can escape, high in the atmosphere. Earth's thin upper atmosphere, the exosphere, is heated by ultraviolet light from the Sun to about 1000 K, much hotter than the ground.

Example 3.15Which gases Earth keeps

Earth's escape speed is 11.19 km/s and its exosphere is at about 1000 K. Which of hydrogen atoms (1.008 u), helium (4.003 u), nitrogen molecules (28.01 u) and oxygen molecules (32.00 u) can Earth keep? One atomic mass unit u is $1.661 \times 10^{- 27}$ kg.

Solution

By (S.26) at 1000 K, the root-mean-square speeds are 4.97 km/s for hydrogen, 2.50 km/s for helium, 0.94 km/s for nitrogen and 0.88 km/s for oxygen. Six times these are 29.8 km/s, 15.0 km/s, 5.66 km/s and 5.30 km/s.

AnswerBy (3.22), Earth loses hydrogen and helium, whose six-fold speeds are above 11.19 km/s, and keeps nitrogen and oxygen, which are well below it. This is why Earth's air holds only a trace of helium, and why the hydrogen in water stays on Earth only while it is bound into water molecules low in the atmosphere.

Example 3.16Why the Moon has no airfinish it

The Moon has a mass of $7.342 \times 10^{22}$ kg and a radius of 1737 km, and its sunlit side reaches about 390 K. Could the Moon keep nitrogen, of molecular mass 28.01 u?

The start

The Moon's escape speed is $\sqrt{2 G M / R} = 2.38$ km/s. At 390 K, nitrogen molecules have $v_{rms} = \sqrt{3 k T / m} = 0.589$ km/s.

Your stepNow find six times the root-mean-square speed and compare it with the escape speed.

km/s

Common errorsSection 3.10

The temperature to use is that of the upper atmosphere, where molecules escape. For Earth this is about 1000 K, much hotter than the ground.
The mass in $v_{rms} = \sqrt{3 k T / m}$ is the mass of one molecule in kilograms. A molecular mass in atomic mass units, or a molar mass in grams per mole, must be converted first, with $1 u = 1.661 \times 10^{- 27}$ kg.
A gas whose average speed is below the escape speed can still be lost. The fastest molecules leak away steadily, which is why the rule asks for the escape speed to be six times the root-mean-square speed.
Heavier molecules move more slowly at the same temperature, as $1 / \sqrt{m}$ . A molecule four times as heavy moves half as fast, and is far easier to keep.

ExercisesSection 3.104 questions

3.10.1
What is the root-mean-square speed of hydrogen atoms, of mass 1.008 u, at 1000 K?
km/s
3.10.2
Helium atoms are four times as heavy as hydrogen atoms. At the same temperature, how does their root-mean-square speed compare?
3.10.3
Mars has an escape speed of 5.03 km/s. Take its upper atmosphere at 210 K. What is six times the root-mean-square speed of carbon dioxide, of molecular mass 44.01 u, there, and can Mars keep it?
km/s
3.10.4
Jupiter's atmosphere is mostly hydrogen, while Earth has lost its hydrogen. What is the main reason?

Olympiad problemsPast IOAA problems, restated2 problems

2016Which gases Titan can keep10 points

Adapted from IOAA 2016 (Bhubaneswar), theory problem 2. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The particles of a gas in a planet's atmosphere move at a wide spread of speeds. If the root-mean-square thermal speed of a gas is more than a sixth of the escape speed, most of that gas will escape. What is the smallest atomic weight $A_{min}$ of an ideal monatomic gas that can stay in the atmosphere of Titan? Titan's mass is $1.23 \times 10^{23}$ kg, its radius 2575 km and its surface temperature 93.7 K.

I solved it without the solution

AnswerThe smallest atomic weight is $A_{min} = 13.2$ . Titan keeps nitrogen, at 28, and loses hydrogen and helium. The problem uses the surface temperature. Titan's upper atmosphere is warmer, at about 160 K, which raises $A_{min}$ to about 23, so the conclusion for nitrogen still holds.

2009Jumping off a small world10 points

Adapted from IOAA 2009 (Tehran), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Estimate the radius of a planet from which a person could escape by jumping straight up. Assume the planet has the same density as Earth, 5514 kg/m³.

I solved it without the solution

AnswerThe planet's radius would have to be at most about 1.8 km. Escape speed grows in proportion to the radius at a given density, which is why small asteroids cannot hold anything loose on their surfaces.