Atmospheric Escape: Thermal Velocity vs Escape Velocity

Understanding the atmospheric escape calculator — concept and principle

Whether a planet or moon can keep a gas in its atmosphere depends on a race between two speeds: how fast the gas molecules move because of heat, and how fast something must move to escape the body's gravity. This calculator compares them and applies a widely used rule of thumb for retention over billions of years.

Escape speed must be about 6× the typical molecule speed to keep a gas for billions of years.

How it is calculated

RMS thermal speed: v_rms = √(3RT / M), with R = 8.314462618 J/(mol·K), T the exosphere temperature in kelvin and M the molar mass in kg/mol (the input in g/mol is divided by 1,000). The result is converted to km/s.
Ratio: v_esc / v_rms, using the escape velocity you enter in km/s.
Verdict: if the ratio is above 6, the gas is marked as retained; otherwise it is marked as lost over time.

The principle

Molecules in a gas do not all move at the same speed; they follow the Maxwell–Boltzmann distribution, whose root-mean-square value is √(3kT/m). At the exobase, where collisions become rare, molecules in the fast tail of the distribution that are moving upward with more than the escape speed leave for good. This slow leak is called Jeans escape. Because the tail falls off roughly exponentially with the square of the speed, the loss rate drops extremely quickly as the ratio grows. A ratio of about 6 makes the loss negligible over the age of the solar system, while ratios of 2 to 4 let the gas escape over geological time. Light molecules are fastest, which is why hydrogen and helium escape first.

Worked example

Inputs: T = 1000 K, M = 1.008 g/mol (atomic hydrogen), v_esc = 11.19 km/s (Earth).
v_rms = √(3 × 8.314462618 × 1000 / 0.001008) = 4,974.5 m/s = 4.974477305 km/s.
Ratio = 11.19 / 4.974477 = 2.249482571.
2.25 is below 6, so the result is lost over time. This matches reality: Earth continuously loses hydrogen to space.

Each point is the calculator re-run with only “Molecular mass (g/mol)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Molecular mass (g/mol)	RMS thermal speed (km/s)
0.252	9.94895461
0.504	7.034973271
0.756	5.744031622
1.01	4.974477305
1.26	4.449307763
1.51	4.061643711
2.02	3.517486635

How to read the result

Where does your result fall?

Above 6: retained for billions of years. Between about 4 and 6: borderline, depends on details. Below about 4: escapes. For the same 1000 K on Earth, helium (4.003 g/mol) gives about 4.5, while water vapour (18.015) gives about 9.5 and nitrogen (28.014) about 11.9, which is why Earth keeps N₂ and O₂ but not free hydrogen.

Common mistakes

Using surface temperature instead of the much hotter exosphere temperature.
Entering molar mass in kg/mol when the field expects g/mol.
Using escape velocity from the surface when the exobase is far above it; the difference is small for Earth but not for small bodies.

Keep in mind: The threshold of 6 is a rule of thumb, and some textbooks apply it to the mean or most probable speed instead of the RMS speed. Real atmospheric loss also includes non-thermal processes (solar wind stripping, photochemical and ion escape) that this model ignores.

Frequently asked questions

Why does the Moon have almost no atmosphere?

Its escape velocity is only about 2.4 km/s, so even heavy gases have a low ratio when the sunlit surface is hot.

Why can Jupiter keep hydrogen?

Its escape velocity is about 60 km/s, giving a ratio far above 6 even for hydrogen.

Sources: J. H. Jeans, The Dynamical Theory of Gases, Cambridge University Press · D. C. Catling and J. F. Kasting, Atmospheric Evolution on Inhabited and Lifeless Worlds, Cambridge University Press

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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