Planetary Atmospheres: Scale Height, Greenhouse and Escape Velocity

Scale height: how "tall" is an atmosphere?

An atmosphere does not end at a hard boundary — it thins out exponentially with altitude, because at every height the gas below has to support the weight of everything above it. The rate of thinning is captured by the scale height, the altitude increase over which pressure drops by a factor of e (about 2.718):

H = k·T / (m·g)

k = Boltzmann constant, T = temperature, m = mean molecular mass, g = surface gravity
P(altitude) = P0 · exp(-altitude / H)

Earth   H ≈ 8.5 km    (cool, moderate gravity, N2/O2 — moderate)
Venus   H ≈ 15.9 km   (hot, similar gravity, dense CO2 atmosphere — tall column but heavy gas)
Mars    H ≈ 11.1 km   (cold, low gravity — but far thinner overall, only ~0.6% of Earth's surface pressure)

Notice the two competing effects buried in the formula: higher temperature puffs an atmosphere up (larger H), but higher gravity or heavier molecules compress it (smaller H). Mars has weak gravity, which should inflate its scale height, but it barely matters because Mars simply does not have much atmosphere left to inflate — a point that leads directly to the escape story below.

The greenhouse effect: trapping outgoing heat

Sunlight arrives mostly as visible light, which passes through a CO2-rich atmosphere largely unimpeded and warms the surface. The surface then re-radiates that energy as infrared, and CO2, water vapour and methane are strong infrared absorbers — they catch that outgoing radiation and re-emit part of it back down, raising the equilibrium surface temperature above what direct sunlight alone would produce. Earth's modest greenhouse effect (about 33°C of warming versus a bare rock) supports liquid water. Venus's atmosphere is 96% CO2 at 92 times Earth's surface pressure, and the resulting greenhouse warming is so severe that the surface sits at roughly 465°C — hot enough to melt lead — despite Venus's thick cloud deck reflecting away more sunlight than Earth receives at the surface.

Why Venus went runaway and Earth did not

The decisive difference is not the CO2 itself but what happened to it. Earth's oceans dissolve atmospheric CO2, which then combines with dissolved minerals to form solid carbonate rock on the sea floor — a long-term geological thermostat called the carbonate-silicate cycle that has kept Earth's CO2, and therefore its temperature, in a relatively stable range for billions of years. Venus, closer to the Sun and hot enough that any early ocean would have evaporated, never developed this sink. Without oceans to lock CO2 away, volcanic outgassing kept adding it to the atmosphere with nothing removing it, and the greenhouse effect and the temperature climbed together until they locked into the extreme, stable-but-brutal state observed today — the textbook example of a runaway greenhouse.

Escape velocity, Jeans escape, and why Mars lost its air

A planet keeps a gas only if that gas's typical thermal speed stays well below the planet's escape velocity — the rule of thumb used by planetary scientists is that a gas escapes significantly over geological time if its average molecular speed exceeds roughly one-sixth of escape velocity (this gradual thermal leak from the high-speed tail of the velocity distribution is called Jeans escape). Lighter molecules move faster at a given temperature, which is why hydrogen and helium escape from every rocky planet in the solar system over time, while heavier CO2 and N2 are retained far more easily by Earth's and Venus's gravity.

v_escape = sqrt(2GM / r)

Earth   v_escape ≈ 11.2 km/s   — retains N2, O2, CO2, H2O easily
Mars    v_escape ≈ 5.0 km/s    — weaker gravity, AND no global magnetic field
Venus   v_escape ≈ 10.4 km/s   — comparable to Earth, retains its dense CO2 atmosphere

Mars's low escape velocity is only half the story. The bigger factor is that Mars lost its global magnetic field early in its history, leaving the upper atmosphere directly exposed to the solar wind, which strips ions out of it continuously — a non-thermal loss process entirely separate from Jeans escape. Earth's magnetosphere deflects most of this stripping. Over roughly the last four billion years, the combination of weaker gravity and no magnetic shielding let Mars lose the great majority of an atmosphere current evidence suggests was once thick enough to support surface liquid water.

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