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Thin air

Rockets and landing · level 7, lesson 2 of 6

A question to start

The pressure of Mars's air at the ground is less than 1% of Earth's, but its density is about 1.6% of ours. Why are the two numbers different? And how does the air change as a lander falls through it?

The idea

Density from pressure. For a gas, \(\rho = \dfrac{pM}{RT}\), with \(M\) the molar mass and \(T\) the temperature in kelvin.

Mars Earth
Pressure at the ground (mean) about 610 Pa 101,325 Pa
Main gas carbon dioxide (95%), \(M\) = 44 g/mol nitrogen and oxygen, \(M\) = 29 g/mol
Mean temperature at the ground about 210 K (−63 °C) about 288 K (15 °C)
Density at the ground about 0.015 to 0.020 kg/m³ 1.225 kg/m³
Scale height \(H\) about 11 km about 8.5 km
Speed of sound near the ground about 240 m/s 343 m/s

Pressure is 0.6% of Earth's, but the gas is 1.5 times heavier per molecule and about 0.73 times as warm, which together raise the density ratio to about \(0.006 \times 1.5 / 0.73 \approx 1.2\%\) for the mean, and more in cold places and in low places. Landers often quote 0.020 kg/m³ for low, cold sites.

With height. In an atmosphere at a single temperature, density falls exponentially:

\[\rho(h) = \rho_0\, e^{-h/H}, \qquad H = \frac{RT}{Mg}.\]

\(H\) is the scale height, the height over which the density falls by a factor \(e\) (2.72). Mars's \(H\) is larger than Earth's because its gravity is weaker, even though its gas is heavier and colder.

Height above Mars's datum 0 km 10 km 20 km 40 km
\(\rho\), model with \(\rho_0 = 0.020\), \(H = 11.1\) km 0.020 0.0081 0.0033 0.00054

It changes with the seasons. Each winter, about a quarter of the air freezes out as carbon dioxide frost and ice on the winter pole, and the pressure across the planet drops by up to about 25 to 30%. The Viking landers measured the pressure rising and falling through the Mars year. Landers are designed for the air they will meet in their season and place.

Mach number. \(M = v/c\), the speed divided by the speed of sound. Sound is slower in Mars's cold carbon dioxide, so the same speed is a higher Mach number there: Perseverance's parachute opened at about 420 m/s, about Mach 1.7 or more in that air.

Worked examples

1. Scale height. \(H = RT/(Mg) = 8.314 \times 210 / (0.044 \times 3.71) = 10{,}700\) m, close to the 11 km measured. For Earth: \(8.314 \times 288/(0.029 \times 9.81) = 8{,}400\) m.

2. A low landing site. Jezero crater lies about 2.6 km below Mars's datum. There, \(\rho\) is about \(e^{2.6/11.1} = 1.26\) times the datum value. Low sites give a parachute more air to work with, which is why landers prefer them.

3. Mach number. A capsule at 1,000 m/s on Mars (\(c = 240\)) is at Mach 4.2; on Earth (\(c = 343\)), Mach 2.9.

Try it

  1. Use \(\rho = pM/(RT)\) to find the density of Mars air at 600 Pa and 200 K.
  2. Find \(\rho\) at 30 km above the datum with the model in the table.
  3. Why is Mars's scale height larger than Earth's?
  4. Why is the speed of sound lower on Mars?
  5. Why do landing sites at low elevation help a lander?
Answers
  1. \(600 \times 0.044/(8.314 \times 200) = 0.0159\) kg/m³.
  2. \(0.020 \times e^{-30/11.1} = 0.020 \times 0.067 = 0.0013\) kg/m³.
  3. Its gravity is weaker, which outweighs its heavier, colder gas: \(H = RT/(Mg)\).
  4. Its gas is colder and heavier: \(c = \sqrt{\gamma RT/M}\) is smaller.
  5. There is more air above the ground, so more time and drag to slow down before reaching it.

Common mistake

Using the pressure ratio for drag. Drag depends on density, not pressure. The 0.6% pressure figure underestimates how much the air of Mars can do.

Where it leads