Atmospheric Scale Height Calculator Formula

Understand the math behind the atmospheric scale height calculator. Each variable explained with a worked example.

Use the Atmospheric Scale Height Calculator

Formulas Used

Scale Height

scale_height = 1.381e-23 * temperature / (mean_molecular_mass * surface_gravity)

Scale Height (km)

scale_height_km = 1.381e-23 * temperature / (mean_molecular_mass * surface_gravity) / 1000

Variables

VariableDescriptionDefault
temperatureTemperature(K)288
mean_molecular_massMean Molecular Mass(kg)4.81e-26
surface_gravitySurface Gravity(m/s2)9.81

How It Works

Atmospheric Scale Height

The scale height is the vertical distance over which atmospheric pressure decreases by a factor of *e* (about 2.718).

Formula

H = k_B T / (m g)

  • *k_B* = 1.381 x 10 to the minus 23 J/K (Boltzmann constant)
  • *T* = temperature in kelvin
  • *m* = mean molecular mass of the atmosphere (kg per molecule)
  • *g* = surface gravity

    • For Earth, H is about 8.5 km.

    Worked Example

    Earth: T = 288 K, m = 4.81e-26 kg (N2/O2 mix), g = 9.81 m/s2.

    temperature = 288mean_molecular_mass = 4.81e-26surface_gravity = 9.81
    1. 01H = k_B T / (m g)
    2. 02H = 1.381e-23 * 288 / (4.81e-26 * 9.81)
    3. 03H = 3.977e-21 / 4.719e-25
    4. 04H = 8 428 m = 8.43 km

    Frequently Asked Questions

    Why does a hotter atmosphere have a larger scale height?

    Higher temperature means molecules move faster and can support a thicker atmospheric column against gravity.

    How does scale height affect habitability?

    A very small scale height means a thin atmosphere; a large one helps retain warmth. Scale height influences atmospheric escape rates too.

    What is the scale height of Mars?

    About 11.1 km, larger than Earth's despite lower temperature because Mars has weaker gravity and a lighter (CO2-dominated) atmosphere.

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