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

    Ready to run the numbers?

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