Engineering & Science
Planetary Atmospheric Scale Height (H) & Barometric Altitude Pressure
Calculate isothermal planetary atmospheric scale height H = (R * T) / (M * g) (km) and exponential barometric pressure P(z) = P_0 * EXP(-z / H) (kPa) across Earth, Mars, Venus, and Titan.
Why Mars has a TALLER atmospheric scale height than Earth despite being freezing cold!
Combining hydrostatic equilibrium dP / dz = -rho * g with the ideal gas law rho = (P * M) / (R * T) reveals that planetary atmospheric pressure drops by a factor of e (~2.718) over every Scale Height H = (R * T) / (M * g), where R = 8.31446 J/(mol*K), T is mean atmospheric temperature (K), M is mean molar mass (kg/mol), and g is surface gravity (m/s^2)!
Compare Earth (T = 255 K equilibrium, N2/O2 molar mass M = 0.028964 kg/mol, g = 9.80665 m/s^2 -> H = 7.46 km, so climbing to 5.50 km altitude drops sea-level 101.325 kPa down to 48.50 kPa!) against Mars (cold T = 210 K, heavier CO2 atmosphere M = 0.04334 kg/mol, but low gravity g = 3.72076 m/s^2): Mars’s weaker gravity wins out, stretching its atmospheric scale height to H = 10.83 km—critical for NASA Mars parachute and heat-shield entry trajectories!
| Planetary Atmosphere | Mean T (K) & Molar Mass M (kg/mol) | Gravity g (m/s^2) | Scale Height H (km) |
|---|---|---|---|
| Earth Mean Troposphere/Stratosphere | 255 K, M = 0.028964 (N2/O2) | 9.80665 m/s^2 | 7.46 km (P at 5.5 km = 48.50 kPa) |
| Mars CO2 Atmosphere | 210 K, M = 0.043340 (CO2) | 3.72076 m/s^2 | 10.83 km (More Extended!) |
| Invalid Gravity or Molar Mass <= 0 | Guarded Singularity | N/A | "Invalid Planetary Inputs" |
Isothermal Atmospheric Scale Height H (km) = (8.31446 * T) / (M * g * 1000)
Computes e-folding altitude H in kilometers from mean temperature B2 (K), molar mass B3 (kg/mol), and gravity B4 (m/s^2).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 < 0), "Invalid Planetary Inputs", ROUND((8.31446 * B2) / (B3 * B4 * 1000), 2))Barometric Pressure at Altitude z (km): P(z) = P_0 * EXP(-z / H) (kPa)
Evaluates hydrostatic pressure decay from surface pressure B5 (kPa) at altitude B6 (km).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 < 0), "Invalid Planetary Inputs", ROUND(B5 * EXP(-B6 / ((8.31446 * B2) / (B3 * B4 * 1000))), 2))Try the example
Use the original values or change them to test your understanding.