Formula Used
The scale height H describes how pressure or number density decays with altitude in an isothermal, hydrostatic atmosphere. A common exponential model is P(z) = P₀ · exp(-z/H).
- Ideal gas (molar mass): H = (R · T) / (M · g)
- Ideal gas (particle mass): H = (kB · T) / (m · g)
- Pressure and density: H = P / (ρ · g) (useful for isothermal snapshots)
R is the molar gas constant, kB is Boltzmann’s constant, T is temperature, g is gravitational acceleration, M is molar mass, m is mean particle mass, and ρ is mass density.
How to Use This Calculator
- Select a calculation method that matches your available data.
- Pick a gravity preset or enter a custom g value.
- For ideal-gas methods, enter temperature and a mass choice.
- For pressure-density, enter absolute pressure and mass density.
- Choose the output unit, then press the calculate button.
- Use CSV or PDF export to save your result summary.
Tip: When using molar mass, gas presets can quickly fill typical values.
Example Data Table
These examples use the ideal-gas molar-mass method with common planetary gravities.
| Scenario | T (K) | g (m/s²) | M (g/mol) | H (m) | Interpretation |
|---|---|---|---|---|---|
| Earth, dry air | 288 | 9.80665 | 28.9647 | 8430 | Pressure drops by e over ~8.4 km. |
| Mars, CO₂ rich | 210 | 3.711 | 44.01 | 10691 | Thinner gravity increases the scale height. |
| Jupiter, H₂ (illustrative) | 1500 | 24.79 | 2.01588 | 249565 | High temperature and light gas inflate H. |
Real atmospheres vary with altitude, temperature, and composition, so H can change with height.
Scale Height Guide for Atmospheric Calculations
1) What scale height means
Scale height (H) is the distance over which atmospheric pressure or density drops by a factor of e (about 2.718) in an isothermal, ideal-gas layer. If you move upward by one scale height, pressure becomes roughly 37% of its previous value.
2) Exponential pressure profile
A common model uses P(z) = P0 · exp(−z/H) and ρ(z) = ρ0 · exp(−z/H). Estimating H helps you compare how quickly different atmospheres “thin out” with altitude, and it gives quick ratios for any height step Δz.
3) Key inputs that control H
H increases with temperature and decreases with stronger gravity and heavier molecules. Hot layers can therefore have larger scale heights, while CO2-rich or otherwise heavy mixtures have smaller scale heights at the same temperature and gravity.
4) Two common physics forms
Using the specific gas constant, H = RspecificT / g. Using molar quantities, H = R T / (M g), where R is the universal gas constant and M is molar mass. Both match when units are consistent.
5) A practical Earth example
For dry air near sea level, molecular mass is about 28.97 g/mol and g is about 9.81 m/s². At 288 K, the isothermal scale height is roughly 8.4–8.6 km. Real conditions are not perfectly isothermal, so “effective” values vary by altitude and weather.
6) Planet-to-planet comparisons
Lower gravity tends to increase H, while heavier composition decreases it. Use this calculator to explore scenarios by changing g, temperature, and molecular mass (or the gas constant) to see which factor dominates. Doubling temperature doubles H, while doubling gravity halves it.
7) When the simple model breaks
The exponential form assumes hydrostatic balance, an ideal gas, and a single temperature. In reality, temperature changes with altitude and composition can shift, so treat results as a baseline for learning and first-pass engineering estimates.
8) How to use results in analysis
Once you have H, estimate pressure ratios with P/P0 = exp(−Δz/H). For example, at Δz = 2H, the pressure ratio is about 0.135; at 3H it is about 0.050. These ratios are independent of the starting pressure. This supports quick checks for sensor calibration, balloon planning, wind-tunnel test setups, and planetary comparisons. For higher accuracy, use layered temperatures or standard-atmosphere tables.
FAQs
1) Is scale height the same as atmospheric thickness?
No. Scale height is a decay length. An atmosphere can extend many scale heights, but pressure becomes extremely small after several H values.
2) Why does my result change a lot with temperature?
Because H is proportional to temperature. Warmer gas needs a larger height for pressure to drop by the same exponential factor.
3) Which method should I choose: molar mass or gas constant?
Use molar mass if you know composition in g/mol or kg/mol. Use the specific gas constant if you already have it for the mixture. Both are equivalent with consistent units.
4) What pressure ratio corresponds to one scale height?
One scale height means P/P0 = e−1 ≈ 0.367, which is about a 63% drop in pressure.
5) Can I use this for liquids?
Not directly. The usual scale height definition assumes gases and the ideal gas law. Liquids are nearly incompressible and follow a different pressure-depth relationship.
6) Does gravity need to be constant?
This calculator assumes constant g. Over very large altitude ranges, g decreases with height. You can approximate by entering an average g for your range.
7) Why might real measurements differ from this estimate?
Temperature gradients, humidity, winds, and composition changes affect density and pressure profiles. The isothermal model is best for clean comparisons and quick estimates.