Atmospheric Scale Height Formula
Computes the vertical scale height H over which the barometric atmospheric pressure and gas density decrease by a factor of e (approx. 2.718).
\[ H = \frac{k_B \cdot T}{m_{mol} \cdot g} = \frac{R_{univ} \cdot T}{M_{molar} \cdot g} \implies P(z) = P_0 \cdot e^{-z/H} \]
Plain Text Formula:
H = (R_univ * T) / (M_molar * g) ==> P(z) = P0 * exp(-z / H)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ H \] | Scale Height | meters (m) or km | Vertical altitude increase producing a 1/e drop in pressure |
| \[ T \] | Atmospheric Temperature | Kelvin (K) | Mean kinetic gas temperature |
| \[ M_{molar} \] | Mean Molecular Mass | kg/mol | Mean molecular weight of air (Earth ~0.02897 kg/mol) |
| \[ g \] | Surface Gravity | m/s² | Gravitational acceleration (Earth 9.81 m/s²) |
| \[ R_{univ} \] | Universal Gas Constant | 8.314 J/(mol·K) | Molar gas constant |
💡 Worked Engineering Example
Problem Statement:
Calculate Earth's scale height at T = 288 K (M_molar = 0.02897 kg/mol, g = 9.81 m/s²).
Step-by-Step Solution:
- Numerator: 8.314 * 288 = 2,394.43
- Denominator: 0.02897 * 9.81 = 0.2842
- H = 2,394.43 / 0.2842 = 8,425 meters ≈ 8.43 km
🎯 Result: Earth's atmospheric scale height is approximately 8.43 kilometers.