✈️ Standard Atmosphere Calculator (ISA 1976)
Compute air thermodynamic, transport, and kinetic properties up to 86 km according to U.S. Standard Atmosphere ISA 1976 standard. Supports temperature deviations, density altitude, and interactive SVG vertical profile tracing.
📋 Input Settings
🚀 Core Parameters
📊 Calculated Air Properties
| Property Name | Symbol | SI Metric Value | Imperial Value |
|---|---|---|---|
| Geometric Altitude | z | - | - |
| Geopotential Altitude | h | - | - |
| Temperature (Standard) | T_std | - | - |
| Temperature (Actual) | T | - | - |
| Atmospheric Pressure | P | - | - |
| Density of Air | ρ | - | - |
| Speed of Sound | a | - | - |
| Dynamic Viscosity | μ | - | - |
| Kinematic Viscosity | ν | - | - |
| Thermal Conductivity | k | - | - |
| Mean Free Path | λ | - | - |
| Local Gravity Acceleration | g | - | - |
| Pressure Altitude | z_p | - | - |
| Density Altitude | z_d | - | - |
| Pressure Ratio | δ | - | |
| Temperature Ratio | θ | - | |
| Density Ratio | σ | - | |
📘 Standard Atmosphere Mathematical Models
1. Geopotential Height & Gravity
The standard model integrates hydrostatic balance on a geopotential height $h$ to simplify the gravity variation $g(z)$ at geometric altitude $z$:
$$h = \frac{r_e \cdot z}{r_e + z}$$ $$g(z) = g_0 \cdot \left(\frac{r_e}{r_e + z}\right)^2$$Where the nominal Earth radius is $r_e = 6,356,766\text{ m}$, and gravity at sea level is $g_0 = 9.80665\text{ m/s}^2$.
2. Thermal Profile & Pressure
Within any atmospheric layer $b$ characterized by a constant temperature lapse rate $L_b = \frac{dT_{std}}{dh}$:
- If $L_b \neq 0$ (Linear lapse rate): $$T_{std} = T_b + L_b(h - h_b)$$ $$P = P_b \cdot \left[1 + \frac{L_b(h - h_b)}{T_b}\right]^{-\frac{g_0 M}{R^* L_b}}$$
- If $L_b = 0$ (Isothermal layer): $$T_{std} = T_b$$ $$P = P_b \cdot \exp\left[-\frac{g_0 M (h - h_b)}{R^* T_b}\right]$$
Where $M = 0.0289644\text{ kg/mol}$ is the molecular weight of air, and $R^* = 8.31432\text{ J/(mol K)}$ is the universal gas constant.
3. Transport & Kinetic Properties
Standard viscosity is governed by Sutherland's law, and thermal conductivity utilizes an empirical formulation:
$$\mu = 1.458 \times 10^{-6} \cdot \frac{T^{1.5}}{T + 110.4}$$ $$k = \frac{2.64638 \times 10^{-3} \cdot T^{1.5}}{T + 245.4 \times 10^{-12 / T}}$$The mean free path $\lambda$ (average distance traveled between molecular collisions) is computed based on effective collision diameter $d = 3.65\text{ Å}$:
$$\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P} \approx \frac{2.33228 \times 10^{-5} \cdot T}{P}$$📘 Calculation Methodology: US Standard Atmosphere 1976 Altitude Model
Mathematical Model & Theory
Models atmospheric temperature, pressure, density, and speed of sound from sea level up to the mesosphere ($86\text{ km}$) using geopotential altitude $H$ and hydrostatic barometric lapse rates:
Assumptions
- Troposphere lapse rate $L_b = -6.5 ext{ K/km}$ up to $H = 11.0 ext{ km}$ (Tropopause).
- Sea level reference: $T_0 = 288.15 ext{ K}$ ($15^\circ ext{C}$), $P_0 = 101.325 ext{ kPa}$, $ ho_0 = 1.225 ext{ kg/m}^3$.
Academic References
- NOAA, NASA, & USAF (1976): U.S. Standard Atmosphere, 1976, NASA-TM-X-74335.
- Anderson, J. D.: Introduction to Flight, Ch. 3.
Worked Engineering Example
Calculate atmospheric pressure, temperature, and density at commercial jet cruising altitude $H = 10,000\text{ m}$ (10 km).
Step-by-step Solution:
1. Temperature: $T = 288.15 - 0.0065 \times 10,000 = 288.15 - 65.0 = 223.15\text{ K}$ ($-50.0^\circ\text{C}$).
2. Pressure: Exponent $g M / (R L) = 9.80665 \times 0.0289644 / (8.31432 \times 0.0065) \approx 5.25588$.
$$P = 101.325 \times (223.15 / 288.15)^{5.25588} = 101.325 \times (0.77442)^{5.25588} = 101.325 \times 0.2608 \approx 26.43\text{ kPa}$$
3. Density: $\rho = P / (R_{air} T) = 26,430 / (287.05 \times 223.15) \approx 0.4127\text{ kg/m}^3$.
Final Result:
At 10 km altitude: $T = \mathbf{-50.0^\circ\text{C}}$, $P = \mathbf{26.43\text{ kPa}}$, $\rho = \mathbf{0.413\text{ kg/m}^3}$ (speed of sound $a = \mathbf{299.5\text{ m/s}}$).