Model overview

Constants

  • \(g_0\): Standard gravitational acceleration [m/s²]

  • \(M_0\): Sea level mean molar mass [kg/mol]

  • \(N_A\): Avogadro constant [mol⁻¹]

  • \(P_0\): Sea level atmospheric pressure [Pa]

  • \(R^{*}\): Universal gas constant [J/(K·mol)]

  • \(R\): Specific gas constant [J/(K·kg)]

  • \(S\): Sutherland’s empirical constant in the equation for dynamic viscosity [K]

  • \(T_i\): Temperature of the ice point at mean sea level [K]

  • \(T_0\): Sea level temperature [K]

  • \(t_i\): Celsius temperature of the ice point at mean sea level [°C]

  • \(t_0\): Celsius sea level temperature [°C]

  • \(\beta_s\): Sutherland’s empirical constant in the equation for dynamic viscosity [kg/(m·s·K^(1/2))]

  • \(\kappa\): Adiabatic index [1]

  • \(\rho_0\): Sea level atmospheric density [kg/m³]

  • \(\sigma\): Effective collision diameter of an air molecule [m]

  • \(r\): Nominal Earth’s radius [m]

  • \(\beta\): Temperature gradient (layer-specific) [K/m]

Variables

  • \(\omega\): Collision frequency [Hz]

  • \(\rho\): Density [kg/m³]

  • \(\mu\): Dynamic viscosity [Pa·s]

  • \(g\): Gravitational acceleration [m/s²]

  • \(\nu\): Kinematic viscosity [m²/s]

  • \(l\): Mean free path [m]

  • \(\bar{\nu}\): Mean particle speed [m/s]

  • \(n\): Number density [m⁻³]

  • \(p\): Pressure [Pa]

  • \(H_p\): Pressure scale height [m]

  • \(\gamma\): Specific weight [N/m³]

  • \(a\): Speed of sound [m/s]

  • \(T\): Temperature [K]

  • \(t\): Temperature (Celsius) [°C]

  • \(\lambda\): Thermal conductivity [W/(m·K)]

Plots and equations

Collision frequency

collision_frequency

Equation: \(\omega = 4 \sigma^2 N_A \left( \frac{\pi}{R^{*} M_0} \right)^{1/2} \frac{p}{\sqrt{T}}\)

Density

density

Equation: \(\rho = \frac{p}{R T}\)

Dynamic viscosity

dynamic_viscosity

Equation: \(\mu = \frac{\beta_s T^{3/2}}{T + S}\)

Gravitational acceleration

grav_accel

Equation: \(g = g_0 \left( \frac{r}{r + h} \right)^2\)

Kinematic viscosity

kinematic_viscosity

Equation: \(\nu = \frac{\mu}{\rho}\)

Mean free path

mean_free_path

Equation: \(l = \frac{1}{\sqrt{2} \pi \sigma^2 n}\)

Mean particle speed

mean_particle_speed

Equation: \(\bar{\nu} = \left( \frac{8}{\pi} R T \right)^{1/2}\)

Number density

number_density

Equation: \(n = \frac{N_A p}{R^{*} T}\)

Pressure

pressure

Equation:

\(p = p_b \exp \left[ - \frac{g_0}{R T} (H - H_b) \right] \quad \text{for} \quad \beta = 0\)

\(p = p_b \left[ 1 + \frac{\beta}{T_b} (H - H_b) \right]^{-g_0/(\beta R)} \quad \text{for} \quad \beta \neq 0\)

Pressure scale height

pressure_scale_height

Equation: \(H_p = \frac{R T}{g}\)

Specific weight

specific_weight

Equation: \(\gamma = \rho g\)

Speed of sound

speed_of_sound

Equation: \(a = \sqrt{\kappa R T}\)

Temperature

temperature

Equation: \(T = T_b + \beta (H - H_b)\)

Temperature (Celsius)

temperature_in_celsius

Equation: \(t = T - T_i\)

Thermal conductivity

thermal_conductivity

Equation: \(\lambda = \frac{2.648151 \cdot 10^{-3} T^{3/2}}{T + (245.4 \cdot 10^{-12/T})}\)