← Back to Part 1: Atmosphere Fundamentals

1.1 Atmosphere Structure

Earth's atmosphere is divided into distinct layers based on temperature variations with altitude. Each layer has unique characteristics that determine weather, climate, and the propagation of electromagnetic radiation.

Vertical Layers of the Atmosphere

Troposphere (0-12 km)

Contains 75% of atmospheric mass. Temperature decreases with altitude at ~6.5°C/km (lapse rate). All weather occurs here.

Stratosphere (12-50 km)

Temperature increases with altitude due to ozone absorption of UV. Very stable, few clouds. Contains the ozone layer.

Mesosphere (50-85 km)

Temperature decreases again. Coldest part of the atmosphere (~-90°C at mesopause). Meteors burn up here.

Thermosphere (85-600 km)

Temperature increases dramatically (up to 2000°C) but low density. Aurora occur here. ISS orbits in this region.

Hydrostatic Equation

The fundamental balance between gravity and the vertical pressure gradient in the atmosphere:

$$\frac{dp}{dz} = -\rho \, g$$

where p is pressure, z is altitude, ρ is air density, and g is gravitational acceleration

Lapse Rate

The lapse rate describes the rate of temperature decrease with altitude:

$$\Gamma = -\frac{dT}{dz}$$

The environmental lapse rate in the troposphere averages $\Gamma \approx 6.5$ K/km

Number Density from the Ideal Gas Law

The number of molecules per unit volume at a given pressure and temperature:

$$n = \frac{p}{k_B T}$$

where $k_B = 1.381 \times 10^{-23}$ J/K is Boltzmann's constant. At sea level, $n \approx 2.55 \times 10^{25}$ m⁻³.

Barometric Formula

$$p(z) = p_0 \exp\left(-\frac{z}{H}\right)$$

where $H$ = scale height $\approx 8.5$ km

The scale height is defined as:

$$H = \frac{R_d T}{g} = \frac{k_B T}{mg}$$

$R_d = 287$ J/(kg·K) for dry air, $g = 9.81$ m/s², $T$ = temperature

Hypsometric Equation

Relates the thickness of an atmospheric layer to the mean virtual temperature between two pressure levels:

$$Z_2 - Z_1 = \frac{R_d \, \bar{T}_v}{g} \ln\!\left(\frac{p_1}{p_2}\right)$$

where $\bar{T}_v$ is the mean virtual temperature of the layer, and $p_1 > p_2$

Standard Atmosphere: Tropospheric Temperature Profile

In the troposphere, temperature decreases linearly with altitude:

$$T(z) = T_0 - \Gamma \, z = 288.15 - 6.5 \, z \quad \text{[K, with } z \text{ in km]}$$

Valid for $0 \le z \le 11$ km. The tropopause height varies from ~8 km at the poles to ~18 km at the equator.

Standard Atmosphere Values

Altitude (km)	Temperature (K)	Pressure (hPa)	Density (kg/m³)
0	288.15	1013.25	1.225
5	255.65	540.20	0.736
10	223.15	264.36	0.413
20	216.65	55.29	0.089
50	270.65	0.80	0.001

Interactive Simulation: Standard Atmosphere Profile

Python

Computes and plots the International Standard Atmosphere (ISA) temperature, pressure, and density profiles from 0-100 km. Shows all major atmospheric layers and boundaries.

standard_atmosphere.py70 lines

import numpy as np
import matplotlib.pyplot as plt

# ISA Constants
g = 9.81; R = 287.0; p0 = 101325.0; T0 = 288.15
layers = [(0,11,-6.5),(11,20,0),(20,32,1.0),(32,47,2.8),(47,51,0),(51,71,-2.8),(71,85,-2.0)]

def isa(z_km):
    T, p = T0, p0
    for zb, zt, L in layers:
        if z_km <= zb: break
        dz = (min(z_km, zt) - zb) * 1000.0
        Lm = L / 1000.0
        if Lm == 0:
            p *= np.exp(-g * dz / (R * T))
        else:
            Tn = T + Lm * dz
            p *= (Tn / T) ** (-g / (R * Lm))
            T = Tn
        if z_km <= zt: break
    rho = p / (R * T)
    return T, p, rho

alt = np.linspace(0, 100, 600)
T_arr = np.zeros_like(alt); P_arr = np.zeros_like(alt); rho_arr = np.zeros_like(alt)
for i, z in enumerate(alt):
    T_arr[i], P_arr[i], rho_arr[i] = isa(z)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 7), sharey=True)
fig.patch.set_facecolor('#0f172a')
for ax in (ax1, ax2):
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1'); ax.xaxis.label.set_color('#cbd5e1'); ax.yaxis.label.set_color('#cbd5e1')
    ax.title.set_color('#cbd5e1'); ax.grid(True, color='#334155', alpha=0.6)
    for sp in ax.spines.values(): sp.set_color('#334155')

boundaries = [(11,'Tropopause','#38bdf8'),(50,'Stratopause','#818cf8'),(85,'Mesopause','#c084fc')]
for bz, nm, c in boundaries:
    ax1.axhline(bz, color=c, ls='--', alpha=0.7, lw=1)
    ax1.text(-95, bz+1.5, nm, color=c, fontsize=8)
    ax2.axhline(bz, color=c, ls='--', alpha=0.7, lw=1)

labels = [(5.5,'Troposphere'),(30,'Stratosphere'),(67,'Mesosphere'),(92,'Thermosphere')]
for y, lbl in labels:
    ax1.text(-95, y, lbl, color='#94a3b8', fontsize=9, fontstyle='italic')

ax1.plot(T_arr - 273.15, alt, color='#f97316', lw=2.5)
ax1.set_xlabel('Temperature (\u00b0C)', fontsize=11); ax1.set_ylabel('Altitude (km)', fontsize=11)
ax1.set_title('Temperature Profile', fontsize=13, fontweight='bold')
ax1.set_xlim(-100, 10); ax1.set_ylim(0, 100)

ax2.semilogx(P_arr / 100, alt, color='#38bdf8', lw=2.5, label='Pressure')
ax2t = ax2.twiny()
ax2t.semilogx(rho_arr, alt, color='#4ade80', lw=2, ls='--', label='Density')
ax2t.set_facecolor('#1e293b'); ax2t.tick_params(colors='#4ade80')
ax2t.set_xlabel('Density (kg/m\u00b3)', fontsize=10, color='#4ade80')
ax2t.spines['top'].set_color('#4ade80')
ax2.set_xlabel('Pressure (hPa)', fontsize=11); ax2.set_title('Pressure & Density', fontsize=13, fontweight='bold')
ax2.set_ylim(0, 100)
lines1, labels1 = ax2.get_legend_handles_labels()
lines2, labels2 = ax2t.get_legend_handles_labels()
ax2.legend(lines1+lines2, labels1+labels2, loc='upper right', facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("=== International Standard Atmosphere Profile ===")
for z in [0, 5, 11, 20, 50, 85]:
    t, p, r = isa(z)
    print(f"  z={z:3d} km : T={t-273.15:7.1f} \u00b0C, P={p/100:9.2f} hPa, rho={r:.4f} kg/m\u00b3")
print("\nPlot saved as output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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