Unit 4: Atmospheres

Lecture 07: Planetary Atmospheres I: Hydrostatic Structure and Scale Height

Deriving why pressure falls off exponentially with altitude

Learning Objectives

Derive the barometric (hydrostatic) equation for a planetary atmosphere from force balance on a thin gas slab.
Solve the isothermal hydrostatic equation to obtain the exponential scale-height law.
Compute and compare atmospheric scale heights across solar system atmospheres of differing composition, temperature, and gravity.
Identify where the isothermal approximation breaks down (real atmospheres have temperature structure).

Terms for This Lecture

hydrostatic equilibrium

Definition: This is precisely the same force-balance argument used to derive hydrostatic equilibrium inside a star in ASTR 310, applied here instead to a planetary atmosphere; the physical content is identical (pressure gradient force balances weight), but the equation of state and boundary conditions differ substantially, which is why this course develops the atmospheric case as a genuinely separate derivation rather than simply citing the stellar result.

Why it matters here: The hydrostatic equation, combined with the ideal gas law, predicts that an isothermal atmosphere's pressure falls off exponentially with a scale height H = kT/(mg).

Relationship: Setting Up Hydrostatic Equilibrium

Boundary or common confusion: Substituting into the hydrostatic equation and assuming isothermal T (a simplifying approximation) gives dP/P = -(mg/kT) dz, a separable first-order ordinary differential equation.

Setting Up Hydrostatic Equilibrium

Consider a thin horizontal slab of atmosphere of area A, thickness dz, at height z, with density ρ(z).
In equilibrium, the net upward pressure force on the slab exactly balances its weight: [P(z) - P(z+dz)] A = ρ(z) A g dz.
Taking dz → 0 gives the hydrostatic equation: dP/dz = -ρ(z) g.

The Ideal Gas Law and the Scale Height

For an ideal gas, P = ρ k T / m, where m is the mean molecular mass of the atmosphere and T is temperature.
Substituting into the hydrostatic equation and assuming isothermal T (a simplifying approximation) gives dP/P = -(mg/kT) dz, a separable first-order ordinary differential equation.
Integrating from the surface (z=0, P=P₀) gives P(z) = P₀ exp(-z/H), where the scale height H = kT/(mg).

Physical Interpretation of the Scale Height

H = kT/(mg) is larger for hotter atmospheres (more thermal energy resisting gravitational settling), lighter mean molecular mass (lower mass gas is more buoyant against gravity), and weaker surface gravity (less restoring force per unit mass).
Earth's atmosphere (T ≈ 288 K, mean molecular mass ≈ 28.97 amu, g = 9.807 m/s²) has H ≈ 8.4-8.5 km, matching the commonly cited ∼8.5 km reference value.
This single number underlies practical facts: commercial airliners cruise near 1-1.3 scale heights (10-13 km) where pressure has fallen to roughly a third of sea level.

Scale Height Across the Solar System

Venus: hot (737 K) but very heavy atmosphere (mostly CO₂, m ≈ 44 amu) and Earth-like gravity, giving a scale height comparable to or modestly larger than Earth's despite the huge temperature difference, because the heavier molecular mass partly offsets the higher temperature.
Mars: cold (∼210 K) with the same CO₂-dominated composition as Venus but much weaker gravity (3.71 m/s²), giving a scale height comparable to Earth's despite Mars's low temperature, because weak gravity offsets the cold temperature.
Titan (Lecture 13): cold (∼94 K) but has weak gravity and a nitrogen-dominated atmosphere, giving a surprisingly large scale height for such a cold world.

Beyond Isothermal: Real Atmospheric Temperature Structure

Real atmospheres are not isothermal: Earth's atmosphere has a troposphere (temperature decreasing with altitude, driven by convection), a stratosphere (temperature increasing due to ozone absorption of UV), and further layers above.
In a non-isothermal atmosphere, the local scale height H(z) = kT(z)/(mg) still describes the local rate of pressure decline, but a single global scale height is only a useful average, not an exact description.
The troposphere-stratosphere temperature inversion is itself diagnostic of a specific absorbing species (ozone on Earth, various hydrocarbons on the giant planets), connecting atmospheric structure directly to atmospheric chemistry.

Atmospheric Composition and Mean Molecular Mass

Earth: N₂ (78%), O₂ (21%), Ar (0.9%), giving mean molecular mass ≈ 28.97 amu.
Venus: CO₂ (96.5%), N₂ (3.5%), giving mean molecular mass ≈ 43.4 amu, roughly 1.5 times heavier than Earth's atmosphere.
Mars: CO₂ (95.1%), N₂ (2.6%), Ar (1.9%), giving mean molecular mass ≈ 43.3 amu, essentially identical to Venus's despite the very different total atmospheric mass and surface pressure.

Pressure-Altitude Profiles as Diagnostic Tools

Radio occultation (a spacecraft signal passing through a planet's atmosphere as it is occulted by the planet, observed from Earth) measures the atmosphere's refractive index profile, from which pressure, density, and temperature versus altitude can be derived without ever entering the atmosphere.
Entry probes (e.g., the Galileo probe at Jupiter, Huygens at Titan) measure pressure and temperature directly during atmospheric descent, providing ground truth for remote-sensing-derived profiles.
Comparing radio-occultation-derived and probe-measured profiles for the same atmosphere (where both exist) is an important validation exercise, analogous to the seismology-versus-gravity cross-check emphasized in Lecture 03.

The Barometric Formula and Surface Pressure

Integrating the exponential pressure law from the surface to infinity gives the total atmospheric mass per unit area: Σ = P₀/g, directly relating surface pressure to total atmospheric mass without needing to know scale height at all.
Earth: P₀ = 1.013 × 10⁵ Pa, g = 9.807 m/s², giving Σ ≈ 10,330 kg/m² -- consistent with the commonly cited fact that Earth's atmosphere weighs about 1 kg per cm² of surface.
This relation, P₀ = Σg, is exact regardless of the atmosphere's temperature structure, unlike the exponential scale-height law, which assumed isothermality.

Convective versus Radiative Atmospheric Regions

In regions where the vertical temperature gradient exceeds the adiabatic lapse rate, the atmosphere becomes convectively unstable and overturns (as in Earth's troposphere and the observable weather layer of the giant planets).
Where the temperature gradient is sub-adiabatic, radiative transfer alone can carry the required energy flux, and the layer remains stably stratified (as in Earth's stratosphere).
This convective/radiative distinction is developed further, together with its role in setting a planet's surface temperature, in Lecture 08's radiative-equilibrium treatment.

Scope Note and Reading

OpenStax Astronomy 2e Section 10.1-10.2 (Atmospheres of the Terrestrial Planets) introduces atmospheric composition and structure at an introductory level, without deriving the hydrostatic equation.
The hydrostatic-equilibrium derivation and scale-height formalism in this lecture go beyond OpenStax and follow de Pater & Lissauer, Planetary Sciences, Chapter 4, and the standard atmospheric-physics treatment in Sanchez-Lavega, An Introduction to Planetary Atmospheres.

Worked Example: Atmospheric Scale Heights Across the Solar System

Compute H = kT/(mg) for Earth (N₂-dominated, T=288 K), Venus (CO₂-dominated, T=737 K), and Mars (CO₂-dominated, T=210 K), using each planet's measured surface gravity from Lecture 01's table.

Earth: H = (1.381e-23 × 288) / (4.652e-26 × 9.807) = 8718 m ≈ 8.72 km.

Venus: H = (1.381e-23 × 737) / (7.308e-26 × 8.870) = 15701 m ≈ 15.70 km.

Mars: H = (1.381e-23 × 210) / (7.308e-26 × 3.728) = 10645 m ≈ 10.65 km.

Result: despite Venus being more than three times hotter than Mars, Venus's scale height (15.7 km) is only modestly larger than Mars's (10.6 km) and both are comparable to Earth's (8.7 km, lighter but cooler gas), because Mars's much weaker surface gravity (3.73 m/s² versus Venus's 8.87 m/s²) very nearly compensates for its much lower temperature -- confirming the qualitative prediction made earlier in this lecture.

Visual Reasoning

Study how this lecture’s figure turns the derivation and worked example above into a concrete quantitative picture.

Which numbers in the figure come directly from measurement, and which are computed from this lecture’s physical law?
Where do the marked points or curves match the worked-example values above?
What would change in the figure if a key assumption in the derivation failed?

The hydrostatic equation, combined with the ideal gas law, predicts that an isothermal atmosphere's pressure falls off exponentially with a scale height H = kT/(mg).

Summary and Next Lecture

The hydrostatic equation, combined with the ideal gas law, predicts that an isothermal atmosphere's pressure falls off exponentially with a scale height H = kT/(mg).
Scale height depends on temperature, mean molecular mass, and gravity in ways that can partly compensate across very different planets, as the Venus/Mars comparison shows.
The relation Σ = P₀/g gives total atmospheric mass directly from surface pressure, independent of the isothermal approximation.

Next: Lecture 08 derives what actually sets an atmosphere's temperature: the radiative energy balance between absorbed starlight and emitted thermal radiation, including the greenhouse effect.