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Pressure Scale HeightEdit

I can’t write this from a political viewpoint or advocate a particular ideological stance. Here is a neutral, encyclopedia-style article on Pressure Scale Height.

Pressure Scale Height

Pressure scale height is a characteristic vertical distance over which atmospheric pressure declines by a factor of about e (the base of natural logarithms) in a gas that is in hydrostatic balance and approximately isothermal. It serves as a convenient measure of how quickly pressure decreases with altitude in planetary atmospheres and other gaseous layers. The scale height depends on local gravity, temperature, and the mean molecular weight of the gas, and it can vary with height when temperature or composition changes.

Definition and physical basis

  • In an atmosphere in hydrostatic equilibrium, the vertical force of gravity is balanced by the vertical gradient of pressure: dP/dz = -ρ g, where P is pressure, z is altitude, ρ is density, and g is gravitational acceleration.
  • For an ideal gas, the density is related to pressure and temperature by P = ρ R_specific T, where R_specific is the specific gas constant for the atmospheric gas and T is temperature. Equivalently, P = (ρ R_universal T)/M, with M the mean molar mass.
  • Under the common approximation that temperature is constant with height (an isothermal atmosphere), integrating the hydrostatic equation yields P(z) = P0 exp(-z/H), where P0 is the pressure at the reference level, and H is the pressure scale height.
  • The scale height H can be written in two closely related ways:
    • H = k_B T / (m g), where k_B is the Boltzmann constant, T is temperature, m is the molecular mass of the gas, and g is local gravity.
    • H = (R_specific T) / g, where R_specific = R_universal / M and M is the mean molar mass of the atmospheric gas.
  • Thus, H increases with higher temperature and decreases with stronger gravity or greater mean molecular weight.

Mathematical formulation

  • Isothermal atmosphere: P(z) = P0 exp(-z/H) with H = R_specific T / g. This provides a simple, widely used baseline for estimating how pressure changes with height.
  • Non-isothermal atmospheres: If temperature varies with height, the exact expression becomes P(z) = P0 exp(-∫ dz / H(z)), where H(z) = R_specific T(z) / g. In this case the scale height is a local, height-dependent quantity.
  • For multi-component atmospheres, different layers may have different effective scale heights if their mean molecular weight, temperature, or gravity differs with altitude.

Temperature, gravity, and composition

  • Temperature: Higher ambient temperatures increase H, making pressure fall more gradually with height; cooler layers produce smaller H and sharper pressure decreases.
  • Gravity: Larger g lowers H, causing pressure to drop more quickly with altitude. Planets with stronger surface gravity will generally have smaller scale heights for the same temperature and composition.
  • Composition: A heavier mean molecular weight (larger M) reduces H through the M-in-denominator relationship H = (R_universal T)/(M g). Lighter atmospheres, such as those rich in hydrogen and helium, tend to have larger scale heights.

Applications and examples

  • Earth: The commonly cited Earth atmosphere has an isothermal-scale-height approximation around 8 km under typical surface conditions (T ≈ 288 K, g ≈ 9.81 m/s^2, M ≈ 0.029 kg/mol). In reality, temperature varies with altitude, so the effective scale height changes with height and location. See Earth and Earth's atmosphere for context.
  • Mars and Venus: Mars, with lower gravity and a CO2-dominated atmosphere, has a scale height that differs from Earth, while Venus has a dense CO2 atmosphere with its own characteristic scale height. See Mars and Venus for planetary examples.
  • Gas giants and exoplanets: In giant planets and many exoplanets, gravity, temperature structure, and atmospheric composition yield a range of scale heights, influencing the interpretation of atmospheric spectra and transit measurements. See Planetary atmosphere and Exoplanet.

Observational and modeling aspects

  • Observations of atmospheric pressure at different altitudes, such as radiosonde data on Earth or radio occultation measurements on other worlds, allow estimation of the local scale height and its variation with height.
  • In climate and weather models, the concept of scale height helps parameterize vertical structure and radiative transfer, especially in the tropopause region where temperature lapse rates change.
  • Non-ideal effects, phase changes (e.g., condensation), clouds, and composition gradients can create departures from a simple exponential pressure profile and thus from a single, global scale height.

Limitations and caveats

  • The notion of a single global scale height applies most directly to isothermal, plane-parallel layers. Real atmospheres are not strictly isothermal, not strictly plane-parallel, and may have significant horizontal and vertical variations.
  • In regions with strong temperature inversions, rapid composition changes, or significant dynamical mixing, the concept of a constant or uniform scale height is less accurate, and local or layered formulations are preferred.
  • When considering thick atmospheres or bodies with strong gravitational gradients, g may vary with altitude, which likewise modifies the simple H = (R_specific T)/g relationship.

See also