Barometric FormulaEdit

Barometric formulae describe how atmospheric pressure falls with altitude. They emerge from a straightforward combination of hydrostatics and the ideal gas law, capturing a core piece of how Earth’s atmosphere is structured. In practice, these relations underlie how pilots read altitude, how weather balloons infer atmospheric profiles, and how altimeters in airplanes or smartphones estimate height above sea level. Because they tie together gravity, temperature, and the composition of air, barometric formulae are a staple in atmospheric science and aviation physics, and they sit at the foundation of more elaborate atmospheric models such as the International Standard Atmosphere and various meteorological profiles. The subject sits comfortably within classical physics, and its predictions remain robust across a wide range of practical conditions, even as real-world atmospheres introduce complications like moisture and weather systems. For a concise mathematical route, see how the pressure field P changes with height h and how this ties into the broader framework of hydrostatic equilibrium and the Ideal gas law.

In essence, pressure decreases with height because the weight of the air above presses down, and because the air itself is a gas whose density depends on temperature and composition. The barometric formula offers a compact way to quantify that decrease and to translate surface conditions into an understanding of conditions at higher levels of the atmosphere. This is not only a matter of academic interest: it informs practical tasks from calibrating instruments to modeling flight dynamics and understanding high-altitude environments. The classical derivations begin with the hydrostatic balance dP/dz = -ρ g and the ideal gas relation P = ρ R_specific T, where z is geometric height, ρ is air density, g is gravity, and R_specific is the specific gas constant for dry air. From these relations one obtains expressions that connect surface pressure P0, surface temperature T0, altitude h, and the temperature profile T(h). For more on the basic variables and the general theory, see hydrostatic equilibrium and Ideal gas law.

Mathematical forms

Isothermal atmosphere

If the temperature is assumed constant with height (an isothermal atmosphere), the barometric formula reduces to an exponentially decaying pressure: P(h) = P0 exp(-h/H). Here P0 is the surface pressure, and the scale height H = R_specific T0 / g0, with T0 the surface temperature and g0 the standard gravity near Earth’s surface. This form highlights the intuitive idea that pressure falls off roughly exponentially with height, at a rate set by the balance between thermal energy in the air and the weight of the overlying column. See pressure and Scale height for related concepts.

Lapse-rate atmosphere (non-isothermal)

In the real troposphere, temperature typically decreases with height at a roughly linear rate L (the lapse rate). If T(h) = T0 - L h and L is treated as approximately constant over a regime of interest, one obtains a more accurate closed form: P(h) = P0 [1 - (L h)/T0]^(g0/(R_specific L)). This expression reduces to the isothermal form in the limit L → 0 and uses the same constants as above. The corresponding temperature profile is T(h) = T0 - L h, and the pressure decline reflects both gravity and the changing temperature with height. For background on temperature structure and lapse rates, see lapse rate.

General framework and the hypsometric relation

More generally, pressure as a function of height can be written in terms of the temperature profile T(z) by integrating the hydrostatic and ideal gas relations. In many atmospheric contexts, a compact form that is widely used is the hypsometric equation, which relates vertical separation between pressure surfaces to the mean temperature of the air between them: Δz = (R_specific / g0) × T_mean × ln(P1/P2), where Δz is the geopotential height difference between two pressure surfaces P1 and P2, and T_mean is an appropriate average temperature of the layer between them. This relation reinforces how higher temperatures yield greater atmospheric thickness for a given pressure change. See Hypsometric equation and Geopotential height for related ideas.

Applications and practical use

Aviation and navigation rely on altimeters that interpret local barometric pressure to estimate altitude. Aviation practices distinguish between different reference pressures (for example, QNH or QFE) to set altimeters according to the desired reference level. See altimeter and barometer for instrument-specific details.
Weather reconnaissance and meteorology make use of vertical pressure profiles to infer atmospheric stability, layering, and moisture effects. The pressure profile, together with temperature and humidity data, feeds into models that forecast flight conditions, precipitation, and storm development. See Atmosphere and pressure for broader context.
In climate science and remote sensing, the barometric formula underpins retrievals from radiosondes and satellite instruments, where pressure profiles are combined with temperature and moisture to reconstruct atmospheric structure. See Atmosphere for overview and Hypsometric equation for related methods.

Limitations and debates

The barometric formula rests on simplifying assumptions. Real atmospheres are not perfectly isothermal, nor are they perfectly dry; humidity, phase changes of water, wind, and convection introduce departures from the idealized relations. The lapse rate itself can vary with latitude, season, and weather, and moisture lowers the effective lapse rate in moist air. Consequently, more elaborate models use a layered or continuously varying temperature profile and account for moisture through the concept of virtual temperature. The basic forms remain valuable, however, because they capture the essential physics with transparent and testable predictions. In discussions of atmospheric modeling, some critics emphasize that simple formulae can be insufficient for detailed forecasting, while proponents argue that a solid grasp of these classical relations is indispensable and that complex models ultimately rely on the same fundamental physics. See Ideal gas law and lapse rate for the building blocks, and Hypsometric equation for a link to how temperature structure translates into height differences.

From a practical standpoint, one can still reliably estimate order-of-magnitude changes in pressure with height using these expressions, which is why they appear in aviation manuals, introductory meteorology, and foundational physics texts. The ongoing refinement of atmospheric models—incorporating moisture, radiation, and dynamics—builds on the clarity and simplicity of the barometric relations, while caution remains that simple forms are idealizations of a more complex atmospheric reality. See Barometer and Atmosphere for broader context.