Scale height: characteristic vertical length for exponential decline
Scale height is the characteristic distance over which a quantity (typically atmospheric pressure or density) falls by a factor of e; used in atmospheric physics, planetary science and astrophysics.
Scale height denotes the characteristic length over which a physical quantity decreases by a factor of e (≈2.718). Commonly written as H, it appears whenever a vertical profile follows an exponential law, so that value(z) ∝ exp(−z/H). The concept is widely used for atmospheric pressure or density, for vertical distributions in astrophysical disks, and for gases in stellar atmospheres.
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Definition and simple formula
For an isothermal, hydrostatic atmosphere the scale height follows directly from balance of forces and the ideal gas law. From dP/dz = −ρg and P = ρRT (or P = ρkT/m at the particle level) one obtains an exponential solution with
- H = kT/(m g) (particle form), and equivalently
- H = R_s T/g where R_s is the specific gas constant per unit mass.
When it applies and when it does not
The equality of pressure and density scale heights requires an isothermal ideal gas; if temperature varies with height the profiles are not simple single exponentials and H becomes local (a function of altitude). Composition changes, phase transitions, or strong turbulent mixing also alter the effective decline rate.
Examples and applications
On Earth the visible atmospheric scale height is on the order of a few kilometers (often quoted near ~8 km at mid-latitudes), but it varies with temperature and molecular composition. In astrophysics the term describes the vertical thickness of galactic disks or accretion flows and is a key parameter in models of stellar atmospheres and protoplanetary disks.
Practical uses include simple estimates of column mass, optical depth, satellite drag, and the vertical extent of emitting regions. Observationally, measured scale heights constrain temperatures, mean molecular weights, and gravitational fields.
For further background and derivations see more information. Note that scale height is a descriptive length scale, not a hard boundary: real atmospheres and disks typically show gradual changes in H with altitude and composition.
Scale height of the atmosphere
The scale height in the barometric altitude formula for the altitude-dependent pressure is in the near-earth range:
with R: universal gas constant, T: temperature, M: molar mass, g: gravitational acceleration, kB: Boltzmann constant, N: number of particles, m: mass.
At altitudes greater than 100 km, the temperature increases to 1500 K and the molar mass decreases to 16 g/mol. A good approximation for the height-dependent pressure is an adjusted scale height of H = 26 km.
In the Martian atmosphere, however, the scale height is 11 km.
In astronomy, one speaks of an air mass with the thickness of a scale height, which, at constant ground pressure, has the same light attenuation as the entire atmosphere with exponentially decreasing density and pressure.
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AlegsaOnline.com Scale height: characteristic vertical length for exponential decline Leandro Alegsa
URL: https://en.alegsaonline.com/art/87744