Scale height

The scale height H is an integration constant of an exponential function. For locus-dependent functions, it indicates the height at which the function decreases by the value of the Eulerian number e. Similarly, it is the height to which a constant function would correspond if it did not decrease exponentially, i.e. value of the definite integral over the function:

{\displaystyle p(x)=p_{0}\cdot \mathrm {e} ^{-{\frac {x}{H}}}}

and

{\displaystyle H=\int _{0}^{\infty }\mathrm {e} ^{-{\frac {x}{H}}}\,\mathrm {d} x}

respectively

{\displaystyle \int _{0}^{\infty }p(x)\,\mathrm {d} x=\int _{0}^{\infty }p_{0}\mathrm {e} ^{-{\frac {x}{H}}}\,\mathrm {d} x=p_{0}H}

For exponential functions that describe dependencies other than a height, the following labels apply:

Scale height of the atmosphere

The scale height in the barometric altitude formula for the altitude-dependent pressure is in the near-earth range:

{\displaystyle H={\frac {RT}{Mg}}={\frac {Nk_{\mathrm {B} }T}{mg}}=7{,}8\,\mathrm {km} }

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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