Scale height

The scale height H is a constant of integration of an exponential function . In the case of location-dependent functions, it indicates the height at which the function is reduced by the value of Euler's number e . Likewise, it is the height to which a constant function would correspond if it did not decrease exponentially, i.e. H. Value of the definite integral over the function:

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

and

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

or.

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

The following terms apply to exponential functions that describe dependencies other than those of a height:

• Law of decay : reciprocal of the decay constants , decay time
• Absorption (physics) : reciprocal of the absorption coefficient , penetration depth

Scale height of the atmosphere

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

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

with R : universal gas constant , T : temperature , M : molar mass , g : acceleration due to gravity , k _B : Boltzmann constant , N : number of particles , m : mass .

At an altitude greater than 100 km, the temperature increases to 1500 K and the molar mass decreases to 16 g / mol. A good approximation for the altitude-dependent pressure is an adjusted scale height of H = 26 km.

In the Martian atmosphere, on the other hand, 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 falling density and pressure.