In physics, a scale height, usually denoted by the capital letter H, is a distance (vertical or radial) over which a physical quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718).
Scale height used in a simple atmospheric pressure model For planetary atmospheres, scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by
H = k B T m g , {\displaystyle H={\frac {k_{\text{B}}T}{mg}},}
or equivalently,
H = R T M g , {\displaystyle H={\frac {RT}{Mg}},}
where
kB = Boltzmann constant = 1.381×10−23 J⋅K−1 R = molar gas constant = 8.31446 J⋅K−1⋅mol−1 T = mean atmospheric temperature in kelvins = 250 K for Earth m = mean mass of a molecule M = mean molar mass of atmospheric particles = 0.029 kg/mol for Earth g = acceleration due to gravity at the current location The pressure (force per unit area) at a given altitude is a result of the weight of the overlying atmosphere. If at a height of z the atmosphere has density ρ and pressure P, then moving upwards an infinitesimally small height dz will decrease the pressure by amount dP, equal to the weight of a layer of atmosphere of thickness dz. Thus:
d P d z = − g ρ , {\displaystyle {\frac {dP}{dz}}=-g\rho ,}
where g is the acceleration due to gravity. For small dz it is possible to assume g to be constant; the minus sign indicates that as the height increases the pressure decreases. Therefore, using the equation of state for an ideal gas of mean molecular mass M at temperature T, the density can be expressed as
ρ = M P R T . {\displaystyle \rho ={\frac {MP}{RT}}.}
Combining these equations gives
d P P = − d z k B T / m g , {\displaystyle {\frac {dP}{P}}={\frac {-dz}{{k_{\text{B}}T}/{mg}}},}
which can then be incorporated with the equation for H given above to give
d P P = − d z H , {\displaystyle {\frac {dP}{P}}=-{\frac {dz}{H}},}
which will not change unless the temperature does. Integrating the above and assuming P0 is the pressure at height z = 0 (pressure at sea level), the pressure at height z can be written as
P = P 0 exp ( − z H ) . {\displaystyle P=P_{0}\exp \left(-{\frac {z}{H}}\right).}
This translates as the pressure decreasing exponentially with height. In Earth's atmosphere, the pressure at sea level P0 averages about 1.01×105 Pa, the mean molecular mass of dry air is 28.964 Da, and hence m = 28.964 Da × 1.660×10−27 kg/Da = 4.808×10−26 kg. As a function of temperature, the scale height of Earth's atmosphere is therefore H/T = kB/mg = 1.381×10−23 J⋅K−1 / (4.808×10−26 kg × 9.81 m⋅s−2) = 29.28 m/K. This yields the following scale heights for representative air temperatures:
T = 290 K, H = 8500 m, T = 273 K, H = 8000 m, T = 260 K, H = 7610 m, T = 210 K, H = 6000 m. These figures should be compared with the temperature and density of Earth's atmosphere plotted at NRLMSISE-00, which shows the air density dropping from 1200 g/m3 at sea level to 0.125 g/m3 at 70 km, a factor of 9600, indicating an average scale height of 70 / ln(9600) = 7.64 km, consistent with the indicated average air temperature over that range of close to 260 K. Note:
Density is related to pressure by the ideal gas laws. Therefore, density will also decrease exponentially with height from a sea-level value of ρ0 roughly equal to 1.2 kg⋅m−3. At an altitude over 100 km, the atmosphere is no longer well-mixed, and each chemical species has its own scale height. Here temperature and gravitational acceleration were assumed to be constant, but both may vary over large distances.
Planetary examples Approximate atmospheric scale heights for selected Solar System bodies:
Scale height for a thin disk
For a disk of gas around a condensed central object, such as, for example, a protostar, one can derive a disk scale height which is somewhat analogous to the planetary scale height. We start with a disc of gas that has a mass small relative to the central object. We assume that the disc is in hydrostatic equilibrium with the z component of gravity from the star, where the gravity component is pointing to the midplane of the disk:
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