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A scale height is a term often used in scientific contexts for a distance over which a quantity decreases by a factor of e
E (mathematical constant)

The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x....
 (the base of natural logarithms). It is usually denoted by the capital letter H.

For planetary atmospheres, it is the vertical distance upwards, over which the pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
 of the atmosphere decreases by a factor of e. The scale height remains constant for a particular temperature.






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A scale height is a term often used in scientific contexts for a distance over which a quantity decreases by a factor of e
E (mathematical constant)

The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x....
 (the base of natural logarithms). It is usually denoted by the capital letter H.

For planetary atmospheres, it is the vertical distance upwards, over which the pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
 of the atmosphere decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by

where:

  • k = Boltzmann constant
    Boltzmann constant

    The Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant NA:...
     = 1.38 x 10−23 J·K−1
  • T = mean planetary surface temperature
    Temperature

    In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
     in kelvin
    Kelvin

    The kelvin is a Units of measurement of temperature and is one of the seven SI base units. The Kelvin scale is a Thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero ....
    s
  • M = mean molecular mass
    Molecular mass

    The molecular mass of a chemical compound, frequently referred by the older term molecular weight and abbreviated as MW, is the mass of one molecule of that substance, relative to the unified atomic mass unit u ....
     of dry air (units kg)
  • g = acceleration
    Acceleration

    File:Acceleration.JPGFile:Acceleration components.JPGIn physics, and more specifically kinematics, acceleration is the change in velocity over time....
     due to gravity on planetary surface (m/s²)


The pressure in the atmosphere is caused by the weight on the atmosphere of the overlying atmosphere [force per unit area]. If at a height of z the atmosphere has density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
 ρ and pressure P, then moving upwards at 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:

where g
G

G is the seventh letter in the Latin alphabet. Its name in English language is spelled gee....
 is used to denote 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
Equation of state

In physics and thermodynamics, an equation of state is a relation between thermodynamic variables. More specifically, an equation of state is a thermodynamic equations describing the state of matter under a given set of physical conditions....
 for a perfect gas of mean molecular mass M at temperature T, the density can be expressed as such:

Therefore combining the equations gives

which can then be incorporated with the equation for H given above to give:

which will not change unless the temperature does. Integrating the above and assuming where P0 is the pressure at height z = 0 (pressure at sea level
Sea level

Mean sea level is the average height of the sea, with reference to a suitable reference surface. Defining the reference level , however, involves complex measurement, and accurately determining MSL can prove difficult....
) the pressure at height z can be written as:

This translates as the pressure decreasing exponentially
Exponential decay

A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and ? is a negative and non-negative numbers called the decay constant....
 with height.

In the Earth's atmosphere
Earth's atmosphere

The Earth's atmosphere is a layer of gases surrounding the planet Earth that is retained by the Earth's gravity. Dry air contains roughly 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.038% Carbon dioxide in the Earth's atmosphere, and trace amounts of other gases....
, the pressure at sea level P0 averages about 1.01×105Pa, the mean molecular mass of dry air is 28.964 u and hence 28.964 × 1.660×10−27 = 4.808×10−26 kg, and g = 9.81 m/s². As a function of temperature the scale height of the Earth's atmosphere is therefore 1.38/(4.808×9.81)×103 = 29.26 m/deg. 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 the Earth's atmosphere plotted at NRLMSISE-00
NRLMSISE-00

NRLMSISE-00 is an empirical, global Mathematical model of the Earth's atmosphere from ground to space. It models the temperatures and density of the atmosphere's components....
, which shows the air density dropping from 1200 g/m3 at sea level to 0.53 = .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:
  1. Density is related to pressure by the ideal gas
    Ideal gas

    The ideal gas model is a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy....
     laws. Therefore with some departures caused by varying temperature—density will also decrease exponentially with height from a sea level value of ρ0 roughly equal to 1.2 kg m−3
  2. At heights over 100 km, molecular diffusion
    Diffusion

    Molecular diffusion, often called simply diffusion, is a net transport of molecules from a region of higher concentration to one of lower concentration by random molecular motion....
     means that each molecular atomic species has its own scale height.