Scale height

In atmospheric, earth, and planetary sciences, a scale height, usually denoted by the capital letter H, is a distance over which a physical quantity decreases by a factor of e.

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
or equivalently,
where
The pressure 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:
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
Combining these 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 P₀ is the pressure at height z = 0, the pressure at height z can be written as
This translates as the pressure decreasing exponentially with height.
In Earth's atmosphere, the pressure at sea level P₀ averages about, the mean molecular mass of dry air is, and hence m = × =. As a function of temperature, the scale height of Earth's atmosphere is therefore H/''T = k''_B/mg = / =. This yields the following scale heights for representative air temperatures:
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 = 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.
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:

Solar System body	Atmospheric scale height	Mean temperature	Mean molecular weight	Surface gravity
Venus	15.9	229	44.01	0.91
Earth	8.5	225	28.96	1.00
Mars	11.1	210	44.01	0.38
Jupiter	27	124	2.22	2.48
Saturn	59.5	95	2.14	1.02
Titan	21	85	28.67	0.13
Uranus	27.7	59	2.30	0.90
Neptune	19.1–20.3	59	2.30	1.13
Pluto	~50

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:
where
In the thin disk approximation,, and the hydrostatic equilibrium, the equation is
To determine the gas pressure, one can use the ideal gas law:
with
Using the ideal gas law and the hydrostatic equilibrium equation, gives
which has the solution
where is the gas mass density at the midplane of the disk at a distance r from the center of the star, and is the disk scale height with
with the solar mass, the astronomical unit, and the dalton.
As an illustrative approximation, if we ignore the radial variation in the temperature, we see that and that the disk increases in altitude as one moves radially away from the central object.
Due to the assumption that the gas temperature T in the disk is independent of z, is sometimes known as the isothermal disk scale height.

Disk scale height in a magnetic field

A magnetic field in a thin gas disk around a central object can change the scale height of the disk. For example, if a non-perfectly conducting disk is rotating through a poloidal magnetic field, then a toroidal magnetic field will be produced within the disk, which will pinch and compress the disk. In this case, the gas density of the disk is
where the cut-off density has the form
where
These formulae give the maximum height of the magnetized disk as
while the e-folding magnetic scale height is