In atmospheric, earth, and planetary sciences, 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 or equivalently where: 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: 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 (pressure at sea level) 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 1.01 Pa, the mean molecular mass of dry air is 28.964 Da and hence m = 28.964 Da × 1.66 kg/Da = 4.808 kg. As a function of temperature, the scale height of Earth's atmosphere is therefore H/T = k/mg = 1.381 J.K-1 / (4.808 kg × 9.81 m.s-2) = 29.28 m/K. 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(9600) = 7.64 km, consistent with the indicated average air temperature over that range of close to 260 K. Note:

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 which is 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, z \ll r and the hydrostatic equilibrium 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 \rho_0 is the gas mass density at the midplane of the disk at a distance r from the center of the star and h_D is the disk scale height with with M_\odot the solar mass, \text{au} the astronomical unit and \text{amu} the atomic mass unit. As an illustrative approximation, if we ignore the radial variation in the temperature, T, 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 in the disk, T, is independent of z, h_D 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 (i.e., the initial magnetic field is perpendicular to the plane of the disk), then a toroidal (i.e., parallel to the disk plane) 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, H_B, of the magnetized disk as while the e-folding magnetic scale height, h_B, is