A scale height is a term often used in scientific contexts for a distance over which a quantity decreases by a factor of e. It is usually denoted by the capital letter H. e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

For planetary atmospheres, it is the vertical distance upwards, over which the pressure of the atmosphere decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by This article is about pressure in the physical sciences. ...

where:

• k = gas constant = 8.314 J·(mol K)⁻¹
• T = mean planetary surface temperature in kelvins
• M = mean molecular mass of dry air (units kg·mol⁻¹)
• g = acceleration 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 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. The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... For other uses, see Temperature (disambiguation). ... For other uses, see Kelvin (disambiguation). ... The molecular mass (abbreviated Mr) of a substance, formerly also called molecular weight and abbreviated as MW, is the mass of one molecule of that substance, relative to the unified atomic mass unit u (equal to 1/12 the mass of one atom of carbon-12). ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... Gravity is a force of attraction that acts between bodies that have mass. ... For other uses, see Density (disambiguation). ...

Thus:

where g 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 for a perfect gas of mean molecular mass m at temperature T, the density can be expressed as such: For other uses, see G (disambiguation). ... In physics and thermodynamics, an equation of state is a relation between state variables. ... An ideal gas (also called a perfect gas) is a hypothetical fluid consisting of particles that are identical to each other, occupy negligible volume and undergo perfect elastic collisions with each other, with no intermolecular forces and no intramolecular storage of energy, as opposed to a real gas, a gas...

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) the pressure at height z can be written as: For considerations of sea level change, in particular rise associated with possible global warming, see sea level rise. ...

This translates as the pressure decreasing exponentially with height. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

In the Earth's atmosphere, the pressure at sea level P₀ roughly equals 1.01×10⁵Pa, and the mean molecular mass of dry air is 28.964 u (1 u = 1.660×10⁻²⁷ kg). The atomic mass unit (amu), unified atomic mass unit (u), or dalton (Da), is a small unit of mass used to express atomic masses and molecular masses. ...

For example:

T = 290 K, H = 8500 m

T = 210 K, H = 6000 m

Note:

1. Density is related to pressure by the ideal gas 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 means that each molecular atomic species has its own scale height.