Scale height
Encyclopedia
In various scientific contexts, a scale height is a distance over which a quantity decreases by a factor of e
(the base of natural logarithms). It is usually denoted by the capital letter H.
of the atmosphere changes by a factor of e (decreasing upward). The scale height remains constant for a particular temperature. It can be calculated by
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 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 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 where P_{0} 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 the Earth's atmosphere
, the pressure at sea level P_{0} averages about 1.01×10^{5}Pa, 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)×10^{3} = 29.26 m/deg. This yields the following scale heights for representative air temperatures.
These figures should be compared with the temperature and density of the Earth's atmosphere plotted at NRLMSISE-00
, which shows the air density dropping from 1200 g/m^{3} at sea level to 0.5^{3} = .125 g/m^{3} 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:
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
(the base of natural logarithms). It is usually denoted by the capital letter H.
Scale Height used in a Simple Atmospheric Pressure Model
For planetary atmospheres, scale height is the vertical distance over which the pressurePressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
of the atmosphere changes by a factor of e (decreasing upward). The scale height remains constant for a particular temperature. It can be calculated by
where:
- k = Boltzmann constant = 1.38 x 10^{−23} J·K^{−1}
- T = mean planetary surface temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
in kelvinKelvinThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
s - M = mean molecular massMolecular massThe molecular mass of a substance is the mass of one molecule of that substance, in unified atomic mass unit u...
of dry air (units kg) - g = accelerationAccelerationIn physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
due to gravity on planetary surface (m/s²)
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
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
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 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
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
for an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
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 where P_{0} is the pressure at height z = 0 (pressure at sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...
) the pressure at height z can be written as:
This translates as the pressure decreasing exponentially with height.
In the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
, the pressure at sea level P_{0} averages about 1.01×10^{5}Pa, 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
Standard gravity
Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...
= 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)×10^{3} = 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 model of the Earth's atmosphere from ground to space. It models the temperatures and densities of the atmosphere's components. A primary use of this model is to aid predictions of satellite orbital decay due to atmospheric drag...
, which shows the air density dropping from 1200 g/m^{3} at sea level to 0.5^{3} = .125 g/m^{3} 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 gasIdeal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
laws. Therefore—with some departures caused by varying temperature—density will also decrease exponentially with height from a sea level value of ρ_{0} roughly equal to 1.2 kg m^{−3} - At heights over 100 km, molecular diffusionDiffusionMolecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
means that each molecular atomic species has its own scale height.