Information about Scale Height
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.
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
where:
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:
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:
This translates as the pressure decreasing exponentially with height.
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).
For example:
Note:
Diffusion is the net movement of particles from an area of high concentration to an area of low concentration.
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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
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)
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:
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:
This translates as the pressure decreasing exponentially with height.
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).
For example:
- T = 290 K, H = 8500 m
- T = 210 K, H = 6000 m
Note:
- 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
- At heights over 100 km, molecular diffusion means that each molecular atomic species has its own scale height.
e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1.
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Pressure (symbol: p) is the force per unit area applied on a surface in a direction perpendicular to that surface.
Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.
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Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.
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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.
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trillion fold).]]
Temperature is a physical property of a system that underlies the common notions of hot and cold; something that is hotter generally has the greater temperature. Temperature is one of the principal parameters of thermodynamics.
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Temperature is a physical property of a system that underlies the common notions of hot and cold; something that is hotter generally has the greater temperature. Temperature is one of the principal parameters of thermodynamics.
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The kelvin (symbol: K) is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero — the coldest possible temperature — is zero kelvins
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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).
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acceleration is defined as the rate of change of velocity, or, equivalently, as the second derivative of position. It is thus a vector quantity with dimension length/time². In SI units, acceleration is measured in metres/second² (m·s-²).
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Gravitation is a natural phenomenon by which all objects with mass attract each other. In everyday life, gravitation is most familiar as the agency that endows objects with weight.
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In physics, density is mass m per unit volume V—how heavy something is compared to its size. A small, heavy object, such as a rock or a lump of lead, is denser than a lighter object of the same size or a larger object of the same weight, such as pieces of
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G is the seventh letter in the Latin alphabet. Its name in English is spelled gee or occasionally ge (IPA /dʒiː/).
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equation of state is a relation between state variables.<ref name="Perrot" >Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6.
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An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. Additionally, the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container.
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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 λ is a positive number called the decay constant.
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The unified atomic mass unit (u), or dalton (Da), is a small unit of mass used to express atomic and molecular masses. It is defined to be one twelfth of the mass of an unbound atom of the carbon-12 nuclide, at rest and in its ground state.
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An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. Additionally, the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container.
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- This article is about the physical mechanism of diffusion. For alternative meanings, see diffusion (disambiguation).
Diffusion is the net movement of particles from an area of high concentration to an area of low concentration.
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