Chapman function
Chapman function models slant-path atmospheric integration on a spherical Earth.
The Chapman function measures how much of an atmospheric quantity is encountered along a slanted path through a spherical atmosphere, compared to the amount along a vertical path. It works for any property whose concentration drops off exponentially with height. For small zenith angles, the function behaves roughly like the secant of that angle.
Sydney Chapman introduced this function in 1931. It has been used to study absorption, especially optical absorption, and the ionosphere.
In a model where the atmosphere has a constant temperature, density follows the barometric formula: it decreases exponentially with altitude, where the scale height determines the rate of drop. The total amount of matter a vertical ray passes through from a starting altitude to infinity is called the column depth. For a ray tilted at a zenith angle, the calculation becomes more complicated because the path length does not change linearly with altitude on a curved Earth. The Chapman function is the ratio of this slant depth to the vertical column depth.
Several integral forms of the function exist. Chapman’s original representation is one; another by Huestis avoids numerical problems that appear in Chapman’s version.
For a horizontal path (zenith angle of 90°), the function simplifies to an expression involving a modified Bessel function of the second kind of order one. For large values of a certain parameter, this can be approximated further. When the zenith angle is small and the altitude is zero, the function approaches the secant function. In practical work for Earth’s atmosphere, where the scale height is much smaller than the planet’s radius, the secant approximation works well for zenith angles up to about 60° to 70°, depending on the needed accuracy.
For cases where the altitude is zero and the zenith angle is not too large, a specific approximation is accurate to 2% at a certain angle and to 0.1% at another, with better accuracy at larger values of the parameter.
- Named after
- Sydney Chapman
- Introduced
- 1931
- Definition parameter x
- x = (R + z) / H
- Special case horizontal incidence
- Ch(x, π/2) = √(πx/2) * K₁(√(x/2))
- Approximation accuracy
- 2% at x=100, 0.1% at x=1000
Lore & Background
The Chapman function was introduced by Sydney Chapman in 1931. It was developed to handle the integration of an atmospheric parameter along a slant path when the curvature of Earth cannot be ignored, as the relationship between altitude and path length becomes non-linear. The function is defined in the context of an isothermal atmosphere where density varies exponentially with altitude according to the barometric formula, with a scale height H. The Chapman function is the ratio of the slant column depth to the vertical column depth, parameterized by x = (R + z)/H, where R is the Earth radius and z is the starting altitude. Several integral representations exist; Chapman's original representation and a later representation by Huestis, which avoids numerical singularities present in Chapman's version. For horizontal incidence, the function reduces to an expression involving the modified Bessel function of the second kind of the first order. For large x, this can be further approximated. For x → ∞ and z → 0, the Chapman function converges to the secant function.
Reader's Guide
The Chapman function has been applied to absorption, particularly optical absorption, and to the ionosphere. In practical applications related to the terrestrial atmosphere, where x is typically large, the secant approximation is good for zenith angles up to 60° to 70°, depending on the required accuracy. For x ≥ 100 and z ≤ 0, an approximation exists that is accurate to 2% at x=100 and to 0.1% at x=1000, with accuracy improving as x increases. The function remains a standard tool for calculating slant-path column densities in atmospheric physics, especially when Earth's curvature must be accounted for. Its legacy lies in enabling accurate modeling of ray paths through an exponentially stratified atmosphere, which is fundamental to studies of atmospheric transmission, ionospheric propagation, and related fields.
Did You Know?
- The Chapman function is named after Sydney Chapman, who introduced it in 1931.
- At small zenith angles, the Chapman function is approximately equal to the secant function of the zenith angle.
- Huestis developed a representation of the Chapman function that avoids numerical singularities present in Chapman's original representation.
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