300px, Graph of ch(x, z)
A Chapman function describes the integration of atmospheric
absorption
Absorption may refer to:
Chemistry and biology
* Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which ...
along a slant path on a spherical earth, relative to the vertical case. It applies to any quantity with a concentration
decreasing exponentially with increasing
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
. To a first approximation, valid at small
zenith
The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
s, the Chapman function for
optical absorption
In physics, absorption of electromagnetic radiation is how matter (typically electrons bound in atoms) takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber (for example, thermal energy). A ...
is equal to
:
where ''z'' is the
zenith
The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...
angle and sec denotes the
secant function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
.
The Chapman function is named after
Sydney Chapman, who introduced the function in
1931
Events
January
* January 2 – South Dakota native Ernest Lawrence invents the cyclotron, used to accelerate particles to study nuclear physics.
* January 4 – German pilot Elly Beinhorn begins her flight to Africa.
* January 22 – Sir I ...
.
Definition
In an isothermal model of the atmosphere, the density
varies exponentially with altitude
according to the
Barometric formula
The barometric formula, sometimes called the ''exponential atmosphere'' or ''isothermal atmosphere'', is a formula used to model how the pressure (or density) of the air changes with altitude. The pressure drops approximately by 11.3 pascals pe ...
:
:
,
where
denotes the density at sea level (
) and
the so-called
scale height
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, approximate ...
.
The total amount of matter traversed by a vertical ray starting at altitude
towards infinity is given by the integrated density ("column depth")
:
.
For inclined rays having a zenith angle
, the integration is not straight-forward due to the non-linear relationship between altitude and path length when considering the
curvature of Earth. Here, the integral reads
:
,
where we defined
(
denotes the
Earth radius
Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...
).
The Chapman function
is defined as the ratio between ''slant depth''
and vertical column depth
. Defining
, it can be written as
:
.
Representations
A number of different integral representations have been developed in the literature. Chapman's original representation reads
:
.
Huestis
developed the representation
:
,
which does not suffer from numerical singularities present in Chapman's representation.
Special cases
For
(horizontal incidence), the Chapman function reduces to
:
.
Here,
refers to the modified
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of the second kind of the first order. For large values of
, this can further be approximated by
:
.
For
and
, the Chapman function converges to the secant function:
:
.
In practical applications related to the terrestrial atmosphere, where
,
is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.
See also
*
Air mass
In meteorology, an air mass is a volume of air defined by its temperature and humidity. Air masses cover many hundreds or thousands of square miles, and adapt to the characteristics of the surface below them. They are classified according to la ...
*
Atmospheric physics
Within the atmospheric sciences, atmospheric physics is the application of physics to the study of the atmosphere. Atmospheric physicists attempt to model Earth's atmosphere and the atmospheres of the other planets using fluid flow equations, che ...
*
Ionosphere
The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays an ...
References
External links
Chapman function at Science World* {{cite journal , last1=Smith , first1=F. L. , last2=Smith , first2=Cody , title=Numerical evaluation of Chapman's grazing incidence integral ch(X,χ) , journal=J. Geophys. Res. , year=1972 , volume=77 , issue=19 , pages=3592–3597 , doi=10.1029/JA077i019p03592, bibcode=1972JGR....77.3592S
Radio frequency propagation
Special functions