![2012 Transit of Venus from SF](https://upload.wikimedia.org/wikipedia/commons/4/4d/2012_Transit_of_Venus_from_SF.jpg)
Limb darkening is an optical effect seen in stars (including the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
), where the central part of the disk appears brighter than the edge, or ''
limb
Limb may refer to:
Science and technology
*Limb (anatomy), an appendage of a human or animal
*Limb, a large or main branch of a tree
*Limb, in astronomy, the curved edge of the apparent disk of a celestial body, e.g. lunar limb
*Limb, in botany, ...
''. Its understanding offered early solar astronomers an opportunity to construct models with such gradients. This encouraged the development of the theory of
radiative transfer
Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The equation of radiative tran ...
.
Basic theory
Optical depth
In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to ''transmitted'' radiant power through a material.
Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through ...
, a measure of the opacity of an object or part of an object, combines with
effective temperature
The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation. Effective temperature is often used as an estimate of a body's surface temperature ...
gradients inside the star to produce limb darkening. The light seen is approximately the integral of all emission along the line of sight modulated by the optical depth to the viewer (i.e. 1/e times the emission at 1 optical depth, 1/e
2 times the emission at 2 optical depths, etc.). Near the center of the star, optical depth is effectively infinite, causing approximately constant brightness. However, the effective optical depth decreases with increasing radius due to lower gas density and a shorter line of sight distance through the star, producing a gradual dimming, until it becomes zero at the apparent edge of the star.
The
effective temperature
The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation. Effective temperature is often used as an estimate of a body's surface temperature ...
of the
photosphere
The photosphere is a star's outer shell from which light is radiated.
The term itself is derived from Ancient Greek roots, φῶς, φωτός/''phos, photos'' meaning "light" and σφαῖρα/''sphaira'' meaning "sphere", in reference to it ...
also decreases with increasing distance from the center of the star. The radiation emitted from a gas is approximately
black-body radiation
Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spect ...
, the intensity of which is proportional to the fourth power of the temperature. Therefore, even in line of sight directions where the optical depth is effectively infinite, the emitted energy comes from cooler parts of the photosphere, resulting in less total energy reaching the viewer.
The temperature in the
atmosphere of a star does not always decrease with increasing height. For certain
spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
s, the optical depth is greatest in regions of increasing temperature. In this scenario, the phenomenon of "limb brightening" is seen instead. In the Sun, the existence of a
temperature minimum region means that limb brightening should start to dominate at
far-infrared
Far infrared (FIR) is a region in the infrared spectrum of electromagnetic radiation. Far infrared is often defined as any radiation with a wavelength of 15 micrometers (μm) to 1 mm (corresponding to a range of about 20 THz to ...
or
radio
Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmit ...
wavelengths. Above the lower atmosphere, and well above the temperature-minimum region, the Sun is surrounded by the million-
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
solar corona
A corona ( coronas or coronae) is the outermost layer of a star's atmosphere. It consists of plasma.
The Sun's corona lies above the chromosphere and extends millions of kilometres into outer space. It is most easily seen during a total solar e ...
.
For most wavelengths this region is optically thin, i.e. has small optical depth, and must, therefore, be limb-brightened if it is spherically symmetric.
Calculation of limb darkening
![Limb darkening geometry](https://upload.wikimedia.org/wikipedia/commons/c/c1/Limb_darkening_geometry.svg)
In the figure shown here, as long as the observer at point P is outside the stellar atmosphere, the intensity seen in the direction θ will be a function only of the angle of incidence ψ. This is most conveniently approximated as a polynomial in cos ψ:
:
where ''I''(ψ) is the intensity seen at P along a line of sight forming angle ψ with respect to the stellar radius, and ''I''(0) is the central intensity. In order that the ratio be unity for ψ = 0, we must have
:
For example, for a
Lambertian radiator (no limb darkening) we will have all ''a''
k = 0 except ''a''
1 = 1. As another example, for the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
at , the limb darkening is well expressed by ''N'' = 2 and
:
:
:
(See Cox, 2000). The equation for limb darkening is sometimes more conveniently written as
:
which now has ''N'' independent coefficients rather than ''N'' + 1 coefficients that must sum to unity.
The ''a''
k constants can be related to the ''A''
k constants. For N = 2,
:
:
For the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
at 550 nm, we then have
:
:
This model gives an intensity at the edge of the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
's disk of only 30% of the intensity at the center of the disk.
We can convert these formulas to functions of θ by using the substitution
:
where Ω is the angle from the observer to the limb of the star. For small θ we have
:
We see that the derivative of cos ψ is infinite at the edge.
The above approximation can be used to derive an
analytic expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
for the ratio of the mean intensity to the central intensity. The mean intensity ''I''
m is the integral of the intensity over the disk of the star divided by the solid angle subtended by the disk:
:
where dω = sin θ dθ dφ is a solid angle element, and the integrals are over the disk: 0 ≤ φ ≤ 2π and 0 ≤ θ ≤ Ω. We may rewrite this as
:
Although this equation can be solved analytically, it is rather cumbersome. However, for an observer at infinite distance from the star,
can be replaced by
, so we have
:
which gives
:
For the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
at 550 nm, this says that the average intensity is 80.5% of the intensity at the center.
References
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Stellar phenomena
Solar phenomena