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A filtered image of the Sun in visible light, showing the limb-darkening effect as a dimmer luminosity towards the edge or limb of the solar disk. The image was taken during the 2012 transit of Venus (seen here as the dark spot at the upper right).

Limb darkening is an optical effect seen in stars (including the Sun), where the center part of the disk appears brighter than the edge or limb of the image. Its understanding offered early solar astronomers an opportunity to construct models with such gradients. This encouraged the development of the theory of radiative transfer.

Basic theoryEdit

 
An idealized case of limb darkening. The outer boundary is the radius at which photons emitted from the star are no longer absorbed. L is a distance for which the optical depth is unity. High-temperature photons emitted at A will just barely escape from the star, as will the low-temperature photons emitted at B. This drawing is not to scale. E.g., for the Sun, L would be only a few hundred km.

Optical depth combines with effective 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/e2 times the emission at 2 optical depths, etc.). Near the center of the star optical depth is effectively infinite, causing approximately constant brightness. Near the edge of the star the optical depth decreases due to lower gas density and shorter distance through the sun until the effective optical depth becomes zero at the apparent edge of the sun.

The effective temperature of the photosphere also decreases for an increasing distance from the center of the star. The radiation emitted from a gas is approximately black-body radiation, which is proportional to the fourth power of the temperature. Thus, even before the optical depth in a direction becomes finite, the emitted energy comes more from cooler parts of the photosphere, resulting in less total energy reaching the viewer.

The temperature in the atmosphere of a star is not always decreasing with increasing height. For certain spectral lines, the optical depth is greatest in a region of increasing temperature. In that case the phenomenon of "limb brightening" is seen instead; for the Sun the existence of a temperature minimum region means that limb brightening should start to dominate at far-infrared or radio wavelengths. Outside the lower atmosphere, and well above the temperature-minimum region, lies the million-kelvin solar corona. For most wavelengths this region is optically thin, i.e. has small optical depth, and must therefore be limb-brightened if spherically symmetric.

Calculation of limb darkeningEdit

 
Limb darkening geometry. The star is centered at O  and has radius R . The observer is at point P  a distance r  from the center of the star, and is looking at point S  on the surface of the star. From the point of view of the observer, S  is at an angle θ from a line through the center of the star, and the edge or limb of the star is at angle Ω.

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 ak = 0 except a0 = 1. As another example, for the sun at 550 nm, 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 ak constants can be related to the Ak constants. For N = 2,

 
 

For the sun at 550 nm, we then have

 
 

This model gives an intensity at the edge of the sun's disk of only 30% of the intensity at the centre 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 for the ratio of the mean intensity to the central intensity. The mean intensity Im 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 at 550 nm, this says that the average intensity is 80.5% of the intensity at the centre.

ReferencesEdit

  • Billings, Donald E. (1966). A Guide to the Solar Corona. Academic Press, New York.
  • Cox, Arthur N. (ed) (2000). Allen's Astrophysical Quantities (14th ed.). Springer-Verlag, NY. ISBN 0-387-98746-0.CS1 maint: extra text: authors list (link)
  • Milne, E.A. (1921). "Radiative Equilibrium in the Outer Layers of a Star: the Temperature Distribution and the Law of Darkening" (PDF). MNRAS. 81 (5): 361–375. Bibcode:1921MNRAS..81..361M. doi:10.1093/mnras/81.5.361.
  • Minnaert, M. (1930). "On the Continuous Spectrum of the Corona and its Polarisation". Zeitschrift für Astrophysik. 1: 209. Bibcode:1930ZA......1..209M.
  • Neckel, H.; Labs, D. (1994). "Solar Limb Darkening 1986-1990". Solar Physics. 153 (1–2): 91–114. Bibcode:1994SoPh..153...91N. doi:10.1007/BF00712494.
  • van de Hulst; H. C. (1950). "The Electron Density of the Solar Corona". Bulletin of the Astronomical Institutes of the Netherlands. 11 (410): 135. Bibcode:1950BAN....11..135V.
  • Mariska, John (1993). The Solar Transition Region. Cambridge University Press, Cambridge. ISBN 0521382610.
  • Steiner, O. (2007). "Photospheric processes and magnetic flux tubes". AIP Conference Proceedings. 919: 74–121. arXiv:0709.0081. Bibcode:2007AIPC..919...74S. doi:10.1063/1.2756784.