Limb darkening

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 theory

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. Note that for a typical star, this drawing is not to scale. E.g., for the Sun, L would be only a few hundred km.

Crucial to understanding limb darkening is the idea of optical depth. A distance equal to one optical depth is the thickness of the absorbing gas from which a fraction of 1/e photons can escape. This is what defines the visible edge of a star, since it is at a few optical depths that the star becomes opaque. The radiation reaching us is closely approximated by the sum of all the emission along the entire line of sight, up to that point where the optical depth is unity. In particular, if the intensity of radiation in the star varies linearly with optical depth, then the radiation reaching us will be of the intensity at an optical depth of unity. When we look near the edge of a star, we cannot "see" to the same depth as when we look at the center because the line of sight must travel at an oblique angle through the stellar gas when looking near the limb. In other words, the solar radius at which we see the optical depth as being unity increases as we move our line of sight towards the limb.

The second effect is the fact that the effective temperature of the stellar atmosphere is (usually) decreasing for an increasing distance from the center of the star. The radiation emitted from a gas is a strong function of temperature. For a black body, for example, the spectrally integrated intensity is proportional to the fourth power of the temperature (Stefan–Boltzmann law). Since when we look at a star, at first approximation, the radiation comes from the point at which the optical depth is unity, and that point is deeper in when looking at the center, the temperature will be higher, and the intensity will be greater, than when we look at the limb.

In fact, the temperature in the atmosphere of a star does not always decrease with increasing height, and for certain spectral lines, the optical depth is unity in a region of increasing temperature. In this case we see the phenomenon of "limb brightening"; 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, we find 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.

Further complication comes from the existence of rough (three-dimensional) structure. The classical analysis of stellar limb darkening, as described below, assumes the existence of a smooth hydrostatic equilibrium, and at some level of precision this assumption must fail (most obviously in sunspots and faculae, but generally everywhere). Instead, the boundary between the chromosphere and the corona consists of a very complicated transition region, best observed at ultraviolet wavelengths only detectable from space.

Calculation of limb darkening

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:

{\frac {I(\psi )}{I(0)}}=\sum _{k=0}^{N}a_{k}\cos ^{k}\psi ,

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

\sum _{k=0}^{N}a_{k}=1.

For example, for a Lambertian radiator (no limb darkening) we will have all a_k = 0 except a₀ = 1. As another example, for the sun at 550 nm, the limb darkening is well expressed by N = 2 and

a_{0}=1-a_{1}-a_{2}=0.3,

a_{1}=0.93,

a_{2}=-0.23

(See Cox, 2000). The equation for limb darkening is sometimes more conveniently written as

{\frac {I(\psi )}{I(0)}}=1+\sum _{k=1}^{N}A_{k}(1-\cos \psi )^{k},

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,

A_{1}=-(a_{1}+2*a_{2}),

A_{2}=a_{2}.

For the sun at 550 nm, we then have

A_{1}=-0.47,

A_{2}=-0.23.

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

\cos \psi ={\frac {\sqrt {\cos ^{2}\theta -\cos ^{2}\Omega }}{\sin \Omega }}={\sqrt {1-\left({\frac {\sin \theta }{\sin \Omega }}\right)^{2}}},

where Ω is the angle from the observer to the limb of the star. For small θ we have

\cos \psi \approx {\sqrt {1-\left({\frac {\theta }{\sin \Omega }}\right)^{2}}}.

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 I_m is the integral of the intensity over the disk of the star divided by the solid angle subtended by the disk:

I_{m}={\frac {\int I(\psi )\,d\omega }{\int d\omega }},

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

I_{m}={\frac {\int _{\cos \Omega }^{1}I(\psi )\,d\cos \theta }{\int _{\cos \Omega }^{1}d\cos \theta }}={\frac {\int _{\cos \Omega }^{1}I(\psi )\,d\cos \theta }{1-\cos \Omega }}.

Although this equation can be solved analytically, it is rather cumbersome. However, for an observer at infinite distance from the star, $d\cos \theta$ can be replaced by $\sin ^{2}\Omega \cos \psi \,d\cos \psi$ , so we have

I_{m}={\frac {\int _{0}^{1}I(\psi )\cos \psi \,d\cos \psi }{\int _{0}^{1}\cos \psi \,d\cos \psi }}=2\int _{0}^{1}I(\psi )\cos \psi \,d\cos \psi ,

which gives

{\frac {I_{m}}{I(0)}}=2\sum _{k=0}^{N}{\frac {a_{k}}{k+2}}.

For the sun at 550 nm, this says that the average intensity is 80.5% of the intensity at the centre.

References

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.
Milne, E.A. (1921). "Radiative Equilibrium in the Outer Layers of a Star: the Temperature Distribution and the Law of Darkening". MNRAS. 81: 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.
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". Bull. Astron. Inst. Netherlands. 11 (410): 135.
Mariska, John (1993). The Solar Transition Region. Cambridge University Press, Cambridge. ISBN 0521382610.
Steiner, O., Photospheric processes and magnetic flux tubes, (2007)

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