Limb darkening refers to the diminishing of intensity in the image of a star as one moves from the center of the image to the edge or "limb" of the image. The effect occurs as the result of two effects:

• The density of the star diminishes as distance from the center increases

• The temperature of the star diminishes as distance from the center increases.

Crucial to understanding limb darkening is the idea of optical depth. An optical depth of unity is that thickness of absorbing gas from which a fraction of 1/e photons can escape. It is at this point that the star becomes opaque. The radiation reaching us is closely approximated by the integral of the emission along the entire line of sight, up to that point where the optical depth is 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 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 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 strongly increasing function of temperature. For a black body, for example, the intensity is proportional to the fourth power of the temperature (Stefan-Boltzmann law). This means that when we look at a star, the part of the star that we see is that part along the line of sight which is visible, and which is at the highest temperature, which will be the point at which the optical depth is unity. Since that point is closer 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 for the sun does not uniformly drop as radius increases, 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"

Calculation of Limb Darkening

In the figure on the right, 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. It can be seen that 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 a₀=1. As another example, for the sun at 550 nm, the limb darkening is well expressed by N=2 and

a₀ = 1 - a₁ - a₂

a₁ = 0.93

a₂ = - 0.23

We can convert from ψ to θ using the relationship:

where Ω is the angle from the observer to the limb of the star.

references

• Billings, Donald E., "A Guide to the Solar Corona", Academic Press, New York 1966.

• Cox, Arthur N., "Allen's Astrophysical Quantities/editor, Arthur Cox", Springer-Verlag, 14th ed., NY, 2000.

• Milne, E.A., 1921, MNRAS 81, pp361-375.

• Minnaert, M., Z.f. Ap. 1, 209, 1930.

• Neckel, H. and Labs, D., "Solar Limb Darkening 1986-1990 Solar Physics 153:91-114, 1994.

• van de Hulst, H. C. 1950, Bull. Astron. Inst. Netherlands, 11 (410), 135