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 a1 = 1. As another example, for the Sun at , the limb darkening is well expressed by N = 2 and 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 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 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 center. References Stellar phenomena Solar phenomena