## Centerto Limb Variation

Neglecting the center-to-limb variation (CLV) is equivalent to assuming that the stellar disks have uniform brightness. There is ample evidence from the Sun and other stars, however, that surface brightness varies from mid-disk to the limb. Stellar surface imaging by microlensing30 [cf. Sasselov (Sasselov 1998a, b)] is used to measure stellar CLV. CLV, as the term is used in this section, is the dependence of intensity on angular distance from the surface normal (see Fig. 3.17). It arises because temperature increases with depth in stellar atmospheres, and the line-of-sight at the limb does not penetrate into high-temperature regions as does the line-of-sight through the disk center. Therefore, in order to compute the intensity at an arbitrary point, a factor D(m) needs to be computed. Let y denote the angle (sometimes called the aspect angle) between the surface normal n(rs ) in point rs and the arbitrary direction e, in which radiation is emitted, so that

M := cos y = cos y(rs) = n(rs) ■ e(rs, 0), 0 < m < 1. (3.2.22)

Limb brightening is important only forchromospheric and coronal emission and far-ultraviolet light curves, so we will concentrate in this section only on limb darkening, which affects the visible radiation from a stellar photosphere. The simple and traditional monochromatic limb-darkening law is

with a limb-darkening coefficient xa. Note the wavelength dependence of x indicated by the subscript A. As discussed at the end of this section, similar coefficients

30 Microlensing occurs if the light of a star is refracted and amplified by the gravity field of another star just moving through the line-of-sight.

Fig. 3.17 Center-to-limb variation. This figure shows the aspect angle y (angle between normal vector n and radiation emission direction e) appearing in the mathematical formulation of the limb darkening. The right part of the figure illustrates that the depth of the atmosphere region (and thus temperature) accessible to an observer varies with the aspect angle y

Fig. 3.17 Center-to-limb variation. This figure shows the aspect angle y (angle between normal vector n and radiation emission direction e) appearing in the mathematical formulation of the limb darkening. The right part of the figure illustrates that the depth of the atmosphere region (and thus temperature) accessible to an observer varies with the aspect angle y and laws can be established for the bolometric case. For notational convenience we drop the wavelength dependence in the rest of this section. The associated (monochromatic or bolometric) flux F received from the (spherical) stellar disk with radius R that is displayed in Fig. E.8 can be computed as p 2n p R