Limb darkening

Definition

Consider a spherical star seen from far away. For a point on its visible disk, let $\theta$ be the angle between the line of sight and the outward normal to the surface there. Let $\mu = \cos\theta \in [0, 1]$ . Let $I(\mu)$ be the specific intensity of the emergent radiation in the direction of the observer. The star shows limb darkening when $I(\mu)$ is an increasing function of $\mu$ , so that $I(\mu) < I(1)$ for $\mu < 1$ ; the ratio $I(\mu)/I(1)$ is the limb-darkening law.

Remarks

Example: in a grey atmosphere in the Eddington approximation, the source function is linear in the optical depth, $S(\tau) = a + b\tau$ with $b = 3a/2 > 0$ . The emergent intensity is then $I(\mu) = a + b\mu$ , so $I(\mu)/I(1) = (2 + 3\mu)/5$ . This increases with $\mu$ and equals $2/5$ at the limb. Non-example: a semi-infinite atmosphere with constant source function $S$ gives $I(\mu) = S$ for all $\mu$ , so there is no limb darkening. A common empirical form is the linear law $I(\mu)/I(1) = 1 - u(1 - \mu)$ with a coefficient $u$ that depends on wavelength. The Eddington–Barbier relation $I(\mu) \approx S(\tau = \mu)$ explains the effect: the observer sees down to optical depth about $\mu$ along the normal.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Optical depth concept
Limb darkening definition

Tag	`9QR0AAW4YE`
URI	`m2://9QR0AAW4YE`
Kind	definition
Writing	W0 stub
MSC	85A25, 85A15
Defines	Limb darkening
Uses	Optical depth
Changed	3 November 2026

Limb darkening

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here