Limb Darkening and Luminosity

The Sun photographed on 8 May 2019 in white light by Matúš Motlo
showing sunspots, faculae, and limb darkening

The photosphere of our Sun and most other stars exhibit a phenomenon called limb darkening where the disk is brighter at the center than at the edges at optical wavelengths. This effect is more pronounced towards the violet end of the visible spectrum than it is towards the red end.

Limb darkening occurs because there is a strong temperature gradient within the photosphere (deeper is hotter) and we see deeper into the Sun at the center of the disk then we do toward the edges. The deeper, hotter regions of the photosphere produce more visible light than do the shallower, cooler regions.

Does this non-uniformity of light emitted from the disk of a star mean we are “missing” some light in measuring a star’s brightness that would then affect our ability to accurately calculate the star’s total luminosity? Not at all. Here’s why.

Stars are almost always isotropic emitters of light. That means they emit light uniformly in all directions. At a given distance from the star, an observer would measure the same brightness of the star no matter what their direction from it. Even though the edges of the stellar disk are darker, the center is brighter, and the total integrated brightness is the same as it would be if all parts of the disk were emitting uniformly.

We calculate the luminosity of the star by measuring the amount of light we receive across our collecting area (whether that be the human eye or the telescope aperture), and then dividing this collecting area into the total surface area of a sphere centered on the star and having a radius that is our distance from the star. We then take that quotient times the amount of light we detect in our small collecting area to get the total amount of light emitted by the star in all directions.