I was wondering if anyone knew of any resources to understand the linear limb darkening coefficient, $u$. That is to say, how the $\theta$-dependent coefficient $u$ (or sometimes $b_{\nu}$) varies with wavelength and with stellar effective temperature.
I've looked at a number of sources, such as Schwarzschild (1906), Milne (1921) and my own lecture notes and others, but nothing on this coefficient? Does anyone know where to even start looking for information of the properties of $u$?
Limb darkening laws are an attempt to parameterise the fall in specific intensity that is observed on the projected disc of a star as you move from the centre of the disc to the limb. Limb darkening is just a consequence of the temperature gradient in a stellar atmosphere and the viewing geometry - a line of sight towards the limb of a star penetrates less deeply into the stellar atmosphere than one towards the centre.
A limb darkening law is specified in terms of the cosine of an angle $\theta$ between the line of sight to the observer and a line from the centre of the star to the point on the stellar surface considered.
The simplest limb darkening law is where one assumes that the dependence on $\cos \theta$ is linear - the so-called linear limb darkening law. It is usually written as $$\frac{I_\theta}{I_0} = 1 - u\left(1 - \cos\theta\right)\ , $$ where $I_\theta$ is the observed intensity at (a circle of) points defined by the angle $\theta$, $I_0$ is the intensity at the centre of the disc and $u$ is the linear limb darkening coefficient.
The limb darkening coefficient is wavelength-dependent and appropriate values need to be obtained empirically (from observations of the Sun for example) or from theoretical stellar atmosphere models.
The website constructed by John Taylor (Southworth) at Keele University is an excellent resource, which includes software to interpolate tabulated values of limb darkening coefficients (for linear and more complex formulations) from a variety of primary sources.
