The transfer equation, Eq. (9.34), is written in terms of the distance, s, measured along the path of a light ray. In different coordinate systems, the transfer equation will look slightly different, and care must be taken to include all of the necessary terms.
(a) Show that in a spherical coordinate system, with the center of the star at the origin, the transfer equation has the form
−κλρcosθ′drdIλ=Iλ−Sλ,
where θ′ is the angle between the ray and the outward radial direction. Note that you cannot simply replace s with r!
(b) Use this form of the transfer equation to derive Eq. (9.31).