In atmospheric radiation, Chandrasekhar's X- and Y-function appears as the solutions of problems involving diffusive reflection and transmission, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar.
[1][2][3][4][5] The Chandrasekhar's X- and Y-function
defined in the interval
, satisfies the pair of nonlinear integral equations where the characteristic function
generally satisfying the condition and
is the optical thickness of the atmosphere.
If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative.
These functions are related to Chandrasekhar's H-function as and also The
can be approximated up to nth order as where
are two basic polynomials of order n (Refer Chandrasekhar chapter VIII equation (97)[6]),
are the zeros of Legendre polynomials and
are the positive, non vanishing roots of the associated characteristic equation where
are the quadrature weights given by