Chandrasekhar's X- and Y-function

${\displaystyle \begin{array}{cc}X(\mu )& =1+\mu {\int }_0^1\frac{\mathrm{\Psi }({\mu }^{\prime })}{\mu +{\mu }^{\prime }}[X(\mu )X({\mu }^{\prime })-Y(\mu )Y({\mu }^{\prime })]d{\mu }^{\prime },\\ Y(\mu )& =e^{-{\tau }_1/\mu }+\mu {\int }_0^1\frac{\mathrm{\Psi }({\mu }^{\prime })}{\mu -{\mu }^{\prime }}[Y(\mu )X({\mu }^{\prime })-X(\mu )Y({\mu }^{\prime })]d{\mu }^{\prime }\end{array}}$

where the characteristic function ${\displaystyle \mathrm{\Psi }(\mu )}$ is an even polynomial in ${\displaystyle \mu }$ generally satisfying the condition

${\displaystyle {\int }_0^1\mathrm{\Psi }(\mu )d\mu \le \frac{1}{2},}$

and ${\displaystyle 0<{\tau }_1<\mathrm{\infty }}$ 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

${\displaystyle X(\mu )\to H(\mu ),Y(\mu )\to 0\text{as}{\tau }_1\to \mathrm{\infty }}$

and also

${\displaystyle X(\mu )\to 1,Y(\mu )\to e^{-{\tau }_1/\mu }\text{as}{\tau }_1\to 0.}$

Approximation

The ${\displaystyle X}$ and ${\displaystyle Y}$ can be approximated up to nth order as

${\displaystyle \begin{array}{cc}X(\mu )& =\frac{(-1{)}^n}{{\mu }_1\cdots {\mu }_n}\frac{1}{[C_0^2(0)-C_1^2(0){]}^{1/2}}\frac{1}{W(\mu )}[P(-\mu )C_0(-\mu )-e^{-{\tau }_1/\mu }P(\mu )C_1(\mu )],\\ Y(\mu )& =\frac{(-1{)}^n}{{\mu }_1\cdots {\mu }_n}\frac{1}{[C_0^2(0)-C_1^2(0){]}^{1/2}}\frac{1}{W(\mu )}[e^{-{\tau }_1/\mu }P(\mu )C_0(\mu )-P(-\mu )C_1(-\mu )]\end{array}}$

where ${\displaystyle C_0}$ and ${\displaystyle C_1}$ are two basic polynomials of order n (Refer Chandrasekhar chapter VIII equation (97)), ${\displaystyle P(\mu )=\prod _{i=1}^n(\mu -{\mu }_i)}$ where ${\displaystyle {\mu }_i}$ are the zeros of Legendre polynomials and ${\displaystyle W(\mu )=\prod _{\alpha =1}^n(1-k_{\alpha }^2{\mu }^2)}$, where ${\displaystyle k_{\alpha }}$ are the positive, non vanishing roots of the associated characteristic equation

${\displaystyle 1=2\sum _{j=1}^n\frac{a_j\mathrm{\Psi }({\mu }_j)}{1-k^2{\mu }_j^2}}$

where ${\displaystyle a_j}$ are the quadrature weights given by

${\displaystyle a_j=\frac{1}{P_{2n}^{\prime }({\mu }_j)}{\int }_{-1}^1\frac{P_{2n}({\mu }_j)}{\mu -{\mu }_j}d{\mu }_j}$