Chandrasekhar's X- and Y-function

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. The Chandrasekhar's X- and Y-function $X(\mu ),Y(\mu )$ defined in the interval $0\le \mu \le 1$, satisfies the pair of nonlinear integral equations

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

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

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

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

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

and also

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

Approximation

The $X$ and $Y$ can be approximated up to nth order as

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

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

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

where $a_j$ are the quadrature weights given by

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

Properties

• If $X(\mu ,{\tau }_1),Y(\mu ,{\tau }_1)$ are the solutions for a particular value of ${\tau }_1$, then solutions for other values of ${\tau }_1$ are obtained from the following integro-differential equations

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

• ${\int }_0^{1}X(\mu )\mathrm{\Psi }(\mu )d\mu =1-{[1-2{\int }_0^1\mathrm{\Psi }(\mu )d\mu +{\{{\int }_0^{1}Y(\mu )\mathrm{\Psi }(\mu )d\mu \}}^2]}^{1/2}.$ For conservative case, this integral property reduces to ${\int }_0^1[X(\mu )+Y(\mu )]\mathrm{\Psi }(\mu )d\mu =1.$
• If the abbreviations $x_n={\int }_0^{1}X(\mu )\mathrm{\Psi }(\mu ){\mu }^{n}d\mu ,y_n={\int }_0^{1}Y(\mu )\mathrm{\Psi }(\mu ){\mu }^{n}d\mu ,{\alpha }_n={\int }_0^{1}X(\mu ){\mu }^{n}d\mu ,{\beta }_n={\int }_0^{1}Y(\mu ){\mu }^{n}d\mu$ for brevity are introduced, then we have a relation stating $(1-x_0)x_2+y_0{y}_2+\frac{1}{2}(x_1^2-y_1^2)={\int }_0^1\mathrm{\Psi }(\mu ){\mu }^{2}d\mu .$ In the conservative, this reduces to $y_0(x_2+y_2)+\frac{1}{2}(x_1^2-y_1^2)={\int }_0^1\mathrm{\Psi }(\mu ){\mu }^{2}d\mu$
• If the characteristic function is $\mathrm{\Psi }(\mu )=a+b{\mu }^2$, where $a,b$ are two constants, then we have ${\alpha }_0=1+\frac{1}{2}[a({\alpha }_0^2-{\beta }_0^2)+b({\alpha }_1^2-{\beta }_1^2)]$.
• For conservative case, the solutions are not unique. If $X(\mu ),Y(\mu )$ are solutions of the original equation, then so are these two functions $F(\mu )=X(\mu )+Q\mu [X(\mu )+Y(\mu )],G(\mu )=Y(\mu )+Q\mu [X(\mu )+Y(\mu )]$, where $Q$ is an arbitrary constant.

See also

• Chandrasekhar's H-function