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.^{[1][2][3][4][5]} The Chandrasekhar's X- and Y-function ${\displaystyle X(\mu ),\ Y(\mu )}$ defined in the interval ${\displaystyle 0\leq \mu \leq 1}$, satisfies the pair of nonlinear integral equations

${\displaystyle {\begin{aligned}X(\mu )&=1+\mu \int _{0}^{1}{\frac {\Psi (\mu ')}{\mu +\mu '}}[X(\mu )X(\mu ')-Y(\mu )Y(\mu ')]\,d\mu ',\\[5pt]Y(\mu )&=e^{-\tau _{1}/\mu }+\mu \int _{0}^{1}{\frac {\Psi (\mu ')}{\mu -\mu '}}[Y(\mu )X(\mu ')-X(\mu )Y(\mu ')]\,d\mu '\end{aligned}}}$

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

${\displaystyle \int _{0}^{1}\Psi (\mu )\,d\mu \leq {\frac {1}{2}},}$

and ${\displaystyle 0<\tau _{1}<\infty }$ 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

${\displaystyle X(\mu )\rightarrow H(\mu ),\quad Y(\mu )\rightarrow 0\ {\text{as}}\ \tau _{1}\rightarrow \infty }$

and also

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

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

${\displaystyle {\begin{aligned}X(\mu )&={\frac {(-1)^{n}}{\mu _{1}\cdots \mu _{n}}}{\frac {1}{[C_{0}^{2}(0)-C_{1}^{2}(0)]^{1/2}}}{\frac {1}{W(\mu )}}[P(-\mu )C_{0}(-\mu )-e^{-\tau _{1}/\mu }P(\mu )C_{1}(\mu )],\\[5pt]Y(\mu )&={\frac {(-1)^{n}}{\mu _{1}\cdots \mu _{n}}}{\frac {1}{[C_{0}^{2}(0)-C_{1}^{2}(0)]^{1/2}}}{\frac {1}{W(\mu )}}[e^{-\tau _{1}/\mu }P(\mu )C_{0}(\mu )-P(-\mu )C_{1}(-\mu )]\end{aligned}}}$

where ${\displaystyle C_{0}}$ and ${\displaystyle C_{1}}$ are two basic polynomials of order n (Refer Chandrasekhar chapter VIII equation (97)^[6]), ${\displaystyle P(\mu )=\prod _{i=1}^{n}(\mu -\mu _{i})}$ 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

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

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

${\displaystyle a_{j}={\frac {1}{P_{2n}'(\mu _{j})}}\int _{-1}^{1}{\frac {P_{2n}(\mu _{j})}{\mu -\mu _{j}}}\,d\mu _{j}}$

• If ${\displaystyle X(\mu ,\tau _{1}),\ Y(\mu ,\tau _{1})}$ are the solutions for a particular value of ${\displaystyle \tau _{1}}$, then solutions for other values of ${\displaystyle \tau _{1}}$ are obtained from the following integro-differential equations

${\displaystyle {\begin{aligned}{\frac {\partial X(\mu ,\tau _{1})}{\partial \tau _{1}}}&=Y(\mu ,\tau _{1})\int _{0}^{1}{\frac {d\mu '}{\mu '}}\Psi (\mu ')Y(\mu ',\tau _{1}),\\{\frac {\partial Y(\mu ,\tau _{1})}{\partial \tau _{1}}}+{\frac {Y(\mu ,\tau _{1})}{\mu }}&=X(\mu ,\tau _{1})\int _{0}^{1}{\frac {d\mu '}{\mu '}}\Psi (\mu ')Y(\mu ',\tau _{1})\end{aligned}}}$

• ${\displaystyle \int _{0}^{1}X(\mu )\Psi (\mu )\,d\mu =1-\left[1-2\int _{0}^{1}\Psi (\mu )\,d\mu +\left\{\int _{0}^{1}Y(\mu )\Psi (\mu )\,d\mu \right\}^{2}\right]^{1/2}.}$ For conservative case, this integral property reduces to ${\displaystyle \int _{0}^{1}[X(\mu )+Y(\mu )]\Psi (\mu )\,d\mu =1.}$
• If the abbreviations ${\displaystyle x_{n}=\int _{0}^{1}X(\mu )\Psi (\mu )\mu ^{n}\,d\mu ,\ y_{n}=\int _{0}^{1}Y(\mu )\Psi (\mu )\mu ^{n}\,d\mu ,\ \alpha _{n}=\int _{0}^{1}X(\mu )\mu ^{n}\,d\mu ,\ \beta _{n}=\int _{0}^{1}Y(\mu )\mu ^{n}\,d\mu }$ for brevity are introduced, then we have a relation stating ${\displaystyle (1-x_{0})x_{2}+y_{0}y_{2}+{\frac {1}{2}}(x_{1}^{2}-y_{1}^{2})=\int _{0}^{1}\Psi (\mu )\mu ^{2}\,d\mu .}$ In the conservative, this reduces to ${\displaystyle y_{0}(x_{2}+y_{2})+{\frac {1}{2}}(x_{1}^{2}-y_{1}^{2})=\int _{0}^{1}\Psi (\mu )\mu ^{2}\,d\mu }$
• If the characteristic function is ${\displaystyle \Psi (\mu )=a+b\mu ^{2}}$, where ${\displaystyle a,b}$ are two constants, then we have ${\displaystyle \alpha _{0}=1+{\frac {1}{2}}[a(\alpha _{0}^{2}-\beta _{0}^{2})+b(\alpha _{1}^{2}-\beta _{1}^{2})]}$.
• For conservative case, the solutions are not unique. If ${\displaystyle X(\mu ),\ Y(\mu )}$ are solutions of the original equation, then so are these two functions ${\displaystyle F(\mu )=X(\mu )+Q\mu [X(\mu )+Y(\mu )],\ G(\mu )=Y(\mu )+Q\mu [X(\mu )+Y(\mu )]}$, where ${\displaystyle Q}$ is an arbitrary constant.

• Chandrasekhar's H-function