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

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

Approximation

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

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

Properties

• 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.

See also

• Chandrasekhar's H-function

References

1. Chandrasekhar, Subrahmanyan. Radiative transfer. Courier Corporation, 2013.
2. Howell, John R., M. Pinar Menguc, and Robert Siegel. Thermal radiation heat transfer. CRC press, 2010.
3. Modest, Michael F. Radiative heat transfer. Academic press, 2013.
4. Hottel, Hoyt Clarke, and Adel F. Sarofim. Radiative transfer. McGraw-Hill, 1967.
5. Sparrow, Ephraim M., and Robert D. Cess. "Radiation heat transfer." Series in Thermal and Fluids Engineering, New York: McGraw-Hill, 1978, Augmented ed. (1978).
6. Chandrasekhar, Subrahmanyan. Radiative transfer. Courier Corporation, 2013.