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 $$0\leq\mu\leq 1$$, satisfies the pair of nonlinear integral equations


 * $$\begin{align}

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{align}$$

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


 * $$\int_0^1\Psi(\mu) \, d\mu \leq \frac{1}{2}, $$

and $$0<\tau_1<\infty$$ 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


 * $$X(\mu)\rightarrow H(\mu), \quad Y(\mu)\rightarrow 0 \ \text{as} \ \tau_1\rightarrow\infty$$

and also


 * $$X(\mu)\rightarrow 1, \quad Y(\mu)\rightarrow e^{-\tau_1/\mu} \ \text{as} \ \tau_1\rightarrow 0.$$

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


 * $$\begin{align}

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{align}$$

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


 * $$1 = 2 \sum_{j=1}^n \frac{a_j\Psi(\mu_j)}{1-k^2\mu_j^2}$$

where $$a_j$$ are the quadrature weights given by


 * $$a_j = \frac 1 {P_{2n}'(\mu_j)} \int_{-1}^1 \frac{P_{2n}(\mu_j)}{\mu-\mu_j} \, d\mu_j$$

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{align}

\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{align}$$ [a(\alpha_0^2-\beta_0^2)+b(\alpha_1^2-\beta_1^2)]$$.
 * $$\int_0^1 X(\mu)\Psi(\mu) \, d\mu = 1- \left[1-2\int_0^1 \Psi(\mu)\,d\mu + \left\{\int_0^1 Y(\mu) \Psi(\mu) \,d\mu\right\}^2\right]^{1/2}.$$ For conservative case, this integral property reduces to $$\int_0^1 [X(\mu)+Y(\mu)]\Psi(\mu) \, d\mu = 1.$$
 * If the abbreviations $$x_n = \int_0^1 X(\mu) \Psi(\mu) \mu^n \, d\mu, \ y_n = \int_0^1 Y(\mu)\Psi(\mu) \mu^n \, d\mu, \ \alpha_n = \int_0^1 X(\mu)\mu^n \, d\mu, \ \beta_n = \int_0^1 Y(\mu) \mu^n \, d\mu$$ for brevity are introduced, then we have a relation stating $$(1-x_0)x_2 + y_0y_2 + \frac{1}{2} (x_1^2-y_1^2) = \int_0^1 \Psi(\mu)\mu^2 \, d\mu.$$ In the conservative, this reduces to $$y_0(x_2+y_2) + \frac{1}{2}(x_1^2-y_1^2)=\int_0^1 \Psi(\mu)\mu^2 \, d\mu$$
 * If the characteristic function is $$\Psi(\mu)=a+b\mu^2$$, where $$a, b $$ are two constants, then we have $$\alpha_0=1+\frac{1}{2}
 * For conservative case, the solutions are not unique. If $$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.