Physics: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 [math]\displaystyle{ X(\mu),\ Y(\mu) }[/math] defined in the interval [math]\displaystyle{ 0\leq\mu\leq 1 }[/math], satisfies the pair of nonlinear integral equations

[math]\displaystyle{ \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} }[/math]

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

[math]\displaystyle{ \int_0^1\Psi(\mu) \, d\mu \leq \frac{1}{2}, }[/math]

and [math]\displaystyle{ 0\lt \tau_1\lt \infty }[/math] 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

[math]\displaystyle{ X(\mu)\rightarrow H(\mu), \quad Y(\mu)\rightarrow 0 \ \text{as} \ \tau_1\rightarrow\infty }[/math]

and also

[math]\displaystyle{ X(\mu)\rightarrow 1, \quad Y(\mu)\rightarrow e^{-\tau_1/\mu} \ \text{as} \ \tau_1\rightarrow 0. }[/math]

Approximation

The [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] can be approximated up to nth order as

[math]\displaystyle{ \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} }[/math]

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

[math]\displaystyle{ 1 = 2 \sum_{j=1}^n \frac{a_j\Psi(\mu_j)}{1-k^2\mu_j^2} }[/math]

where [math]\displaystyle{ a_j }[/math] are the quadrature weights given by

[math]\displaystyle{ a_j = \frac 1 {P_{2n}'(\mu_j)} \int_{-1}^1 \frac{P_{2n}(\mu_j)}{\mu-\mu_j} \, d\mu_j }[/math]

Properties

• If [math]\displaystyle{ X(\mu,\tau_1), \ Y(\mu,\tau_1) }[/math] are the solutions for a particular value of [math]\displaystyle{ \tau_1 }[/math], then solutions for other values of [math]\displaystyle{ \tau_1 }[/math] are obtained from the following integro-differential equations

[math]\displaystyle{ \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} }[/math]

• [math]\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}. }[/math] For conservative case, this integral property reduces to [math]\displaystyle{ \int_0^1 [X(\mu)+Y(\mu)]\Psi(\mu) \, d\mu = 1. }[/math]
• If the abbreviations [math]\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 }[/math] for brevity are introduced, then we have a relation stating [math]\displaystyle{ (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. }[/math] In the conservative, this reduces to [math]\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 }[/math]
• If the characteristic function is [math]\displaystyle{ \Psi(\mu)=a+b\mu^2 }[/math], where [math]\displaystyle{ a, b }[/math] are two constants, then we have [math]\displaystyle{ \alpha_0=1+\frac{1}{2} [a(\alpha_0^2-\beta_0^2)+b(\alpha_1^2-\beta_1^2)] }[/math].
• For conservative case, the solutions are not unique. If [math]\displaystyle{ X(\mu), \ Y(\mu) }[/math] are solutions of the original equation, then so are these two functions [math]\displaystyle{ F(\mu)=X(\mu) + Q\mu [X(\mu) + Y(\mu)],\ G(\mu)=Y(\mu) + Q\mu[X(\mu)+Y(\mu)] }[/math], where [math]\displaystyle{ Q }[/math] is an arbitrary constant.

See also

• Chandrasekhar's H-function