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
X(\mu) &= 1+ \mu \int_0^1 \frac (\mu)X(\mu')-Y(\mu)Y(\mu')\, d\mu',\\ ptY(\mu) &= e^ + \mu \int_0^1 \frac (\mu)X(\mu')-X(\mu)Y(\mu')\, d\mu'
\end

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

:$\int_0^1\Psi(\mu) \, d\mu \leq \frac,$

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

: $X(\mu)\rightarrow H(\mu), \quad Y(\mu)\rightarrow 0 \ \text \ \tau_1\rightarrow\infty$

and also

:$X(\mu)\rightarrow 1, \quad Y(\mu)\rightarrow e^ \ \text \ \tau_1\rightarrow 0.$

Approximation

The $X$ and $Y$ can be approximated up to ''n''th order as

:\begin
X(\mu) &= \frac\frac \frac (-\mu) C_0(-\mu)-e^P(\mu)C_1(\mu)\\ ptY(\mu) &= \frac\frac \frac ^P(\mu) C_0(\mu)-P(-\mu)C_1(-\mu)\end

where $C_0$ and $C_1$ are two basic polynomials of order n (Refer Chandrasekhar chapter VIII equation (97)Chandrasekhar, Subrahmanyan. Radiative transfer. Courier Corporation, 2013.), $P(\mu) = \prod_^n (\mu-\mu_i)$ where $\mu_i$ are the zeros of Legendre polynomials and $W(\mu)= \prod_^n (1-k_\alpha^2\mu^2)$, where $k_\alpha$ are the positive, non vanishing roots of the associated characteristic equation

:$1 = 2 \sum_^n \frac$

where $a_j$ are the quadrature weights given by

:$a_j = \frac 1 \int_^1 \frac \, d\mu_j$

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
:$\begin \frac &= Y(\mu,\tau_1)\int_0^1 \frac \Psi(\mu') Y(\mu',\tau_1),\\ \frac + \frac&= X(\mu,\tau_1)\int_0^1 \frac \Psi(\mu') Y(\mu',\tau_1) \end$
* \int_0^1 X(\mu)\Psi(\mu) \, d\mu = 1- \left -2\int_0^1 \Psi(\mu)\,d\mu + \left\^2\right. For conservative case, this integral property reduces to \int_0^1 (\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 $(1-x_0)x_2 + y_0y_2 + \frac (x_1^2-y_1^2) = \int_0^1 \Psi(\mu)\mu^2 \, d\mu.$ In the conservative, this reduces to $y_0(x_2+y_2) + \frac(x_1^2-y_1^2)=\int_0^1 \Psi(\mu)\mu^2 \, d\mu$
*If the characteristic function is $\Psi(\mu)=a+b\mu^2$, where $a, b$ are two constants, then we have \alpha_0=1+\frac
(\alpha_0^2-\beta_0^2)+b(\alpha_1^2-\beta_1^2)/math>.
*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 (\mu) + Y(\mu)\ G(\mu)=Y(\mu) + Q\mu (\mu)+Y(\mu)/math>, where $Q$ is an arbitrary constant.

See also

*Chandrasekhar's H-function

References

{{Reflist

Special functions
Integral equations
Scattering
Scattering, absorption and radiative transfer (optics)