Chandrasekhar's H-function



In atmospheric radiation, Chandrasekhar's H-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandrasekhar's H-function $$H(\mu)$$ defined in the interval $$0\leq\mu\leq 1$$, satisfies the following nonlinear integral equation


 * $$H(\mu) = 1+\mu H(\mu)\int_0^1 \frac{\Psi(\mu')}{\mu + \mu'}H(\mu') \, d\mu'$$

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


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

If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative. Albedo is given by $$\omega_o= 2\Psi(\mu) = \text{constant}$$. An alternate form which would be more useful in calculating the H function numerically by iteration was derived by Chandrasekhar as,


 * $$\frac{1}{H(\mu)} = \left[1-2\int_0^1\Psi(\mu) \, d\mu\right]^{1/2} + \int_0^1 \frac{\mu'\Psi(\mu')}{\mu + \mu'}H(\mu') \, d\mu'$$.

In conservative case, the above equation reduces to


 * $$\frac{1}{H(\mu)}= \int_0^1 \frac{\mu' \Psi(\mu')}{\mu+\mu'}H(\mu')d\mu'$$.

Approximation
The H function can be approximated up to an order $$n$$ as


 * $$H(\mu) = \frac{1}{\mu_1 \cdots \mu_n}\frac{\prod_{i=1}^n (\mu+\mu_i)}{\prod_\alpha (1+k_\alpha\mu)}$$

where $$\mu_i$$ are the zeros of Legendre polynomials $$P_{2n}$$ and $$k_\alpha$$ 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


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

Explicit solution in the complex plane
In complex variable $$z$$ the H equation is


 * $$ H(z) = 1- \int_0^1 \frac z {z+\mu} H(\mu)\Psi(\mu) \, d\mu, \quad \int_0^1 |\Psi(\mu)| \, d\mu \leq \frac{1}{2}, \quad \int_0^\delta |\Psi(\mu)| \, d\mu \rightarrow 0, \ \delta\rightarrow 0$$

then for $$\Re (z)>0$$, a unique solution is given by


 * $$\ln H(z) = \frac{1}{2\pi i} \int_{-i\infty}^{+ i\infty} \ln T(w) \frac{z}{w^2-z^2} \, dw$$

where the imaginary part of the function $$T(z)$$ can vanish if $$z^2$$ is real i.e., $$z^2 = u+iv = u\ (v=0)$$. Then we have


 * $$T(z) = 1- 2 \int_0^1 \Psi(\mu) \, d\mu - 2 \int_0^1 \frac{\mu^2 \Psi(\mu)}{u-\mu^2} \, d\mu$$

The above solution is unique and bounded in the interval $$0\leq z\leq 1$$ for conservative cases. In non-conservative cases, if the equation $$T(z)=0$$ admits the roots $$\pm 1/k$$, then there is a further solution given by


 * $$H_1(z) = H(z) \frac{1+kz}{1-kz}$$

Properties

 * $$\int_0^1 H(\mu)\Psi(\mu) \, d\mu = 1-\left[1-2\int_0^1\Psi(\mu) \, d\mu \right]^{1/2}$$. For conservative case, this reduces to $$\int_0^1 \Psi(\mu)d\mu=\frac{1}{2}$$.
 * $$\left[1-2\int_0^1\Psi(\mu) \, d\mu\right]^{1/2} \int_0^1 H(\mu) \Psi(\mu) \mu^2 \, d\mu + \frac{1}{2} \left[\int_0^1 H(\mu)\Psi(\mu)\mu \, d\mu\right]^2 = \int_0^1 \Psi(\mu)\mu^2 \, d\mu$$. For conservative case, this reduces to $$\int_0^1 H(\mu)\Psi(\mu) \mu d\mu = \left[2\int_0^1 \Psi(\mu)\mu^2d\mu\right]^{1/2}$$.
 * If the characteristic function is $$\Psi(\mu)=a+b\mu^2$$, where $$a, b $$ are two constants(have to satisfy $$a+b/3\leq 1/2$$) and if $$\alpha_n = \int_0^1 H(\mu)\mu^n \, d\mu, \ n\geq 1$$ is the nth moment of the H function, then we have
 * $$\alpha_0 = 1 + \frac{1}{2} (a\alpha_0^2 + b \alpha_1^2)$$

and
 * $$(a+b\mu^2) \int_0^1\frac{H(\mu')}{\mu+\mu'}\,d\mu'=\frac{H(\mu)-1}{\mu H(\mu)}-b(\alpha_1-\mu\alpha_0)$$