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 ${\displaystyle H(\mu )}$ defined in the interval ${\displaystyle 0\leq \mu \leq 1}$, satisfies the following nonlinear integral equation

${\displaystyle H(\mu )=1+\mu H(\mu )\int _{0}^{1}{\frac {\Psi (\mu ')}{\mu +\mu '}}H(\mu ')\,d\mu '}$

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

${\displaystyle \int _{0}^{1}\Psi (\mu )\,d\mu \leq {\frac {1}{2}}}$.

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

${\displaystyle {\frac {1}{H(\mu )}}=\left[1-2\int _{0}^{1}\Psi (\mu )\,d\mu \right]^{1/2}+\int _{0}^{1}{\frac {\mu '\Psi (\mu ')}{\mu +\mu '}}H(\mu ')\,d\mu '}$.

In conservative case, the above equation reduces to

${\displaystyle {\frac {1}{H(\mu )}}=\int _{0}^{1}{\frac {\mu '\Psi (\mu ')}{\mu +\mu '}}H(\mu ')d\mu '}$.