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.^{[1][2][3][4][5]} 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

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

The H function can be approximated up to an order ${\displaystyle n}$ as

${\displaystyle H(\mu )={\frac {1}{\mu _{1}\cdots \mu _{n}}}{\frac {\prod _{i=1}^{n}(\mu +\mu _{i})}{\prod _{\alpha }(1+k_{\alpha }\mu )}}}$

where ${\displaystyle \mu _{i}}$ are the zeros of Legendre polynomials ${\displaystyle P_{2n}}$ and ${\displaystyle k_{\alpha }}$ are the positive, non vanishing roots of the associated characteristic equation

${\displaystyle 1=2\sum _{j=1}^{n}{\frac {a_{j}\Psi (\mu _{j})}{1-k^{2}\mu _{j}^{2}}}}$

where ${\displaystyle a_{j}}$ are the quadrature weights given by

${\displaystyle a_{j}={\frac {1}{P_{2n}'(\mu _{j})}}\int _{-1}^{1}{\frac {P_{2n}(\mu _{j})}{\mu -\mu _{j}}}\,d\mu _{j}}$

In complex variable ${\displaystyle z}$ the H equation is

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

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

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

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

${\displaystyle T(z)=1-2\int _{0}^{1}\Psi (\mu )\,d\mu -2\int _{0}^{1}{\frac {\mu ^{2}\Psi (\mu )}{u-\mu ^{2}}}\,d\mu }$

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

${\displaystyle H_{1}(z)=H(z){\frac {1+kz}{1-kz}}}$

• ${\displaystyle \int _{0}^{1}H(\mu )\Psi (\mu )\,d\mu =1-\left[1-2\int _{0}^{1}\Psi (\mu )\,d\mu \right]^{1/2}}$. For conservative case, this reduces to ${\displaystyle \int _{0}^{1}\Psi (\mu )d\mu ={\frac {1}{2}}}$.
• ${\displaystyle \left[1-2\int _{0}^{1}\Psi (\mu )\,d\mu \right]^{1/2}\int _{0}^{1}H(\mu )\Psi (\mu )\mu ^{2}\,d\mu +{\frac {1}{2}}\left[\int _{0}^{1}H(\mu )\Psi (\mu )\mu \,d\mu \right]^{2}=\int _{0}^{1}\Psi (\mu )\mu ^{2}\,d\mu }$. For conservative case, this reduces to ${\displaystyle \int _{0}^{1}H(\mu )\Psi (\mu )\mu d\mu =\left[2\int _{0}^{1}\Psi (\mu )\mu ^{2}d\mu \right]^{1/2}}$.
• If the characteristic function is ${\displaystyle \Psi (\mu )=a+b\mu ^{2}}$, where ${\displaystyle a,b}$ are two constants(have to satisfy ${\displaystyle a+b/3\leq 1/2}$) and if ${\displaystyle \alpha _{n}=\int _{0}^{1}H(\mu )\mu ^{n}\,d\mu ,\ n\geq 1}$ is the nth moment of the H function, then we have

${\displaystyle \alpha _{0}=1+{\frac {1}{2}}(a\alpha _{0}^{2}+b\alpha _{1}^{2})}$

and

${\displaystyle (a+b\mu ^{2})\int _{0}^{1}{\frac {H(\mu ')}{\mu +\mu '}}\,d\mu '={\frac {H(\mu )-1}{\mu H(\mu )}}-b(\alpha _{1}-\mu \alpha _{0})}$

• Chandrasekhar's X- and Y-function

• MATLAB function to calculate the H function https://www.mathworks.com/matlabcentral/fileexchange/29333-chandrasekhar-s-h-function