Chandrasekhar's H-function

Chandrasekhar's H-function for different albedo

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

$H(\mu )={\frac {1}{\mu _{1}\cdots \mu _{n}}}{\frac {\prod _{i=0}^{n}(\mu +\mu _{i})}{\prod _{\alpha }(1+k_{\alpha }\mu )}}$

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 iff $z^{2}$ 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 $\int _{0}^{1}H(\mu )\Psi (\mu )\mu d\mu =\left[2\int _{0}^{1}\Psi (\mu )\mu ^{2}d\mu \right]^{1/2}$.
• 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

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

See also

• Chandrasekhar's X- and Y-function

External links

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