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.^{[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

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

Approximation

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

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

Explicit solution in the complex plane

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

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

Properties

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

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

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

References

1. Chandrasekhar, Subrahmanyan. Radiative transfer. Courier Corporation, 2013.
2. Howell, John R., M. Pinar Menguc, and Robert Siegel. Thermal radiation heat transfer. CRC press, 2010.
3. Modest, Michael F. Radiative heat transfer. Academic press, 2013.
4. Hottel, Hoyt Clarke, and Adel F. Sarofim. Radiative transfer. McGraw-Hill, 1967.
5. Sparrow, Ephraim M., and Robert D. Cess. "Radiation heat transfer." Series in Thermal and Fluids Engineering, New York: McGraw-Hill, 1978, Augmented ed. (1978).