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
defined in the interval
, 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 '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7e9017c641f9ee471d0efa96c27cdeb3ef7316)
where the characteristic function
is an even polynomial in
satisfying the following condition
.
If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative. Albedo is given by
. An alternate form which would be more useful in calculating the H function numerically by iteration was derived by Chandrasekhar as,
.
In conservative case, the above equation reduces to
.
Approximation
The H function can be approximated up to an order
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 )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6635592e6162d356315c29c3abaa003b23d2678)
where
are the zeros of Legendre polynomials
and
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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dea68164e4c2e22bcc8cf3257eba0c1758cc68bd)
where
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffee52cf4ab65326952a874b8de018f5876dec69)
Explicit solution in the complex plane
In complex variable
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ada59add74a1b69ebd74d3c6bbbccd5f4a96492)
then for
, 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35098c49b258137cff85a46677fb97dddc60f931)
where the imaginary part of the function
can vanish if
is real i.e.,
. 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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6db30fb468da9cdc0915594fce9cf372fc16715)
The above solution is unique and bounded in the interval
for conservative cases. In non-conservative cases, if the equation
admits the roots
, then there is a further solution given by
![{\displaystyle H_{1}(z)=H(z){\frac {1+kz}{1-kz}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd9e77e19edb7511fdb094e7575498950ff885a)
Properties
. For conservative case, this reduces to
.
. For conservative case, this reduces to
.
- If the characteristic function is
, where
are two constants(have to satisfy
) and if
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2e018d1fd057139db5d30c7d899f175ddb4e6e)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/711cb9b684619ca3d97af5647cc2345dd0266861)
See also
External links
References
- ^ Chandrasekhar, Subrahmanyan. Radiative transfer. Courier Corporation, 2013.
- ^ Howell, John R., M. Pinar Menguc, and Robert Siegel. Thermal radiation heat transfer. CRC press, 2010.
- ^ Modest, Michael F. Radiative heat transfer. Academic press, 2013.
- ^ Hottel, Hoyt Clarke, and Adel F. Sarofim. Radiative transfer. McGraw-Hill, 1967.
- ^ Sparrow, Ephraim M., and Robert D. Cess. "Radiation heat transfer." Series in Thermal and Fluids Engineering, New York: McGraw-Hill, 1978, Augmented ed. (1978).