- Approximation
- Explicit solution in the complex plane
- Properties
- See also
- External links
- References
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
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
where 
are the zeros of Legendre polynomials 
and 
are the positive, non vanishing roots of the associated characteristic equation
where 
are the quadrature weights given by
Explicit solution in the complex plane
In complex variable 
the H equation is
then for 
, a unique solution is given by
where the imaginary part of the function 
can vanish iff 
is real i.e., 
. Then we have
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
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
and
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
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).
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