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
.
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
In complex variable the H equation is
then for , a unique solution is given by
where the imaginary part of the function can vanish if 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
. 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
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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