Chandrasekhar's X- and Y-function

In atmospheric radiation, Chandrasekhar's X- and Y-function appears as the solutions of problems involving diffusive reflection and transmission, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar.[1][2][3][4][5] The Chandrasekhar's X- and Y-function defined in the interval , satisfies the pair of nonlinear integral equations

where the characteristic function is an even polynomial in generally satisfying the condition

and is the optical thickness of the atmosphere. If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative. These functions are related to Chandrasekhar's H-function as

and also

Approximation

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The   and   can be approximated up to nth order as

 

where   and   are two basic polynomials of order n (Refer Chandrasekhar chapter VIII equation (97)[6]),   where   are the zeros of Legendre polynomials and  , where   are the positive, non vanishing roots of the associated characteristic equation

 

where   are the quadrature weights given by

 

Properties

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  • If   are the solutions for a particular value of  , then solutions for other values of   are obtained from the following integro-differential equations
 
  •   For conservative case, this integral property reduces to  
  • If the abbreviations   for brevity are introduced, then we have a relation stating   In the conservative, this reduces to  
  • If the characteristic function is  , where   are two constants, then we have  .
  • For conservative case, the solutions are not unique. If   are solutions of the original equation, then so are these two functions  , where   is an arbitrary constant.

See also

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References

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  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).
  6. ^ Chandrasekhar, Subrahmanyan. Radiative transfer. Courier Corporation, 2013.