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
![{\displaystyle {\begin{aligned}X(\mu )&=1+\mu \int _{0}^{1}{\frac {\Psi (\mu ')}{\mu +\mu '}}[X(\mu )X(\mu ')-Y(\mu )Y(\mu ')]\,d\mu ',\\[5pt]Y(\mu )&=e^{-\tau _{1}/\mu }+\mu \int _{0}^{1}{\frac {\Psi (\mu ')}{\mu -\mu '}}[Y(\mu )X(\mu ')-X(\mu )Y(\mu ')]\,d\mu '\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acece5b1d39d913fff4aa1c9df24b3c538b15bc4)
where the characteristic function
is an even polynomial in
generally satisfying the condition
![{\displaystyle \int _{0}^{1}\Psi (\mu )\,d\mu \leq {\frac {1}{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc79555be5291b5a90f822f6720aeaf2f23edd9)
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
![{\displaystyle X(\mu )\rightarrow H(\mu ),\quad Y(\mu )\rightarrow 0\ {\text{as}}\ \tau _{1}\rightarrow \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6920a4748c56b4aa18a1a7466cd2293a407bde95)
and also
![{\displaystyle X(\mu )\rightarrow 1,\quad Y(\mu )\rightarrow e^{-\tau _{1}/\mu }\ {\text{as}}\ \tau _{1}\rightarrow 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb312d25fd78e22aae24650f99104df99fdb6d3)
Approximation
The
and
can be approximated up to nth order as
![{\displaystyle {\begin{aligned}X(\mu )&={\frac {(-1)^{n}}{\mu _{1}\cdots \mu _{n}}}{\frac {1}{[C_{0}^{2}(0)-C_{1}^{2}(0)]^{1/2}}}{\frac {1}{W(\mu )}}[P(-\mu )C_{0}(-\mu )-e^{-\tau _{1}/\mu }P(\mu )C_{1}(\mu )],\\[5pt]Y(\mu )&={\frac {(-1)^{n}}{\mu _{1}\cdots \mu _{n}}}{\frac {1}{[C_{0}^{2}(0)-C_{1}^{2}(0)]^{1/2}}}{\frac {1}{W(\mu )}}[e^{-\tau _{1}/\mu }P(\mu )C_{0}(\mu )-P(-\mu )C_{1}(-\mu )]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c4225f1afe95129375c5984ecd5567715c0cb7)
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
![{\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)
Properties
- If
are the solutions for a particular value of
, then solutions for other values of
are obtained from the following integro-differential equations
![{\displaystyle {\begin{aligned}{\frac {\partial X(\mu ,\tau _{1})}{\partial \tau _{1}}}&=Y(\mu ,\tau _{1})\int _{0}^{1}{\frac {d\mu '}{\mu '}}\Psi (\mu ')Y(\mu ',\tau _{1}),\\{\frac {\partial Y(\mu ,\tau _{1})}{\partial \tau _{1}}}+{\frac {Y(\mu ,\tau _{1})}{\mu }}&=X(\mu ,\tau _{1})\int _{0}^{1}{\frac {d\mu '}{\mu '}}\Psi (\mu ')Y(\mu ',\tau _{1})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d040227c9d83bdd1c7f3aeabd2e66516931d72cd)
For conservative case, this integral property reduces to ![{\displaystyle \int _{0}^{1}[X(\mu )+Y(\mu )]\Psi (\mu )\,d\mu =1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6ee7edef7eebb53c6446e1a2bff952f09095de)
- If the abbreviations
for brevity are introduced, then we have a relation stating
In the conservative, this reduces to ![{\displaystyle y_{0}(x_{2}+y_{2})+{\frac {1}{2}}(x_{1}^{2}-y_{1}^{2})=\int _{0}^{1}\Psi (\mu )\mu ^{2}\,d\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d31e35215c9e7458fad3521edf38f58787c1664c)
- 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
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).
- ^ Chandrasekhar, Subrahmanyan. Radiative transfer. Courier Corporation, 2013.