Английская Википедия: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 <math>X(\mu),\ Y(\mu) </math> defined in the interval <math>0\leq\mu\leq 1</math>, satisfies the pair of nonlinear integral equations
- <math>\begin{align}
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{align}</math>
where the characteristic function <math>\Psi(\mu)</math> is an even polynomial in <math>\mu</math> generally satisfying the condition
- <math>\int_0^1\Psi(\mu) \, d\mu \leq \frac{1}{2}, </math>
and <math>0<\tau_1<\infty</math> 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
- <math>X(\mu)\rightarrow H(\mu), \quad Y(\mu)\rightarrow 0 \ \text{as} \ \tau_1\rightarrow\infty</math>
and also
- <math>X(\mu)\rightarrow 1, \quad Y(\mu)\rightarrow e^{-\tau_1/\mu} \ \text{as} \ \tau_1\rightarrow 0.</math>
Approximation
The <math>X</math> and <math>Y</math> can be approximated up to nth order as
- <math>\begin{align}
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{align}</math>
where <math>C_0</math> and <math>C_1</math> are two basic polynomials of order n (Refer Chandrasekhar chapter VIII equation (97)[6]), <math>P(\mu) = \prod_{i=1}^n (\mu-\mu_i)</math> where <math>\mu_i</math> are the zeros of Legendre polynomials and <math>W(\mu)= \prod_{\alpha=1}^n (1-k_\alpha^2\mu^2)</math>, where <math>k_\alpha</math> are the positive, non vanishing roots of the associated characteristic equation
- <math>1 = 2 \sum_{j=1}^n \frac{a_j\Psi(\mu_j)}{1-k^2\mu_j^2}</math>
where <math>a_j</math> are the quadrature weights given by
- <math>a_j = \frac 1 {P_{2n}'(\mu_j)} \int_{-1}^1 \frac{P_{2n}(\mu_j)}{\mu-\mu_j} \, d\mu_j</math>
Properties
- If <math>X(\mu,\tau_1), \ Y(\mu,\tau_1)</math> are the solutions for a particular value of <math>\tau_1</math>, then solutions for other values of <math>\tau_1</math> are obtained from the following integro-differential equations
- <math>\begin{align}
\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{align}</math>
- <math>\int_0^1 X(\mu)\Psi(\mu) \, d\mu = 1- \left[1-2\int_0^1 \Psi(\mu)\,d\mu + \left\{\int_0^1 Y(\mu) \Psi(\mu) \,d\mu\right\}^2\right]^{1/2}.</math> For conservative case, this integral property reduces to <math>\int_0^1 [X(\mu)+Y(\mu)]\Psi(\mu) \, d\mu = 1.</math>
- If the abbreviations <math>x_n = \int_0^1 X(\mu) \Psi(\mu) \mu^n \, d\mu, \ y_n = \int_0^1 Y(\mu)\Psi(\mu) \mu^n \, d\mu, \ \alpha_n = \int_0^1 X(\mu)\mu^n \, d\mu, \ \beta_n = \int_0^1 Y(\mu) \mu^n \, d\mu</math> for brevity are introduced, then we have a relation stating <math>(1-x_0)x_2 + y_0y_2 + \frac{1}{2} (x_1^2-y_1^2) = \int_0^1 \Psi(\mu)\mu^2 \, d\mu.</math> In the conservative, this reduces to <math>y_0(x_2+y_2) + \frac{1}{2}(x_1^2-y_1^2)=\int_0^1 \Psi(\mu)\mu^2 \, d\mu</math>
- If the characteristic function is <math>\Psi(\mu)=a+b\mu^2</math>, where <math>a, b </math> are two constants, then we have <math>\alpha_0=1+\frac{1}{2}
[a(\alpha_0^2-\beta_0^2)+b(\alpha_1^2-\beta_1^2)]</math>.
- For conservative case, the solutions are not unique. If <math>X(\mu), \ Y(\mu)</math> are solutions of the original equation, then so are these two functions <math>F(\mu)=X(\mu) + Q\mu [X(\mu) + Y(\mu)],\ G(\mu)=Y(\mu) + Q\mu[X(\mu)+Y(\mu)]</math>, where <math>Q</math> 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.
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