Physics:Chandrasekhar's H-function
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 [math]\displaystyle{ H(\mu) }[/math] defined in the interval [math]\displaystyle{ 0\leq\mu\leq 1 }[/math], satisfies the following nonlinear integral equation
- [math]\displaystyle{ H(\mu) = 1+\mu H(\mu)\int_0^1 \frac{\Psi(\mu')}{\mu + \mu'}H(\mu') \, d\mu' }[/math]
where the characteristic function [math]\displaystyle{ \Psi(\mu) }[/math] is an even polynomial in [math]\displaystyle{ \mu }[/math] satisfying the following condition
- [math]\displaystyle{ \int_0^1\Psi(\mu) \, d\mu \leq \frac{1}{2} }[/math].
If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative. Albedo is given by [math]\displaystyle{ \omega_o= 2\Psi(\mu) = \text{constant} }[/math]. An alternate form which would be more useful in calculating the H function numerically by iteration was derived by Chandrasekhar as,
- [math]\displaystyle{ \frac{1}{H(\mu)} = \left[1-2\int_0^1\Psi(\mu) \, d\mu\right]^{1/2} + \int_0^1 \frac{\mu'\Psi(\mu')}{\mu + \mu'}H(\mu') \, d\mu' }[/math].
In conservative case, the above equation reduces to
- [math]\displaystyle{ \frac{1}{H(\mu)}= \int_0^1 \frac{\mu' \Psi(\mu')}{\mu+\mu'}H(\mu')d\mu' }[/math].
Approximation
The H function can be approximated up to an order [math]\displaystyle{ n }[/math] as
- [math]\displaystyle{ H(\mu) = \frac{1}{\mu_1 \cdots \mu_n}\frac{\prod_{i=1}^n (\mu+\mu_i)}{\prod_\alpha (1+k_\alpha\mu)} }[/math]
where [math]\displaystyle{ \mu_i }[/math] are the zeros of Legendre polynomials [math]\displaystyle{ P_{2n} }[/math] and [math]\displaystyle{ k_\alpha }[/math] are the positive, non vanishing roots of the associated characteristic equation
- [math]\displaystyle{ 1 = 2 \sum_{j=1}^n \frac{a_j\Psi(\mu_j)}{1-k^2\mu_j^2} }[/math]
where [math]\displaystyle{ a_j }[/math] are the quadrature weights given by
- [math]\displaystyle{ a_j = \frac{1}{P_{2n}'(\mu_j)}\int_{-1}^1 \frac{P_{2n}(\mu_j)}{\mu-\mu_j} \, d\mu_j }[/math]
Explicit solution in the complex plane
In complex variable [math]\displaystyle{ z }[/math] the H equation is
- [math]\displaystyle{ H(z) = 1- \int_0^1 \frac z {z+\mu} H(\mu)\Psi(\mu) \, d\mu, \quad \int_0^1 |\Psi(\mu)| \, d\mu \leq \frac{1}{2}, \quad \int_0^\delta |\Psi(\mu)| \, d\mu \rightarrow 0, \ \delta\rightarrow 0 }[/math]
then for [math]\displaystyle{ \Re (z)\gt 0 }[/math], a unique solution is given by
- [math]\displaystyle{ \ln H(z) = \frac{1}{2\pi i} \int_{-i\infty}^{+ i\infty} \ln T(w) \frac{z}{w^2-z^2} \, dw }[/math]
where the imaginary part of the function [math]\displaystyle{ T(z) }[/math] can vanish if [math]\displaystyle{ z^2 }[/math] is real i.e., [math]\displaystyle{ z^2 = u+iv = u\ (v=0) }[/math]. Then we have
- [math]\displaystyle{ T(z) = 1- 2 \int_0^1 \Psi(\mu) \, d\mu - 2 \int_0^1 \frac{\mu^2 \Psi(\mu)}{u-\mu^2} \, d\mu }[/math]
The above solution is unique and bounded in the interval [math]\displaystyle{ 0\leq z\leq 1 }[/math] for conservative cases. In non-conservative cases, if the equation [math]\displaystyle{ T(z)=0 }[/math] admits the roots [math]\displaystyle{ \pm 1/k }[/math], then there is a further solution given by
- [math]\displaystyle{ H_1(z) = H(z) \frac{1+kz}{1-kz} }[/math]
Properties
- [math]\displaystyle{ \int_0^1 H(\mu)\Psi(\mu) \, d\mu = 1-\left[1-2\int_0^1\Psi(\mu) \, d\mu \right]^{1/2} }[/math]. For conservative case, this reduces to [math]\displaystyle{ \int_0^1 \Psi(\mu)d\mu=\frac{1}{2} }[/math].
- [math]\displaystyle{ \left[1-2\int_0^1\Psi(\mu) \, d\mu\right]^{1/2} \int_0^1 H(\mu) \Psi(\mu) \mu^2 \, d\mu + \frac{1}{2} \left[\int_0^1 H(\mu)\Psi(\mu)\mu \, d\mu\right]^2 = \int_0^1 \Psi(\mu)\mu^2 \, d\mu }[/math]. For conservative case, this reduces to [math]\displaystyle{ \int_0^1 H(\mu)\Psi(\mu) \mu d\mu = \left[2\int_0^1 \Psi(\mu)\mu^2d\mu\right]^{1/2} }[/math].
- If the characteristic function is [math]\displaystyle{ \Psi(\mu)=a+b\mu^2 }[/math], where [math]\displaystyle{ a, b }[/math] are two constants(have to satisfy [math]\displaystyle{ a+b/3\leq 1/2 }[/math]) and if [math]\displaystyle{ \alpha_n = \int_0^1 H(\mu)\mu^n \, d\mu, \ n\geq 1 }[/math] is the nth moment of the H function, then we have
- [math]\displaystyle{ \alpha_0 = 1 + \frac{1}{2} (a\alpha_0^2 + b \alpha_1^2) }[/math]
and
- [math]\displaystyle{ (a+b\mu^2) \int_0^1\frac{H(\mu')}{\mu+\mu'}\,d\mu'=\frac{H(\mu)-1}{\mu H(\mu)}-b(\alpha_1-\mu\alpha_0) }[/math]
See also
External links
- MATLAB function to calculate the H function https://www.mathworks.com/matlabcentral/fileexchange/29333-chandrasekhar-s-h-function
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).
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