Chandrasekhar's H-function for different albedo

In atmospheric radiation, Chandrasekhar's H-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandrasekhar's H-function H ( μ ) {\displaystyle H(\mu )} defined in the interval 0 ≤ μ ≤ 1 {\displaystyle 0\leq \mu \leq 1}, satisfies the following nonlinear integral equation

H ( μ ) = 1 + μ H ( μ ) ∫ 0 1 Ψ ( μ ′ ) μ + μ ′ H ( μ ′ ) d μ ′ {\displaystyle H(\mu )=1+\mu H(\mu )\int _{0}^{1}{\frac {\Psi (\mu ')}{\mu +\mu '}}H(\mu ')\,d\mu '}

where the characteristic function Ψ ( μ ) {\displaystyle \Psi (\mu )} is an even polynomial in μ {\displaystyle \mu } satisfying the following condition

∫ 0 1 Ψ ( μ ) d μ ≤ 1 2 {\displaystyle \int _{0}^{1}\Psi (\mu )\,d\mu \leq {\frac {1}{2}}}.

If the equality is satisfied in the above condition, it is called conservative case, otherwise non-conservative. Albedo is given by ω o = 2 Ψ ( μ ) = constant {\displaystyle \omega _{o}=2\Psi (\mu )={\text{constant}}}. An alternate form which would be more useful in calculating the H function numerically by iteration was derived by Chandrasekhar as,

1 H ( μ ) = [ 1 − 2 ∫ 0 1 Ψ ( μ ) d μ ] 1 / 2 + ∫ 0 1 μ ′ Ψ ( μ ′ ) μ + μ ′ H ( μ ′ ) d μ ′ {\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 '}.

In conservative case, the above equation reduces to

1 H ( μ ) = ∫ 0 1 μ ′ Ψ ( μ ′ ) μ + μ ′ H ( μ ′ ) d μ ′ {\displaystyle {\frac {1}{H(\mu )}}=\int _{0}^{1}{\frac {\mu '\Psi (\mu ')}{\mu +\mu '}}H(\mu ')d\mu '}.

Approximation

The H function can be approximated up to an order n {\displaystyle n} as

H ( μ ) = 1 μ 1 ⋯ μ n ∏ i = 1 n ( μ + μ i ) ∏ α ( 1 + k α μ ) {\displaystyle H(\mu )={\frac {1}{\mu _{1}\cdots \mu _{n}}}{\frac {\prod _{i=1}^{n}(\mu +\mu _{i})}{\prod _{\alpha }(1+k_{\alpha }\mu )}}}

where μ i {\displaystyle \mu _{i}} are the zeros of Legendre polynomials P 2 n {\displaystyle P_{2n}} and k α {\displaystyle k_{\alpha }} are the positive, non vanishing roots of the associated characteristic equation

1 = 2 ∑ j = 1 n a j Ψ ( μ j ) 1 − k 2 μ j 2 {\displaystyle 1=2\sum _{j=1}^{n}{\frac {a_{j}\Psi (\mu _{j})}{1-k^{2}\mu _{j}^{2}}}}

where a j {\displaystyle a_{j}} are the quadrature weights given by

a j = 1 P 2 n ′ ( μ j ) ∫ − 1 1 P 2 n ( μ j ) μ − μ j d μ j {\displaystyle a_{j}={\frac {1}{P_{2n}'(\mu _{j})}}\int _{-1}^{1}{\frac {P_{2n}(\mu _{j})}{\mu -\mu _{j}}}\,d\mu _{j}}

Explicit solution in the complex plane

In complex variable z {\displaystyle z} the H equation is

H ( z ) = 1 − ∫ 0 1 z z + μ H ( μ ) Ψ ( μ ) d μ , ∫ 0 1 | Ψ ( μ ) | d μ ≤ 1 2 , ∫ 0 δ | Ψ ( μ ) | d μ → 0 , δ → 0 {\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}

then for ℜ ( z ) > 0 {\displaystyle \Re (z)>0}, a unique solution is given by

ln ⁡ H ( z ) = 1 2 π i ∫ − i ∞ + i ∞ ln ⁡ T ( w ) z w 2 − z 2 d w {\displaystyle \ln H(z)={\frac {1}{2\pi i}}\int _{-i\infty }^{+i\infty }\ln T(w){\frac {z}{w^{2}-z^{2}}}\,dw}

where the imaginary part of the function T ( z ) {\displaystyle T(z)} can vanish if z 2 {\displaystyle z^{2}} is real i.e., z 2 = u + i v = u ( v = 0 ) {\displaystyle z^{2}=u+iv=u\ (v=0)}. Then we have

T ( z ) = 1 − 2 ∫ 0 1 Ψ ( μ ) d μ − 2 ∫ 0 1 μ 2 Ψ ( μ ) u − μ 2 d μ {\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 }

The above solution is unique and bounded in the interval 0 ≤ z ≤ 1 {\displaystyle 0\leq z\leq 1} for conservative cases. In non-conservative cases, if the equation T ( z ) = 0 {\displaystyle T(z)=0} admits the roots ± 1 / k {\displaystyle \pm 1/k}, then there is a further solution given by

H 1 ( z ) = H ( z ) 1 + k z 1 − k z {\displaystyle H_{1}(z)=H(z){\frac {1+kz}{1-kz}}}

Properties

  • ∫ 0 1 H ( μ ) Ψ ( μ ) d μ = 1 − [ 1 − 2 ∫ 0 1 Ψ ( μ ) d μ ] 1 / 2 {\displaystyle \int _{0}^{1}H(\mu )\Psi (\mu )\,d\mu =1-\left[1-2\int _{0}^{1}\Psi (\mu )\,d\mu \right]^{1/2}}. For conservative case, this reduces to ∫ 0 1 Ψ ( μ ) d μ = 1 2 {\displaystyle \int _{0}^{1}\Psi (\mu )d\mu ={\frac {1}{2}}}.
  • [ 1 − 2 ∫ 0 1 Ψ ( μ ) d μ ] 1 / 2 ∫ 0 1 H ( μ ) Ψ ( μ ) μ 2 d μ + 1 2 [ ∫ 0 1 H ( μ ) Ψ ( μ ) μ d μ ] 2 = ∫ 0 1 Ψ ( μ ) μ 2 d μ {\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 }. For conservative case, this reduces to ∫ 0 1 H ( μ ) Ψ ( μ ) μ d μ = [ 2 ∫ 0 1 Ψ ( μ ) μ 2 d μ ] 1 / 2 {\displaystyle \int _{0}^{1}H(\mu )\Psi (\mu )\mu d\mu =\left[2\int _{0}^{1}\Psi (\mu )\mu ^{2}d\mu \right]^{1/2}}.
  • If the characteristic function is Ψ ( μ ) = a + b μ 2 {\displaystyle \Psi (\mu )=a+b\mu ^{2}}, where a , b {\displaystyle a,b} are two constants(have to satisfy a + b / 3 ≤ 1 / 2 {\displaystyle a+b/3\leq 1/2}) and if α n = ∫ 0 1 H ( μ ) μ n d μ , n ≥ 1 {\displaystyle \alpha _{n}=\int _{0}^{1}H(\mu )\mu ^{n}\,d\mu ,\ n\geq 1} is the nth moment of the H function, then we have

α 0 = 1 + 1 2 ( a α 0 2 + b α 1 2 ) {\displaystyle \alpha _{0}=1+{\frac {1}{2}}(a\alpha _{0}^{2}+b\alpha _{1}^{2})}

and

( a + b μ 2 ) ∫ 0 1 H ( μ ′ ) μ + μ ′ d μ ′ = H ( μ ) − 1 μ H ( μ ) − b ( α 1 − μ α 0 ) {\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})}

See also

External links

  • MATLAB function to calculate the H function