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 .[1] [2] [3] [4] [5] The Chandrasekhar's H -function
H
(
μ μ -->
)
{\displaystyle H(\mu )}
defined in the interval
0
≤ ≤ -->
μ μ -->
≤ ≤ -->
1
{\displaystyle 0\leq \mu \leq 1}
, satisfies the following nonlinear integral equation
H
(
μ μ -->
)
=
1
+
μ μ -->
H
(
μ μ -->
)
∫ ∫ -->
0
1
Ψ Ψ -->
(
μ μ -->
′
)
μ μ -->
+
μ μ -->
′
H
(
μ μ -->
′
)
d
μ 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 Ψ -->
(
μ μ -->
)
{\displaystyle \Psi (\mu )}
is an even polynomial in
μ μ -->
{\displaystyle \mu }
satisfying the following condition
∫ ∫ -->
0
1
Ψ Ψ -->
(
μ μ -->
)
d
μ 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 ω -->
o
=
2
Ψ Ψ -->
(
μ 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 μ -->
)
=
[
1
− − -->
2
∫ ∫ -->
0
1
Ψ Ψ -->
(
μ μ -->
)
d
μ μ -->
]
1
/
2
+
∫ ∫ -->
0
1
μ μ -->
′
Ψ Ψ -->
(
μ μ -->
′
)
μ μ -->
+
μ μ -->
′
H
(
μ μ -->
′
)
d
μ 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 μ -->
)
=
∫ ∫ -->
0
1
μ μ -->
′
Ψ Ψ -->
(
μ μ -->
′
)
μ μ -->
+
μ μ -->
′
H
(
μ μ -->
′
)
d
μ 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 μ -->
)
=
1
μ μ -->
1
⋯ ⋯ -->
μ μ -->
n
∏ ∏ -->
i
=
1
n
(
μ μ -->
+
μ μ -->
i
)
∏ ∏ -->
α α -->
(
1
+
k
α α -->
μ 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 μ -->
i
{\displaystyle \mu _{i}}
are the zeros of Legendre polynomials
P
2
n
{\displaystyle P_{2n}}
and
k
α α -->
{\displaystyle k_{\alpha }}
are the positive, non vanishing roots of the associated characteristic equation
1
=
2
∑ ∑ -->
j
=
1
n
a
j
Ψ Ψ -->
(
μ μ -->
j
)
1
− − -->
k
2
μ 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 μ -->
j
)
∫ ∫ -->
− − -->
1
1
P
2
n
(
μ μ -->
j
)
μ μ -->
− − -->
μ μ -->
j
d
μ 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 μ -->
H
(
μ μ -->
)
Ψ Ψ -->
(
μ μ -->
)
d
μ μ -->
,
∫ ∫ -->
0
1
|
Ψ Ψ -->
(
μ μ -->
)
|
d
μ μ -->
≤ ≤ -->
1
2
,
∫ ∫ -->
0
δ δ -->
|
Ψ Ψ -->
(
μ μ -->
)
|
d
μ μ -->
→ → -->
0
,
δ 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 ℜ -->
(
z
)
>
0
{\displaystyle \Re (z)>0}
, a unique solution is given by
ln
-->
H
(
z
)
=
1
2
π 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 Ψ -->
(
μ μ -->
)
d
μ μ -->
− − -->
2
∫ ∫ -->
0
1
μ μ -->
2
Ψ Ψ -->
(
μ μ -->
)
u
− − -->
μ μ -->
2
d
μ 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 μ -->
)
Ψ Ψ -->
(
μ μ -->
)
d
μ μ -->
=
1
− − -->
[
1
− − -->
2
∫ ∫ -->
0
1
Ψ Ψ -->
(
μ μ -->
)
d
μ 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 Ψ -->
(
μ μ -->
)
d
μ μ -->
=
1
2
{\displaystyle \int _{0}^{1}\Psi (\mu )d\mu ={\frac {1}{2}}}
.
[
1
− − -->
2
∫ ∫ -->
0
1
Ψ Ψ -->
(
μ μ -->
)
d
μ μ -->
]
1
/
2
∫ ∫ -->
0
1
H
(
μ μ -->
)
Ψ Ψ -->
(
μ μ -->
)
μ μ -->
2
d
μ μ -->
+
1
2
[
∫ ∫ -->
0
1
H
(
μ μ -->
)
Ψ Ψ -->
(
μ μ -->
)
μ μ -->
d
μ μ -->
]
2
=
∫ ∫ -->
0
1
Ψ Ψ -->
(
μ μ -->
)
μ μ -->
2
d
μ 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 μ -->
)
Ψ Ψ -->
(
μ μ -->
)
μ μ -->
d
μ μ -->
=
[
2
∫ ∫ -->
0
1
Ψ Ψ -->
(
μ μ -->
)
μ μ -->
2
d
μ 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 Ψ -->
(
μ μ -->
)
=
a
+
b
μ 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 α -->
n
=
∫ ∫ -->
0
1
H
(
μ μ -->
)
μ μ -->
n
d
μ 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 α -->
0
=
1
+
1
2
(
a
α α -->
0
2
+
b
α α -->
1
2
)
{\displaystyle \alpha _{0}=1+{\frac {1}{2}}(a\alpha _{0}^{2}+b\alpha _{1}^{2})}
and
(
a
+
b
μ μ -->
2
)
∫ ∫ -->
0
1
H
(
μ μ -->
′
)
μ μ -->
+
μ μ -->
′
d
μ μ -->
′
=
H
(
μ μ -->
)
− − -->
1
μ μ -->
H
(
μ μ -->
)
− − -->
b
(
α α -->
1
− − -->
μ μ -->
α 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
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).