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
)
{\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 '}
Chandrasekhar's H -function for different albedo
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 '}
.