= Solution
Consider a ray bundle from projected source area $dA_s\cos\theta_s$ into solid angle $d\Omega_s$. In empty space its power is conserved, while geometric propagation preserves the étendue
$$
dA_s\cos\theta_s\,d\Omega_s
=dA_o\cos\theta_o\,d\Omega_o.
$$
Since <specific intensity> is power divided by this étendue and by frequency interval,
$$
\boxed{I_\nu/\nu^3=\text{constant along a ray}}.
$$
In a static medium with no redshift, frequency is unchanged and $I_\nu$ itself is independent of distance. The apparent solid angle shrinks as distance squared while the physical beam area grows by the same factor.
For a full plane-parallel angular field linear in direction cosine,
$$
I(\mu)=I_0+I_1\mu,
$$
the <radiation-field moments> are
$$
\boxed{J=I_0,
\qquad H=\frac{I_1}{3},
\qquad K=\frac{I_0}{3}=\frac J3},
$$
and $F=4\pi H=4\pi I_1/3$. At a surface with no incoming intensity and the same law only for $0<\mu<1$,
$$
J=\frac{I_0}{2}+\frac{I_1}{4},
\quad
F=\pi I_0+\frac{2\pi I_1}{3},
\quad
K=\frac{I_0}{6}+\frac{I_1}{8}.
$$
Solved by gpt-5.6-sol high.
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