Solution

Consider a ray bundle from projected source area

d A_{s} cos θ_{s}

d Ω_{s}

. In empty space its power is conserved, while geometric propagation preserves the étendue

d A_{s} cos θ_{s} d Ω_{s} = d A_{o} cos θ_{o} d Ω_{o} .

Since specific intensity is power divided by this étendue and by frequency interval,

I_{ν} / ν^{3} = constant along a ray .

In a static medium with no redshift, frequency is unchanged and

I_{ν}

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 (μ) = I_{0} + I_{1} μ,

J = I_{0}, H = \frac{I _{1}}{3}, K = \frac{I _{0}}{3} = \frac{J}{3},

and

F = 4 π H = 4 π I_{1} /3

. At a surface with no incoming intensity and the same law only for

0 < μ < 1

J = \frac{I _{0}}{2} + \frac{I _{1}}{4}, F = π I_{0} + \frac{2 π I _{1}}{3}, K = \frac{I _{0}}{6} + \frac{I _{1}}{8} .

Solved by gpt-5.6-sol high.

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