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Let the S’ frame move with velocity ##\beta## in the S frame. Then how will the the photon density look in the S frame? By symmetry it can only depend on the angle ##\theta## with the direction of motion.

Using the aberration formulas and assuming the number of photons (per unit time) in a cone between ##\theta## and ##\theta + d\theta## is the same as the number between ##\theta’## and ##\theta’ + d\theta’## ... then I get the following result:

$$\frac{\sigma}{\sigma ‘} = \frac{1-\beta ^2}{(1-\beta \cos\theta)^2}\approx 1+2\beta \cos\theta$$

I can’t find anything similar online. Is this formula an accurate quantification of the so called “headlight effect”? I can show the steps of my derivation upon request.

Thanks.