How this calculator works
A shell contains 4 pi r^2 n dr stars, while each contributes L/(4 pi r^2), leaving n L dr. Equal path-length shells therefore add equally in a transparent static model. The finite model integrates to a chosen depth. Extinction supplies an exponential path weight. The expansion option deliberately uses the toy weight exp(-2Hr/c), representing redshift and arrival-rate dimming at constant H while holding source density fixed. It is not a Friedmann cosmology prediction. Dust would heat and re-emit in an eternal equilibrium, so extinction alone does not solve the physical paradox.
Formula
F_finite = n L D; F_extinction = n L/k; F_toy = n L c/(2H). F_surface = L/(4 pi R^2). Infinite transparent independent shells do not converge.
Worked example
In an artificial population with n = 2 stars/pc3, L = 10 W and depth 1 pc, F = 20/pc^2 = about 2.10053e-32 W/m2.
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Frequently asked questions
Is this an article or a calculator?
It is a shell-integration calculator. Changing density, luminosity, depth or attenuation changes a numerical flux.
Can dust alone keep an eternal universe dark?
Absorbed light heats the dust, which radiates too. A permanent sink cannot simply be assumed.
Why does the infinite option have no numerical flux?
Its integral diverges under the stated transparent independent-shell assumptions. Reporting a finite number would hide that failure.