Analytical Resolution of the Dark Night Sky (Olbers') Paradox

Abstract

We derive a spatiotemporal analytical resolution of the dark night sky, or Olbers' paradox, first showing that in an infinitely large universe the cumulative solid angle of the light that is projected upon the celestial sphere by an infinite population of directly observable stars is indeed finite. Using the GAIA DR2 data, we show that the number radial density of the stars that are directly visible has a radial distribution that is quasi‐unimodal, and that such a radial distribution is constrained by the inverse square law, which asymptotically limits the magnitude of the cumulative solid angle of the light that is projected upon the celestial sphere. Second, we show that the temporal summation parameter of the imaging device that samples the celestial sphere projection further limits the cumulative apparent brightness of the image formed on the detector surface of the imaging device. The unaided human eye is a biological imaging device, which in the mode of achromatic dim light sensing known as scotopic vision, has a short temporal summation window of approximately 650 ms, which is determined by retinal photochemistry (Holmes et al., Vision Res., 2017, 140, 33–43; Lamb & Pugh Jr, Prog. Retin. Eye Res., 2004, 23(3), 307–380). This short temporal summation window makes the eye act as a high‐pass power filter, making it incapable of rendering visible those stars that project an apparent brightness below a certain threshold. We show analytically that astronomical objects that individually have an apparent brightness mG > 8 will not contribute to the cumulative apparent brightness of the night sky as seen with the fully dark‐adapted unaided eye under ideal seeing conditions, even when the celestial sphere is tiled completely with an infinite number of such faint objects. Using numerical simulations, we also show that a large‐scale homogeneous star field and a Galactic‐scale inhomogeneous field project collectively a cumulative apparent brightness upon the human retina that is approximately 100 million times fainter than the apparent brightness of the Sun. We show that the cumulative solid angle projected by all the unaided eye visible stars is approximately 5.95×10−13 sr, and the asymptotic cumulative solid angle limiting distance is approximately 2 kpc, which is the radial distance beyond which no significant increase in the cumulative apparent brightness of the night sky can be detected with the unaided eye, even in a large‐scale homogeneous and infinitely large universe with an infinite number of stars.