This answer explains that the shimmering patterns seen on surfaces at the moment before totality of a solar eclipse are called shadow bands, and that the phenomenon is closely related to the twinkling appearance of stars (but not angularly larger planets) through the Earth's atmosphere.

If for some reason there happened to be a bright star close enough to Earth that its nighttime illumination visible on surfaces, but small enough in angular width that it twinkled, would we see shadow bands at night, every night when the star was out?

As there's a good chance the answer is "yes" to the hypothetical, I'd like to ask if this situation is in fact possible, or if a star were close enough to be bright enough to visibly illuminate a white surface at night, would it then be large enough in angular extent that it wouldn't twinkle any more?

The pattern from the Eclipse can be seen in the Smarter Every Day video Space Station Transiting 2017 ECLIPSE, My Brain Stopped Working - Smarter Every Day 175 at about 04:30.

Yes, the nonuniform illumination pattern would be visible for extremely bright stars.

Venus (mag -4.6) is bright enough to cast visible shadows. Let's move Sirius, the brightest star in the northern hemisphere closer, to match the Venus brightness:

  • currently: mag -1.46, 8.6 light years away, 0.006" angular diameter
  • relative brightness difference: $\sqrt[5]{100}^{4.6-1.46} \approx 18$
  • required distance factor: $\sqrt{18} \approx 4.24$
  • at $8.6/4.24 \approx 2$ light years away, Sirius apparent diameter would be $0.006'' \cdot 4.24 \approx 0.026''$ which is still much smaller than planets

Moreover, we can even see planets twinkling sometimes (depending on the seeing), so the "point light source" is not a strict requirement (well, strictly speaking, point light sources do not even exist ;))

Interestingly, there are no mentions of shadowband-like phenomena in the supernova observation history, even though some of them were bright enough to be visible in daylight.

I guess we have to wait for the next Milky Way supernova to be sure...

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