VY Canis Majoris planetary system
Since the Sun is the only star we know to have life in its planetary system, I came to wonder about if it was replaced by, say, VY Canis Majoris How far would we have to be so that we would still be functioning normally as if it was really the Sun? Could such a thing even be possible?
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This is actually pretty simple. We just have to calculate the stellar flux, which is the luminosity of the star divided by the surface area of a shell with the radius of the planet's orbit. In other words: $$F=\frac{L}{4\pi r^2}$$ Now, let's denote the Sun's luminosity as $L_{\odot}$, the current radius of Earth's orbit as $r_i$, the luminosity of VY Canis Majoris as $L_{\text{VY}}$, and the new radius of Earth's orbit as $r_f$. We need to set the fluxes equal, in order to keep conditions on Earth the same. Therefore, $$F_{\odot}=F_{\text{VY}}\to\frac{L_{\odot}}{4\pi r_i^2}=\frac{L_{\text{VY}}}{4\pi r_f^2}$$ The $4\pi$s cancel, and, given that Wikipedia gives VY Canis Majoris's luminosity as about 270,000 times that of the Sun, we get $$\frac{L_{\odot}}{r_i^2}=\frac{270,000}{r_f^2}\to r_f=\sqrt{270,000}r_i\simeq520r_i$$ Therefore, Earth would have to be 520 times as far away - 520 AU - for conditions to be such that we could "function normally", as you put it.
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