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I am learning about photometry to do Physically Based Rendering. As I checked the human eye dynamic range, I was surprised to see it was expressed in terms of luminance ($\mathrm{cd}/\mathrm{m}^2$) : from $10^{-6}\,\mathrm{cd} / \mathrm{m}^2$ to $10^8\, \mathrm{cd}/\mathrm{m}^2$ (see https://en.wikipedia.org/wiki/Human_eye#Dynamic_range, sources 18, 19 and 20).

The reason of my surprise is that, if I understood correctly, luminance of an object is independant of the distance to it. The Sun has a luminance of $10^9\, \mathrm{cd}/\mathrm{m}^2$, so, according to this way of expressing dynamic range, the Sun should blind you from anywhere in the Universe (neglecting extinction).

I was expecting this dynamic range to be expressed in terms of irradiance ($\mathrm{lx} = \mathrm{lm}/\mathrm{m}^2$), as flux per area of retina seems to make more sense to me.

Am I forgetting something ?

Florian Cabot
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2 Answers2

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Ok, so I think I got it. The fact that our eyes don't have an infinite resolution might be the reason we don't get blinded by stars even if dynamic range is expressed in terms of luminance.

If you can resolve the solar disk, then it will lay its light on several of your retina cells, which are getting the same luminance. The further away you get, the less cells are triggered, but still with the same luminance.

This would mean that, as long as you can resolve the solar disk, it will blind you. We can still resolve it from Jupiter, so even if the Sun illuminates Jupiter 25 times less than it illuminates the Earth, looking at the Sun from Jupiter would damage your retina just as bad for each retina cell exposed. It would just damage a smaller spot of it.

(I'm not entirely sure about this last paragraph.) Then, when you can't resolve the disk anymore, it will lay its light on "less than one cell". At that point, perceived luminance should be multiplied by the fraction of the solar disk solid angle over the solid angle covered by one cell. It would still "burn", but burn an area smaller than the cell, so the cell can easily take it.

If nobody has anything to add or correct, I will accept my own answer for now.

Florian Cabot
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The luminance of an object is not a constant. It depends on how far you are from the object. The luminance of the light from the Sun at earth is $10^9 \text{cd/m}^2$ but the luminance of the sun closer to the Sun is more and less further from the Sun.

This is because luminance is power per area and the light energy output by the object gets spread out over more area the further away you get from it. The Sun outputs light energy at a constant rate which then spreads out over a sphere. The further away you are from the sun the bigger the sphere. Since the area of a sphere is proportional to the area of the sphere we get

$$\text{lm}(r) = \frac{\text{lm}_0 r_0^2}{r^2}$$

where $\text{lm}_0$ and $r_0$ are the luminance and distance from the Sun at some reference point (like the surface of the Earth). As $r$ gets bigger the luminance goes down.

So for example if the Sun stops blinding you at $10^8 \text{cd/m}^2$ if you go to a distance where

$$10^8 = 10^9 \times \frac{r^2_{\text{Earth}}}{r^2}$$ $$\Rightarrow r \simeq 3.16 \,r_{\text{Earth}}$$

then the Sun will no longer be blinding.

Luke Pritchett
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