Realistic Visibility

In a typical case of overthinking, I wondered how far you can actually see with a torch. In a dungeon. You know, your typical Monday morning. Things you ask yourself while working on your latest RPG project (Dragon Eye in my case)

Or with a candle. Or an oil lamp. Did all those RPGs just wing it or did they do the research and get it right. Well, only one way to find out — let’s do the math. Fair warning: This rabbit hole went deeper than I anticipated.

Brightness follows the inverse square law. It is measured in lumen, or in lux. Or in candela or in candlefoot or… damn…

After some research, the simple version is: Lumen is how much light comes out of a (point) light source. Lux is how much brightness appears in an area. Candela is an old measurement (the light of one standard candle, about 12.6 lumens) and candlefoot is — like lux, one candle on one square foot of area.

Try as I might, I could not find any numbers for the brightness of a torch. But aside from the 12.6 lumens for a standard candle, I found good numbers for oil lamps which went from 100 lumens all the way up to 1000 lumens though that was one lamp advertised for being very bright. Most of the oil lantern seem to have 200 to 500 lumens in brightness, depending on how you regulate them (i.e. how quickly you burn through oil and wick).

But how bright is that, really?

Well, that is only the light that’s coming out of the candle or lamp. The inverse square law governs how bright a surface appears to be depending on the distance from the light source and its brightness. See the sources below for more details on the math, the short is that the formula is 4πr²

Or, in other words, we transform lumens into lux, the brightness per square metre, more useful for apparent brightness that interests us and taking into account the light spreading out.

At 5 metres that 12.6 lumens candle gives us 0.04 lux. At 10m just 0.01 lux.

Our 500 lumens oil lamp, on the other hand, gives us 0.4 lux at 10m and 0.016 lux at 50m.

But how bright is that, really?

Brightness
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Here, comparisons are easier to find, because astronomy. It turns out that 0.001 lux is about what you get from starlight on a moonless night. A full moon will give you 0.11 lux or 0.27 or 0.4 — depending on which source you trust and what factors you account for. But that gives us a comparison. Oh yeah, none of that compares to daylight in the least — even an overcast day has a brightness of 1,100 lux and a sunny day ten times that, and if you look straight at the sun, that’s about 100,000 lux.

But our eyes are wonderful and adaptable. In darkness, we can spot magnitude 6 stars with the naked eye, at least just about barely. And their apparent brightness is — 8 nanolux. That’s 0.000,000,008 lux.

Because, you see, we’re just touching the surface of the whole question. What we calculated so far, if we look carefully is how much light arrives at the target. If the question was if the torch or lamp is bright enough to let us see the far wall or that door, we didn’t even do half of the math so far.

First, the light needs to travel back to us, scattering again. But before that, much of the light will actually be absorbed by whatever material it hits. Wood, stone, metal — not even mirrors reflect 100% of the light that hits them, so we lose some there. There are some lists of various materials and their absorption in the sources below, for simplicity I will assume 60% absorption as a rough average of different stones and woods.

So of those 500 lumens, 0.4 lux hit the wall at 10m distance. 60% of that is absorbed. The remaining 40% get scattered and need to make their way back to us, again going through the inverse square law. When you run through the math you see a curious thing: It turns out to be about 10% for a variety of different values. In our case, almost 0.04 lux will reach our eyes (0.03979).

Definitely bright enough to see! Yeah, we have a result! How far can we see with that lantern? Actually yes, how far? How good are our eyes?

It turns out that magnitude 6 starlight is messing with us. Stars we see against a dark background, attuned to darkness and we see them as points — we don’t actually “see” anything, we only notice enough light to understand that there’s something there.

Eyes
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There’s a ton of science on this, so it’s easy to find. Again, more details are in the links down below. The simple is that we have two kinds of light receptors in our eyes. Cones are responsible for colour vision. Their minimum activation is at about 0.001 lux — less than that and your cones won’t see a thing. But your rods will — these are responsible for low-light and grayscale vision and they are much more sensible. They go down to 0.000,001 lux. Wait, what about those magnitude 6 stars? Well, theoretically the human eye can detect a single photon. But as I said, that’s not vision, that’s just a feeling like there was something there. It works under laboratory conditions in perfect darkness with test people instructed to see if they notice the faintest of light blips. Let’s go with the rod sensitivity better.

So at what distance does the light from our oil lamp, considering absorption, and return to our eye, drop below that level?

2 kilometres.

I calculated that twice.

That is strange and counter-intuitive, isn’t it? You would never think that even with a very bright lamp — and wait, ours isn’t even that bright! — you could see for two kilometres in a dungeon.

Is our math wrong? No. We just forgot something:

Dynamic Range
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Why can we see faint stars in the sky but when we’re driving by car, everything outside the headlamps cones seems black? Because our eyes realize the amazing feat of giving us vision both in bright sunlight (11,000 lux) and at night with just the moon and the stars (0.1 lux) by adaptation. They can change their sensitivity. Our eyes can see both 11,000 lux and 0.1 lux — but not at the same time.

In fact, the dynamic range of the human eye allows for a contrast of about 1000:1 — or in other words: Anything darker than one thousandth of the brightness our eyes are adapted to appears black to us.

We cannot consciously adapt our eyes, it happens automatically, to the brightest light source (more or less). That is why we can see very faint stars in the sky — because the rest of the sky is also near-black. If you have light pollution, or a full moon nearby, you won’t see those magnitude 6 stars.

For our dungeon, that means we will not be able to see things far ahead not because so little light is returned from there, but because things closer to us are so much brighter that our eyes cannot adapt to perfect darkness. In a very dark night, an oil lamp or torch or campfire or — under good conditions — even a single candle can in fact be seen from two kilometres away. But when we are holding that torch, we cannot see two kilometres into the darkness not just because the light has to travel that distance twice, but also because there will also be light reflect back into our eyes from the nearby wall, floor, ceiling, trees, the warrior’s platemail in front of us, etc. etc. That light is much brighter and our eyes will adapt to it — and whatever light is reflected from the far wall falls outside our dynamic range.

That, btw., is why everyone who’s actually been outdoors with a torch knowns to hold it not in front of himself, but off to the side, above his head or even slightly behind. You definitely, really, don’t want direct light coming into your eyes. At one metre from our oil lamp that would be 40 lux, and with a 1000:1 dynamic range our low detectable end would drop to 0.04 lux — 10 metres.

The Floor, The Floor
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Or the ceiling. Even if you are in a vast hall, there will be something nearby that reflects light. For a simplified calculation, let’s assume that with good positioning, you can have 2m of distance between the lightsource, you, and the first visible surface that reflects light back at you. The floor ahead of you or something. That floor has a brightness of 10 lux. So our dynamic range extends down to 0.01 lux. The distance at which the light returned from a surface with 60% absorption drops below that is about 20m.

And when we do the math, this holds true for not just 500 lumens, but also 250 lumens, and 100 lumens and even our 12.6 lumens candle — but also for 1000 or 2000 lumens. (footnote: We still can’t see 20m with the candle, because while the dynamic range works out, the brightness doesn’t — the reflected light is below the rod activation limit, which is at about 10m).

Dynamic range limits our vision, not the brightness of our light source. Isn’t that interesting?

Scout Ahead!
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Instead of a brighter light, we want to keep the lights with the main party and send the rogue ahead. Because if we manage to up the distance that the light travels between the light source and the floor and then his eyes to, say, 4m instead of 2m, then the range at which he can faintly make out shapes increases to 40m.

At that range, the light is bright enough to make things out if our light source is 500 lumens or 250 lumens, but not if we go down to 100 lumens. Here, again, brightness takes over as the factor that matters and the distance where we can see is about 28 metres.

Conclusion
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So there you have it. Effective visibility in a dungeon with torches or small oil lamps about 20m for those carrying them and 30m for the scout, and with brighter lamps up to 40m, beyond that it doesn’t matter if you make it brighter. Caveat

This is all based on my research and understanding of the sources. I may have made some terrible mistake somewhere and it’s all wrong. Please comment if you think you found a mistake so that I can check it and fix it.