A 20 million candle power lighthouse
June 8, 2010 8:45 AM   Subscribe

I have a candle and its brightness is exactly one candle power or candela. If I buy a 10,000 candle power torch/flashlight, does that mean it's 10,000 times brighter (and what does that mean) or that I can see it 10,000 times further away? Are these the same thing and why do they seem unbelievable?
posted by westerly to Science & Nature (20 answers total) 3 users marked this as a favorite
The Language of Light
posted by jon1270 at 8:51 AM on June 8, 2010 [4 favorites]

A 10,000 candela source of light is 10,000 "brighter" than the 1 candela source, but that's not the full picture.

A source of light, unless focused by mirrors, will shine equally in all directions in a roughly spherical manner. A sphere's surface area is proportionate to the square of the radius, or distance, and thus it follows that the amount of light a source shines on a given surface is inversely proportionate to the square of the distance.

All said, you will be able to see it sqrt(10,000) = 100 times further away. The source is throwing out 10,000 times as much "brightness," but it's throwing the light in every direction.
posted by explosion at 8:54 AM on June 8, 2010

Brightness follows an inverse square law - so if a pojnt source has 10,000 times the absolute brightness, it'll have the same apparent brightness when you're 100 times further away. (This is oversimplifying and not taking account of the eye's colour response, the way a torch's light actually spreads out, etc)
posted by Electric Dragon at 8:54 AM on June 8, 2010 [1 favorite]

Best answer: It means it has 10,000 times as many photons in it. But our perception of brightness doesn't work on a linear scale: if light A is twice as bright as light B, it looks only a little bit brighter to our eyes. If you start with a light of 1000 candles and progressively add light 1000 candles at a time, the difference between each successive step will seem to get smaller as you go.

So your big light is 10,000 times as powerful, but (let's say) a hundred times brighter than the candle as we perceive it.
posted by echo target at 8:56 AM on June 8, 2010

Relatedly an average horse has about 15 horsepower.
posted by The 10th Regiment of Foot at 8:58 AM on June 8, 2010 [6 favorites]

The inverse square law only holds for isotropic radiators, i.e. those that are the same in every direction. For focused beams like in a flashlight it does not -- that's kind of the whole point behind focusing light: in return for trading off less than full coverage in every direction, you get intensity that does not drop off as fast as 1/r^2. So "the way a torch's light actually spreads out" is exactly what explains the 1/r^n drop-off in intensity. In fact if you somehow had a perfectly collimated beam (like in a laser) it does not spread out at all and in principle its intensity does not drop off with distance at all.
posted by Rhomboid at 9:57 AM on June 8, 2010

it does not spread out at all and in principle its intensity does not drop off with distance at all.

...in a perfect vacuum
posted by nomisxid at 10:02 AM on June 8, 2010

The average horse outputs, on average, less than 1 horsepower.

Sorry, 10th regiment.
posted by dmd at 12:24 PM on June 8, 2010

dmd, from your Wikipedia link: "The peak power over a few seconds has been measured to be as high as 14.9 hp. However, for longer periods, an average horse produces less than one horsepower." . So it seems you're both right.
posted by Gomez_in_the_South at 2:17 PM on June 8, 2010

Best answer: Sorry if this is rambly. It's a bit late now and it's been a long day. If I slip up on anything I'll try to rectify it tomorrow...

This is complicated, and you have to take these manufacturer's claims with a pinch of salt. My suspicion is the term "candle power" (two words, the obsolete unit that has become the candela was candlepower) doesn't mean anything whatsoever. When I specify light fittings, I need to know the fitting's candela or lumen output -- the output I can use which, combined with the optical properties of the light and the distance away from the surface I'm interested in will tell me how much light is landing on it (*not* the object's brightness -- that depends on the object's properties itself, and is a fuzzy term in any case. *Light* obeys the inverse square law, *brightness* certainly does not). Good luck getting a figure quoted in candela or lumens from the supplier of a domestic torch.

Anyway -- here's what a 10,000 "candle power" torch would mean if it does emit 10,000 candela.

A torch *doesn't* emit light in all directions (as has been claimed above). Yes the individual point sources will (I say sources, it's probably got an LED cluster) but behind this will be a mirrored lens that reflects all the light out in a useful direction. A candle doesn't do that. So, the candle is emitting 1 candela, but in all directions; while the torch is emitting 10,000 times that but all of it is pointing in the rough direction you point the torch. Note that the candela is the SI unit of 'luminous intensity' or 'luminous flux' and not brightness as such.

To work out how much light will arrive on a surface we need to know a few things including the particular optical properties of your torch -- the beam angle of the light it emits, and the properties of that beam. For instance, the torch might emit more light at a perpendicular angle to its end than it does at 1 degree away from the perpendicular in the x or the y. This information is often conveyed graphically in what is called photometric data.

The amount of light emitted in a particular angle of 1 steradian is measured in lumens. A steradian will form an area on the surface of a sphere of r2 where r is the radius of the sphere. A light source emitting 1 candela spread equally in all directions emits 1 lumen in the solid angle of 1 steradian. Let's keep things simple and say that the beam angle of your torch happens to be 1 steradian. Because the surface area of a sphere is 4πr2 and a staradian has surface area r2, there are 4π steradians in a sphere. Without the lens, the torch would be emitting 10,000 lumens in each steradian, but because all 4π steradians are being focused in that direction, we have 4π the number of lumens -- so we'd have 125,663 lumens being emitted by the torch, which is 125,663 times the amount of light the candle emits in the same direction (but 10,000 times what the candle emits overall, or 10,000 times what the candle would emit if we could contain its light source within the torch).

This does seem unlikely, and there are torches out there claiming to be 1 million, 3 million or even 10 million "candle power".

If I'm catching up on the latest LED performance, I'll take the Phillips Luxeon Rebel's current performance as a benchmark. Their "brightest" LED emits a maximum of 200 lumens. It'd take a cluster of over 600 of these LED's in a torch might meet the claims (if a "foot candle" really is a candela), but the space requirement of the LED's is nothing as to the size of the heat sink you'd need. If your torch had ten of these LED's it'd be emitting 2000 lumens, which would be 2000 times the output of the candle in the equivalent direction. This is much more doable. I'm not saying it isn't possible because I don't work with torches, and aren't familiar with their specs. It'd be great if we could get an inside man to confirm the output.

As has been pointed out, even if it did emit 10,000 times (rather than 2,000 or 125,663 times) as much light per steradian, this doesn't mean you could see 10,000 times as far because light obeys the inverse square law: that steradian becomes a bigger surface area the further you get away from the source. Now what you can see depends how much light is arriving at an object, and that's a function of the amount of light hitting it, the distance away it is, and the angle at which it hits (perpendicular is good). The amount of light hitting an object is the object's illuminance (not luminance, that's the amount of light coming back off it per unit area, and then, finally, we really are getting into the realms of what might be called brightness) and illuminance is measured in lux. 1 lux = 1 lumen / square meter.

Let's define what you can see as what you can see usefully -- let's take the figure of 1 lux, since this equates to the illuminance of the ground in the tropics under a full moon, and is also the escape design average illuminance for an escape route in the UK. Let's keep things simple and say that our angle of incidence is perpendicular.

E = lm/d2

where E = illuminance in lux, lm = luminous flux in lumens and d is the distance of the source to the object.

Our candle is emitting 1 lumen. So at a distance of 1 metre, the illuminance of a surface is 1 lux -- so already we've hit what we've defined as our useful limit. Our torch is illuminating it to 10,000 lux. If you plug the numbers you'll find that because of the inverse square law, you'll be 100 metres away when the light from the torch dissipates to 1 lux, so you *might* say that a torch that is 10000 times as bright as a candle will let you see 100 times as far.

If the light were hitting the surface at an angle, you'd multiple that equation by the cosine of that angle from the normal. If the light hits at a perpendicular angle, the angle from the normal is 0 degrees and the cosine of 0 is 1, so we can omit that step in this special case.
posted by nthdegx at 2:21 PM on June 8, 2010 [1 favorite]

echo target has the right response. The measurements are linear but our eyes are not.
posted by chairface at 2:22 PM on June 8, 2010

You've picked a small element of a much broader question, though.
posted by nthdegx at 2:41 PM on June 8, 2010

Best answer: dmd, that statement on wikipedia was not really consistent with the sources cited: a working horse, averaged over a day, produces just about 1 horsepower. But this isn't a question about horsepower; sorry for continuing a derail.

Here's a history of the SI definition of a candela. Notice that the definition includes the solid angle, which is what makes the candela a unit that's independent of the distance between the source and the observer.

Rhomboid is correct that focusing can increase the luminance from a particular source. However there is no such thing as a perfectly collimated beam, even from a laser; every real focused beam passes through a beam waist and begins to diverge like an inverse square again.

The logarithmic response of the eye is definitely part of what's throwing off your intuition.
posted by fantabulous timewaster at 3:10 PM on June 8, 2010

You're right, fabulous timewaster. 1 lumen = 1 candela * 1 steradian (hey, I said it was late). So by my reckoning, if the output of the torch is 10,000 candela, then it is emitting 10,000 lumens if it is only emitting outwardly through 1 steradian, which was the assumption made later for the illuminance calculations. The candle emits a total of 4π lumens for the same reason.

It's still debatable what "candle power", though the interpretation that it represents the obsolete candlepower seems to be widely accepted. If that's the case then we can definitively state that the torch has a luminous intensity of 10,000 candela. Because this is directional, it seems highly unlikely that multiplying by 4π to calculate the luminous flux is valid, though this would be fair if it did emit light in all directions. Why then they can't just state that the lumen output is 10,000 I don't know.

The preoccupation with the eye and brightness is a bit of a distraction in my opinion since the questioner is clearly asking about the relative light outputs and relative usefulness of the torch and the candle.

Candela and lumens measure properties of light that aren't actually useful until it hits a surface. The illuminance of a surface is the quantifiable usefuleness of light, and so long as we define our useful threshold then it *is* possible to work out a definitive multiplier of how much better the light is, not because 100 lux looks 100 times brighter than 1 lux (it doesn't), but because the torch will achieve (with the assumptions given with the quoted spec) our designated useful threshold at a distance 100 times greater than the candle.
posted by nthdegx at 5:05 PM on June 8, 2010

I'm not following the algebra closely in this thread, but a typical focusing flashlight makes a spot a foot or so across on a wall twenty or thirty feet away. That's a solid angle of π(15 cm)2/10 m = 0.007 steradian, which means our 10k candela flashlight is putting out about 70 lumen.

Probably the advertisers use "candle power" because it looks less exotic than "candela," "lux," or "lumen" and is also a legitimate way to write "ten thousand thingys" on a relatively dinky flashlight that happens to have good focusing.
posted by fantabulous timewaster at 8:24 AM on June 9, 2010

Response by poster: Quoting from jon1270's link, "the candela is the unit used to measure the intensity of light in a particular direction." Doesn't that make the whole thing about direction irrelevant, since the candle puts out 1 candela in the one direction I'm interested in and the torch or lighthouse presumably puts out x quoted candela in the direction it's designed to go... which is I think Rhomboid's point. Am I misunderstanding 'all directions'?

I'm guessing the thing with lighthouses is a much more focused beam.

nthdegx, getting down to your 8th paragraph, the most basic of the things I don't understand is: if you have a number of identical light sources, whether that's 10,000 candles or 600 LEDs — surely they are going to put out the same light intensity each in a cluster as they would on their own. That is, I can see that there's more light overall, but what makes it travel distance?

Running with your definition of 'what we can see usefully' — I just went outside on a cloudy-but-some-stars midnight and measured this — I can make out the text of a page of a book held towards the candle at a distance of about 1.5 metres. I realise this is hugely unscientific. (I lack a tropical moon :( )

Then the 2000 lumen torch lets me read at 150m, and the 125,663 torch at ... I'll work this out when my brain is functioning. And just to add to the factoid-gathering/gloss over my lack of maths, that was based on a geranium-scented soy wax candle. Beeswax is brighter.

I'm 21km from my local lighthouse (output unknown). I can see it shining in my bedroom window at night, but I don't think I could read by it outside.
posted by westerly at 5:17 PM on June 9, 2010

Interesting bit up front: 10,000 candle power sounds like a lot, but if you do the maths it really isn't

fantabulous timewaster, a light manufacturer could never put lux as a measure of output since it's a measure of the amount of light hitting an object and is dependent on variables that the manufacturer can't be expected to know.

70 lumen would be a very low output for a torch. A single LED is capable of spitting out more light than that. If we're saying you're 10m away, then a steradian at that distance would form a circle 100m2. The circle the spotlight makes on your wall we've said has radius of 0.15 metres, making its area 0.07m2. As a proportion of our 100m2, that's 1/1430 the size, and therefore our solid angle is 1/1430 of 1 steradian! This sounds like an incredibly narrow angle to me, and if this is a really life example I'd suggest the output of the torch is not uniform across the beam angle and a lot of light is "lost" on the periphery of the circle you see, apparently (though not actually) doing nothing. Simplistically, we can convert the luminous intensity to luminous flux by multiplying it by the solid angle, just as you did. This would make about 7 lumens, but it does completely ignore the characteristics of the light source. However, this is an extremely instructive calculation because it demonstrates how an apparently high output in candela can translate to a much more modest figure in terms of lumen output.

Where I was confused in my initial response was to assume that the quoted output, converted to lumens would be *increased* per unit area because of the focusing of that amount of light. I see now that what they seem to be doing is measuring their light output for their small angle, and then work out how much would be falling on one steradian, were the torch illuminating this much wider angle at the same number of lumens per 1/1430th (for example) of a steradian. Does this make sense to anyone? If I'm right, it's crucial to the original question, since it effectively makes the answer "10,000 candlepower sounds like a lot, but if you do the maths it really isn't."

westerly, the thing with the beam angle is (and I led you up the garden path accidentally a bit with my first answer, sorry) really important, but I got it backwards because I am not at all used to working with candela. The light sources I deal with have total outputs quoted in lumens, and diagrams and computer files to show you how those lumens are distributed (always unevenly).

Essentially I was attempting to work out the total lumen output for the torch, by applying the same logic I applied to the candle. The candela has a luminous intensity of 1 candela, or a luminous flux of 1 lumen per steradian. The candle emits in all directions, and there are 4π steradians in a sphere, therefore the output of the candle is 4π lumens. Now, I worked out with the same sum that if the torch emitted light in all directions it would be emitting 125,663 lumens total. This is where I confused myself in my late, tired state. This is only true in the theoretical sense that, yes, if the torch could put that amount light for all 4π steradians in a sphere, then the lumen output would be 125,663 lumens. But of course its mirrored reflector is already working hard to put all the light produced by the source into our one steradian, and if the reflector was removed it would be emitting about 795 lumens per steradian.

But it's even worse than that because the actual beam angle of the torch will be much less than one steradian. But, as it turns out, those 10,000 lumens aren't all focused into the small angle. They're just proportionately divided so that that smaller angle, and all the ones around it, get a sum total of 10,000 lumens. Let's say the torch has a cluster of ten 100-lumen LEDs. That means the torch is outputing 1000 lumens of luminous flux regardless of direction. It is incapable of putting out any more. If the same torch emitted light at a solid angle of one tenth of a steradian, then the manufacturer could claim that the torch is outputting 10000 candela (or "candle power") because they could argue that that's what the torch would do if it emitted light at the same "rate" per unit area over a larger area of 1 steradian. i.e. the torch can emit light at 10,000 lumens per steradian, but *only* for angles much less than 1 steradian!

However, from "candle power" figure in isolation, it is *impossible* to work out the luminous flux of the torch, and it's the luminous flux that is the useful measure for working out illuminance of an object at x distance.

"I can make out the text of a page of a book held towards the candle at a distance of about 1.5 metres."

I should mention that my original calculation of illuminance assumed in both cases that, at a distance of 1m, the area of the book was 1 steradian (ie 1m2 - a big book, but not impossible). I said the candle's luminous flux was 1 lumen. I meant 1 lumen per steradian. Let's maintain this assumption, and the theoretical assumption that the torch can emit 10,000 lumens over one steradian.

Recall E = lm/d2.

At 1.5 metres, your candle is lighting the book to 0.44 lux (remember, this is a theoretical candle, but not your actual candle). This is the new illuminance we are defining as useful, since it is the illuminance by which we have ascertained a book can be read.

If we want to know the distance at which the torch can produce the same illuminance, we rearrange the formula to d = √(lm/E) = √(10,000/0.44) = 150 metres. You can see that the torch is still illuminating to our useful illuminance at 100 times the distance.

*Crucially* you need to be the same distance away from the book to appreciate this -- your friend could be holding the torch 150 metres away, but of course the light reflected back to your eyes is also subject to the inverse square law. Illuminance is only a measure of how much light is arriving on the object, remember.

In practice our torch is more likely to emit 1000 lumens and this could not usefully illuminate a book 100 or 150 metres away. At 1 metre, with all the lumens of the torch falling on the book due to a narrow enough optic, then the average illuminance would be 1000 lux, though in practice this might be much higher on a bright spot at the middle of the book, and much dimmer towards the edge. At 1.5 metres, the average illuminance would be 444 lux. But for our "useful" 0.44 lux, the distance comes out at 48 metres. Obviously, again, you still need to be 1.5 metres from the book for this to look as bright as the candle at 1.5 metres.

Clear as mud, eh?
posted by nthdegx at 2:50 AM on June 10, 2010

Let me make an attempt at saying something useful:

As an analogy for what I think the torch manufacturer is doing, let's say a pneumatic press manufacturer says his press can apply a pressure of 1000 newtons per m2. However, the press he is selling is only 0.1m2 in size. The press can therefore only apply a maximum of 100 newtons.

In this case the pressure quoted is analagous to the luminous intensity of the torch -- our 10,000 candela, and the total force applied by the clamp of 100 newtons if analogous to the luminous flux, the total lumens emitted by the torch.

In the case of the clamp, we need to know the size of it to know how much useful work it can do. Likewise, to know how much light the torch can emit, we need to know the angle at which it emits light. In both cases, the specified performance is meaningless without additional data.
posted by nthdegx at 3:07 AM on June 10, 2010

This would make about 7 lumens
You're right, of course; the dimensionless solid angle is (area)/(radius)2, so I was missing a factor of 10 meters. My arithmetic is weak lately.

I agree with your analogy.
posted by fantabulous timewaster at 9:08 AM on June 10, 2010

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