# How many shields, for the Archimedes legend to work

## Main Question or Discussion Point

There is a claim that Archimedes recommended the use of well-polished shields to concentrate light from the sun and onto enemy ships, in order to set them alight from afar.

But Andy Resnick here says this is a misconception. It may well be a misconception for some of us. But does it work?

I reckon that it does, provided a large enough number of shields are accurately pointed onto the sails. And provided the shields are flat like mirrors, not curved. Each shield is like a window through which a 2nd sun shines. If you're just close enough to 100 shields, you can be illuminated by 100 suns, minus losses. If you're twice as far as the limit dictated by the size of the shield and the size of the solar disk in degrees, you need twice the number of shields for the same effect.

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And here's a power generator based on this idea, except the mirrors are closer for cost efficiency.

http://www.rise.org.au/info/Tech/hightemp/image013.jpg [Broken]

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Danger
Gold Member
You can check out the 'Mythbusters' website. I remember seeing them try to duplicate the situation a while back. It seems to me that they ended with a 'plausible' verdict, but they had to fudge the conditions a bit. Details escape me.

Can't we predict whether it works without physically reproducing it?

Danger
Gold Member
I suppose that someone can, but I wouldn't know how. There are a lot of variables, such as actual vs. ideal reflectivity of the shields (Mythbusters used mirrors), atmospheric attenuation of the light, material properties of the target, etc..

Danger
Gold Member
Thanks for clarifying that. I must have been getting it mixed up with a different experiment.

Integral
Staff Emeritus
Gold Member
I failed to be impressed with the Jamie and Adams failure to reproduce Archimedes mirror to burn anything. First problem, it the idea of "pointing" the mirrors. That is NOT the way to achieve the desired results. Archimedes would have computed the parabola needed to focus the suns light at the desired point, each soldiers instructuctions would have been to hold the shield stationary on a prescribed line, not to try an point the reflection. You need a large precise parabolic reflector, not a bunch soldiers randomly flashing the suns reflection.

All Jamie and Adam demonstrated is that they are not Archimedes.

This is from the 2nd episode summary:

"the ship only ignited when it was stationary and positioned at less than half the distance described in the myth."

That's a strange statement, that it only worked at half the distance described, why didn't they try with more shields then?

Or is the number of shields described in the myth too?

This is from the 2nd episode summary:

"the ship only ignited when it was stationary and positioned at less than half the distance described in the myth."

That's a strange statement, that it only worked at half the distance described, why didn't they try with more shields then?

Or is the number of shields described in the myth too?
The thing is, that in that episode, not only did they use incredibly large amounts of mirrors, it took a long time for the boat to ignite.

I am led to believe that no matter how many shields are used, it won't work on any moving boat.

Andy Resnick
Here's a simple way to bust the myth: energy considerations. This avoids any discussion of aberrations, poor mirrors etc. etc.:

An area of 1 cm^2, directly illuminated by the sun, absorbes 0.14 W. If the heat capacity is 1, the height 1 cm, and thermal conductivity infinite, the area heats up 2 deg. per minute, or 120 deg. per hour. Any black body heated up to 120 deg. C radiates 0.14 W per square cm of surface. Therefore, the light of the sun cannot raise the temperature of any body beyond 120 deg. C.

Experiments show that dry wood ignites around 500-700 deg. C. This temperature can only be obtained by a black body if every square centimeter received a luminous flux of 2-5 Watts, or 20-40 times more than the sun provides under the best of circumstances, and the heating would have to occur over a long period of time. Instantaneous ignition requires hundreds of Watts.

So, to set a ship on fire before the crew will require the ship to remain still for an extended period of time. And for motion of the sun to be compensated. And for the wood to be perfectly dry and absorptive. Let's get around all this by going for instantaeous ignition- say 400 times the luminous flux from the sun. That will produce spontaneous combustion in thick oak planks within a few seconds.

So, let's use a lens (or bank of mirrors) to concentrate the luminous flux from the sun. The flux from focused illumination is greater by a factor of (110*D/f)^2, where D is the diameter and f the focal length. It would seem that given a sufficiently large numerical aperture, we could easily acheive it.

Put the ships 1 km offshore. Then the diameter of the lens/mirror must be 0.18 km. So, it would seem that the problem is solved- perhaps cumbersome, but indeed, we can collect sufficient radiant flux to set a ship on fire. However, we neglected to account for the dissipation of radiant energy.

Without getting into too much detail, the Mangen-Chikolvev formula is a statement about conservation of energy (brightness or flux) within an optical system. The calculation for the illumination produced by a source of brightness B, lens area A and focal length f is given as E = AB/l^2 To get an idea of what this means, let's use a high intensity arc lamp as the source, a lens 2 m in diameter, and target distance 1 km. The brightness at the target is 30 times less than the sun's brightness at the surface of the Earth.

The main actual application for all this are lighthouses. And in fact, in 1747 a scientist Buffon constructed a device very similar to what we are talking about. He had a system of 168 mirrors, each 6 inches by 8 inches, and was constructed so that all mirrors could be aimed at a common spot. He did set fire to pinewood at a distance of 158 feet, and it took several minutes to get the fire going. Scaling this up, in order to acheive instantaneous ignition, he would require a diameter of 500 m or more to acheive combistion at 1 km.

Now, I'll take the final paragraph verbatim from "On the Possible and Impossible in Optics":

"But let us imagine that the enemy is not able to prevent the construction of a mirror 1-2 km in diameter, and that it is carried in full view of them,let us say a few km away. It would not be at all difficult to render the weapon harmless. All that need be done is to paint all easily combustable objects of value white or cover them with aluminum or some other foil and any danger of a conflagration is prevented."

Defennder
Homework Helper
Isn't it possible that they might not have ignited dry wood in the first place, but the cloth sails of the invading ships?

He had a system of 168 mirrors, each 6 inches by 8 inches, and was constructed so that all mirrors could be aimed at a common spot. He did set fire to pinewood at a distance of 158 feet, and it took several minutes to get the fire going. Scaling this up, in order to acheive instantaneous ignition, he would require a diameter of 500 m or more to acheive combistion at 1 km.
I think at this particular point you may have got the maths wrong. Those 168 mirrors can be arranged in a rectangle of 13 by 13. The longer side of the rectangle would then be 13 x 8 inches = 8.7 feet. That's a target distance to mirror width ratio of 158/8.7 = 18.2 to 1. Therefore to ignite pinewood at 1 km, it would need a mirror with a width of 1 km / 18.2 = 55 metres. Quite small.

And let us not forget it is humans on the boat, in clothes.

Andy Resnick
"Archimedes, Kircher, Buffon, and the Burning-Mirrors", by W. E. Knowles Middleton
Isis © 1961 The History of Science Society

http://www.jstor.org/view/00211753/ap010120/01a00020/0

Including the small gaps in between the mirrors, the array was equivant to a single reflecting mirror with area 5.9 m^2- we agree so far. The (equivalent) mirror then subtends an angle *from the target* of 3 deg. In order to scale up to speed up the process, the mirror must be made much larger- say a 30 deg. angular subtense, in order to collect that much more light. Recall, the experiment was performed on resinous pinewood (highly flammable), and it took many minutes for combustion to occur. A 30 degree subtense means that at 1 km, the mirror must be 500 m in diameter.

In your calculation, the 'scaled up' mirror still has a 3 degree subtense, which would result in an extremely slow process, just as the original demonstration. Clearly the target's motion from bobbing in the water would be sufficient to prevent any useful result.

Obviously, it's impossible to state with certainty wether or not such a feat occured- AFAIK, nobody here was present at the time, nor are there reliable sources recording any details. Rather, I can confidently state that it was vanishingly likely that such an event occurred.

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You need a large precise parabolic reflector, not a bunch soldiers randomly flashing the suns reflection.
It's the same thing really.
If you have enough mirrors it will work.
Just like a convex lens focusing light onto a cigarette - which also works.

Main variable is the weather, and that's currently impossible to predict with absolute certainty. So whatever you calculate, the whole thing ultimately depends on weather, even today.

Andy Resnick
The reason one can burn things close up easily is becasue the source (the lens or mirror) subtends a larger angle than the 'raw' sun.

The crux of the argument against Archimedes lies in recognizing that the sun is not a point source and that the angular subtense of the reflecting/refracting surface must be hundreds of times larger than the sun in order to concentrate sufficient energy.

The crux of the argument against Archimedes lies in recognizing that the sun is not a point source
If the sun was a point source, the image generated by a convex shield would be a point. But each shield is not convex. It is flat or concave at worst. So it produces no image. Therefore the same number of flat shields would produce the same heat if the sun was a point source. And it would be harder to aim them all at the same point.

The reason one can burn things close up easily is becasue the source (the lens or mirror) subtends a larger angle than the 'raw' sun.
Not so.
I think light travels perfectly well over large distances through air without noticeable loss of intensity (ie without loosing many photons along the way). In a perfect vacuum the light would travel infinitely far with no loss of intensity whatsoever. So nothing really to do with distance travelled.

And absolutely nothing to do with angles whatsoever.

I think it's easier to visualise if you just imagine bouncing the photons (ping pong balls) arriving at each sheild onto the boat.
The more shields you use the more photons will arrive at the boat.

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Andy Resnick
If the sun was a point source, the image generated by a convex shield would be a point. But each shield is not convex. It is flat or concave at worst. So it produces no image. Therefore the same number of flat shields would produce the same heat if the sun was a point source. And it would be harder to aim them all at the same point.
I don't understand what you are saying here.

Andy Resnick
Not so.
I think light travels perfectly well over large distances through air without noticeable loss of intensity (ie without loosing many photons along the way). In a perfect vacuum the light would travel infinitely far with no loss of intensity whatsoever. So nothing really to do with distance travelled.

And absolutely nothing to do with angles whatsoever.

I think it's easier to visualise if you just imagine bouncing the photons (ping pong balls) arriving at each sheild onto the boat.
The more shields you use the more photons will arrive at the boat.

what didn't you like about my analogy?
Maybe it works better for you if you think of reflecting light rays instead of 'ping pong balls' off the shields.

Why do you think astronomical telescopes use large reflectors (which can be thought of as a huge number of small reflectors all reflecting toward the sensor) - just because it looks cool ?

to capture ALL the light coming from the sun you need to completely surround it by a large internally reflecting ball.
That wouldn't be easy to do.

The bigger the reflector the more light you have 'caught'.

OR even simpler example:
The more solar panels you put on your roof, the more light energy you capture.

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I don't understand what you are saying here.
Sorry, swap "convex" with "concave". You must be familiar with these diagrams of geometric optics:

This is a single shield. The sun is the "object". The image is where light is concentrated.

If the sun was a point source, the image would be a point too. A single shield could burn a point on the sail and possibly start a fire.

But the sun is not a point source and each shield is flat. So the image of the sun is behind the flat mirror, at a distance equal to the distance to the sun. It's like a window to a 2nd sun.

Each shield, makes an image of the sun behind it. When enough sun images are created behind shields, and they can all be seen from an observer on the boat, the boat burns.

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Andy Resnick
to capture ALL the light coming from the sun you need to completely surround it by a large internally reflecting ball.
That wouldn't be easy to do.

The bigger the reflector the more light you have 'caught'.

OR even simpler example:
The more solar panels you put on your roof, the more light energy you capture.
Yes. This is the central reason why the Archimedes legend is not reasonably based in fact. The question is: How much sunlight needs to be captured in order to instantaneously ignite dry wood at a distance of 1 km? How big must such a mirror be?