Archimedes Death Ray: Theoretical vs Practical Reasons for Its Failure

[russ_watters]

I don't want to hijack another thread, so I'm moving the discussion here...

A point about Archimedes Death Ray came up. The question is whether it failed for practical or theoretical reasons. I am of the opinion that it failed for practical reasons due to the inability of a bunch of soldiers to accurately aim a mirror when the spots they project overlap. Particularly at a long distance.

Andy, here's a thread where you explain your point: https://www.physicsforums.com/showthread.php?t=224629&highlight=archimedes&page=2

Perhaps the best place to start is I don't understand where this equation comes from:

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 achieve it.

Stepping away from that, the principle, which I agree with from an earlier post, is that you need about 50x the intensity of the sun to get fire. To me, that simply implies 50 mirrors - but to be more realistic (accounting for weather, imperfect mirrors, imperfect focusing) probably a couple hundred.

The problem I see with the idea when I draw a diagram is that the sun in a mirror would subtend the same angle as the sun does to the unaded eye, assuming a small distance between the target and mirror compared to sun and mirror. So for best efficiency, you'd want to match the size of the mirror to the angular size of the sun. For the sun at .53 degrees, projected on a target 1 km away, that's 9m per mirror and a 9m spot. Due to the sun being much further from the mirror than the object is, I don't see how the spot can spread out much, so the spot would still be about 9m.

Thinking about it another way, a parabolic mirror can be approximated via a bunch of flat plates. The size of the spot projected is a function of the focal length only (as I calculated), so "brightness" would be a function of aperture only.

 

[russ_watters]

It seems to me that the crux really is here:

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.

Where I disagree is that it seems to me that it isn't the subtended angle that must be a couple of hundred times larger than the sun, but that the area of the captured sunlight must be a couple of hundred times larger than the normal projected size of the sun.

Using a magnifying glass, the intensity of the focused light is a linear function of area, which is a square function of angle.

Ie, using your logic, it would seem that you'd need a theoretical minimum of 50x the angle of a 9m mirror, or a 450m diameter meter. I'm saying you'd need 50x the area or the equivalent of a 63m mirror.

Another way to think about it using the principles of photography: if you double the aperture while holding the focal length constant, you get an image 4x as bright.

 

[maverick_starstrider]

Have you seen the mythbusters where they attempt to make one (actually they do it twice). I was convinced. http://en.wikipedia.org/wiki/MythBu...de_46_.E2.80.93_.22Archimedes.27_Death_Ray.22

 

[Andy Resnick]

I think the origin and persistence of believing in this is because people place themselves at the mirror looking at the ship, not at the ship looking at the mirror. We have all (well, I have) burned ants with a magnifying glass, so it appears reasonable that if we scale the magnifying glass up by a factor of 'x', we could set fire to objects at greater distances.

If you are on the ship, looking at a parabolic (heck, make it elliptical with one focus at the sun) mirror 10 km wide, what will you see? Recall that the sun is *not* a point object.

Again, it's a thermodynamic argument, not an optics argument. The essence, in terms of geometrical optics, is that an infinitesimal ray (the basic element of geometrical optics) carries zero energy.

Also, the incident energy onto the ship (neglecting relative movement of the beamed energy due to ship motion, etc) must be greater than that re-radiated by the ship in order to generate heating. That in itself sets a lower limit for 'x'.

So, let's step through this together. First, if you are on the ship looking at the mirror, what will you see?

 

[flatmaster]

I'm pretty sure you don't need to worry about the angle in the mirror example. THe same luminous flux that strikes the mirror is the same luminous flux that will strike the target. This is just my tuition, and I have no real way to demonstrate it now. However, having the sun more perpendicular to the mirror will create a larger "image" of each mirror on the target, making the likelihood of multiple mirrors overlapping larger.

 

[Andy Resnick]

I don't want to hijack another thread, so I'm moving the discussion here...

<snip>

Andy, here's a thread where you explain your point: https://www.physicsforums.com/showthread.php?t=224629&highlight=archimedes&page=2

Perhaps the best place to start is I don't understand where this equation comes from: <snip>

Sorry- I just noticed this sentence.

Ok- the question is "where does (110*D/f)^2 come from?". That is a re-statement of the Mangin-Chikolev formula, a delightfully obscure result.

The formula states 'the illumination produced by a source of brightness B, lens area A and focal length f is given as E = BA/f^2' Given a solar irradiance of 1 kW/m^2, a subtended angle of 6*10-5 sr gives a solar radiance (brightness) of 16666 kW/m^2*sr. Since the area is 1/4*pi*D^2, lumping the 0.25*pi in with the solar brightness gives (114 D/f)^2 for the radiance incident on a target.

 

[Pupil]

 

[Andy Resnick]

I don't understand what that video is good for- please explain how it would be feasible (in ancient greece) to scale that up such that the object is now 1 km away.

Please tell us how big that mirror would have to be.

 

[maverick_starstrider]

I don't understand what that video is good for- please explain how it would be feasible (in ancient greece) to scale that up such that the object is now 1 km away.

Please tell us how big that mirror would have to be.

Well that's just a clip from the episode. Go find and watch the episode. They made multiple HUGE archimedes death rays and tested them. They had a competition between teams from Harvard, MIT, etc. to see who (if any) could set a moving boat on fire from a distance (I think it was 100 feet) using only period specific materials. None of them could do it in the end, I think one team succeeds in lighting a stationary boat from 50 feet though. So they declared it busted.

 

[maverick_starstrider]

So there you have a group with a lot of resources who tested a number of different designs drafted by some very bright people and none of the incarnations could remotely do what the myth says. I believe it took something ridiculous in the end, like they coated the ship in pitch, and anchored it at 50 feet and then only ONE of the designs set it alight after like 30 minutes of sustained death raying. Which was good enough for me to believe that the myth is completely bogus.

 

[Andy Resnick]

Which is my point.

 

[Count Iblis]

According to the wiki page, there were some additional arguments why the myth was declared busted:

Syracuse, where the myth was supposed to take place, faced east, and thus could not take advantage of the more intense midday rays, instead relying on less powerful morning rays.

The death ray would not work during cloudy weather.
Enemy ships were likely to be moving targets, and thus the mirrors would need to be constantly refocused.

The historical records: there was no mention of the use of fire during the Battle of Syracuse until 300 years after the event, and no mention of mirrors until 800 years after the event.

The impossibly large numbers of mirrors and personnel needed in order to light a boat with any reasonable speed.

The availability of other weapons that were much more effective: flaming arrows and molotov cocktails were more reliable at setting an enemy ship ablaze, and were more effective over longer distances.

These arguments make a lot of sense to me. But merely saying that: "we tried to do this and we failed", doesn't sound convincing to me. On NGC channel there was a documentary about a battle between the Roman army and Germanic tribes in which the Roman army had build a bridge over river in a very short time. But when scientists tried do this for themselves they failed, so it is still a mystery how the Romans exactly pulled this off (in this case we know for sure that the bridge was really build).

 

[Borg]

Sorry- I just noticed this sentence.

Ok- the question is "where does (110*D/f)^2 come from?". That is a re-statement of the Mangin-Chikolev formula, a delightfully obscure result.

The formula states 'the illumination produced by a source of brightness B, lens area A and focal length f is given as E = BA/f^2' Given a solar irradiance of 1 kW/m^2, a subtended angle of 6*10-5 sr gives a solar radiance (brightness) of 16666 kW/m^2*sr. Since the area is 1/4*pi*D^2, lumping the 0.25*pi in with the solar brightness gives (114 D/f)^2 for the radiance incident on a target.

I couldn't find the Mangin-Chikolev formula anywhere. I did find 50 year old translated Russian article about this exact discussion ( http://www.google.com/url?sa=t&sour...rmula&usg=AFQjCNFk4eHQPGrBqBQFeMpcvNlHNfeEGQ") that put question marks next to the name. BTW, they are definitely in agreement with you on the impossibility of this.

Are there any online links to the formula other than this that you know of? The spelling of Mangin's name seems to be throwing off the searches.

 

[mgb_phys]

the Roman army had build a bridge over river in a very short time. But when scientists tried do this for themselves they failed, so it is still a mystery how the Romans exactly pulled this off

The Royal Engineers did it using rafts with pile drivers driven by counterbalanced weights supported on A frames.
Not only did it work very well (with enough squaddie muscle power) but they reckoned that if you had a dozen barges working in parallel it was almost as fast as a modern bridge laying tank.

 

[Andy Resnick]

I couldn't find the Mangin-Chikolev formula anywhere. I did find 50 year old translated Russian article about this exact discussion ( http://www.google.com/url?sa=t&sour...rmula&usg=AFQjCNFk4eHQPGrBqBQFeMpcvNlHNfeEGQ") that put question marks next to the name. BTW, they are definitely in agreement with you on the impossibility of this.

Are there any online links to the formula other than this that you know of? The spelling of Mangin's name seems to be throwing off the searches.

I got it from the same report as you, I don't have the primary source. The best I could easily find are here:

"Drei grundlegende und gemeinverständliche arbeiten zur scheinwerferfrage", Alphonse François Eugène Mangin; V N Chikolev; August Sonnfeld
http://www.worldcat.org/oclc/7601194?tab=details

Which I am unable to read

And here:

"The range of electric searchlight projectors",By Jean Alexandre Rey, John Henry Johnson:
http://books.google.com/books?id=TZBRAAAAMAAJ&pg=PA59&dq="A.+mangin", Chapter 6, about 1/2 way down, says the formula came from Blondel in 1899 (unpublished works), but also provides a lot primary references by Mangin and "Tchikoleff"