Archimedes death ray: idea feasibility
testing > FAQ
FAQ about Archimedes Death Ray
A comment first...
In the 2.009 experiment, by saying "feasibility estimate confirmed",
we mean that our mathematical feasibility estimate was confirmed by the
experiment and that the myth is at least possible.
This is not '"myth confirmed" in the Mythbusters sense. In the context of the class, the goal was to verify our order-of-magnitude feasibility estimation and accomplish the effect in a very simple manner that might have been possible in Archimedes' day. We were not trying to make a statement that Archimedes did it (which, of course, one could never prove conclusively). However, it is hard for me to say the death ray was impossible.
We have received a lot of questions and statements, so here is a FAQ from the course instructor to facilitate the discussion. And... we really enjoy the Mythbusters, too!
i) Why not set the sails on fire, would this not be easier?
ii ) Archimedes would only have had access to bronze or silver plated mirrors.
iii) A bowshot is easily greater than 100 feet. Are greater distances a problem?
iv) Does the intensity of the reflected light not decrease with the square of the distance?
v) The mirror array would have to have a variable focal length.
vii) The device would only be capable of its peak burning power at midday on a cloudless day.
viii) The target would have to be directly below the sun on the horizon for maximum effectiveness.
xi) Weren't those ships caulked with pitch, which would increase their flammability?
xii) Is the cooling effect of wind significant?
xiii) Does the type of wood the boat is made from matter?
i) Why not set the sails on fire, would
this not be easier?
The raised sails seem like an easy target, but at anchor they would be furled.
Also, I believe that in battle they would be furled since otherwise they
are a great target for other types of projectiles. However,
even if unfurled sails were available as targets, the raised sails would
likely more difficult to set on fire because of the cooling effect of the
wind. See FAQ xii.
ii) Archimedes would only have had access to bronze
or silver plated mirrors.
Yes, my impression is that Archimedes would certainly have had access to
polished bronze at this time. It is not clear if he would have had silver
mirrors. If silver was available, silver mirrors would be about as good
as the mirrors we used. Bronze would not be as good. But polished, it is
actually a fairly good reflector. Its lower reflectivity can be compensated
for by increasing the collector area.
Considering the reflectivity of polished brass/bronze, my initial estimate is that we would need to increase the size of the array by roughly 1.5 times for bronze reflectors to work. The bronze mirror array used for the subsequent test with the Mythbusters was 2.4 times larger, accounting for the bronze and for the shorter time available to heat the wood to ignition temperature (see FAQ iii).
iii) A bowshot is easily greater than 100 feet.
Are greater distances a problem?
A bow shot can definitely be more than 100 feet, but this is actually not
a huge deal assuming that the air is clear. 100 feet was the biggest space
we had at MIT.
The number of 'perfectly flat' mirrors needed with increasing distance will go up because the beam from a flat mirror reflecting sunlight will disperse slightly with distance. In fact, the amount of dispersion from a flat mirror is almost exactly what we observed in our test (1 square ft mirror projects a 3 square ft area at 100 feet), meaning the mirrors we used probably really were very flat. This is because the sun is not a true point source, and actually is about 0.5 degrees wide in the sky. This leads leads to the dispersion that we observed from a flat mirror. However, a quick experiment flexing the mirror tiles by hand (done by one of my colleagues) indicated that it is not too hard to make the mirror slightly concave to compensate for this effect. So, with slightly concave mirrors one would not need to increase the collector area for greater target distances. To understand this last statement more, please see FAQ iv.
The most significant effect of distance between the mirrors and the target is on how quickly the concentrated 'hot spot' moves across the target. Using the 2.009 mirror setup, the mirrors are not repositioned once aimed at the target. However, the earth is spinning and thus the sun travels across the sky each day. As a result of this motion the reflected hot spot will move along the target. During the 100 foot test at MIT, the beam moved horizontally along the target at approximately 6 inches per minute. Given that the hot spot was roughly 6 square feet, this meant that a point on the boat over which the 'death ray' passed was heated for approximately 5 minutes. Thus, the death ray had to have sufficient power to heat the wood to ignition temperature in less than this amount of time.
The rate at which the hot spot moves across the target will increase with the distance between the target and the mirrors. Thus, for example, in the test with the Mythbusters at a distance of 150 feet, the hot spot moved horizontally at roughly 9-10 inches per minute. Given the size of the hot spot, this meant that the death ray had approximately 3 minutes to heat the wood to ignition temperature. Heating the wood faster requires more power.
iv) Does the intensity of the reflected light not decrease
with the square of the distance.
The reflected light does not decrease in intensity
with the square of the distance from the mirror. If this were the case,
there would be no hope whatsoever for the myth (or a laser pointer) to work,
even in modern times. The attenuation of the reflected light from a flat
mirror is only related to how much the beam disperses geometrically before
it hits the target (e.g., our 1 ft square tile's reflection spread to an
area of 3 square feet at a distance of 100ft, so the attenuation was a factor
of 3). Low power laser pointers can project bright points over very long
distances because they have a very tight, parallel (or well-collimated)
beam. As a result, it does not increase in diameter with distance.
The intensity of light (energy per unit of area) falls off with the square of distance from the source when one has a source emitting radiation in all directions (like the sun or a light bulb). Again, this has a purely geometric interpretation. Imagine a 100 W light bulb emitting light in all directions. At a distance of one foot, the available power (100W) is spread over the surface area of a sphere 1 foot in diameter. At a diameter of 2 feet, the same 100W is being spread over the surface of a 2 feet diameter sphere, which is 4 times greater in area than the 1 foot sphere. Hence, the power available per unit of area falls off with the square of distance from the source... in this specific case. An illustration of this idea is at: http://www.astronomynotes.com/starprop/s3.htm.
But, is the sun not just like a very bright light bulb that is very far away from the earth? Yes, it is. The light from the sun has decreased in intensity as it has spread over a growing spherical surface until it reaches the mirror. However, if we compare the distance between the mirror and the sun vs the distance between the target and the sun via the mirror, they are basically the same. Thus, the light will not fall off in intensity with the square of the distance of the target measured from the mirror.
So, if our mirrors were corrected to not disperse the light (see
FAQ iii), we would not need to increase the number of mirrors with target
distance (within the myth's implied range) except to account for the time
available to heat the wood.
Note: One could also adjust each individual mirror's shape to concentrate
sunlight at a distant point, but this would need to be done with a specific
target distance in mind and thereby somewhat compromise the variable focal
length capabilities as described above.
v) The mirror array would have to have a variable
focal length
Variable focal length is needed and I believe that the approach
we took addresses this issue. The 2.009 mirror array can be set up and adjusted
on-the-fly quite quickly (our 5 people needed under 10 minutes, with more
trained people it would be faster). The 2.009 mirror array is focused by
aiming each of the 127 flat mirrors at the same spot on the target. To change
the focus of the array, all one needs to do is to reposition the mirrors
to point at a different spot.
| The array of mirrors can also be arranged to significantly reduce sensitivity to ships at different distances, much as a small aperture gives a camera a large depth of field. The 2.009 array used two tiers of mirrors and formed a long arc around the target. Thus, it would be fairly sensitive if the boat moved closer or farther away (the hot spot would grow quickly in size as the distance changes). However, if the several vertically arranged tiers were used so that same number of mirrors can be placed in a narrow configuration, the hot spot will not grow very quickly as the target moves closer or farther away. |  |
vi) The approaching
boat would have to be completely stationary for a few minutes. Could anchors
at the time achieve this?
My instinct is that if the ship was anchored it would actually be still
enough. If it was say 70-150 feet long (that's the size historians seem
to cite) and it was fairly calm, the boat would be almost completely still
in the vertical direction (irrespective of the anchor). Given that the city
was in a bay, I would surmise that the waves there are typically not too
big, and the Mediterranean is somewhat calm.
The Mythbuster rig in the original show was very small (guessing ~10 feet?) and thus would be much more sensitive to wave action. Consider the difference between riding on a cruise ship vs. a 14 ft Boston whaler. So, that covers vertical motion.
I am not aware of a reason why their anchors could not keep the boat stationary in a horizontal direction.
Additionally the boat can bob around some. For example, in our test the hot spot was roughly 2-3 feet in diameter, so the boat could bob around roughly +/-1.5 feet and still keep at least some common area in the heat all the time. The bigger the hot spot, the less sensitive to boat movement the death ray will be. The bigger the hot spot, the bigger the collector area needed.
vii) The device would only be capable
of its peak burning power at midday on a cloudless day.
Yes, the position of the target and time of day does matter, but at least
the geometry of the region is about right (so the boat would be roughly
between the sun and the mirrors).
One needs to size the collector for the worst case for which you want it to work. The effective area of the mirror is a function of the position of both the target and the sun. This notion is considered in the order of magnitude estimate on the 2.009 page (you need to load the big image of the hand written estimate to read it).
The time of day can be compensated for by adding more mirrors. Of course, at some point the scale gets so big it does not make sense. But, say between 10 AM and 3-4 PM at this time of year in Boston (early October), I feel we could get it to work at a reasonable scale.
We found our 'death ray' was very sensitive to even light cloud between the mirror array and the sun because of how the clouds diffused the sunlight. However, even on our successful day, you can see a lot of cloud in the picture. All we needed was a bit of time between clouds. So cloudless is a bit too strict. On the up side, my Greek graduate student tells me it is sunny there much of the time!
viii) The target would have to be directly
below the sun on the horizon for maximum effectiveness.
This is more or less the same issue as discussed in FAQ
vii. The position of the sun and the target matter since that determines
how the mirror needs to be oriented, and thus the effective area of sunlight
it reflects. The less perpendicular the mirror is to an imaginary line from
the center of the mirror to the sun, the lower its effective area. One just
has to account for this and design for the range of conditions you want
it to work in.
Maximum/peak burn power is not required for the death ray to work. For greater reliability, the death ray should be designed to work over a reasonable range of conditions (for example, between the hours of 10AM-3PM). We also need to know the time of year, latitude and where the ships are most likely to be relative to the collector to decide a reasonable time range and appropriate number of mirrors/collector size. Using the 2.009 approach is is very easy to employ more or fewer mirrors.
If designed to have enough area to work at 10 AM, the death ray would be very effective (much bigger than needed) in ideal conditions, but still be sufficiently effective for a larger range of conditions. In product design speak (i.e., in our 2.009 class), you need to design for the worst case scenarios.
ix) The weapon would only work on targets
to the east, south and west as the sun shifted through the day.
Feasibility is site specific. Looking at a map, and assuming that the modern
Syracuse is in exactly the same location (I don't know if it is), a naval
attack would have only been possible from roughly ESE to SW, which is pretty
ideal.
x) The planking of the boat would have had
a high moisture content after possibly years in the water. Is the wood moisture
content important?
The percent moisture content of wood is the difference in weight
between the wet wood and the dry wood divided by the weight of the dry wood
and then multiplied by 100%.
Below and near the water line the ship would be dependent on the wood being swollen with moisture just to keep the joints tight and not leak. However, in a seaworthy vessel, I suspect that well above the water line the wood would have a much lower moisture content. Growing up, my family had a cedar strip boat that stayed in the water for extended periods and the wood above the chines was reasonably dry. I do not have any measurements, but I would guess that it was not much different than outdoor, seasoned wood (roughly 25% moisture content). Wikipedia suggests that the Roman ships may have been taken out of the water to prevent them from becoming water logged, which would cause them to ride lower in the water and become more difficult to propel and maneuver. Thus, it is probably reasonable to assume that the Roman ships would not have been water logged... and one would certainly want to aim for the driest parts of the boat to reduce the amount of time to achieve a sustaining flame.
Higher moisture content will increase the amount of time needed to raise the wood to ignition temperature. The wood will first heat to the temperature needed to boil the water contained within. It will remain at this temperature until the water has boiled off, just as a boiling pot of water on the stove will remain at 100 C as long as water remains in the pot. Once the water has been boiled out of the wood, the dry wood will then continue to heat to its ignition temperature.
The table below shows estimated ignition times for oak of different moisture content using the mirror array from the 2.009 test at MIT. I also assumed that there is no wind (see FAQ xii for a discussion of the wind's cooling effect).
Thanks go out to one of my Doctoral students, Sittha Sukkasi, for writing the simulation to estimate the time to reach ignition temperature. The simulation agreed well with observations from the MIT test (Oak, 10% moisture content) but it has not been verified against other experimental data. In a more qualitative sense, the predictions for the less well-quantified test with the Mythbusters also seemed to agree with predicted results.
Estimated ignition times for the 2.009 MIT mirror array.
| Oak moisture content | 0% | 10% (similar to test conditions) |
25% (outdoor, seasoned wood) |
45% (green, live wood) |
100% (wood soaked in water for several weeks) |
| estimated ignition time | 3.5 minutes | 4.5 minutes | 5.5 minutes | 8 minutes | 11 minutes |
The estimated ignition time for the 10% Oak (equivalent to the kiln dried lumber that was used) was within the maximum allowable time for a successful experiment (approximately 5 minutes). Please refer to FAQ iii for a discussion of the maximum allowable ignition times.
If the wood had been stored outdoors for an extended period, which would been more representative of and Oak boat, it is border line whether the mirror array would have had sufficient power. Clearly, the size of the mirror array would need to be increased for moisture content higher than 25%.
Similar estimates are provided below for the test performed with the Mythbusters. In this case the fishing boat was made from Douglas Fir.
Estimated ignition times for the Mythbusters/MIT mirror array.
| Douglas Fir moisture content | 0% | 10% (kiln dried) |
25% (outdoor, seasoned wood) |
45% (green, live wood) |
100% (wood soaked in water for several weeks) |
| estimated ignition time | 1.5 minutes | 2.5 minutes | 3 minutes | 4.5 minutes | 7.5 minutes |
For the test with the Mythbusters, the maximum allowable time to ignition was around 3 minutes. Based on the table above I anticipate successful ignition for up to 25% moisture content, which was expected. However, based on my understanding that the boat had spent a period of time submerged underwater in the weeks prior to the test, it is at least plausible to speculate that the moisture content would have been in the 100% or higher range (fully saturated wood that has been under water for an extended period can have a moisture content over 100%, meaning the wet wood is more than twice as heavy as the dry wood. The maximum possible moisture content varies with wood type depending upon the amount of airspace within the dry wood that can fill with water. Eastern white pine has a maximum moisture content that approaches 230%!).
The estimated ignition time for 100% moisture content seems somewhat consistent with what happened during the experiment with the Mythbusters. After 3 passes over the boat (three passes, each with a 3 minute exposure on a given spot) wood was ignited. However, a large flame did not break out likely because only small amount of material was sufficiently dry. The fire smoldered for an extended period after the experiment, slowly drying more wood and increasing the size of the hole in the boat.
Finally, estimates for cedar similar to materials likely used in a Roman ship are below. The bronze mirror array used for the tests with the Mythbusters is assumed in the estimation. The predictions suggest that the mirror array used in the test would be sufficient for Cedar that is fairly wet.
Estimated ignition times for a cedar boat.
| Cedar moisture content | 0% | 10% (kiln dried) |
25% (outdoor, seasoned wood) |
45% (green, live wood) |
100% (wood soaked in water for several weeks) |
| estimated ignition time | 1 minute | 2 minutes | <2.5 minutes | 2.5 minutes | 4 minutes |
xi) Weren't those ships caulked with pitch,
which would increase their flammability?
This is not my area of expertise. There seems to be a fair bit
of inconsistent information, but the planking may have been covered with
pitch or wax coatings. Dark pitch would increase the surfaces' absorption
of energy from the death ray. In our experiment, the fire started in a region
with both raw wood and wood coated with an oil-based stain. We had wax in
the joints and this boiled off well before ignition. Therefore, coated or
not, it seems possible to set the wood on fire.
xii) Is the cooling effect of wind significant?
The energy from the death ray heats the wood until it reaches its
ignition temperature. At the same time, cooling due to convection, both
natural and forced, carries energy away from the wood. Natural convection
is caused by the air near the hot surface of the wood also becoming heated.
This warmed air is less dense than the surrounding air, so it rises and
fresh cooler air flows in behind it. This creates a cooling air current
over the wood. Forced convection refers to a wind blowing across the heated
surface of the wood, which also has a cooling effect.
These cooling effects lower the maximum temperature that the wood will achieve when heated by the death ray and they also increase the amount of time needed to heat the wood to a given temperature below this maximum.
For example, in the 2.009 experiment the wood needed to reach ignition temperature in less than 5 minutes (see FAQ iii). Given the estimated power that was available from the array, an ambient temperature of 20 C, and dry oak (10% moisture content), we anticipate the maximum wind speed for successful ignition would be only 6 mph, or just over 5 knots! In order to be successful in higher winds a larger mirror array would be needed.
In the test with the Mythbusters, which was at a greater distance, the wood needed to reach ignition temperature in under 3 minutes. Given the estimated power of the larger Mythbuster array and assuming the boat was Douglas Fir seasoned outdoors (25% moisture content), we expected the maximum wind speed for successful ignition to be in the vicinity of 7 mph, or 6 knots.
These estimates were based on the assumption that convection cooling occurred on one side of the wood only, and that there was no heat loss through the back side. This is a reasonable assumption since the wood is fairly thick and is also a fairly good insulator. However, if one considered igniting the thin sails (see FAQ i), convection cooling on both sides of the sail must be accounted for. Using our mirror arrays, even a breeze as light as 2 mph might prevent the sail from heating in the available time. I believe that this cooling effect, combined with the lighter, more reflective color of the sail material, explains why the sails did not ignite in the test with the Mythbusters.
xiii) Does the type of wood the boat
is made from matter?
Yes. The type of wood influences the amount of
time that it will take to raise the wood to ignition temperature (see FAQ
iii for a discussion on the significance of ignition time).
Equal volumes of different types of wood require different amounts of energy to raise their temperature an equal amount. Thus, for a given power source, the some types of wood will take longer than others to heat to ignition temperature.
Two important factors that determine the amount of energy needed to raise wood's temperature are its density and its moisture content. Here, I assume that the wood is dry (0% moisture content). Please see FAQ x for a discussion on the significance of wood moisture content and a comparison of woods with different levels of moisture content.
Predicted times for some different types of wood, relative to the time needed to ignite oak (used in the 2.009 test) are below. My understanding is that cedar was most likely used in Roman ships. I expect that dry cedar would ignite in just over 1/2 the amount of time needed to ignite oak when using the same power source.
| Wood type | dry Oak | dry Douglas Fir | dry Cedar |
| Ignition time (relative to oak) |
1 | 0.76 | 0.55 |
xiv) Why bother with all of this when powerful
weapons like the Palintonon or Ballista were available? They work in all
weather conditions and can be accurately and repeatedly targeted.
True, they did have these options, and given the dependence on weather I
personally would not want the death ray as my only choice! Aside from weather
dependency, I think that (using the technique of the 2.009 attempt) we could
accurately and repeatedly target objects to achieve the desired outcome
(although there was a definite learning curve).
On the rationale side, it is fun to speculate. Does it hurt to increase one's options? With appropriate mirrors it might be possible to target ships at significant range. Also, one might simply benefit from the element of surprise. Would the Romans understand enough about optics to realize what was happening and react quickly? One certainly could make people on the ships very uncomfortable. Using the ray does not consume any materials. Why haul all those heavy rocks to use as projectiles when you can just aim a beam of light! (kidding) ...
xv) Did it really happen?
We're not historians and cannot answer this question. Overall, there are
a number of special conditions and important details that need to be considered
to get the death ray to start a fire. However, given the location of the
city, the local conditions, the ease of implementing the idea once one has
worked out the sensitivities, and the brilliance of Archimedes, my own opinion
is that I personally can't rule it out all together (can't say it happened
either). However, our goal was to use Mythbusters to motivate doing a quantitative
estimation of feasibility, to demonstrate the possibility using a sketch
model experiment, and to have some fun in the process.
