MadSci Network: Physics

Re: Can a beam of focused sunlight melt a small iceberg?

Date: Sat Feb 28 09:56:37 2004
Posted By: Michael Richmond, Faculty, Physics, Rochester Institute of Technology
Area of science: Physics
ID: 1077922856.Ph

In case you are in a hurry, the short answer to your question is "no".

Let me explain. First, I'll write about the energy required to melt an iceberg, and I'll compare it to the incoming energy of ordinary sunlight. Then I'll address some of the issues involved in making an orbiting mirror. In all my work below, I deal with approximate numbers for convenience.

Consider this simple model of an iceberg: a sphere of ice of diameter d=30 m. It has a volume of about 14,000 cubic meters, and a mass of roughly 14,000,000 kg, or 14,000 metric tonnes. Since it requires 340 kiloJoules to melt 1 kg of ice, the total energy required to melt this iceberg is about 4.7 x 10^(12) Joules. By a peculiar coincidence, this is just about one kiloton, a unit of energy used in discussing explosives and atomic weapons. That's a sign that it will take a LOT of energy to melt the iceberg.

The sun shines all by itself on an iceberg. Let's see how long it might take to melt the object. The amount of energy in the sunlight that falls on one square meter of the Earth's surface is very roughly one kilowatt, or 1,000 Joules per second. How much area does an iceberg present to the Sun? At most, its cross-section, which is (pi*r^2) = 700 square meters, for our example iceberg. Of the 700,000 Joules which fall upon the ice each second, however, only half are absorbed -- the other half reflects off the snowy surface (the fraction of energy which bounces off the surface, the "albedo", can vary quite a bit, but this is again a representative value for snow). That means that about 350,000 Joules might be absorbed each second. At that rate, it will take a time equalt to 4.7 x 10^(12) Joules divided by 350,000 Joules per second = 13 million seconds to melt the ice. That's about 155 days. Hmmm. That sounds reasonable, doesn't it?

Oh, wait. The Sun only shines during the daytime, not at night. That would double the time. And if it's cloudy, little sunlight reaches the iceberg. To account for these factors, we might triple or quadruple the estimate, making it closer to a couple of years. Right.

Your plan is to put a mirror of some sort into space and reflect enough light onto the iceberg to melt it in a much shorter time -- you said, "in a few days". Let's pick 3 days as a desired timescale. Hmmm. We can estimate the size of the mirror which would be required by asking, "How much power must be delivered to the iceberg to melt in in 3 days?" Using the parameters above, I find an average power of 36 Megawatts sent to the iceberg (half of which reflects off the surface) will do the trick.

Okay. Assume that you can arrange it so that the sunlight reflected from some mirror in space will always strike the iceberg straight on, around the clock, in all weather. How large must the mirror be to collect 36 Megawatts? At roughly 1 kilowatt per square meter, we need 36,000 square meters, corresponding to a square mirror about 190 meters on a side. If the mirror is made of aluminized mylar, a thin, flexible and reflective material, it might have a density of 7 grams per square meter; the entire mirror would then have a mass of about 250 kg.

How can you keep a big mirror pointed at the iceberg for long periods of time? Perhaps you might think that the mirror could be placed in a geosynchronous orbit. Unfortunately, these orbits circle the Earth's equator, not the poles, where the icebergs live. That would make it difficult to send the beam of light towards the iceberg. Since the mirror would appear very low in the sky (as seen from the iceberg), most of its light would be absorbed or scattered by the Earth's atmosphere before it reached the iceberg. So that won't work.

Two possibilities remain. First, you might place the mirror over the North Pole and keep it there, hovering like a kite, by firing a rocket engine. Ordinary rockets would soon run out of fuel, in a matter of minutes or hours, so that won't work. However, a new sort of engine, using "ion propulsion", has been developed for low-thrust, long-term use. Could it meet our needs? Let's see -- the mirror has a mass of about 250 kg, and the associated spacecraft structure would probably have a roughly equal mass. How large is the force of gravity on a 500 kg object floating, say, H = 3000 km above the North Pole? About 2300 Newtons. That means that our ion engine would have to provide at least 2300 Newtons of thrust to keep the mirror in place. Deep Space One was the first American spacecraft to use ion engines. Its engine provided about 0.09 Newtons of thrust for many months. Hmmm. Nope, this isn't going to work.

Second possibility: build a series of N mirrors, and place them all in the same orbit, spaced out evenly so that one mirror is always visible from the iceberg at any time. As one mirror moves below the horizon (as seen from the iceberg), another rises to takes it place. For a typical low earth orbit of altitude H = 800 km, like that of the Iridium satellite system, it takes about 15 satellites to ensure that one will always be visible. In other words, each mirror would shine on the iceberg for about 1.5 hours before the next one took over. Each mirror would start shining at a distance of about 3000 km from the iceberg, reach a minimum distance of 800 km when overhead, and then recede to about 3000 km again before the following mirror took its place. Let's say the average distance from mirror to iceberg is about D = 2000 km.

When light bounces from a mirror, the beam of light spreads out slightly due to a phenomenon called "diffraction." It doesn't make much difference in ordinary life -- you don't have to worry about it as you shave or put on makeup -- but it might become significant over these very large distances. The angle by which light spreads depends on the size of the mirror and the wavelength of the light. For our mirror of diameter 190 meters, and for a typical visible wavelength of 500 nm, the angle is tiny: only 2.6 x 10^(-9) radians, which means that in theory, the beam would spread out by much less than a meter over its journey of D = 2000 km from mirror to iceberg. Good. We don't have to worry about losing power due to diffraction.

But -- wait a minute. The mirror isn't reflecting light from a point source, it is reflecting light from the Sun. The Sun is about one half of a degree in angular size, as seen from the Earth. That means that any beam reflected from the Sun will have an angular spread of about half a degree. Think of it this way: if you build your mirror in just the right way to reflect light from the center of the Sun onto the iceberg, light from the right-hand edge of the Sun will bounce off to the side slightly, and ditto for light from the left-hand edge of the Sun. How much will the light spread out by the time it travels D = 2000 km and reaches the iceberg? The mirror's beam of light will cover an area about 17,000 meters in diameter. Uh-oh. The iceberg is only 30 meters in diameter. Only a tiny, tiny fraction of the sunlight will actually hit the iceberg! Most of it will fall on the waters surrounding the iceberg.

I believe that when you put together all the factors, you must conclude that this is a nice idea, but it just won't work in practice. But keep thinking!

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