Spoiler Alert


Like many people, I went to see the movie Elysium last weekend. The movie’s premise is that the wealthy members of society have relocated to an orbital space station, named Elysium, that circles the Earth while the rest of humanity is stuck on a seemingly dying world.

Focusing on the science of the movie, what can Mathematica and Wolfram|Alpha tell us about the space station and some of the other events portrayed ?

From the production crew, we know that the space station Elysium is 60 kilometers, or about 37.3 miles, in diameter and about 2 kilometers (1.2 miles) thick. If this ring were set upright on the surface of the Earth like a gigantic Ferris wheel, it could carry 25 city blocks side by side into the mesosphere. If a person were to leap off at that point, the person would exceed the height of Felix Baumgartner’s record jump by a factor of 3/2.

In the movie, Elysium does indeed turn like a Ferris wheel, spinning to generate a centripetal acceleration equivalent to Earth’s gravity. But how fast would it need to move to accomplish that ?

Wolfram|Alpha tells us that the rotation speed for a radius of 15 kilometers and centripetal acceleration of standard gravitational acceleration is 0.02556 radians per second, or a full rotation every 246 seconds. That is roughly one turn every four minutes.

What else can we determine about Elysium ? Well, based on its angular diameter as seen from the surface, one can determine its altitude. I was unable to get an exact measurement of its angular diameter, but I estimate that Elysium appeared at least as large as the Sun, which gives us an upper limit on its orbital distance.

The formula for angular diameter is :

,

where δ is the angular diameter, d is the actual diameter of the object, and D is the distance from the viewer. The angular diameter of the Sun is 0.009177 radians. Putting this and the diameter of Elysium into the formula and solving for D yields an upper altitude of 3,269 kilometers, or 2,031 miles.

This is high above where any space station has yet been placed. The International Space Station, for instance, maintains an average altitude of 265 miles, or 412 kilometers. We might ask how large Elysium would look if it traveled at the altitude of the International Space Station. Revisiting our formula we see we get an angular diameter of 0.0727834 radians, eight times larger than the Sun.

This is close to the size that the Statue of Liberty, including the base (93 meters), would appear to someone looking at it from Ellis Island.

Regardless of the exact altitude, Elysium is still fairly close to Earth. Communication times remain in milliseconds. The view of Earth depends on the altitude, but at 3,269 kilometers, the horizon is 7,238 kilometers, or 4,497 miles, away. In the movie, Elysium remains well connected to Earth’s surface, with travel from the surface only taking 19 minutes or so. Based on this we can easily measure the average speed of a vessel heading to Elysium :

This is pretty fast in terms of atmospheric vehicles, with the speed at Mach 8 as measured at sea level. But most of the travel distance would be in relatively airless space. The Space Shuttles routinely traveled at over 17,000 mph, so reaching these speeds is entirely reasonable.

However, building Elysium would take extraordinary effort. Assuming the materials were mined on Earth, we can look at how much energy it would take to lift them into space.

Estimating the mass of Elysium requires some assumptions. Elysium presumably uses similar materials to those used in modern-day space stations : carbon fiber, aluminum, titanium, steel, and plastics. The densities of carbon fiber, aluminum, titanium, and steel are 1.79 g/cm^3, 2.7 g/cm^3, 4.5 g/cm^3, and 7.859 g/cm^3, respectively. Plastics also occupy a large range, but are mostly used for shielding. Let’s assume the main structure consists of steel (a structure of this size requires a great deal of tensile strength to counteract the centripetal forces pulling it apart).

Next we need a thickness for the ring so we can determine its volume. From the movie it looks like the main living areas are nestled between two walls about a fifth of the width of the ring, or about 400 meters high. Most of the space between these walls however is filled with air, and the volume beneath the surface of Elysium’s living areas would likely contain large amounts of air-filled space. Let’s assume however that we can treat the width of the steel itself in the ring as 1 meter thick. This should yield a lower limit on the steel used.

The volume of steel would be approximately :

Which gives us a volume of 1.885*10^8 cubic meters. Multiplying this by the density of steel, we find we need 1.481*10^15 grams, or 1.481*10^9 metric tons.

To put that in perspective, the world steel production in 2010 was under 1.5 x10^6 tonnes. Elysium would require over 1,000 years’ worth of production at that rate.

Lifting that steel into orbit wouldn’t be easy either. The gravitational potential energy is :

,

where G is the gravitational constant, Me is the mass of the Earth, m is the mass of the object, and r is the distance from the center of the Earth. Plugging in the numbers and taking the difference from the potential at the surface yields 3.131*10^19 joules.

But just lifting it into orbit isn’t enough. You also need to accelerate it to sufficient speed for it to maintain an orbit. The orbital velocity at 3,269 kilometers above the Earth is 6.431 kilometers per second, or 14,386 mph. Now we see why the NASA Space Shuttle had to travel so fast.

The kinetic energy of 1.63295*10^9 tonnes at 6.431 kilometers per second is 3.377*10^19 joules. Combining that with the energy needed to lift the steel frame, we arrive at 6.508*10^19 joules, or the equivalent of 15,550 megatons of TNT. This is a more manageable problem than the steel production. It is comparable to the US energy consumption for an entire year.

One thing that was not addressed was how Elysium handled radiation, which is particularly important since it is moving through areas covered by the inner Van Allen belt. This is somewhat mitigated by the fact that Elysium has access to miraculous medical technology that seemingly can heal almost any damage.

This is in fact a major driver of the plot : the main character Max, played by Matt Damon, receives a lethal dose of radiation at his work place and needs treatment soon. Based on the time he is given until he dies from radiation poisoning and the known lethalities of severe radiation exposure, we can estimate that he took between 8 and 30 grays. For those more familiar with the depreciated unit rad, that is 800 to 3,000 rads.

To put this in perspective, we can look at what the equivalent dose of radiation for 30 grays in different sources would be :

If this radiation was due to beta or gamma radiation, then it would be about 30 sieverts, or 3,000 rems. This is similar to the much more targeted levels associated with cancer radiotherapy. However, those cases involve the radiation pointed directly at the cancerous cells, with as little affecting the rest of the patient as possible. Max gets a full body dose. If on the other hand the radiation came from alpha particles or neutrons, then the dose would be 600 sieverts, a much more lethal level. Either way, Max would have only days to live.

Which makes his quest to reach Elysium all the more imperative. While the existence of a space station like Elysium suggests a world far more rich than our own, in the movie the only place with the medical technology necessary to save him is in space. I hope that if we do build such a marvel that we share it with everyone.

Cet article est repris du site http://blog.wolframalpha.com/2013/0...